[380a17b] | 1 | //////////////////////////////////////////////////////////////////////////// |
---|
[3686937] | 2 | version="version primdec.lib 4.0.0.0 Jun_2013 "; // $Id$ |
---|
[0ae4ce] | 3 | category="Commutative Algebra"; |
---|
[5480da] | 4 | info=" |
---|
[8942a5] | 5 | LIBRARY: primdec.lib Primary Decomposition and Radical of Ideals |
---|
[f3c6e5] | 6 | AUTHORS: Gerhard Pfister, pfister@mathematik.uni-kl.de (GTZ)@* |
---|
[768b28] | 7 | Wolfram Decker, decker@math.uni-sb.de (SY)@* |
---|
[f3c6e5] | 8 | Hans Schoenemann, hannes@mathematik.uni-kl.de (SY)@* |
---|
| 9 | Santiago Laplagne, slaplagn@dm.uba.ar (GTZ) |
---|
[f34c37c] | 10 | |
---|
[b9b906] | 11 | OVERVIEW: |
---|
[07c623] | 12 | Algorithms for primary decomposition based on the ideas of |
---|
[367e88] | 13 | Gianni, Trager and Zacharias (implementation by Gerhard Pfister), |
---|
| 14 | respectively based on the ideas of Shimoyama and Yokoyama (implementation |
---|
[7f7c25e] | 15 | by Wolfram Decker and Hans Schoenemann).@* |
---|
| 16 | The procedures are implemented to be used in characteristic 0.@* |
---|
| 17 | They also work in positive characteristic >> 0.@* |
---|
| 18 | In small characteristic and for algebraic extensions, primdecGTZ |
---|
| 19 | may not terminate.@* |
---|
[b9b906] | 20 | Algorithms for the computation of the radical based on the ideas of |
---|
[7f7c25e] | 21 | Krick, Logar, Laplagne and Kemper (implementation by Gerhard Pfister and Santiago Laplagne). |
---|
[f3a046] | 22 | They work in any characteristic.@* |
---|
[b15849d] | 23 | Baserings must have a global ordering and no quotient ideal. |
---|
[3f4e52] | 24 | |
---|
[8942a5] | 25 | |
---|
[f34c37c] | 26 | PROCEDURES: |
---|
[24f458] | 27 | Ann(M); annihilator of R^n/M, R=basering, M in R^n |
---|
[8942a5] | 28 | primdecGTZ(I); complete primary decomposition via Gianni,Trager,Zacharias |
---|
| 29 | primdecSY(I...); complete primary decomposition via Shimoyama-Yokoyama |
---|
[7f7c25e] | 30 | minAssGTZ(I); the minimal associated primes via Gianni,Trager,Zacharias (with modifications by Laplagne) |
---|
[8942a5] | 31 | minAssChar(I...); the minimal associated primes using characteristic sets |
---|
| 32 | testPrimary(L,k); tests the result of the primary decomposition |
---|
[7f7c25e] | 33 | radical(I); computes the radical of I via Krick/Logar (with modifications by Laplagne) and Kemper |
---|
| 34 | radicalEHV(I); computes the radical of I via Eisenbud,Huneke,Vasconcelos |
---|
[8942a5] | 35 | equiRadical(I); the radical of the equidimensional part of the ideal I |
---|
| 36 | prepareAss(I); list of radicals of the equidimensional components of I |
---|
| 37 | equidim(I); weak equidimensional decomposition of I |
---|
| 38 | equidimMax(I); equidimensional locus of I |
---|
| 39 | equidimMaxEHV(I); equidimensional locus of I via Eisenbud,Huneke,Vasconcelos |
---|
| 40 | zerodec(I); zerodimensional decomposition via Monico |
---|
[326dba] | 41 | absPrimdecGTZ(I); the absolute prime components of I |
---|
[8942a5] | 42 | "; |
---|
[e801fe] | 43 | |
---|
| 44 | LIB "general.lib"; |
---|
[67bd4c] | 45 | LIB "elim.lib"; |
---|
[e801fe] | 46 | LIB "poly.lib"; |
---|
| 47 | LIB "random.lib"; |
---|
[8afd58] | 48 | LIB "inout.lib"; |
---|
[7f24dd7] | 49 | LIB "matrix.lib"; |
---|
[24f458] | 50 | LIB "triang.lib"; |
---|
[6fa3af] | 51 | LIB "absfact.lib"; |
---|
[cb980ab] | 52 | LIB "ring.lib"; |
---|
[d6db1f2] | 53 | /////////////////////////////////////////////////////////////////////////////// |
---|
[ebecf83] | 54 | // |
---|
[091424] | 55 | // Gianni/Trager/Zacharias |
---|
[ebecf83] | 56 | // |
---|
| 57 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 58 | |
---|
[07c623] | 59 | static proc sat1 (ideal id, poly p) |
---|
[d2b2a7] | 60 | "USAGE: sat1(id,j); id ideal, j polynomial |
---|
[d6db1f2] | 61 | RETURN: saturation of id with respect to j (= union_(k=1...) of id:j^k) |
---|
| 62 | NOTE: result is a std basis in the basering |
---|
[d2b2a7] | 63 | " |
---|
[d6db1f2] | 64 | { |
---|
[70ab73] | 65 | int @k; |
---|
| 66 | ideal inew=std(id); |
---|
| 67 | ideal iold; |
---|
| 68 | intvec op=option(get); |
---|
| 69 | option(returnSB); |
---|
| 70 | while(specialIdealsEqual(iold,inew)==0 ) |
---|
| 71 | { |
---|
| 72 | iold=inew; |
---|
| 73 | inew=quotient(iold,p); |
---|
| 74 | @k++; |
---|
| 75 | } |
---|
| 76 | @k--; |
---|
| 77 | option(set,op); |
---|
| 78 | list L =inew,p^@k; |
---|
| 79 | return (L); |
---|
[d6db1f2] | 80 | } |
---|
| 81 | |
---|
| 82 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 83 | |
---|
[07c623] | 84 | static proc sat2 (ideal id, ideal h) |
---|
[d2b2a7] | 85 | "USAGE: sat2(id,j); id ideal, j polynomial |
---|
[d6db1f2] | 86 | RETURN: saturation of id with respect to j (= union_(k=1...) of id:j^k) |
---|
| 87 | NOTE: result is a std basis in the basering |
---|
[d2b2a7] | 88 | " |
---|
[d6db1f2] | 89 | { |
---|
[70ab73] | 90 | int @k,@i; |
---|
| 91 | def @P= basering; |
---|
| 92 | if(ordstr(basering)[1,2]!="dp") |
---|
| 93 | { |
---|
| 94 | execute("ring @Phelp=("+charstr(@P)+"),("+varstr(@P)+"),(C,dp);"); |
---|
| 95 | ideal inew=std(imap(@P,id)); |
---|
| 96 | ideal @h=imap(@P,h); |
---|
| 97 | } |
---|
| 98 | else |
---|
| 99 | { |
---|
| 100 | ideal @h=h; |
---|
| 101 | ideal inew=std(id); |
---|
| 102 | } |
---|
| 103 | ideal fac; |
---|
[d6db1f2] | 104 | |
---|
[70ab73] | 105 | for(@i=1;@i<=ncols(@h);@i++) |
---|
| 106 | { |
---|
| 107 | if(deg(@h[@i])>0) |
---|
| 108 | { |
---|
| 109 | fac=fac+factorize(@h[@i],1); |
---|
| 110 | } |
---|
| 111 | } |
---|
| 112 | fac=simplify(fac,6); |
---|
| 113 | poly @f=1; |
---|
| 114 | if(deg(fac[1])>0) |
---|
| 115 | { |
---|
| 116 | ideal iold; |
---|
| 117 | for(@i=1;@i<=size(fac);@i++) |
---|
| 118 | { |
---|
| 119 | @f=@f*fac[@i]; |
---|
| 120 | } |
---|
| 121 | intvec op = option(get); |
---|
| 122 | option(returnSB); |
---|
| 123 | while(specialIdealsEqual(iold,inew)==0 ) |
---|
| 124 | { |
---|
| 125 | iold=inew; |
---|
| 126 | if(deg(iold[size(iold)])!=1) |
---|
[d6db1f2] | 127 | { |
---|
[70ab73] | 128 | inew=quotient(iold,@f); |
---|
[d6db1f2] | 129 | } |
---|
[70ab73] | 130 | else |
---|
| 131 | { |
---|
| 132 | inew=iold; |
---|
| 133 | } |
---|
| 134 | @k++; |
---|
| 135 | } |
---|
| 136 | option(set,op); |
---|
| 137 | @k--; |
---|
| 138 | } |
---|
[d6db1f2] | 139 | |
---|
[70ab73] | 140 | if(ordstr(@P)[1,2]!="dp") |
---|
| 141 | { |
---|
| 142 | setring @P; |
---|
| 143 | ideal inew=std(imap(@Phelp,inew)); |
---|
| 144 | poly @f=imap(@Phelp,@f); |
---|
| 145 | } |
---|
| 146 | list L =inew,@f^@k; |
---|
| 147 | return (L); |
---|
[d6db1f2] | 148 | } |
---|
| 149 | |
---|
| 150 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 151 | |
---|
[24f458] | 152 | |
---|
| 153 | proc minSat(ideal inew, ideal h) |
---|
[d6db1f2] | 154 | { |
---|
[70ab73] | 155 | int i,k; |
---|
| 156 | poly f=1; |
---|
| 157 | ideal iold,fac; |
---|
| 158 | list quotM,l; |
---|
[d6db1f2] | 159 | |
---|
[70ab73] | 160 | for(i=1;i<=ncols(h);i++) |
---|
| 161 | { |
---|
| 162 | if(deg(h[i])>0) |
---|
| 163 | { |
---|
| 164 | fac=fac+factorize(h[i],1); |
---|
| 165 | } |
---|
| 166 | } |
---|
| 167 | fac=simplify(fac,6); |
---|
| 168 | if(size(fac)==0) |
---|
| 169 | { |
---|
| 170 | l=inew,1; |
---|
| 171 | return(l); |
---|
| 172 | } |
---|
| 173 | fac=sort(fac)[1]; |
---|
| 174 | for(i=1;i<=size(fac);i++) |
---|
| 175 | { |
---|
| 176 | f=f*fac[i]; |
---|
| 177 | } |
---|
| 178 | quotM[1]=inew; |
---|
| 179 | quotM[2]=fac; |
---|
| 180 | quotM[3]=f; |
---|
| 181 | f=1; |
---|
| 182 | intvec op = option(get); |
---|
| 183 | option(returnSB); |
---|
| 184 | while(specialIdealsEqual(iold,quotM[1])==0) |
---|
| 185 | { |
---|
| 186 | if(k>0) |
---|
| 187 | { |
---|
| 188 | f=f*quotM[3]; |
---|
| 189 | } |
---|
| 190 | iold=quotM[1]; |
---|
| 191 | quotM=quotMin(quotM); |
---|
| 192 | k++; |
---|
| 193 | } |
---|
| 194 | option(set,op); |
---|
| 195 | l=quotM[1],f; |
---|
| 196 | return(l); |
---|
[18dd47] | 197 | } |
---|
[d6db1f2] | 198 | |
---|
[07c623] | 199 | static proc quotMin(list tsil) |
---|
[d6db1f2] | 200 | { |
---|
[70ab73] | 201 | int i,j,k,action; |
---|
| 202 | ideal verg; |
---|
| 203 | list l; |
---|
| 204 | poly g; |
---|
[d6db1f2] | 205 | |
---|
[70ab73] | 206 | ideal laedi=tsil[1]; |
---|
| 207 | ideal fac=tsil[2]; |
---|
| 208 | poly f=tsil[3]; |
---|
[3939bc] | 209 | |
---|
[70ab73] | 210 | ideal star=quotient(laedi,f); |
---|
[b1d1e8c] | 211 | |
---|
[70ab73] | 212 | if(specialIdealsEqual(star,laedi)) |
---|
| 213 | { |
---|
| 214 | l=star,fac,f; |
---|
| 215 | return(l); |
---|
| 216 | } |
---|
[b9b906] | 217 | |
---|
[70ab73] | 218 | action=1; |
---|
[18dd47] | 219 | |
---|
[70ab73] | 220 | while(action==1) |
---|
| 221 | { |
---|
| 222 | if(size(fac)==1) |
---|
| 223 | { |
---|
| 224 | action=0; |
---|
| 225 | break; |
---|
| 226 | } |
---|
| 227 | for(i=1;i<=size(fac);i++) |
---|
| 228 | { |
---|
| 229 | g=1; |
---|
| 230 | verg=laedi; |
---|
| 231 | for(j=1;j<=size(fac);j++) |
---|
[d6db1f2] | 232 | { |
---|
[70ab73] | 233 | if(i!=j) |
---|
| 234 | { |
---|
| 235 | g=g*fac[j]; |
---|
| 236 | } |
---|
[d6db1f2] | 237 | } |
---|
[70ab73] | 238 | verg=quotient(laedi,g); |
---|
[3939bc] | 239 | |
---|
[70ab73] | 240 | if(specialIdealsEqual(verg,star)==1) |
---|
| 241 | { |
---|
| 242 | f=g; |
---|
| 243 | fac[i]=0; |
---|
| 244 | fac=simplify(fac,2); |
---|
| 245 | break; |
---|
[d6db1f2] | 246 | } |
---|
[70ab73] | 247 | if(i==size(fac)) |
---|
| 248 | { |
---|
| 249 | action=0; |
---|
| 250 | } |
---|
| 251 | } |
---|
| 252 | } |
---|
| 253 | l=star,fac,f; |
---|
| 254 | return(l); |
---|
[d6db1f2] | 255 | } |
---|
| 256 | |
---|
[091424] | 257 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 258 | |
---|
[07c623] | 259 | static proc testFactor(list act,poly p) |
---|
[d6db1f2] | 260 | { |
---|
[e801fe] | 261 | poly keep=p; |
---|
[3939bc] | 262 | |
---|
[70ab73] | 263 | int i; |
---|
| 264 | poly q=act[1][1]^act[2][1]; |
---|
| 265 | for(i=2;i<=size(act[1]);i++) |
---|
| 266 | { |
---|
| 267 | q=q*act[1][i]^act[2][i]; |
---|
| 268 | } |
---|
| 269 | q=1/leadcoef(q)*q; |
---|
| 270 | p=1/leadcoef(p)*p; |
---|
| 271 | if(p-q!=0) |
---|
| 272 | { |
---|
| 273 | "ERROR IN FACTOR, please inform the authors"; |
---|
| 274 | } |
---|
[d6db1f2] | 275 | } |
---|
[091424] | 276 | /////////////////////////////////////////////////////////////////////////////// |
---|
[d6db1f2] | 277 | |
---|
[07c623] | 278 | static proc factor(poly p) |
---|
[d2b2a7] | 279 | "USAGE: factor(p) p poly |
---|
[d6db1f2] | 280 | RETURN: list=; |
---|
[18dd47] | 281 | NOTE: |
---|
[d6db1f2] | 282 | EXAMPLE: example factor; shows an example |
---|
[d2b2a7] | 283 | " |
---|
[d6db1f2] | 284 | { |
---|
| 285 | ideal @i; |
---|
| 286 | list @l; |
---|
| 287 | intvec @v,@w; |
---|
| 288 | int @j,@k,@n; |
---|
| 289 | |
---|
[e801fe] | 290 | @l=factorize(p); |
---|
[70ab73] | 291 | for(@j=1;@j<=size(@l[1]);@j++) |
---|
| 292 | { |
---|
[1e1ec4] | 293 | if(leadcoef(@l[1][@j])==@l[1][@j]) |
---|
[70ab73] | 294 | { |
---|
| 295 | @n++; |
---|
| 296 | } |
---|
| 297 | } |
---|
| 298 | if(@n>0) |
---|
| 299 | { |
---|
| 300 | if(@n==size(@l[1])) |
---|
| 301 | { |
---|
| 302 | @l[1]=ideal(1); |
---|
| 303 | @v=1; |
---|
| 304 | @l[2]=@v; |
---|
| 305 | } |
---|
| 306 | else |
---|
| 307 | { |
---|
| 308 | @k=0; |
---|
| 309 | int pleh; |
---|
| 310 | for(@j=1;@j<=size(@l[1]);@j++) |
---|
[d6db1f2] | 311 | { |
---|
[1e1ec4] | 312 | if(leadcoef(@l[1][@j])!=@l[1][@j]) |
---|
[70ab73] | 313 | { |
---|
| 314 | @k++; |
---|
| 315 | @i=@i+ideal(@l[1][@j]); |
---|
| 316 | if(size(@i)==pleh) |
---|
| 317 | { |
---|
| 318 | "//factorization error"; |
---|
| 319 | @l; |
---|
| 320 | @k--; |
---|
| 321 | @v[@k]=@v[@k]+@l[2][@j]; |
---|
| 322 | } |
---|
| 323 | else |
---|
| 324 | { |
---|
| 325 | pleh++; |
---|
| 326 | @v[@k]=@l[2][@j]; |
---|
| 327 | } |
---|
| 328 | } |
---|
[d6db1f2] | 329 | } |
---|
[70ab73] | 330 | @l[1]=@i; |
---|
| 331 | @l[2]=@v; |
---|
| 332 | } |
---|
| 333 | } |
---|
| 334 | // } |
---|
[d6db1f2] | 335 | return(@l); |
---|
| 336 | } |
---|
| 337 | example |
---|
| 338 | { "EXAMPLE:"; echo = 2; |
---|
| 339 | ring r = 0,(x,y,z),lp; |
---|
| 340 | poly p = (x+y)^2*(y-z)^3; |
---|
| 341 | list l = factor(p); |
---|
| 342 | l; |
---|
| 343 | ring r1 =(0,b,d,f,g),(a,c,e),lp; |
---|
| 344 | poly p =(1*d)*e^2+(1*d*f^2*g); |
---|
| 345 | list l = factor(p); |
---|
| 346 | l; |
---|
| 347 | ring r2 =(0,b,f,g),(a,c,e,d),lp; |
---|
| 348 | poly p =(1*d)*e^2+(1*d*f^2*g); |
---|
| 349 | list l = factor(p); |
---|
| 350 | l; |
---|
| 351 | } |
---|
| 352 | |
---|
[091424] | 353 | /////////////////////////////////////////////////////////////////////////////// |
---|
[d6db1f2] | 354 | |
---|
[50cbdc] | 355 | proc idealsEqual( ideal k, ideal j) |
---|
[18dd47] | 356 | { |
---|
[70ab73] | 357 | return(stdIdealsEqual(std(k),std(j))); |
---|
[d6db1f2] | 358 | } |
---|
| 359 | |
---|
[07c623] | 360 | static proc specialIdealsEqual( ideal k1, ideal k2) |
---|
[d6db1f2] | 361 | { |
---|
[70ab73] | 362 | int j; |
---|
[d6db1f2] | 363 | |
---|
[70ab73] | 364 | if(size(k1)==size(k2)) |
---|
| 365 | { |
---|
| 366 | for(j=1;j<=size(k1);j++) |
---|
| 367 | { |
---|
| 368 | if(leadexp(k1[j])!=leadexp(k2[j])) |
---|
[d6db1f2] | 369 | { |
---|
[70ab73] | 370 | return(0); |
---|
[d6db1f2] | 371 | } |
---|
[70ab73] | 372 | } |
---|
| 373 | return(1); |
---|
| 374 | } |
---|
| 375 | return(0); |
---|
[d6db1f2] | 376 | } |
---|
| 377 | |
---|
[07c623] | 378 | static proc stdIdealsEqual( ideal k1, ideal k2) |
---|
[d6db1f2] | 379 | { |
---|
[70ab73] | 380 | int j; |
---|
[d6db1f2] | 381 | |
---|
[70ab73] | 382 | if(size(k1)==size(k2)) |
---|
| 383 | { |
---|
| 384 | for(j=1;j<=size(k1);j++) |
---|
| 385 | { |
---|
| 386 | if(leadexp(k1[j])!=leadexp(k2[j])) |
---|
[d6db1f2] | 387 | { |
---|
[70ab73] | 388 | return(0); |
---|
[d6db1f2] | 389 | } |
---|
[70ab73] | 390 | } |
---|
| 391 | attrib(k2,"isSB",1); |
---|
| 392 | if(size(reduce(k1,k2,1))==0) |
---|
| 393 | { |
---|
| 394 | return(1); |
---|
| 395 | } |
---|
| 396 | } |
---|
| 397 | return(0); |
---|
[d6db1f2] | 398 | } |
---|
[091424] | 399 | /////////////////////////////////////////////////////////////////////////////// |
---|
[d6db1f2] | 400 | |
---|
[50cbdc] | 401 | proc primaryTest (ideal i, poly p) |
---|
[d6db1f2] | 402 | { |
---|
[70ab73] | 403 | int m=1; |
---|
| 404 | int n=nvars(basering); |
---|
| 405 | int e,f; |
---|
| 406 | poly t; |
---|
| 407 | ideal h; |
---|
| 408 | list act; |
---|
[d6db1f2] | 409 | |
---|
[70ab73] | 410 | ideal prm=p; |
---|
| 411 | attrib(prm,"isSB",1); |
---|
[d6db1f2] | 412 | |
---|
[70ab73] | 413 | while (n>1) |
---|
| 414 | { |
---|
| 415 | n--; |
---|
| 416 | m++; |
---|
[d6db1f2] | 417 | |
---|
[70ab73] | 418 | //search for i[m] which has a power of var(n) as leading term |
---|
| 419 | if (n==1) |
---|
| 420 | { |
---|
| 421 | m=size(i); |
---|
| 422 | } |
---|
| 423 | else |
---|
| 424 | { |
---|
| 425 | while (lead(i[m])/var(n-1)==0) |
---|
[d6db1f2] | 426 | { |
---|
[70ab73] | 427 | m++; |
---|
[d6db1f2] | 428 | } |
---|
[70ab73] | 429 | m--; |
---|
| 430 | } |
---|
| 431 | //check whether i[m] =(c*var(n)+h)^e modulo prm for some |
---|
| 432 | //h in K[var(n+1),...,var(nvars(basering))], c in K |
---|
| 433 | //if not (0) is returned, else var(n)+h is added to prm |
---|
| 434 | |
---|
| 435 | e=deg(lead(i[m])); |
---|
| 436 | if(char(basering)!=0) |
---|
| 437 | { |
---|
| 438 | f=1; |
---|
| 439 | if(e mod char(basering)==0) |
---|
[d6db1f2] | 440 | { |
---|
[70ab73] | 441 | if ( voice >=15 ) |
---|
[d6db1f2] | 442 | { |
---|
[70ab73] | 443 | "// WARNING: The characteristic is perhaps too small to use"; |
---|
| 444 | "// the algorithm of Gianni/Trager/Zacharias."; |
---|
| 445 | "// This may result in an infinte loop"; |
---|
| 446 | "// loop in primaryTest, voice:",voice;""; |
---|
| 447 | } |
---|
| 448 | while(e mod char(basering)==0) |
---|
| 449 | { |
---|
| 450 | f=f*char(basering); |
---|
[85ba0a] | 451 | e=e div char(basering); |
---|
[a3432c] | 452 | } |
---|
[971ba6f] | 453 | } |
---|
[70ab73] | 454 | t=leadcoef(i[m])*e*var(n)^f+(i[m]-lead(i[m]))/var(n)^((e-1)*f); |
---|
| 455 | i[m]=poly(e)^e*leadcoef(i[m])^(e-1)*i[m]; |
---|
| 456 | if (reduce(i[m]-t^e,prm,1) !=0) |
---|
| 457 | { |
---|
| 458 | return(ideal(0)); |
---|
| 459 | } |
---|
| 460 | if(f>1) |
---|
| 461 | { |
---|
| 462 | act=factorize(t); |
---|
| 463 | if(size(act[1])>2) |
---|
| 464 | { |
---|
| 465 | return(ideal(0)); |
---|
| 466 | } |
---|
| 467 | if(deg(act[1][2])>1) |
---|
| 468 | { |
---|
| 469 | return(ideal(0)); |
---|
| 470 | } |
---|
| 471 | t=act[1][2]; |
---|
| 472 | } |
---|
| 473 | } |
---|
| 474 | else |
---|
| 475 | { |
---|
| 476 | t=leadcoef(i[m])*e*var(n)+(i[m]-lead(i[m]))/var(n)^(e-1); |
---|
| 477 | i[m]=poly(e)^e*leadcoef(i[m])^(e-1)*i[m]; |
---|
| 478 | if (reduce(i[m]-t^e,prm,1) !=0) |
---|
[a3432c] | 479 | { |
---|
[70ab73] | 480 | return(ideal(0)); |
---|
[a3432c] | 481 | } |
---|
[70ab73] | 482 | } |
---|
[6ffa84] | 483 | |
---|
[70ab73] | 484 | h=interred(t); |
---|
| 485 | t=h[1]; |
---|
[d6db1f2] | 486 | |
---|
[70ab73] | 487 | prm = prm,t; |
---|
| 488 | attrib(prm,"isSB",1); |
---|
| 489 | } |
---|
| 490 | return(prm); |
---|
[d6db1f2] | 491 | } |
---|
| 492 | |
---|
[d12f079] | 493 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 494 | proc gcdTest(ideal act) |
---|
| 495 | { |
---|
| 496 | int i,j; |
---|
| 497 | if(size(act)<=1) |
---|
| 498 | { |
---|
[70ab73] | 499 | return(0); |
---|
[d12f079] | 500 | } |
---|
| 501 | for (i=1;i<=size(act)-1;i++) |
---|
| 502 | { |
---|
[70ab73] | 503 | for(j=i+1;j<=size(act);j++) |
---|
| 504 | { |
---|
| 505 | if(deg(std(ideal(act[i],act[j]))[1])>0) |
---|
| 506 | { |
---|
| 507 | return(0); |
---|
| 508 | } |
---|
| 509 | } |
---|
[d12f079] | 510 | } |
---|
| 511 | return(1); |
---|
| 512 | } |
---|
[d6db1f2] | 513 | |
---|
| 514 | /////////////////////////////////////////////////////////////////////////////// |
---|
[07c623] | 515 | static proc splitPrimary(list l,ideal ser,int @wr,list sact) |
---|
[d6db1f2] | 516 | { |
---|
[70ab73] | 517 | int i,j,k,s,r,w; |
---|
| 518 | list keepresult,act,keepprime; |
---|
| 519 | poly @f; |
---|
| 520 | int sl=size(l); |
---|
[4173c7] | 521 | for(i=1;i<=sl div 2;i++) |
---|
[70ab73] | 522 | { |
---|
| 523 | if(sact[2][i]>1) |
---|
| 524 | { |
---|
| 525 | keepprime[i]=l[2*i-1]+ideal(sact[1][i]); |
---|
| 526 | } |
---|
| 527 | else |
---|
| 528 | { |
---|
| 529 | keepprime[i]=l[2*i-1]; |
---|
| 530 | } |
---|
| 531 | } |
---|
| 532 | i=0; |
---|
[4173c7] | 533 | while(i<size(l) div 2) |
---|
[70ab73] | 534 | { |
---|
| 535 | i++; |
---|
| 536 | if((size(ser)>0)&&(size(reduce(ser,l[2*i-1],1))==0)) |
---|
| 537 | { |
---|
| 538 | l[2*i-1]=ideal(1); |
---|
| 539 | l[2*i]=ideal(1); |
---|
| 540 | continue; |
---|
| 541 | } |
---|
| 542 | |
---|
| 543 | if(size(l[2*i])==0) |
---|
| 544 | { |
---|
| 545 | if(homog(l[2*i-1])==1) |
---|
[d6db1f2] | 546 | { |
---|
[70ab73] | 547 | l[2*i]=maxideal(1); |
---|
| 548 | continue; |
---|
[d6db1f2] | 549 | } |
---|
[70ab73] | 550 | j=0; |
---|
| 551 | /* |
---|
[4173c7] | 552 | if(i<=sl div 2) |
---|
[d6db1f2] | 553 | { |
---|
[70ab73] | 554 | j=1; |
---|
[d6db1f2] | 555 | } |
---|
[70ab73] | 556 | */ |
---|
| 557 | while(j<size(l[2*i-1])) |
---|
[d6db1f2] | 558 | { |
---|
[70ab73] | 559 | j++; |
---|
| 560 | act=factor(l[2*i-1][j]); |
---|
| 561 | r=size(act[1]); |
---|
| 562 | attrib(l[2*i-1],"isSB",1); |
---|
| 563 | if((r==1)&&(vdim(l[2*i-1])==deg(l[2*i-1][j]))) |
---|
| 564 | { |
---|
| 565 | l[2*i]=std(l[2*i-1],act[1][1]); |
---|
| 566 | break; |
---|
| 567 | } |
---|
| 568 | if((r==1)&&(act[2][1]>1)) |
---|
| 569 | { |
---|
| 570 | keepprime[i]=interred(keepprime[i]+ideal(act[1][1])); |
---|
| 571 | if(homog(keepprime[i])==1) |
---|
| 572 | { |
---|
[d6db1f2] | 573 | l[2*i]=maxideal(1); |
---|
[70ab73] | 574 | break; |
---|
| 575 | } |
---|
| 576 | } |
---|
| 577 | if(gcdTest(act[1])==1) |
---|
| 578 | { |
---|
| 579 | for(k=2;k<=r;k++) |
---|
| 580 | { |
---|
[4173c7] | 581 | keepprime[size(l) div 2+k-1]=interred(keepprime[i]+ideal(act[1][k])); |
---|
[70ab73] | 582 | } |
---|
| 583 | keepprime[i]=interred(keepprime[i]+ideal(act[1][1])); |
---|
| 584 | for(k=1;k<=r;k++) |
---|
| 585 | { |
---|
| 586 | if(@wr==0) |
---|
| 587 | { |
---|
| 588 | keepresult[k]=std(l[2*i-1],act[1][k]^act[2][k]); |
---|
| 589 | } |
---|
| 590 | else |
---|
| 591 | { |
---|
| 592 | keepresult[k]=std(l[2*i-1],act[1][k]); |
---|
| 593 | } |
---|
| 594 | } |
---|
| 595 | l[2*i-1]=keepresult[1]; |
---|
| 596 | if(vdim(keepresult[1])==deg(act[1][1])) |
---|
| 597 | { |
---|
| 598 | l[2*i]=keepresult[1]; |
---|
| 599 | } |
---|
| 600 | if((homog(keepresult[1])==1)||(homog(keepprime[i])==1)) |
---|
| 601 | { |
---|
| 602 | l[2*i]=maxideal(1); |
---|
| 603 | } |
---|
| 604 | s=size(l)-2; |
---|
| 605 | for(k=2;k<=r;k++) |
---|
| 606 | { |
---|
| 607 | l[s+2*k-1]=keepresult[k]; |
---|
[4173c7] | 608 | keepprime[s div 2+k]=interred(keepresult[k]+ideal(act[1][k])); |
---|
[70ab73] | 609 | if(vdim(keepresult[k])==deg(act[1][k])) |
---|
| 610 | { |
---|
| 611 | l[s+2*k]=keepresult[k]; |
---|
| 612 | } |
---|
| 613 | else |
---|
| 614 | { |
---|
| 615 | l[s+2*k]=ideal(0); |
---|
| 616 | } |
---|
[4173c7] | 617 | if((homog(keepresult[k])==1)||(homog(keepprime[s div 2+k])==1)) |
---|
[70ab73] | 618 | { |
---|
| 619 | l[s+2*k]=maxideal(1); |
---|
| 620 | } |
---|
| 621 | } |
---|
| 622 | i--; |
---|
| 623 | break; |
---|
| 624 | } |
---|
| 625 | if(r>=2) |
---|
| 626 | { |
---|
| 627 | s=size(l); |
---|
| 628 | @f=act[1][1]; |
---|
| 629 | act=sat1(l[2*i-1],act[1][1]); |
---|
| 630 | if(deg(act[1][1])>0) |
---|
| 631 | { |
---|
| 632 | l[s+1]=std(l[2*i-1],act[2]); |
---|
| 633 | if(homog(l[s+1])==1) |
---|
| 634 | { |
---|
| 635 | l[s+2]=maxideal(1); |
---|
| 636 | } |
---|
| 637 | else |
---|
| 638 | { |
---|
| 639 | l[s+2]=ideal(0); |
---|
[d6db1f2] | 640 | } |
---|
[4173c7] | 641 | keepprime[s div 2+1]=interred(keepprime[i]+ideal(@f)); |
---|
| 642 | if(homog(keepprime[s div 2+1])==1) |
---|
[18dd47] | 643 | { |
---|
[70ab73] | 644 | l[s+2]=maxideal(1); |
---|
[d6db1f2] | 645 | } |
---|
[70ab73] | 646 | keepprime[i]=act[1]; |
---|
| 647 | l[2*i-1]=act[1]; |
---|
| 648 | attrib(l[2*i-1],"isSB",1); |
---|
| 649 | if(homog(l[2*i-1])==1) |
---|
[d6db1f2] | 650 | { |
---|
[70ab73] | 651 | l[2*i]=maxideal(1); |
---|
[d6db1f2] | 652 | } |
---|
[70ab73] | 653 | i--; |
---|
| 654 | break; |
---|
| 655 | } |
---|
| 656 | } |
---|
[d6db1f2] | 657 | } |
---|
[70ab73] | 658 | } |
---|
| 659 | } |
---|
| 660 | if(sl==size(l)) |
---|
| 661 | { |
---|
| 662 | return(l); |
---|
| 663 | } |
---|
[4173c7] | 664 | for(i=1;i<=size(l) div 2;i++) |
---|
[70ab73] | 665 | { |
---|
| 666 | attrib(l[2*i-1],"isSB",1); |
---|
[3939bc] | 667 | |
---|
[70ab73] | 668 | if((size(ser)>0)&&(size(reduce(ser,l[2*i-1],1))==0)&&(deg(l[2*i-1][1])>0)) |
---|
| 669 | { |
---|
| 670 | "Achtung in split"; |
---|
[3939bc] | 671 | |
---|
[70ab73] | 672 | l[2*i-1]=ideal(1); |
---|
| 673 | l[2*i]=ideal(1); |
---|
| 674 | } |
---|
| 675 | if((size(l[2*i])==0)&&(specialIdealsEqual(keepprime[i],l[2*i-1])!=1)) |
---|
| 676 | { |
---|
| 677 | keepprime[i]=std(keepprime[i]); |
---|
| 678 | if(homog(keepprime[i])==1) |
---|
| 679 | { |
---|
| 680 | l[2*i]=maxideal(1); |
---|
[d6db1f2] | 681 | } |
---|
[70ab73] | 682 | else |
---|
| 683 | { |
---|
| 684 | act=zero_decomp(keepprime[i],ideal(0),@wr,1); |
---|
| 685 | if(size(act)==2) |
---|
| 686 | { |
---|
| 687 | l[2*i]=act[2]; |
---|
| 688 | } |
---|
| 689 | } |
---|
| 690 | } |
---|
| 691 | } |
---|
| 692 | return(l); |
---|
[d6db1f2] | 693 | } |
---|
| 694 | example |
---|
| 695 | { "EXAMPLE:"; echo=2; |
---|
| 696 | ring r = 32003,(x,y,z),lp; |
---|
| 697 | ideal i1=x*(x+1),yz,(z+1)*(z-1); |
---|
| 698 | ideal i2=xy,yz,(x-2)*(x+3); |
---|
| 699 | list l=i1,ideal(0),i2,ideal(0),i2,ideal(1); |
---|
| 700 | list l1=splitPrimary(l,ideal(0),0); |
---|
| 701 | l1; |
---|
| 702 | } |
---|
[651953] | 703 | /////////////////////////////////////////////////////////////////////////////// |
---|
[07c623] | 704 | static proc splitCharp(list l) |
---|
[651953] | 705 | { |
---|
| 706 | if((char(basering)==0)||(npars(basering)>0)) |
---|
| 707 | { |
---|
[70ab73] | 708 | return(l); |
---|
[651953] | 709 | } |
---|
| 710 | def P=basering; |
---|
[24f458] | 711 | int i,j,k,m,q,d,o; |
---|
[651953] | 712 | int n=nvars(basering); |
---|
| 713 | ideal s,t,u,sact; |
---|
| 714 | poly ni; |
---|
| 715 | string minp,gnir,va; |
---|
[24f458] | 716 | list sa,keep,rp,keep1; |
---|
[4173c7] | 717 | for(i=1;i<=size(l) div 2;i++) |
---|
[651953] | 718 | { |
---|
| 719 | if(size(l[2*i])==0) |
---|
| 720 | { |
---|
[70ab73] | 721 | if(deg(l[2*i-1][1])==vdim(l[2*i-1])) |
---|
| 722 | { |
---|
| 723 | l[2*i]=l[2*i-1]; |
---|
| 724 | } |
---|
[651953] | 725 | } |
---|
| 726 | } |
---|
[4173c7] | 727 | for(i=1;i<=size(l) div 2;i++) |
---|
[651953] | 728 | { |
---|
| 729 | if(size(l[2*i])==0) |
---|
| 730 | { |
---|
[24f458] | 731 | s=factorize(l[2*i-1][1],1); //vermeiden!!! |
---|
[651953] | 732 | t=l[2*i-1]; |
---|
| 733 | m=size(t); |
---|
| 734 | ni=s[1]; |
---|
| 735 | if(deg(ni)>1) |
---|
| 736 | { |
---|
| 737 | va=varstr(P); |
---|
| 738 | j=size(va); |
---|
| 739 | while(va[j]!=","){j--;} |
---|
| 740 | va=va[1..j-1]; |
---|
[24f458] | 741 | gnir="ring RL=("+string(char(P))+","+string(var(n))+"),("+va+"),lp;"; |
---|
[651953] | 742 | execute(gnir); |
---|
| 743 | minpoly=leadcoef(imap(P,ni)); |
---|
| 744 | ideal act; |
---|
| 745 | ideal t=imap(P,t); |
---|
[24f458] | 746 | |
---|
[651953] | 747 | for(k=2;k<=m;k++) |
---|
[b9b906] | 748 | { |
---|
[70ab73] | 749 | act=factorize(t[k],1); |
---|
| 750 | if(size(act)>1){break;} |
---|
[651953] | 751 | } |
---|
| 752 | setring P; |
---|
| 753 | sact=imap(RL,act); |
---|
[24f458] | 754 | |
---|
[651953] | 755 | if(size(sact)>1) |
---|
| 756 | { |
---|
[70ab73] | 757 | sa=sat1(l[2*i-1],sact[1]); |
---|
| 758 | keep[size(keep)+1]=std(l[2*i-1],sa[2]); |
---|
| 759 | l[2*i-1]=std(sa[1]); |
---|
| 760 | l[2*i]=primaryTest(sa[1],sa[1][1]); |
---|
[651953] | 761 | } |
---|
[24f458] | 762 | if((size(sact)==1)&&(m==2)) |
---|
| 763 | { |
---|
[70ab73] | 764 | l[2*i]=l[2*i-1]; |
---|
| 765 | attrib(l[2*i],"isSB",1); |
---|
[24f458] | 766 | } |
---|
| 767 | if((size(sact)==1)&&(m>2)) |
---|
| 768 | { |
---|
[70ab73] | 769 | setring RL; |
---|
| 770 | option(redSB); |
---|
| 771 | t=std(t); |
---|
| 772 | |
---|
| 773 | list sp=zero_decomp(t,0,0); |
---|
| 774 | |
---|
| 775 | setring P; |
---|
| 776 | rp=imap(RL,sp); |
---|
| 777 | for(o=1;o<=size(rp);o++) |
---|
| 778 | { |
---|
| 779 | rp[o]=interred(simplify(rp[o],1)+ideal(ni)); |
---|
| 780 | } |
---|
| 781 | l[2*i-1]=rp[1]; |
---|
| 782 | l[2*i]=rp[2]; |
---|
| 783 | rp=delete(rp,1); |
---|
| 784 | rp=delete(rp,1); |
---|
| 785 | keep1=keep1+rp; |
---|
| 786 | option(noredSB); |
---|
[24f458] | 787 | } |
---|
| 788 | kill RL; |
---|
[651953] | 789 | } |
---|
| 790 | } |
---|
| 791 | } |
---|
| 792 | if(size(keep)>0) |
---|
| 793 | { |
---|
| 794 | for(i=1;i<=size(keep);i++) |
---|
| 795 | { |
---|
[70ab73] | 796 | if(deg(keep[i][1])>0) |
---|
| 797 | { |
---|
| 798 | l[size(l)+1]=keep[i]; |
---|
| 799 | l[size(l)+1]=primaryTest(keep[i],keep[i][1]); |
---|
| 800 | } |
---|
[651953] | 801 | } |
---|
| 802 | } |
---|
[24f458] | 803 | l=l+keep1; |
---|
[651953] | 804 | return(l); |
---|
| 805 | } |
---|
[d6db1f2] | 806 | |
---|
[091424] | 807 | /////////////////////////////////////////////////////////////////////////////// |
---|
[d6db1f2] | 808 | |
---|
[24f458] | 809 | proc zero_decomp (ideal j,ideal ser,int @wr,list #) |
---|
[d2b2a7] | 810 | "USAGE: zero_decomp(j,ser,@wr); j,ser ideals, @wr=0 or 1 |
---|
[7f7c25e] | 811 | (@wr=0 for primary decomposition, @wr=1 for computation of associated |
---|
[d6db1f2] | 812 | primes) |
---|
| 813 | RETURN: list = list of primary ideals and their radicals (at even positions |
---|
| 814 | in the list) if the input is zero-dimensional and a standardbases |
---|
| 815 | with respect to lex-ordering |
---|
| 816 | If ser!=(0) and ser is contained in j or if j is not zero-dimen- |
---|
| 817 | sional then ideal(1),ideal(1) is returned |
---|
[7b3971] | 818 | NOTE: Algorithm of Gianni/Trager/Zacharias |
---|
[d6db1f2] | 819 | EXAMPLE: example zero_decomp; shows an example |
---|
[d2b2a7] | 820 | " |
---|
[d6db1f2] | 821 | { |
---|
| 822 | def @P = basering; |
---|
[20057b] | 823 | int uytrewq; |
---|
[d6db1f2] | 824 | int nva = nvars(basering); |
---|
[e801fe] | 825 | int @k,@s,@n,@k1,zz; |
---|
[a39a07] | 826 | list primary,lres0,lres1,act,@lh,@wh; |
---|
[e801fe] | 827 | map phi,psi,phi1,psi1; |
---|
| 828 | ideal jmap,jmap1,jmap2,helpprim,@qh,@qht,ser1; |
---|
[d6db1f2] | 829 | intvec @vh,@hilb; |
---|
| 830 | string @ri; |
---|
| 831 | poly @f; |
---|
| 832 | if (dim(j)>0) |
---|
| 833 | { |
---|
[70ab73] | 834 | primary[1]=ideal(1); |
---|
| 835 | primary[2]=ideal(1); |
---|
| 836 | return(primary); |
---|
[d6db1f2] | 837 | } |
---|
[a90eb0] | 838 | intvec save=option(get); |
---|
| 839 | option(redSB); |
---|
[3939bc] | 840 | j=interred(j); |
---|
[0bcebab] | 841 | |
---|
[d6db1f2] | 842 | attrib(j,"isSB",1); |
---|
[24f458] | 843 | |
---|
[d6db1f2] | 844 | if(vdim(j)==deg(j[1])) |
---|
[3939bc] | 845 | { |
---|
[70ab73] | 846 | act=factor(j[1]); |
---|
| 847 | for(@k=1;@k<=size(act[1]);@k++) |
---|
| 848 | { |
---|
| 849 | @qh=j; |
---|
| 850 | if(@wr==0) |
---|
| 851 | { |
---|
| 852 | @qh[1]=act[1][@k]^act[2][@k]; |
---|
| 853 | } |
---|
| 854 | else |
---|
| 855 | { |
---|
| 856 | @qh[1]=act[1][@k]; |
---|
| 857 | } |
---|
| 858 | primary[2*@k-1]=interred(@qh); |
---|
| 859 | @qh=j; |
---|
| 860 | @qh[1]=act[1][@k]; |
---|
| 861 | primary[2*@k]=interred(@qh); |
---|
| 862 | attrib( primary[2*@k-1],"isSB",1); |
---|
[3939bc] | 863 | |
---|
[70ab73] | 864 | if((size(ser)>0)&&(size(reduce(ser,primary[2*@k-1],1))==0)) |
---|
| 865 | { |
---|
| 866 | primary[2*@k-1]=ideal(1); |
---|
| 867 | primary[2*@k]=ideal(1); |
---|
| 868 | } |
---|
| 869 | } |
---|
[a90eb0] | 870 | option(set,save); |
---|
[70ab73] | 871 | return(primary); |
---|
[d6db1f2] | 872 | } |
---|
| 873 | |
---|
[a90eb0] | 874 | option(set,save); |
---|
[d6db1f2] | 875 | if(homog(j)==1) |
---|
| 876 | { |
---|
[70ab73] | 877 | primary[1]=j; |
---|
| 878 | if((size(ser)>0)&&(size(reduce(ser,j,1))==0)) |
---|
| 879 | { |
---|
| 880 | primary[1]=ideal(1); |
---|
| 881 | primary[2]=ideal(1); |
---|
| 882 | return(primary); |
---|
| 883 | } |
---|
| 884 | if(dim(j)==-1) |
---|
| 885 | { |
---|
| 886 | primary[1]=ideal(1); |
---|
| 887 | primary[2]=ideal(1); |
---|
| 888 | } |
---|
| 889 | else |
---|
| 890 | { |
---|
| 891 | primary[2]=maxideal(1); |
---|
| 892 | } |
---|
| 893 | return(primary); |
---|
[d6db1f2] | 894 | } |
---|
[18dd47] | 895 | |
---|
[d6db1f2] | 896 | //the first element in the standardbase is factorized |
---|
| 897 | if(deg(j[1])>0) |
---|
| 898 | { |
---|
| 899 | act=factor(j[1]); |
---|
| 900 | testFactor(act,j[1]); |
---|
| 901 | } |
---|
| 902 | else |
---|
| 903 | { |
---|
[70ab73] | 904 | primary[1]=ideal(1); |
---|
| 905 | primary[2]=ideal(1); |
---|
| 906 | return(primary); |
---|
[d6db1f2] | 907 | } |
---|
| 908 | |
---|
[9050ca] | 909 | //with the factors new ideals (hopefully the primary decomposition) |
---|
[d6db1f2] | 910 | //are created |
---|
| 911 | if(size(act[1])>1) |
---|
| 912 | { |
---|
[70ab73] | 913 | if(size(#)>1) |
---|
| 914 | { |
---|
| 915 | primary[1]=ideal(1); |
---|
| 916 | primary[2]=ideal(1); |
---|
| 917 | primary[3]=ideal(1); |
---|
| 918 | primary[4]=ideal(1); |
---|
| 919 | return(primary); |
---|
| 920 | } |
---|
| 921 | for(@k=1;@k<=size(act[1]);@k++) |
---|
| 922 | { |
---|
| 923 | if(@wr==0) |
---|
| 924 | { |
---|
| 925 | primary[2*@k-1]=std(j,act[1][@k]^act[2][@k]); |
---|
| 926 | } |
---|
| 927 | else |
---|
| 928 | { |
---|
| 929 | primary[2*@k-1]=std(j,act[1][@k]); |
---|
| 930 | } |
---|
| 931 | if((act[2][@k]==1)&&(vdim(primary[2*@k-1])==deg(act[1][@k]))) |
---|
| 932 | { |
---|
[a36e78] | 933 | primary[2*@k] = primary[2*@k-1]; |
---|
[70ab73] | 934 | } |
---|
| 935 | else |
---|
| 936 | { |
---|
| 937 | primary[2*@k] = primaryTest(primary[2*@k-1],act[1][@k]); |
---|
| 938 | } |
---|
| 939 | } |
---|
[d6db1f2] | 940 | } |
---|
| 941 | else |
---|
[3939bc] | 942 | { |
---|
[70ab73] | 943 | primary[1]=j; |
---|
| 944 | if((size(#)>0)&&(act[2][1]>1)) |
---|
| 945 | { |
---|
| 946 | act[2]=1; |
---|
| 947 | primary[1]=std(primary[1],act[1][1]); |
---|
| 948 | } |
---|
| 949 | if(@wr!=0) |
---|
| 950 | { |
---|
| 951 | primary[1]=std(j,act[1][1]); |
---|
| 952 | } |
---|
| 953 | if((act[2][1]==1)&&(vdim(primary[1])==deg(act[1][1]))) |
---|
| 954 | { |
---|
| 955 | primary[2]=primary[1]; |
---|
| 956 | } |
---|
| 957 | else |
---|
| 958 | { |
---|
| 959 | primary[2]=primaryTest(primary[1],act[1][1]); |
---|
| 960 | } |
---|
[d6db1f2] | 961 | } |
---|
[50cbdc] | 962 | |
---|
[d6db1f2] | 963 | if(size(#)==0) |
---|
| 964 | { |
---|
[70ab73] | 965 | primary=splitPrimary(primary,ser,@wr,act); |
---|
[d6db1f2] | 966 | } |
---|
[24f458] | 967 | |
---|
| 968 | if((voice>=6)&&(char(basering)<=181)) |
---|
| 969 | { |
---|
[70ab73] | 970 | primary=splitCharp(primary); |
---|
[24f458] | 971 | } |
---|
| 972 | |
---|
| 973 | if((@wr==2)&&(npars(basering)>0)&&(voice>=6)&&(char(basering)>0)) |
---|
| 974 | { |
---|
| 975 | //the prime decomposition of Yokoyama in characteristic p |
---|
[70ab73] | 976 | list ke,ek; |
---|
| 977 | @k=0; |
---|
[4173c7] | 978 | while(@k<size(primary) div 2) |
---|
[70ab73] | 979 | { |
---|
| 980 | @k++; |
---|
| 981 | if(size(primary[2*@k])==0) |
---|
| 982 | { |
---|
| 983 | ek=insepDecomp(primary[2*@k-1]); |
---|
| 984 | primary=delete(primary,2*@k); |
---|
| 985 | primary=delete(primary,2*@k-1); |
---|
| 986 | @k--; |
---|
| 987 | } |
---|
| 988 | ke=ke+ek; |
---|
| 989 | } |
---|
| 990 | for(@k=1;@k<=size(ke);@k++) |
---|
| 991 | { |
---|
| 992 | primary[size(primary)+1]=ke[@k]; |
---|
| 993 | primary[size(primary)+1]=ke[@k]; |
---|
| 994 | } |
---|
[24f458] | 995 | } |
---|
| 996 | |
---|
[b15849d] | 997 | if(voice>=8){primary=extF(primary);}; |
---|
[24f458] | 998 | |
---|
[d6db1f2] | 999 | //test whether all ideals in the decomposition are primary and |
---|
| 1000 | //in general position |
---|
| 1001 | //if not after a random coordinate transformation of the last |
---|
| 1002 | //variable the corresponding ideal is decomposed again. |
---|
[24f458] | 1003 | if((npars(basering)>0)&&(voice>=8)) |
---|
| 1004 | { |
---|
[70ab73] | 1005 | poly randp; |
---|
| 1006 | for(zz=1;zz<nvars(basering);zz++) |
---|
| 1007 | { |
---|
| 1008 | randp=randp |
---|
[24f458] | 1009 | +(random(0,5)*par(1)^2+random(0,5)*par(1)+random(0,5))*var(zz); |
---|
[70ab73] | 1010 | } |
---|
| 1011 | randp=randp+var(nvars(basering)); |
---|
[24f458] | 1012 | } |
---|
[d6db1f2] | 1013 | @k=0; |
---|
[4173c7] | 1014 | while(@k<(size(primary) div 2)) |
---|
[d6db1f2] | 1015 | { |
---|
| 1016 | @k++; |
---|
| 1017 | if (size(primary[2*@k])==0) |
---|
| 1018 | { |
---|
[70ab73] | 1019 | for(zz=1;zz<size(primary[2*@k-1])-1;zz++) |
---|
| 1020 | { |
---|
| 1021 | attrib(primary[2*@k-1],"isSB",1); |
---|
| 1022 | if(vdim(primary[2*@k-1])==deg(primary[2*@k-1][zz])) |
---|
| 1023 | { |
---|
| 1024 | primary[2*@k]=primary[2*@k-1]; |
---|
| 1025 | } |
---|
| 1026 | } |
---|
[67bd4c] | 1027 | } |
---|
| 1028 | } |
---|
[3939bc] | 1029 | |
---|
[67bd4c] | 1030 | @k=0; |
---|
[e801fe] | 1031 | ideal keep; |
---|
[4173c7] | 1032 | while(@k<(size(primary) div 2)) |
---|
[67bd4c] | 1033 | { |
---|
| 1034 | @k++; |
---|
| 1035 | if (size(primary[2*@k])==0) |
---|
| 1036 | { |
---|
[70ab73] | 1037 | jmap=randomLast(100); |
---|
| 1038 | jmap1=maxideal(1); |
---|
| 1039 | jmap2=maxideal(1); |
---|
| 1040 | @qht=primary[2*@k-1]; |
---|
| 1041 | if((npars(basering)>0)&&(voice>=10)) |
---|
| 1042 | { |
---|
| 1043 | jmap[size(jmap)]=randp; |
---|
| 1044 | } |
---|
[67bd4c] | 1045 | |
---|
[70ab73] | 1046 | for(@n=2;@n<=size(primary[2*@k-1]);@n++) |
---|
| 1047 | { |
---|
| 1048 | if(deg(lead(primary[2*@k-1][@n]))==1) |
---|
| 1049 | { |
---|
| 1050 | for(zz=1;zz<=nva;zz++) |
---|
[d6db1f2] | 1051 | { |
---|
[70ab73] | 1052 | if(lead(primary[2*@k-1][@n])/var(zz)!=0) |
---|
| 1053 | { |
---|
| 1054 | jmap1[zz]=-1/leadcoef(primary[2*@k-1][@n])*primary[2*@k-1][@n] |
---|
[a36e78] | 1055 | +2/leadcoef(primary[2*@k-1][@n])*lead(primary[2*@k-1][@n]); |
---|
[70ab73] | 1056 | jmap2[zz]=primary[2*@k-1][@n]; |
---|
| 1057 | @qht[@n]=var(zz); |
---|
| 1058 | } |
---|
[d6db1f2] | 1059 | } |
---|
[70ab73] | 1060 | jmap[nva]=subst(jmap[nva],lead(primary[2*@k-1][@n]),0); |
---|
| 1061 | } |
---|
| 1062 | } |
---|
| 1063 | if(size(subst(jmap[nva],var(1),0)-var(nva))!=0) |
---|
| 1064 | { |
---|
| 1065 | // jmap[nva]=subst(jmap[nva],var(1),0); |
---|
| 1066 | //hier geaendert +untersuchen!!!!!!!!!!!!!! |
---|
| 1067 | } |
---|
| 1068 | phi1=@P,jmap1; |
---|
| 1069 | phi=@P,jmap; |
---|
| 1070 | for(@n=1;@n<=nva;@n++) |
---|
| 1071 | { |
---|
| 1072 | jmap[@n]=-(jmap[@n]-2*var(@n)); |
---|
| 1073 | } |
---|
| 1074 | psi=@P,jmap; |
---|
| 1075 | psi1=@P,jmap2; |
---|
| 1076 | @qh=phi(@qht); |
---|
[24f458] | 1077 | |
---|
| 1078 | //=================== the new part ============================ |
---|
| 1079 | |
---|
[8992ed] | 1080 | if (npars(basering)>1) { @qh=groebner(@qh,"par2var"); } |
---|
| 1081 | else { @qh=groebner(@qh); } |
---|
[24f458] | 1082 | |
---|
| 1083 | //============================================================= |
---|
| 1084 | // if(npars(@P)>0) |
---|
| 1085 | // { |
---|
| 1086 | // @ri= "ring @Phelp =" |
---|
| 1087 | // +string(char(@P))+", |
---|
| 1088 | // ("+varstr(@P)+","+parstr(@P)+",@t),(C,dp);"; |
---|
| 1089 | // } |
---|
| 1090 | // else |
---|
| 1091 | // { |
---|
| 1092 | // @ri= "ring @Phelp =" |
---|
| 1093 | // +string(char(@P))+",("+varstr(@P)+",@t),(C,dp);"; |
---|
| 1094 | // } |
---|
| 1095 | // execute(@ri); |
---|
| 1096 | // ideal @qh=homog(imap(@P,@qht),@t); |
---|
| 1097 | // |
---|
| 1098 | // ideal @qh1=std(@qh); |
---|
| 1099 | // @hilb=hilb(@qh1,1); |
---|
| 1100 | // @ri= "ring @Phelp1 =" |
---|
| 1101 | // +string(char(@P))+",("+varstr(@Phelp)+"),(C,lp);"; |
---|
| 1102 | // execute(@ri); |
---|
| 1103 | // ideal @qh=homog(imap(@P,@qh),@t); |
---|
| 1104 | // kill @Phelp; |
---|
| 1105 | // @qh=std(@qh,@hilb); |
---|
| 1106 | // @qh=subst(@qh,@t,1); |
---|
| 1107 | // setring @P; |
---|
| 1108 | // @qh=imap(@Phelp1,@qh); |
---|
| 1109 | // kill @Phelp1; |
---|
| 1110 | // @qh=clearSB(@qh); |
---|
| 1111 | // attrib(@qh,"isSB",1); |
---|
| 1112 | //============================================================= |
---|
| 1113 | |
---|
[70ab73] | 1114 | ser1=phi1(ser); |
---|
| 1115 | @lh=zero_decomp (@qh,phi(ser1),@wr); |
---|
[18dd47] | 1116 | |
---|
[70ab73] | 1117 | kill lres0; |
---|
| 1118 | list lres0; |
---|
| 1119 | if(size(@lh)==2) |
---|
| 1120 | { |
---|
| 1121 | helpprim=@lh[2]; |
---|
| 1122 | lres0[1]=primary[2*@k-1]; |
---|
| 1123 | ser1=psi(helpprim); |
---|
| 1124 | lres0[2]=psi1(ser1); |
---|
| 1125 | if(size(reduce(lres0[2],lres0[1],1))==0) |
---|
| 1126 | { |
---|
| 1127 | primary[2*@k]=primary[2*@k-1]; |
---|
| 1128 | continue; |
---|
| 1129 | } |
---|
| 1130 | } |
---|
| 1131 | else |
---|
| 1132 | { |
---|
| 1133 | lres1=psi(@lh); |
---|
| 1134 | lres0=psi1(lres1); |
---|
| 1135 | } |
---|
[d6db1f2] | 1136 | |
---|
[24f458] | 1137 | //=================== the new part ============================ |
---|
[d6db1f2] | 1138 | |
---|
[70ab73] | 1139 | primary=delete(primary,2*@k-1); |
---|
| 1140 | primary=delete(primary,2*@k-1); |
---|
| 1141 | @k--; |
---|
| 1142 | if(size(lres0)==2) |
---|
| 1143 | { |
---|
[a36e78] | 1144 | lres0[2]=groebner(lres0[2]); |
---|
[70ab73] | 1145 | } |
---|
| 1146 | else |
---|
| 1147 | { |
---|
[4173c7] | 1148 | for(@n=1;@n<=size(lres0) div 2;@n++) |
---|
[70ab73] | 1149 | { |
---|
| 1150 | if(specialIdealsEqual(lres0[2*@n-1],lres0[2*@n])==1) |
---|
[d6db1f2] | 1151 | { |
---|
[a36e78] | 1152 | lres0[2*@n-1]=groebner(lres0[2*@n-1]); |
---|
[70ab73] | 1153 | lres0[2*@n]=lres0[2*@n-1]; |
---|
| 1154 | attrib(lres0[2*@n],"isSB",1); |
---|
[d6db1f2] | 1155 | } |
---|
[70ab73] | 1156 | else |
---|
| 1157 | { |
---|
[a36e78] | 1158 | lres0[2*@n-1]=groebner(lres0[2*@n-1]); |
---|
| 1159 | lres0[2*@n]=groebner(lres0[2*@n]); |
---|
[70ab73] | 1160 | } |
---|
| 1161 | } |
---|
| 1162 | } |
---|
| 1163 | primary=primary+lres0; |
---|
[18dd47] | 1164 | |
---|
[24f458] | 1165 | //============================================================= |
---|
| 1166 | // if(npars(@P)>0) |
---|
| 1167 | // { |
---|
| 1168 | // @ri= "ring @Phelp =" |
---|
| 1169 | // +string(char(@P))+", |
---|
| 1170 | // ("+varstr(@P)+","+parstr(@P)+",@t),(C,dp);"; |
---|
| 1171 | // } |
---|
| 1172 | // else |
---|
| 1173 | // { |
---|
| 1174 | // @ri= "ring @Phelp =" |
---|
| 1175 | // +string(char(@P))+",("+varstr(@P)+",@t),(C,dp);"; |
---|
| 1176 | // } |
---|
| 1177 | // execute(@ri); |
---|
| 1178 | // list @lvec; |
---|
| 1179 | // list @lr=imap(@P,lres0); |
---|
| 1180 | // ideal @lr1; |
---|
| 1181 | // |
---|
| 1182 | // if(size(@lr)==2) |
---|
| 1183 | // { |
---|
| 1184 | // @lr[2]=homog(@lr[2],@t); |
---|
| 1185 | // @lr1=std(@lr[2]); |
---|
| 1186 | // @lvec[2]=hilb(@lr1,1); |
---|
| 1187 | // } |
---|
| 1188 | // else |
---|
| 1189 | // { |
---|
[4173c7] | 1190 | // for(@n=1;@n<=size(@lr) div 2;@n++) |
---|
[24f458] | 1191 | // { |
---|
| 1192 | // if(specialIdealsEqual(@lr[2*@n-1],@lr[2*@n])==1) |
---|
| 1193 | // { |
---|
| 1194 | // @lr[2*@n-1]=homog(@lr[2*@n-1],@t); |
---|
| 1195 | // @lr1=std(@lr[2*@n-1]); |
---|
| 1196 | // @lvec[2*@n-1]=hilb(@lr1,1); |
---|
| 1197 | // @lvec[2*@n]=@lvec[2*@n-1]; |
---|
| 1198 | // } |
---|
| 1199 | // else |
---|
| 1200 | // { |
---|
| 1201 | // @lr[2*@n-1]=homog(@lr[2*@n-1],@t); |
---|
| 1202 | // @lr1=std(@lr[2*@n-1]); |
---|
| 1203 | // @lvec[2*@n-1]=hilb(@lr1,1); |
---|
| 1204 | // @lr[2*@n]=homog(@lr[2*@n],@t); |
---|
| 1205 | // @lr1=std(@lr[2*@n]); |
---|
| 1206 | // @lvec[2*@n]=hilb(@lr1,1); |
---|
| 1207 | // |
---|
| 1208 | // } |
---|
| 1209 | // } |
---|
| 1210 | // } |
---|
| 1211 | // @ri= "ring @Phelp1 =" |
---|
| 1212 | // +string(char(@P))+",("+varstr(@Phelp)+"),(C,lp);"; |
---|
| 1213 | // execute(@ri); |
---|
| 1214 | // list @lr=imap(@Phelp,@lr); |
---|
| 1215 | // |
---|
| 1216 | // kill @Phelp; |
---|
| 1217 | // if(size(@lr)==2) |
---|
| 1218 | // { |
---|
| 1219 | // @lr[2]=std(@lr[2],@lvec[2]); |
---|
| 1220 | // @lr[2]=subst(@lr[2],@t,1); |
---|
| 1221 | // } |
---|
| 1222 | // else |
---|
| 1223 | // { |
---|
[4173c7] | 1224 | // for(@n=1;@n<=size(@lr) div 2;@n++) |
---|
[24f458] | 1225 | // { |
---|
| 1226 | // if(specialIdealsEqual(@lr[2*@n-1],@lr[2*@n])==1) |
---|
| 1227 | // { |
---|
| 1228 | // @lr[2*@n-1]=std(@lr[2*@n-1],@lvec[2*@n-1]); |
---|
| 1229 | // @lr[2*@n-1]=subst(@lr[2*@n-1],@t,1); |
---|
| 1230 | // @lr[2*@n]=@lr[2*@n-1]; |
---|
| 1231 | // attrib(@lr[2*@n],"isSB",1); |
---|
| 1232 | // } |
---|
| 1233 | // else |
---|
| 1234 | // { |
---|
| 1235 | // @lr[2*@n-1]=std(@lr[2*@n-1],@lvec[2*@n-1]); |
---|
| 1236 | // @lr[2*@n-1]=subst(@lr[2*@n-1],@t,1); |
---|
| 1237 | // @lr[2*@n]=std(@lr[2*@n],@lvec[2*@n]); |
---|
| 1238 | // @lr[2*@n]=subst(@lr[2*@n],@t,1); |
---|
| 1239 | // } |
---|
| 1240 | // } |
---|
| 1241 | // } |
---|
| 1242 | // kill @lvec; |
---|
| 1243 | // setring @P; |
---|
| 1244 | // lres0=imap(@Phelp1,@lr); |
---|
| 1245 | // kill @Phelp1; |
---|
| 1246 | // for(@n=1;@n<=size(lres0);@n++) |
---|
| 1247 | // { |
---|
| 1248 | // lres0[@n]=clearSB(lres0[@n]); |
---|
| 1249 | // attrib(lres0[@n],"isSB",1); |
---|
| 1250 | // } |
---|
| 1251 | // |
---|
| 1252 | // primary[2*@k-1]=lres0[1]; |
---|
| 1253 | // primary[2*@k]=lres0[2]; |
---|
[4173c7] | 1254 | // @s=size(primary) div 2; |
---|
| 1255 | // for(@n=1;@n<=size(lres0) div 2-1;@n++) |
---|
[24f458] | 1256 | // { |
---|
| 1257 | // primary[2*@s+2*@n-1]=lres0[2*@n+1]; |
---|
| 1258 | // primary[2*@s+2*@n]=lres0[2*@n+2]; |
---|
| 1259 | // } |
---|
| 1260 | // @k--; |
---|
| 1261 | //============================================================= |
---|
[70ab73] | 1262 | } |
---|
[d6db1f2] | 1263 | } |
---|
| 1264 | return(primary); |
---|
| 1265 | } |
---|
| 1266 | example |
---|
| 1267 | { "EXAMPLE:"; echo = 2; |
---|
| 1268 | ring r = 0,(x,y,z),lp; |
---|
| 1269 | poly p = z2+1; |
---|
| 1270 | poly q = z4+2; |
---|
| 1271 | ideal i = p^2*q^3,(y-z3)^3,(x-yz+z4)^4; |
---|
| 1272 | i=std(i); |
---|
| 1273 | list pr= zero_decomp(i,ideal(0),0); |
---|
| 1274 | pr; |
---|
| 1275 | } |
---|
[24f458] | 1276 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1277 | proc extF(list l,list #) |
---|
| 1278 | { |
---|
| 1279 | //zero_dimensional primary decomposition after finite field extension |
---|
[70ab73] | 1280 | def R=basering; |
---|
| 1281 | int p=char(R); |
---|
[24f458] | 1282 | |
---|
[70ab73] | 1283 | if((p==0)||(p>13)||(npars(R)>0)){return(l);} |
---|
[24f458] | 1284 | |
---|
[70ab73] | 1285 | int ex=3; |
---|
| 1286 | if(size(#)>0){ex=#[1];} |
---|
[24f458] | 1287 | |
---|
[70ab73] | 1288 | list peek,peek1; |
---|
| 1289 | while(size(l)>0) |
---|
| 1290 | { |
---|
| 1291 | if(size(l[2])==0) |
---|
| 1292 | { |
---|
| 1293 | peek[size(peek)+1]=l[1]; |
---|
| 1294 | } |
---|
| 1295 | else |
---|
| 1296 | { |
---|
| 1297 | peek1[size(peek1)+1]=l[1]; |
---|
| 1298 | peek1[size(peek1)+1]=l[2]; |
---|
| 1299 | } |
---|
| 1300 | l=delete(l,1); |
---|
| 1301 | l=delete(l,1); |
---|
| 1302 | } |
---|
| 1303 | if(size(peek)==0){return(peek1);} |
---|
| 1304 | |
---|
| 1305 | string gnir="ring RH=("+string(p)+"^"+string(ex)+",a),("+varstr(R)+"),lp;"; |
---|
| 1306 | execute(gnir); |
---|
| 1307 | string mp="minpoly="+string(minpoly)+";"; |
---|
| 1308 | gnir="ring RL=("+string(p)+",a),("+varstr(R)+"),lp;"; |
---|
| 1309 | execute(gnir); |
---|
| 1310 | execute(mp); |
---|
| 1311 | list L=imap(R,peek); |
---|
| 1312 | list pr, keep; |
---|
| 1313 | int i; |
---|
| 1314 | for(i=1;i<=size(L);i++) |
---|
| 1315 | { |
---|
| 1316 | attrib(L[i],"isSB",1); |
---|
| 1317 | pr=zero_decomp(L[i],0,0); |
---|
| 1318 | keep=keep+pr; |
---|
| 1319 | } |
---|
| 1320 | for(i=1;i<=size(keep);i++) |
---|
| 1321 | { |
---|
| 1322 | keep[i]=simplify(keep[i],1); |
---|
| 1323 | } |
---|
| 1324 | mp="poly pp="+string(minpoly)+";"; |
---|
[24f458] | 1325 | |
---|
[70ab73] | 1326 | string gnir1="ring RS="+string(p)+",("+varstr(R)+",a),lp;"; |
---|
| 1327 | execute(gnir1); |
---|
| 1328 | execute(mp); |
---|
| 1329 | list L=imap(RL,keep); |
---|
[24f458] | 1330 | |
---|
[70ab73] | 1331 | for(i=1;i<=size(L);i++) |
---|
| 1332 | { |
---|
| 1333 | L[i]=eliminate(L[i]+ideal(pp),a); |
---|
| 1334 | } |
---|
| 1335 | i=0; |
---|
| 1336 | int j; |
---|
[4173c7] | 1337 | while(i<size(L) div 2-1) |
---|
[70ab73] | 1338 | { |
---|
| 1339 | i++; |
---|
| 1340 | j=i; |
---|
[4173c7] | 1341 | while(j<size(L) div 2) |
---|
[70ab73] | 1342 | { |
---|
| 1343 | j++; |
---|
| 1344 | if(idealsEqual(L[2*i-1],L[2*j-1])) |
---|
| 1345 | { |
---|
| 1346 | L=delete(L,2*j-1); |
---|
| 1347 | L=delete(L,2*j-1); |
---|
| 1348 | j--; |
---|
[24f458] | 1349 | } |
---|
[70ab73] | 1350 | } |
---|
| 1351 | } |
---|
| 1352 | setring R; |
---|
| 1353 | list re=imap(RS,L); |
---|
| 1354 | re=re+peek1; |
---|
[24f458] | 1355 | |
---|
[70ab73] | 1356 | return(extF(re,ex+1)); |
---|
[24f458] | 1357 | } |
---|
| 1358 | |
---|
| 1359 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1360 | proc zeroSp(ideal i) |
---|
| 1361 | { |
---|
| 1362 | //preparation for the separable closure |
---|
| 1363 | //decomposition into ideals of special type |
---|
| 1364 | //i.e. the minimal polynomials of every variable mod i are irreducible |
---|
| 1365 | //returns a list of 2 lists: rr=pe,qe |
---|
| 1366 | //the ideals in pe[l] are special, their special elements are in qe[l] |
---|
| 1367 | //pe[l] is a dp-Groebnerbasis |
---|
| 1368 | //the radical of the intersection of the pe[l] is equal to the radical of i |
---|
| 1369 | |
---|
[70ab73] | 1370 | def R=basering; |
---|
[24f458] | 1371 | |
---|
[70ab73] | 1372 | //i has to be a reduced groebner basis |
---|
| 1373 | ideal F=finduni(i); |
---|
[24f458] | 1374 | |
---|
[70ab73] | 1375 | int j,k,l,ready; |
---|
| 1376 | list fa; |
---|
| 1377 | fa[1]=factorize(F[1],1); |
---|
| 1378 | poly te,ti; |
---|
| 1379 | ideal tj; |
---|
| 1380 | //avoid factorization of the same polynomial |
---|
| 1381 | for(j=2;j<=size(F);j++) |
---|
| 1382 | { |
---|
| 1383 | for(k=1;k<=j-1;k++) |
---|
| 1384 | { |
---|
| 1385 | ti=F[k]; |
---|
| 1386 | te=subst(ti,var(k),var(j)); |
---|
| 1387 | if(te==F[j]) |
---|
[24f458] | 1388 | { |
---|
[70ab73] | 1389 | tj=fa[k]; |
---|
| 1390 | fa[j]=subst(tj,var(k),var(j)); |
---|
| 1391 | ready=1; |
---|
| 1392 | break; |
---|
[24f458] | 1393 | } |
---|
[70ab73] | 1394 | } |
---|
| 1395 | if(!ready) |
---|
| 1396 | { |
---|
| 1397 | fa[j]=factorize(F[j],1); |
---|
| 1398 | } |
---|
| 1399 | ready=0; |
---|
| 1400 | } |
---|
| 1401 | execute( "ring P=("+charstr(R)+"),("+varstr(R)+"),(C,dp);"); |
---|
| 1402 | ideal i=imap(R,i); |
---|
| 1403 | if(npars(basering)==0) |
---|
| 1404 | { |
---|
| 1405 | ideal J=fglm(R,i); |
---|
| 1406 | } |
---|
| 1407 | else |
---|
| 1408 | { |
---|
[a36e78] | 1409 | ideal J=groebner(i); |
---|
[70ab73] | 1410 | } |
---|
| 1411 | list fa=imap(R,fa); |
---|
| 1412 | list qe=J; //collects a dp-Groebnerbasis of the special ideals |
---|
| 1413 | list keep=ideal(0); //collects the special elements |
---|
[24f458] | 1414 | |
---|
[70ab73] | 1415 | list re,em,ke; |
---|
| 1416 | ideal K,L; |
---|
[24f458] | 1417 | |
---|
[70ab73] | 1418 | for(j=1;j<=nvars(basering);j++) |
---|
| 1419 | { |
---|
| 1420 | for(l=1;l<=size(qe);l++) |
---|
| 1421 | { |
---|
| 1422 | for(k=1;k<=size(fa[j]);k++) |
---|
[24f458] | 1423 | { |
---|
[70ab73] | 1424 | L=std(qe[l],fa[j][k]); |
---|
| 1425 | K=keep[l],fa[j][k]; |
---|
| 1426 | if(deg(L[1])>0) |
---|
| 1427 | { |
---|
| 1428 | re[size(re)+1]=L; |
---|
| 1429 | ke[size(ke)+1]=K; |
---|
| 1430 | } |
---|
[24f458] | 1431 | } |
---|
| 1432 | } |
---|
[70ab73] | 1433 | qe=re; |
---|
| 1434 | re=em; |
---|
| 1435 | keep=ke; |
---|
| 1436 | ke=em; |
---|
| 1437 | } |
---|
| 1438 | |
---|
| 1439 | setring R; |
---|
| 1440 | list qe=imap(P,keep); |
---|
| 1441 | list pe=imap(P,qe); |
---|
| 1442 | for(l=1;l<=size(qe);l++) |
---|
| 1443 | { |
---|
| 1444 | qe[l]=simplify(qe[l],2); |
---|
| 1445 | } |
---|
| 1446 | list rr=pe,qe; |
---|
| 1447 | return(rr); |
---|
[24f458] | 1448 | } |
---|
| 1449 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1450 | |
---|
| 1451 | proc zeroSepClos(ideal I,ideal F) |
---|
| 1452 | { |
---|
| 1453 | //computes the separable closure of the special ideal I |
---|
| 1454 | //F is the set of special elements of I |
---|
| 1455 | //returns the separable closure sc(I) of I and an intvec v |
---|
| 1456 | //such that sc(I)=preimage(frobenius definde by v) |
---|
| 1457 | //i.e. var(i)----->var(i)^(p^v[i]) |
---|
| 1458 | |
---|
[70ab73] | 1459 | if(homog(I)==1){return(maxideal(1));} |
---|
[24f458] | 1460 | |
---|
[70ab73] | 1461 | //assume F[i] irreducible in I and depending only on var(i) |
---|
[24f458] | 1462 | |
---|
[70ab73] | 1463 | def R=basering; |
---|
| 1464 | int n=nvars(R); |
---|
| 1465 | int p=char(R); |
---|
| 1466 | intvec v; |
---|
| 1467 | v[n]=0; |
---|
| 1468 | int i,k; |
---|
| 1469 | list l; |
---|
[24f458] | 1470 | |
---|
[70ab73] | 1471 | for(i=1;i<=n;i++) |
---|
| 1472 | { |
---|
| 1473 | l[i]=sep(F[i],i); |
---|
| 1474 | F[i]=l[i][1]; |
---|
| 1475 | if(l[i][2]>k){k=l[i][2];} |
---|
| 1476 | } |
---|
[24f458] | 1477 | |
---|
[70ab73] | 1478 | if(k==0){return(list(I,v));} //the separable case |
---|
| 1479 | ideal m; |
---|
[24f458] | 1480 | |
---|
[70ab73] | 1481 | for(i=1;i<=n;i++) |
---|
| 1482 | { |
---|
| 1483 | m[i]=var(i)^(p^l[i][2]); |
---|
| 1484 | v[i]=l[i][2]; |
---|
| 1485 | } |
---|
| 1486 | map phi=R,m; |
---|
| 1487 | ideal J=preimage(R,phi,I); |
---|
| 1488 | return(list(J,v)); |
---|
[24f458] | 1489 | } |
---|
| 1490 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1491 | |
---|
| 1492 | proc insepDecomp(ideal i) |
---|
| 1493 | { |
---|
| 1494 | //decomposes i into special ideals |
---|
| 1495 | //computes the prime decomposition of the special ideals |
---|
| 1496 | //and transforms it back to a decomposition of i |
---|
| 1497 | |
---|
[70ab73] | 1498 | def R=basering; |
---|
| 1499 | list pr=zeroSp(i); |
---|
| 1500 | int l,k; |
---|
| 1501 | list re,wo,qr; |
---|
| 1502 | ideal m=maxideal(1); |
---|
| 1503 | ideal K; |
---|
| 1504 | map phi=R,m; |
---|
| 1505 | int p=char(R); |
---|
| 1506 | intvec op=option(get); |
---|
| 1507 | |
---|
| 1508 | for(l=1;l<=size(pr[1]);l++) |
---|
| 1509 | { |
---|
| 1510 | wo=zeroSepClos(pr[1][l],pr[2][l]); |
---|
| 1511 | for(k=1;k<=nvars(basering);k++) |
---|
| 1512 | { |
---|
| 1513 | m[k]=var(k)^(p^wo[2][k]); |
---|
| 1514 | } |
---|
| 1515 | phi=R,m; |
---|
| 1516 | qr=decomp(wo[1],2); |
---|
[24f458] | 1517 | |
---|
[70ab73] | 1518 | option(redSB); |
---|
[4173c7] | 1519 | for(k=1;k<=size(qr) div 2;k++) |
---|
[70ab73] | 1520 | { |
---|
| 1521 | K=qr[2*k]; |
---|
| 1522 | K=phi(K); |
---|
| 1523 | K=groebner(K); |
---|
| 1524 | re[size(re)+1]=zeroRad(K); |
---|
| 1525 | } |
---|
| 1526 | option(noredSB); |
---|
| 1527 | } |
---|
| 1528 | option(set,op); |
---|
| 1529 | return(re); |
---|
[24f458] | 1530 | } |
---|
| 1531 | |
---|
| 1532 | |
---|
[67bd4c] | 1533 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1534 | |
---|
[07c623] | 1535 | static proc clearSB (ideal i,list #) |
---|
[d2b2a7] | 1536 | "USAGE: clearSB(i); i ideal which is SB ordered by monomial ordering |
---|
[d6db1f2] | 1537 | RETURN: ideal = minimal SB |
---|
[18dd47] | 1538 | NOTE: |
---|
[d6db1f2] | 1539 | EXAMPLE: example clearSB; shows an example |
---|
[d2b2a7] | 1540 | " |
---|
[d6db1f2] | 1541 | { |
---|
| 1542 | int k,j; |
---|
| 1543 | poly m; |
---|
| 1544 | int c=size(i); |
---|
[18dd47] | 1545 | |
---|
[d6db1f2] | 1546 | if(size(#)==0) |
---|
| 1547 | { |
---|
| 1548 | for(j=1;j<c;j++) |
---|
| 1549 | { |
---|
| 1550 | if(deg(i[j])==0) |
---|
| 1551 | { |
---|
| 1552 | i=ideal(1); |
---|
| 1553 | return(i); |
---|
[18dd47] | 1554 | } |
---|
[d6db1f2] | 1555 | if(deg(i[j])>0) |
---|
| 1556 | { |
---|
| 1557 | m=lead(i[j]); |
---|
| 1558 | for(k=j+1;k<=c;k++) |
---|
| 1559 | { |
---|
[70ab73] | 1560 | if(size(lead(i[k])/m)>0) |
---|
| 1561 | { |
---|
| 1562 | i[k]=0; |
---|
| 1563 | } |
---|
[d6db1f2] | 1564 | } |
---|
| 1565 | } |
---|
| 1566 | } |
---|
| 1567 | } |
---|
| 1568 | else |
---|
| 1569 | { |
---|
| 1570 | j=0; |
---|
| 1571 | while(j<c-1) |
---|
| 1572 | { |
---|
| 1573 | j++; |
---|
| 1574 | if(deg(i[j])==0) |
---|
| 1575 | { |
---|
| 1576 | i=ideal(1); |
---|
| 1577 | return(i); |
---|
[18dd47] | 1578 | } |
---|
[d6db1f2] | 1579 | if(deg(i[j])>0) |
---|
| 1580 | { |
---|
| 1581 | m=lead(i[j]); |
---|
| 1582 | for(k=j+1;k<=c;k++) |
---|
| 1583 | { |
---|
[70ab73] | 1584 | if(size(lead(i[k])/m)>0) |
---|
| 1585 | { |
---|
| 1586 | if((leadexp(m)!=leadexp(i[k]))||(#[j]<=#[k])) |
---|
| 1587 | { |
---|
| 1588 | i[k]=0; |
---|
| 1589 | } |
---|
| 1590 | else |
---|
| 1591 | { |
---|
| 1592 | i[j]=0; |
---|
| 1593 | break; |
---|
| 1594 | } |
---|
| 1595 | } |
---|
[d6db1f2] | 1596 | } |
---|
| 1597 | } |
---|
| 1598 | } |
---|
| 1599 | } |
---|
| 1600 | return(simplify(i,2)); |
---|
| 1601 | } |
---|
| 1602 | example |
---|
| 1603 | { "EXAMPLE:"; echo = 2; |
---|
| 1604 | ring r = (0,a,b),(x,y,z),dp; |
---|
| 1605 | ideal i=ax2+y,a2x+y,bx; |
---|
| 1606 | list l=1,2,1; |
---|
| 1607 | ideal j=clearSB(i,l); |
---|
| 1608 | j; |
---|
| 1609 | } |
---|
| 1610 | |
---|
[f54c83] | 1611 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1612 | static proc clearSBNeu (ideal i,list #) |
---|
| 1613 | "USAGE: clearSB(i); i ideal which is SB ordered by monomial ordering |
---|
| 1614 | RETURN: ideal = minimal SB |
---|
| 1615 | NOTE: |
---|
| 1616 | EXAMPLE: example clearSB; shows an example |
---|
| 1617 | " |
---|
| 1618 | { |
---|
[a36e78] | 1619 | int k,j; |
---|
| 1620 | intvec m,n,v,w; |
---|
| 1621 | int c=size(i); |
---|
| 1622 | w=leadexp(0); |
---|
| 1623 | v[size(i)]=0; |
---|
| 1624 | |
---|
| 1625 | j=0; |
---|
| 1626 | while(j<c-1) |
---|
| 1627 | { |
---|
| 1628 | j++; |
---|
| 1629 | if(deg(i[j])>=0) |
---|
| 1630 | { |
---|
[f54c83] | 1631 | m=leadexp(i[j]); |
---|
| 1632 | for(k=j+1;k<=c;k++) |
---|
| 1633 | { |
---|
| 1634 | n=leadexp(i[k]); |
---|
| 1635 | if(n!=w) |
---|
| 1636 | { |
---|
[a36e78] | 1637 | if(((m==n)&&(#[j]>#[k]))||((teilt(n,m))&&(n!=m))) |
---|
| 1638 | { |
---|
| 1639 | i[j]=0; |
---|
| 1640 | v[j]=1; |
---|
| 1641 | break; |
---|
| 1642 | } |
---|
| 1643 | if(((m==n)&&(#[j]<=#[k]))||((teilt(m,n))&&(n!=m))) |
---|
| 1644 | { |
---|
| 1645 | i[k]=0; |
---|
| 1646 | v[k]=1; |
---|
| 1647 | } |
---|
[f54c83] | 1648 | } |
---|
| 1649 | } |
---|
| 1650 | } |
---|
| 1651 | } |
---|
| 1652 | return(v); |
---|
| 1653 | } |
---|
| 1654 | |
---|
| 1655 | static proc teilt(intvec a, intvec b) |
---|
| 1656 | { |
---|
[70ab73] | 1657 | int i; |
---|
| 1658 | for(i=1;i<=size(a);i++) |
---|
| 1659 | { |
---|
| 1660 | if(a[i]>b[i]){return(0);} |
---|
| 1661 | } |
---|
| 1662 | return(1); |
---|
[f54c83] | 1663 | } |
---|
[d6db1f2] | 1664 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1665 | |
---|
[07c623] | 1666 | static proc independSet (ideal j) |
---|
[d2b2a7] | 1667 | "USAGE: independentSet(i); i ideal |
---|
[d6db1f2] | 1668 | RETURN: list = new varstring with the independent set at the end, |
---|
| 1669 | ordstring with the corresponding block ordering, |
---|
| 1670 | the integer where the independent set starts in the varstring |
---|
[18dd47] | 1671 | NOTE: |
---|
[d6db1f2] | 1672 | EXAMPLE: example independentSet; shows an example |
---|
[d2b2a7] | 1673 | " |
---|
[d6db1f2] | 1674 | { |
---|
[70ab73] | 1675 | int n,k,di; |
---|
| 1676 | list resu,hilf; |
---|
| 1677 | string var1,var2; |
---|
| 1678 | list v=indepSet(j,1); |
---|
[18dd47] | 1679 | |
---|
[70ab73] | 1680 | for(n=1;n<=size(v);n++) |
---|
| 1681 | { |
---|
| 1682 | di=0; |
---|
| 1683 | var1=""; |
---|
| 1684 | var2=""; |
---|
| 1685 | for(k=1;k<=size(v[n]);k++) |
---|
| 1686 | { |
---|
| 1687 | if(v[n][k]!=0) |
---|
| 1688 | { |
---|
| 1689 | di++; |
---|
| 1690 | var2=var2+"var("+string(k)+"),"; |
---|
[d6db1f2] | 1691 | } |
---|
| 1692 | else |
---|
| 1693 | { |
---|
[70ab73] | 1694 | var1=var1+"var("+string(k)+"),"; |
---|
[d6db1f2] | 1695 | } |
---|
[70ab73] | 1696 | } |
---|
| 1697 | if(di>0) |
---|
| 1698 | { |
---|
| 1699 | var1=var1+var2; |
---|
| 1700 | var1=var1[1..size(var1)-1]; |
---|
| 1701 | hilf[1]=var1; |
---|
| 1702 | hilf[2]="lp"; |
---|
| 1703 | //"lp("+string(nvars(basering)-di)+"),dp("+string(di)+")"; |
---|
| 1704 | hilf[3]=di; |
---|
| 1705 | resu[n]=hilf; |
---|
| 1706 | } |
---|
| 1707 | else |
---|
| 1708 | { |
---|
| 1709 | resu[n]=varstr(basering),ordstr(basering),0; |
---|
| 1710 | } |
---|
| 1711 | } |
---|
| 1712 | return(resu); |
---|
[d6db1f2] | 1713 | } |
---|
| 1714 | example |
---|
| 1715 | { "EXAMPLE:"; echo = 2; |
---|
| 1716 | ring s1=(0,x,y),(a,b,c,d,e,f,g),lp; |
---|
| 1717 | ideal i=ea-fbg,fa+be,ec-fdg,fc+de; |
---|
| 1718 | i=std(i); |
---|
| 1719 | list l=independSet(i); |
---|
| 1720 | l; |
---|
| 1721 | i=i,g; |
---|
| 1722 | l=independSet(i); |
---|
| 1723 | l; |
---|
| 1724 | |
---|
| 1725 | ring s=0,(x,y,z),lp; |
---|
| 1726 | ideal i=z,yx; |
---|
| 1727 | list l=independSet(i); |
---|
| 1728 | l; |
---|
| 1729 | |
---|
| 1730 | |
---|
| 1731 | } |
---|
| 1732 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1733 | |
---|
[07c623] | 1734 | static proc maxIndependSet (ideal j) |
---|
[d2b2a7] | 1735 | "USAGE: maxIndependentSet(i); i ideal |
---|
[d6db1f2] | 1736 | RETURN: list = new varstring with the maximal independent set at the end, |
---|
| 1737 | ordstring with the corresponding block ordering, |
---|
| 1738 | the integer where the independent set starts in the varstring |
---|
[18dd47] | 1739 | NOTE: |
---|
[d6db1f2] | 1740 | EXAMPLE: example maxIndependentSet; shows an example |
---|
[d2b2a7] | 1741 | " |
---|
[d6db1f2] | 1742 | { |
---|
[70ab73] | 1743 | int n,k,di; |
---|
| 1744 | list resu,hilf; |
---|
| 1745 | string var1,var2; |
---|
| 1746 | list v=indepSet(j,0); |
---|
[18dd47] | 1747 | |
---|
[70ab73] | 1748 | for(n=1;n<=size(v);n++) |
---|
| 1749 | { |
---|
| 1750 | di=0; |
---|
| 1751 | var1=""; |
---|
| 1752 | var2=""; |
---|
| 1753 | for(k=1;k<=size(v[n]);k++) |
---|
| 1754 | { |
---|
| 1755 | if(v[n][k]!=0) |
---|
| 1756 | { |
---|
| 1757 | di++; |
---|
| 1758 | var2=var2+"var("+string(k)+"),"; |
---|
[d6db1f2] | 1759 | } |
---|
| 1760 | else |
---|
| 1761 | { |
---|
[70ab73] | 1762 | var1=var1+"var("+string(k)+"),"; |
---|
[d6db1f2] | 1763 | } |
---|
[70ab73] | 1764 | } |
---|
| 1765 | if(di>0) |
---|
| 1766 | { |
---|
| 1767 | var1=var1+var2; |
---|
| 1768 | var1=var1[1..size(var1)-1]; |
---|
| 1769 | hilf[1]=var1; |
---|
| 1770 | hilf[2]="lp"; |
---|
| 1771 | hilf[3]=di; |
---|
| 1772 | resu[n]=hilf; |
---|
| 1773 | } |
---|
| 1774 | else |
---|
| 1775 | { |
---|
| 1776 | resu[n]=varstr(basering),ordstr(basering),0; |
---|
| 1777 | } |
---|
| 1778 | } |
---|
| 1779 | return(resu); |
---|
[d6db1f2] | 1780 | } |
---|
| 1781 | example |
---|
| 1782 | { "EXAMPLE:"; echo = 2; |
---|
| 1783 | ring s1=(0,x,y),(a,b,c,d,e,f,g),lp; |
---|
| 1784 | ideal i=ea-fbg,fa+be,ec-fdg,fc+de; |
---|
| 1785 | i=std(i); |
---|
| 1786 | list l=maxIndependSet(i); |
---|
| 1787 | l; |
---|
| 1788 | i=i,g; |
---|
| 1789 | l=maxIndependSet(i); |
---|
| 1790 | l; |
---|
| 1791 | |
---|
| 1792 | ring s=0,(x,y,z),lp; |
---|
| 1793 | ideal i=z,yx; |
---|
| 1794 | list l=maxIndependSet(i); |
---|
| 1795 | l; |
---|
| 1796 | |
---|
| 1797 | |
---|
| 1798 | } |
---|
| 1799 | |
---|
| 1800 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1801 | |
---|
[07c623] | 1802 | static proc prepareQuotientring (int nnp) |
---|
[d2b2a7] | 1803 | "USAGE: prepareQuotientring(nnp); nnp int |
---|
[d6db1f2] | 1804 | RETURN: string = to define Kvar(nnp+1),...,var(nvars)[..rest ] |
---|
[18dd47] | 1805 | NOTE: |
---|
[d6db1f2] | 1806 | EXAMPLE: example independentSet; shows an example |
---|
[d2b2a7] | 1807 | " |
---|
[18dd47] | 1808 | { |
---|
[d6db1f2] | 1809 | ideal @ih,@jh; |
---|
| 1810 | int npar=npars(basering); |
---|
| 1811 | int @n; |
---|
[18dd47] | 1812 | |
---|
[d6db1f2] | 1813 | string quotring= "ring quring = ("+charstr(basering); |
---|
| 1814 | for(@n=nnp+1;@n<=nvars(basering);@n++) |
---|
| 1815 | { |
---|
[a36e78] | 1816 | quotring=quotring+",var("+string(@n)+")"; |
---|
| 1817 | @ih=@ih+var(@n); |
---|
[d6db1f2] | 1818 | } |
---|
[18dd47] | 1819 | |
---|
[d6db1f2] | 1820 | quotring=quotring+"),(var(1)"; |
---|
| 1821 | @jh=@jh+var(1); |
---|
| 1822 | for(@n=2;@n<=nnp;@n++) |
---|
| 1823 | { |
---|
| 1824 | quotring=quotring+",var("+string(@n)+")"; |
---|
| 1825 | @jh=@jh+var(@n); |
---|
| 1826 | } |
---|
[e801fe] | 1827 | quotring=quotring+"),(C,lp);"; |
---|
[18dd47] | 1828 | |
---|
[d6db1f2] | 1829 | return(quotring); |
---|
| 1830 | |
---|
| 1831 | } |
---|
| 1832 | example |
---|
| 1833 | { "EXAMPLE:"; echo = 2; |
---|
| 1834 | ring s1=(0,x),(a,b,c,d,e,f,g),lp; |
---|
| 1835 | def @Q=basering; |
---|
| 1836 | list l= prepareQuotientring(3); |
---|
| 1837 | l; |
---|
[2d2cad9] | 1838 | execute(l[1]); |
---|
| 1839 | execute(l[2]); |
---|
[d6db1f2] | 1840 | basering; |
---|
| 1841 | phi; |
---|
| 1842 | setring @Q; |
---|
[18dd47] | 1843 | |
---|
[d6db1f2] | 1844 | } |
---|
| 1845 | |
---|
[091424] | 1846 | /////////////////////////////////////////////////////////////////////////////// |
---|
[07c623] | 1847 | static proc cleanPrimary(list l) |
---|
[d6db1f2] | 1848 | { |
---|
[a36e78] | 1849 | int i,j; |
---|
| 1850 | list lh; |
---|
[4173c7] | 1851 | for(i=1;i<=size(l) div 2;i++) |
---|
[a36e78] | 1852 | { |
---|
| 1853 | if(deg(l[2*i-1][1])>0) |
---|
| 1854 | { |
---|
| 1855 | j++; |
---|
| 1856 | lh[j]=l[2*i-1]; |
---|
| 1857 | j++; |
---|
| 1858 | lh[j]=l[2*i]; |
---|
| 1859 | } |
---|
| 1860 | } |
---|
| 1861 | return(lh); |
---|
[d6db1f2] | 1862 | } |
---|
| 1863 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1864 | |
---|
[840745] | 1865 | |
---|
| 1866 | proc minAssPrimesold(ideal i, list #) |
---|
[d2b2a7] | 1867 | "USAGE: minAssPrimes(i); i ideal |
---|
[d6db1f2] | 1868 | minAssPrimes(i,1); i ideal (to use also the factorizing Groebner) |
---|
| 1869 | RETURN: list = the minimal associated prime ideals of i |
---|
| 1870 | EXAMPLE: example minAssPrimes; shows an example |
---|
[d2b2a7] | 1871 | " |
---|
[d6db1f2] | 1872 | { |
---|
[a36e78] | 1873 | def @P=basering; |
---|
| 1874 | if(size(i)==0){return(list(ideal(0)));} |
---|
| 1875 | list qr=simplifyIdeal(i); |
---|
| 1876 | map phi=@P,qr[2]; |
---|
| 1877 | i=qr[1]; |
---|
[3939bc] | 1878 | |
---|
[a36e78] | 1879 | execute ("ring gnir = ("+charstr(basering)+"),("+varstr(basering)+"),(" |
---|
[2d2cad9] | 1880 | +ordstr(basering)+");"); |
---|
[67bd4c] | 1881 | |
---|
| 1882 | |
---|
[a36e78] | 1883 | ideal i=fetch(@P,i); |
---|
| 1884 | if(size(#)==0) |
---|
| 1885 | { |
---|
| 1886 | int @wr; |
---|
| 1887 | list tluser,@res; |
---|
| 1888 | list primary=decomp(i,2); |
---|
[d6db1f2] | 1889 | |
---|
[a36e78] | 1890 | @res[1]=primary; |
---|
[d6db1f2] | 1891 | |
---|
[a36e78] | 1892 | tluser=union(@res); |
---|
| 1893 | setring @P; |
---|
| 1894 | list @res=imap(gnir,tluser); |
---|
| 1895 | return(phi(@res)); |
---|
| 1896 | } |
---|
| 1897 | list @res,empty; |
---|
| 1898 | ideal ser; |
---|
| 1899 | option(redSB); |
---|
| 1900 | list @pr=facstd(i); |
---|
| 1901 | //if(size(@pr)==1) |
---|
[17407e] | 1902 | // { |
---|
| 1903 | // attrib(@pr[1],"isSB",1); |
---|
| 1904 | // if((dim(@pr[1])==0)&&(homog(@pr[1])==1)) |
---|
| 1905 | // { |
---|
| 1906 | // setring @P; |
---|
| 1907 | // list @res=maxideal(1); |
---|
| 1908 | // return(phi(@res)); |
---|
| 1909 | // } |
---|
| 1910 | // if(dim(@pr[1])>1) |
---|
| 1911 | // { |
---|
| 1912 | // setring @P; |
---|
| 1913 | // // kill gnir; |
---|
| 1914 | // execute ("ring gnir1 = ("+charstr(basering)+"), |
---|
| 1915 | // ("+varstr(basering)+"),(C,lp);"); |
---|
| 1916 | // ideal i=fetch(@P,i); |
---|
| 1917 | // list @pr=facstd(i); |
---|
| 1918 | // // ideal ser; |
---|
| 1919 | // setring gnir; |
---|
| 1920 | // @pr=fetch(gnir1,@pr); |
---|
| 1921 | // kill gnir1; |
---|
| 1922 | // } |
---|
| 1923 | // } |
---|
[a36e78] | 1924 | option(noredSB); |
---|
| 1925 | int j,k,odim,ndim,count; |
---|
| 1926 | attrib(@pr[1],"isSB",1); |
---|
| 1927 | if(#[1]==77) |
---|
| 1928 | { |
---|
| 1929 | odim=dim(@pr[1]); |
---|
| 1930 | count=1; |
---|
| 1931 | intvec pos; |
---|
| 1932 | pos[size(@pr)]=0; |
---|
| 1933 | for(j=2;j<=size(@pr);j++) |
---|
| 1934 | { |
---|
| 1935 | attrib(@pr[j],"isSB",1); |
---|
| 1936 | ndim=dim(@pr[j]); |
---|
| 1937 | if(ndim>odim) |
---|
[80b3cd] | 1938 | { |
---|
[a36e78] | 1939 | for(k=count;k<=j-1;k++) |
---|
| 1940 | { |
---|
| 1941 | pos[k]=1; |
---|
| 1942 | } |
---|
| 1943 | count=j; |
---|
| 1944 | odim=ndim; |
---|
[80b3cd] | 1945 | } |
---|
[a36e78] | 1946 | if(ndim<odim) |
---|
| 1947 | { |
---|
| 1948 | pos[j]=1; |
---|
| 1949 | } |
---|
| 1950 | } |
---|
| 1951 | for(j=1;j<=size(@pr);j++) |
---|
| 1952 | { |
---|
| 1953 | if(pos[j]!=1) |
---|
| 1954 | { |
---|
| 1955 | @res[j]=decomp(@pr[j],2); |
---|
| 1956 | } |
---|
| 1957 | else |
---|
| 1958 | { |
---|
| 1959 | @res[j]=empty; |
---|
| 1960 | } |
---|
| 1961 | } |
---|
| 1962 | } |
---|
| 1963 | else |
---|
| 1964 | { |
---|
| 1965 | ser=ideal(1); |
---|
| 1966 | for(j=1;j<=size(@pr);j++) |
---|
| 1967 | { |
---|
[e801fe] | 1968 | //@pr[j]; |
---|
[917fb5] | 1969 | //pause(); |
---|
[a36e78] | 1970 | @res[j]=decomp(@pr[j],2); |
---|
[e801fe] | 1971 | // @res[j]=decomp(@pr[j],2,@pr[j],ser); |
---|
| 1972 | // for(k=1;k<=size(@res[j]);k++) |
---|
| 1973 | // { |
---|
[d950c5] | 1974 | // ser=intersect(ser,@res[j][k]); |
---|
[e801fe] | 1975 | // } |
---|
[a36e78] | 1976 | } |
---|
| 1977 | } |
---|
[d6db1f2] | 1978 | |
---|
[a36e78] | 1979 | @res=union(@res); |
---|
| 1980 | setring @P; |
---|
| 1981 | list @res=imap(gnir,@res); |
---|
| 1982 | return(phi(@res)); |
---|
[d6db1f2] | 1983 | } |
---|
| 1984 | example |
---|
| 1985 | { "EXAMPLE:"; echo = 2; |
---|
| 1986 | ring r = 32003,(x,y,z),lp; |
---|
| 1987 | poly p = z2+1; |
---|
| 1988 | poly q = z4+2; |
---|
| 1989 | ideal i = p^2*q^3,(y-z3)^3,(x-yz+z4)^4; |
---|
[091424] | 1990 | list pr= minAssPrimes(i); pr; |
---|
| 1991 | |
---|
[9050ca] | 1992 | minAssPrimes(i,1); |
---|
[d6db1f2] | 1993 | } |
---|
| 1994 | |
---|
[24f458] | 1995 | static proc primT(ideal i) |
---|
| 1996 | { |
---|
[a36e78] | 1997 | //assumes that all generators of i are irreducible |
---|
| 1998 | //i is standard basis |
---|
[840745] | 1999 | |
---|
[a36e78] | 2000 | attrib(i,"isSB",1); |
---|
| 2001 | int j=size(i); |
---|
| 2002 | int k; |
---|
| 2003 | while(j>0) |
---|
| 2004 | { |
---|
| 2005 | if(deg(i[j])>1){break;} |
---|
| 2006 | j--; |
---|
| 2007 | } |
---|
| 2008 | if(j==0){return(1);} |
---|
| 2009 | if(deg(i[j])==vdim(i)){return(1);} |
---|
| 2010 | return(0); |
---|
[24f458] | 2011 | } |
---|
[840745] | 2012 | |
---|
| 2013 | static proc minAssPrimes(ideal i, list #) |
---|
| 2014 | "USAGE: minAssPrimes(i); i ideal |
---|
[808a9f3] | 2015 | Optional parameters in list #: (can be entered in any order) |
---|
| 2016 | 0, "facstd" -> uses facstd to first decompose the ideal |
---|
| 2017 | 1, "noFacstd" -> does not use facstd (default) |
---|
| 2018 | "SL" -> the new algorithm is used (default) |
---|
| 2019 | "GTZ" -> the old algorithm is used |
---|
[840745] | 2020 | RETURN: list = the minimal associated prime ideals of i |
---|
| 2021 | EXAMPLE: example minAssPrimes; shows an example |
---|
| 2022 | " |
---|
| 2023 | { |
---|
[70ab73] | 2024 | if(size(i) == 0){return(list(ideal(0)));} |
---|
| 2025 | string algorithm; // Algorithm to be used |
---|
| 2026 | string facstdOption; // To uses proc facstd |
---|
| 2027 | int j; // Counter |
---|
| 2028 | def P0 = basering; |
---|
| 2029 | list Pl=ringlist(P0); |
---|
| 2030 | intvec dp_w; |
---|
| 2031 | for(j=nvars(P0);j>0;j--) {dp_w[j]=1;} |
---|
| 2032 | Pl[3]=list(list("dp",dp_w),list("C",0)); |
---|
| 2033 | def P=ring(Pl); |
---|
| 2034 | setring P; |
---|
| 2035 | ideal i=imap(P0,i); |
---|
[24f458] | 2036 | |
---|
[70ab73] | 2037 | // Set input parameters |
---|
| 2038 | algorithm = "SL"; // Default: SL algorithm |
---|
[fc1526c] | 2039 | facstdOption = "Facstd"; // Default: facstd is not used |
---|
[70ab73] | 2040 | if(size(#) > 0) |
---|
| 2041 | { |
---|
| 2042 | int valid; |
---|
| 2043 | for(j = 1; j <= size(#); j++) |
---|
| 2044 | { |
---|
| 2045 | valid = 0; |
---|
| 2046 | if((typeof(#[j]) == "int") or (typeof(#[j]) == "number")) |
---|
[f54c83] | 2047 | { |
---|
[70ab73] | 2048 | if (#[j] == 0) {facstdOption = "noFacstd"; valid = 1;} // If #[j] == 0, facstd is not used. |
---|
| 2049 | if (#[j] == 1) {facstdOption = "facstd"; valid = 1;} // If #[j] == 1, facstd is used. |
---|
[f54c83] | 2050 | } |
---|
[70ab73] | 2051 | if(typeof(#[j]) == "string") |
---|
[f54c83] | 2052 | { |
---|
[70ab73] | 2053 | if(#[j] == "GTZ" || #[j] == "SL") |
---|
| 2054 | { |
---|
| 2055 | algorithm = #[j]; |
---|
| 2056 | valid = 1; |
---|
| 2057 | } |
---|
| 2058 | if(#[j] == "noFacstd" || #[j] == "facstd") |
---|
| 2059 | { |
---|
| 2060 | facstdOption = #[j]; |
---|
| 2061 | valid = 1; |
---|
| 2062 | } |
---|
[f54c83] | 2063 | } |
---|
[70ab73] | 2064 | if(valid == 0) |
---|
[24a90ca] | 2065 | { |
---|
[70ab73] | 2066 | dbprint(1, "Warning! The following input parameter was not recognized:", #[j]); |
---|
[7f7c25e] | 2067 | } |
---|
[70ab73] | 2068 | } |
---|
| 2069 | } |
---|
[4d63da] | 2070 | |
---|
[70ab73] | 2071 | list q = simplifyIdeal(i); |
---|
| 2072 | list re = maxideal(1); |
---|
| 2073 | int a, k; |
---|
| 2074 | intvec op = option(get); |
---|
| 2075 | map phi = P,q[2]; |
---|
| 2076 | |
---|
| 2077 | list result; |
---|
| 2078 | |
---|
| 2079 | if(npars(P) == 0){option(redSB);} |
---|
| 2080 | |
---|
| 2081 | if(attrib(i,"isSB")!=1) |
---|
| 2082 | { |
---|
| 2083 | i=groebner(q[1]); |
---|
| 2084 | } |
---|
| 2085 | else |
---|
| 2086 | { |
---|
| 2087 | for(j=1;j<=nvars(basering);j++) |
---|
| 2088 | { |
---|
| 2089 | if(q[2][j]!=var(j)){k=1;break;} |
---|
| 2090 | } |
---|
| 2091 | if(k) |
---|
| 2092 | { |
---|
| 2093 | i=groebner(q[1]); |
---|
| 2094 | } |
---|
| 2095 | } |
---|
| 2096 | |
---|
| 2097 | if(dim(i) == -1){setring P0;return(ideal(1));} |
---|
| 2098 | if((dim(i) == 0) && (npars(P) == 0)) |
---|
| 2099 | { |
---|
| 2100 | int di = vdim(i); |
---|
| 2101 | execute ("ring gnir=("+charstr(P)+"),("+varstr(P)+"),lp;"); |
---|
| 2102 | ideal J = std(imap(P,i)); |
---|
| 2103 | attrib(J, "isSB", 1); |
---|
| 2104 | if(vdim(J) != di) |
---|
| 2105 | { |
---|
| 2106 | J = fglm(P, i); |
---|
| 2107 | } |
---|
[b0db25] | 2108 | // list pr = triangMH(J,2); HIER KOENNEN verschiedene Mengen zu gleichen |
---|
| 2109 | // asoziierten Primidealen fuehren |
---|
| 2110 | // Aenderung |
---|
[85e68dd] | 2111 | list pr = triangMH(J,2); |
---|
[70ab73] | 2112 | list qr, re; |
---|
| 2113 | for(k = 1; k <= size(pr); k++) |
---|
| 2114 | { |
---|
[fc1526c] | 2115 | if(primT(pr[k])&&(0)) |
---|
[840745] | 2116 | { |
---|
[70ab73] | 2117 | re[size(re) + 1] = pr[k]; |
---|
[840745] | 2118 | } |
---|
| 2119 | else |
---|
| 2120 | { |
---|
[70ab73] | 2121 | attrib(pr[k], "isSB", 1); |
---|
| 2122 | // Lines changed |
---|
| 2123 | if (algorithm == "GTZ") |
---|
| 2124 | { |
---|
| 2125 | qr = decomp(pr[k], 2); |
---|
| 2126 | } |
---|
| 2127 | else |
---|
| 2128 | { |
---|
| 2129 | qr = minAssSL(pr[k]); |
---|
| 2130 | } |
---|
[4173c7] | 2131 | for(j = 1; j <= size(qr) div 2; j++) |
---|
[70ab73] | 2132 | { |
---|
[fc1526c] | 2133 | re[size(re) + 1] = std(qr[2 * j]); |
---|
[70ab73] | 2134 | } |
---|
[840745] | 2135 | } |
---|
[70ab73] | 2136 | } |
---|
| 2137 | setring P; |
---|
| 2138 | re = imap(gnir, re); |
---|
| 2139 | re=phi(re); |
---|
| 2140 | option(set, op); |
---|
| 2141 | setring(P0); |
---|
| 2142 | list re=imap(P,re); |
---|
| 2143 | return(re); |
---|
| 2144 | } |
---|
| 2145 | |
---|
| 2146 | // Lines changed |
---|
| 2147 | if ((facstdOption == "noFacstd") || (dim(i) == 0)) |
---|
| 2148 | { |
---|
| 2149 | if (algorithm == "GTZ") |
---|
| 2150 | { |
---|
| 2151 | re[1] = decomp(i, 2); |
---|
| 2152 | } |
---|
| 2153 | else |
---|
| 2154 | { |
---|
| 2155 | re[1] = minAssSL(i); |
---|
| 2156 | } |
---|
| 2157 | re = union(re); |
---|
| 2158 | option(set, op); |
---|
| 2159 | re=phi(re); |
---|
| 2160 | setring(P0); |
---|
| 2161 | list re=imap(P,re); |
---|
| 2162 | return(re); |
---|
| 2163 | } |
---|
| 2164 | q = facstd(i); |
---|
| 2165 | |
---|
| 2166 | /* |
---|
| 2167 | if((size(q) == 1) && (dim(i) > 1)) |
---|
| 2168 | { |
---|
| 2169 | execute ("ring gnir=("+charstr(P)+"),("+varstr(P)+"),lp;"); |
---|
| 2170 | list p = facstd(fetch(P, i)); |
---|
| 2171 | if(size(p) > 1) |
---|
| 2172 | { |
---|
| 2173 | a = 1; |
---|
| 2174 | setring P; |
---|
| 2175 | q = fetch(gnir,p); |
---|
| 2176 | } |
---|
| 2177 | else |
---|
| 2178 | { |
---|
| 2179 | setring P; |
---|
| 2180 | } |
---|
| 2181 | kill gnir; |
---|
| 2182 | } |
---|
[f54c83] | 2183 | */ |
---|
[70ab73] | 2184 | option(set,op); |
---|
| 2185 | // Debug |
---|
| 2186 | dbprint(printlevel - voice, "Components returned by facstd", size(q), q); |
---|
| 2187 | for(j = 1; j <= size(q); j++) |
---|
| 2188 | { |
---|
| 2189 | if(a == 0){attrib(q[j], "isSB", 1);} |
---|
| 2190 | // Debug |
---|
| 2191 | dbprint(printlevel - voice, "We compute the decomp of component", j); |
---|
| 2192 | // Lines changed |
---|
| 2193 | if (algorithm == "GTZ") |
---|
| 2194 | { |
---|
| 2195 | re[j] = decomp(q[j], 2); |
---|
| 2196 | } |
---|
| 2197 | else |
---|
| 2198 | { |
---|
| 2199 | re[j] = minAssSL(q[j]); |
---|
| 2200 | } |
---|
| 2201 | // Debug |
---|
[4173c7] | 2202 | dbprint(printlevel - voice, "Number of components obtained for this component:", size(re[j]) div 2); |
---|
[70ab73] | 2203 | dbprint(printlevel - voice, "re[j]:", re[j]); |
---|
| 2204 | } |
---|
| 2205 | re = union(re); |
---|
| 2206 | re=phi(re); |
---|
| 2207 | setring(P0); |
---|
| 2208 | list re=imap(P,re); |
---|
| 2209 | return(re); |
---|
[840745] | 2210 | } |
---|
| 2211 | example |
---|
| 2212 | { "EXAMPLE:"; echo = 2; |
---|
| 2213 | ring r = 32003,(x,y,z),lp; |
---|
| 2214 | poly p = z2+1; |
---|
| 2215 | poly q = z4+2; |
---|
| 2216 | ideal i = p^2*q^3,(y-z3)^3,(x-yz+z4)^4; |
---|
| 2217 | list pr= minAssPrimes(i); pr; |
---|
| 2218 | |
---|
| 2219 | minAssPrimes(i,1); |
---|
| 2220 | } |
---|
| 2221 | |
---|
[07c623] | 2222 | static proc union(list li) |
---|
[d6db1f2] | 2223 | { |
---|
| 2224 | int i,j,k; |
---|
[67bd4c] | 2225 | |
---|
| 2226 | def P=basering; |
---|
| 2227 | |
---|
[2d2cad9] | 2228 | execute("ring ir = ("+charstr(basering)+"),("+varstr(basering)+"),(C,lp);"); |
---|
[67bd4c] | 2229 | list l=fetch(P,li); |
---|
[d6db1f2] | 2230 | list @erg; |
---|
| 2231 | |
---|
| 2232 | for(k=1;k<=size(l);k++) |
---|
| 2233 | { |
---|
[4173c7] | 2234 | for(j=1;j<=size(l[k]) div 2;j++) |
---|
[a36e78] | 2235 | { |
---|
| 2236 | if(deg(l[k][2*j][1])!=0) |
---|
| 2237 | { |
---|
| 2238 | i++; |
---|
| 2239 | @erg[i]=l[k][2*j]; |
---|
| 2240 | } |
---|
| 2241 | } |
---|
[d6db1f2] | 2242 | } |
---|
| 2243 | |
---|
| 2244 | list @wos; |
---|
| 2245 | i=0; |
---|
| 2246 | ideal i1,i2; |
---|
| 2247 | while(i<size(@erg)-1) |
---|
| 2248 | { |
---|
[a36e78] | 2249 | i++; |
---|
| 2250 | k=i+1; |
---|
| 2251 | i1=lead(@erg[i]); |
---|
| 2252 | attrib(i1,"isSB",1); |
---|
| 2253 | attrib(@erg[i],"isSB",1); |
---|
[d6db1f2] | 2254 | |
---|
[a36e78] | 2255 | while(k<=size(@erg)) |
---|
| 2256 | { |
---|
| 2257 | if(deg(@erg[i][1])==0) |
---|
| 2258 | { |
---|
| 2259 | break; |
---|
| 2260 | } |
---|
| 2261 | i2=lead(@erg[k]); |
---|
| 2262 | attrib(@erg[k],"isSB",1); |
---|
| 2263 | attrib(i2,"isSB",1); |
---|
[d6db1f2] | 2264 | |
---|
[a36e78] | 2265 | if(size(reduce(i1,i2,1))==0) |
---|
[d6db1f2] | 2266 | { |
---|
[a36e78] | 2267 | if(size(reduce(@erg[i],@erg[k],1))==0) |
---|
| 2268 | { |
---|
| 2269 | @erg[k]=ideal(1); |
---|
| 2270 | i2=ideal(1); |
---|
| 2271 | } |
---|
[d6db1f2] | 2272 | } |
---|
[a36e78] | 2273 | if(size(reduce(i2,i1,1))==0) |
---|
[d6db1f2] | 2274 | { |
---|
[a36e78] | 2275 | if(size(reduce(@erg[k],@erg[i],1))==0) |
---|
| 2276 | { |
---|
| 2277 | break; |
---|
| 2278 | } |
---|
[d6db1f2] | 2279 | } |
---|
[a36e78] | 2280 | k++; |
---|
| 2281 | if(k>size(@erg)) |
---|
| 2282 | { |
---|
| 2283 | @wos[size(@wos)+1]=@erg[i]; |
---|
| 2284 | } |
---|
| 2285 | } |
---|
[d6db1f2] | 2286 | } |
---|
| 2287 | if(deg(@erg[size(@erg)][1])!=0) |
---|
| 2288 | { |
---|
[a36e78] | 2289 | @wos[size(@wos)+1]=@erg[size(@erg)]; |
---|
[d6db1f2] | 2290 | } |
---|
[67bd4c] | 2291 | setring P; |
---|
| 2292 | list @ser=fetch(ir,@wos); |
---|
| 2293 | return(@ser); |
---|
[d6db1f2] | 2294 | } |
---|
| 2295 | /////////////////////////////////////////////////////////////////////////////// |
---|
[d8d3af] | 2296 | proc equidim(ideal i,list #) |
---|
[b9b906] | 2297 | "USAGE: equidim(i) or equidim(i,1) ; i ideal |
---|
[7b3971] | 2298 | RETURN: list of equidimensional ideals a[1],...,a[s] with: |
---|
[25c431] | 2299 | - a[s] the equidimensional locus of i, i.e. the intersection |
---|
| 2300 | of the primary ideals of dimension of i |
---|
[367e88] | 2301 | - a[1],...,a[s-1] the lower dimensional equidimensional loci. |
---|
| 2302 | NOTE: An embedded component q (primary ideal) of i can be replaced in the |
---|
[7b3971] | 2303 | decomposition by a primary ideal q1 with the same radical as q. @* |
---|
| 2304 | @code{equidim(i,1)} uses the algorithm of Eisenbud/Huneke/Vasconcelos. |
---|
| 2305 | |
---|
[07c623] | 2306 | EXAMPLE:example equidim; shows an example |
---|
[ba94539] | 2307 | " |
---|
| 2308 | { |
---|
[d88470] | 2309 | if(attrib(basering,"global")!=1) |
---|
[07c623] | 2310 | { |
---|
[cb980ab] | 2311 | ERROR( |
---|
| 2312 | "// Not implemented for this ordering, please change to global ordering." |
---|
| 2313 | ); |
---|
[07c623] | 2314 | } |
---|
[cb980ab] | 2315 | intvec op ; |
---|
| 2316 | def P = basering; |
---|
[ba94539] | 2317 | list eq; |
---|
| 2318 | intvec w; |
---|
[4d68980] | 2319 | int n,m; |
---|
[6d6ed5b] | 2320 | int g=size(i); |
---|
[ba94539] | 2321 | int a=attrib(i,"isSB"); |
---|
| 2322 | int homo=homog(i); |
---|
[d8d3af] | 2323 | if(size(#)!=0) |
---|
| 2324 | { |
---|
[4d68980] | 2325 | m=1; |
---|
| 2326 | } |
---|
| 2327 | |
---|
[ba94539] | 2328 | if(((homo==1)||(a==1))&&(find(ordstr(basering),"l")==0) |
---|
| 2329 | &&(find(ordstr(basering),"s")==0)) |
---|
| 2330 | { |
---|
[a36e78] | 2331 | execute("ring gnir = ("+charstr(basering)+"),("+varstr(basering)+"),(" |
---|
[2d2cad9] | 2332 | +ordstr(basering)+");"); |
---|
[a36e78] | 2333 | ideal i=imap(P,i); |
---|
| 2334 | ideal j=i; |
---|
| 2335 | if(a==1) |
---|
| 2336 | { |
---|
| 2337 | attrib(j,"isSB",1); |
---|
| 2338 | } |
---|
| 2339 | else |
---|
| 2340 | { |
---|
| 2341 | j=groebner(i); |
---|
| 2342 | } |
---|
[ba94539] | 2343 | } |
---|
| 2344 | else |
---|
| 2345 | { |
---|
[a36e78] | 2346 | execute("ring gnir = ("+charstr(basering)+"),("+varstr(basering)+"),dp;"); |
---|
| 2347 | ideal i=imap(P,i); |
---|
| 2348 | ideal j=groebner(i); |
---|
[b9b906] | 2349 | } |
---|
[ba94539] | 2350 | if(homo==1) |
---|
| 2351 | { |
---|
[a36e78] | 2352 | for(n=1;n<=nvars(basering);n++) |
---|
| 2353 | { |
---|
| 2354 | w[n]=ord(var(n)); |
---|
| 2355 | } |
---|
| 2356 | intvec hil=hilb(j,1,w); |
---|
[ba94539] | 2357 | } |
---|
[4d68980] | 2358 | |
---|
[6d6ed5b] | 2359 | if ((dim(j)==-1)||(size(j)==0)||(nvars(basering)==1) |
---|
| 2360 | ||(dim(j)==0)||(dim(j)+g==nvars(basering))) |
---|
[ba94539] | 2361 | { |
---|
| 2362 | setring P; |
---|
[6d6ed5b] | 2363 | eq[1]=i; |
---|
[ba94539] | 2364 | return(eq); |
---|
| 2365 | } |
---|
| 2366 | |
---|
[4d68980] | 2367 | if(m==0) |
---|
[ba94539] | 2368 | { |
---|
[a36e78] | 2369 | ideal k=equidimMax(j); |
---|
[ba94539] | 2370 | } |
---|
| 2371 | else |
---|
| 2372 | { |
---|
[a36e78] | 2373 | ideal k=equidimMaxEHV(j); |
---|
[ba94539] | 2374 | } |
---|
[6d6ed5b] | 2375 | if(size(reduce(k,j,1))==0) |
---|
| 2376 | { |
---|
| 2377 | setring P; |
---|
| 2378 | eq[1]=i; |
---|
| 2379 | kill gnir; |
---|
| 2380 | return(eq); |
---|
| 2381 | } |
---|
[466f80] | 2382 | op=option(get); |
---|
[b9b906] | 2383 | option(returnSB); |
---|
[651953] | 2384 | j=quotient(j,k); |
---|
[02335e] | 2385 | option(set,op); |
---|
[d8d3af] | 2386 | |
---|
[b9b906] | 2387 | list equi=equidim(j); |
---|
[4d68980] | 2388 | if(deg(equi[size(equi)][1])<=0) |
---|
[a9cf54] | 2389 | { |
---|
[a36e78] | 2390 | equi[size(equi)]=k; |
---|
[a9cf54] | 2391 | } |
---|
| 2392 | else |
---|
| 2393 | { |
---|
[4d68980] | 2394 | equi[size(equi)+1]=k; |
---|
[a9cf54] | 2395 | } |
---|
[ba94539] | 2396 | setring P; |
---|
[4d68980] | 2397 | eq=imap(gnir,equi); |
---|
[ba94539] | 2398 | kill gnir; |
---|
| 2399 | return(eq); |
---|
| 2400 | } |
---|
| 2401 | example |
---|
| 2402 | { "EXAMPLE:"; echo = 2; |
---|
| 2403 | ring r = 32003,(x,y,z),dp; |
---|
[7b3971] | 2404 | ideal i = intersect(ideal(z),ideal(x,y),ideal(x2,z2),ideal(x5,y5,z5)); |
---|
[ba94539] | 2405 | equidim(i); |
---|
| 2406 | } |
---|
[6d6ed5b] | 2407 | |
---|
[03f29c] | 2408 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 2409 | proc equidimMax(ideal i) |
---|
[b9b906] | 2410 | "USAGE: equidimMax(i); i ideal |
---|
[07c623] | 2411 | RETURN: ideal of equidimensional locus (of maximal dimension) of i. |
---|
| 2412 | EXAMPLE: example equidimMax; shows an example |
---|
[03f29c] | 2413 | " |
---|
| 2414 | { |
---|
[d88470] | 2415 | if(attrib(basering,"global")!=1) |
---|
[07c623] | 2416 | { |
---|
[cb980ab] | 2417 | ERROR( |
---|
| 2418 | "// Not implemented for this ordering, please change to global ordering." |
---|
| 2419 | ); |
---|
[07c623] | 2420 | } |
---|
[cb980ab] | 2421 | def P = basering; |
---|
[03f29c] | 2422 | ideal eq; |
---|
| 2423 | intvec w; |
---|
| 2424 | int n; |
---|
[6d6ed5b] | 2425 | int g=size(i); |
---|
[03f29c] | 2426 | int a=attrib(i,"isSB"); |
---|
| 2427 | int homo=homog(i); |
---|
[b9b906] | 2428 | |
---|
[03f29c] | 2429 | if(((homo==1)||(a==1))&&(find(ordstr(basering),"l")==0) |
---|
| 2430 | &&(find(ordstr(basering),"s")==0)) |
---|
| 2431 | { |
---|
[a36e78] | 2432 | execute("ring gnir = ("+charstr(basering)+"),("+varstr(basering)+"),(" |
---|
[2d2cad9] | 2433 | +ordstr(basering)+");"); |
---|
[a36e78] | 2434 | ideal i=imap(P,i); |
---|
| 2435 | ideal j=i; |
---|
| 2436 | if(a==1) |
---|
| 2437 | { |
---|
| 2438 | attrib(j,"isSB",1); |
---|
| 2439 | } |
---|
| 2440 | else |
---|
| 2441 | { |
---|
| 2442 | j=groebner(i); |
---|
| 2443 | } |
---|
[03f29c] | 2444 | } |
---|
| 2445 | else |
---|
| 2446 | { |
---|
[a36e78] | 2447 | execute("ring gnir = ("+charstr(basering)+"),("+varstr(basering)+"),dp;"); |
---|
| 2448 | ideal i=imap(P,i); |
---|
| 2449 | ideal j=groebner(i); |
---|
[03f29c] | 2450 | } |
---|
| 2451 | list indep; |
---|
| 2452 | ideal equ,equi; |
---|
| 2453 | if(homo==1) |
---|
| 2454 | { |
---|
[a36e78] | 2455 | for(n=1;n<=nvars(basering);n++) |
---|
| 2456 | { |
---|
| 2457 | w[n]=ord(var(n)); |
---|
| 2458 | } |
---|
| 2459 | intvec hil=hilb(j,1,w); |
---|
[03f29c] | 2460 | } |
---|
[6d6ed5b] | 2461 | if ((dim(j)==-1)||(size(j)==0)||(nvars(basering)==1) |
---|
| 2462 | ||(dim(j)==0)||(dim(j)+g==nvars(basering))) |
---|
[03f29c] | 2463 | { |
---|
| 2464 | setring P; |
---|
[a9cf54] | 2465 | return(i); |
---|
[03f29c] | 2466 | } |
---|
| 2467 | |
---|
| 2468 | indep=maxIndependSet(j); |
---|
[a9cf54] | 2469 | |
---|
[2d2cad9] | 2470 | execute("ring gnir1 = ("+charstr(basering)+"),("+indep[1][1]+"),(" |
---|
| 2471 | +indep[1][2]+");"); |
---|
[03f29c] | 2472 | if(homo==1) |
---|
| 2473 | { |
---|
[a36e78] | 2474 | ideal j=std(imap(gnir,j),hil,w); |
---|
[03f29c] | 2475 | } |
---|
| 2476 | else |
---|
| 2477 | { |
---|
[a36e78] | 2478 | ideal j=groebner(imap(gnir,j)); |
---|
[03f29c] | 2479 | } |
---|
| 2480 | string quotring=prepareQuotientring(nvars(basering)-indep[1][3]); |
---|
[2d2cad9] | 2481 | execute(quotring); |
---|
[03f29c] | 2482 | ideal j=imap(gnir1,j); |
---|
| 2483 | kill gnir1; |
---|
| 2484 | j=clearSB(j); |
---|
| 2485 | ideal h; |
---|
| 2486 | for(n=1;n<=size(j);n++) |
---|
| 2487 | { |
---|
[a36e78] | 2488 | h[n]=leadcoef(j[n]); |
---|
[03f29c] | 2489 | } |
---|
| 2490 | setring gnir; |
---|
| 2491 | ideal h=imap(quring,h); |
---|
| 2492 | kill quring; |
---|
[6d6ed5b] | 2493 | |
---|
[03f29c] | 2494 | list l=minSat(j,h); |
---|
[b9b906] | 2495 | |
---|
[b1d1e8c] | 2496 | if(deg(l[2])>0) |
---|
| 2497 | { |
---|
| 2498 | equ=l[1]; |
---|
| 2499 | attrib(equ,"isSB",1); |
---|
| 2500 | j=std(j,l[2]); |
---|
[6d6ed5b] | 2501 | |
---|
[b1d1e8c] | 2502 | if(dim(equ)==dim(j)) |
---|
| 2503 | { |
---|
| 2504 | equi=equidimMax(j); |
---|
| 2505 | equ=interred(intersect(equ,equi)); |
---|
| 2506 | } |
---|
| 2507 | } |
---|
| 2508 | else |
---|
[03f29c] | 2509 | { |
---|
[b1d1e8c] | 2510 | equ=i; |
---|
[03f29c] | 2511 | } |
---|
[b1d1e8c] | 2512 | |
---|
[03f29c] | 2513 | setring P; |
---|
| 2514 | eq=imap(gnir,equ); |
---|
| 2515 | kill gnir; |
---|
| 2516 | return(eq); |
---|
| 2517 | } |
---|
| 2518 | example |
---|
| 2519 | { "EXAMPLE:"; echo = 2; |
---|
| 2520 | ring r = 32003,(x,y,z),dp; |
---|
[7b3971] | 2521 | ideal i = intersect(ideal(z),ideal(x,y),ideal(x2,z2),ideal(x5,y5,z5)); |
---|
[03f29c] | 2522 | equidimMax(i); |
---|
| 2523 | } |
---|
[24f458] | 2524 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 2525 | static proc islp() |
---|
| 2526 | { |
---|
[a36e78] | 2527 | string s=ordstr(basering); |
---|
| 2528 | int n=find(s,"lp"); |
---|
| 2529 | if(!n){return(0);} |
---|
| 2530 | int k=find(s,","); |
---|
| 2531 | string t=s[k+1..size(s)]; |
---|
| 2532 | int l=find(t,","); |
---|
| 2533 | t=s[1..k-1]; |
---|
| 2534 | int m=find(t,","); |
---|
| 2535 | if(l+m){return(0);} |
---|
| 2536 | return(1); |
---|
[24f458] | 2537 | } |
---|
| 2538 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 2539 | |
---|
| 2540 | proc algeDeco(ideal i, int w) |
---|
| 2541 | { |
---|
| 2542 | //reduces primery decomposition over algebraic extensions to |
---|
| 2543 | //the other cases |
---|
[a36e78] | 2544 | def R=basering; |
---|
| 2545 | int n=nvars(R); |
---|
[fc5095] | 2546 | |
---|
| 2547 | //---Anfang Provisorium |
---|
[a36e78] | 2548 | if((size(i)==2) && (w==2)) |
---|
| 2549 | { |
---|
| 2550 | option(redSB); |
---|
| 2551 | ideal J=std(i); |
---|
| 2552 | option(noredSB); |
---|
| 2553 | if((size(J)==2)&&(deg(J[1])==1)) |
---|
| 2554 | { |
---|
| 2555 | ideal keep; |
---|
| 2556 | poly f; |
---|
| 2557 | int j; |
---|
| 2558 | for(j=1;j<=nvars(basering);j++) |
---|
| 2559 | { |
---|
| 2560 | f=J[2]; |
---|
| 2561 | while((f/var(j))*var(j)-f==0) |
---|
| 2562 | { |
---|
| 2563 | f=f/var(j); |
---|
| 2564 | keep=keep,var(j); |
---|
| 2565 | } |
---|
| 2566 | J[2]=f; |
---|
| 2567 | } |
---|
| 2568 | ideal K=factorize(J[2],1); |
---|
| 2569 | if(deg(K[1])==0){K=0;} |
---|
| 2570 | K=K+std(keep); |
---|
| 2571 | ideal L; |
---|
| 2572 | list resu; |
---|
| 2573 | for(j=1;j<=size(K);j++) |
---|
| 2574 | { |
---|
| 2575 | L=J[1],K[j]; |
---|
| 2576 | resu[j]=L; |
---|
| 2577 | } |
---|
| 2578 | return(resu); |
---|
[70ab73] | 2579 | } |
---|
[a36e78] | 2580 | } |
---|
[fc5095] | 2581 | //---Ende Provisorium |
---|
[a36e78] | 2582 | string mp="poly p="+string(minpoly)+";"; |
---|
| 2583 | string gnir="ring RH="+string(char(R))+",("+varstr(R)+","+string(par(1)) |
---|
[24f458] | 2584 | +"),dp;"; |
---|
[a36e78] | 2585 | execute(gnir); |
---|
| 2586 | execute(mp); |
---|
| 2587 | ideal i=imap(R,i); |
---|
| 2588 | ideal I=subst(i,var(nvars(basering)),0); |
---|
| 2589 | int j; |
---|
| 2590 | for(j=1;j<=ncols(i);j++) |
---|
| 2591 | { |
---|
| 2592 | if(i[j]!=I[j]){break;} |
---|
| 2593 | } |
---|
| 2594 | if((j>ncols(i))&&(deg(p)==1)) |
---|
| 2595 | { |
---|
| 2596 | setring R; |
---|
| 2597 | kill RH; |
---|
| 2598 | kill gnir; |
---|
| 2599 | string gnir="ring RH="+string(char(R))+",("+varstr(R)+"),dp;"; |
---|
| 2600 | execute(gnir); |
---|
| 2601 | ideal i=imap(R,i); |
---|
| 2602 | ideal J; |
---|
| 2603 | } |
---|
| 2604 | else |
---|
| 2605 | { |
---|
| 2606 | i=i,p; |
---|
| 2607 | } |
---|
| 2608 | list pr; |
---|
[24f458] | 2609 | |
---|
[a36e78] | 2610 | if(w==0) |
---|
| 2611 | { |
---|
| 2612 | pr=decomp(i); |
---|
| 2613 | } |
---|
| 2614 | if(w==1) |
---|
| 2615 | { |
---|
| 2616 | pr=prim_dec(i,1); |
---|
| 2617 | pr=reconvList(pr); |
---|
| 2618 | } |
---|
| 2619 | if(w==2) |
---|
| 2620 | { |
---|
| 2621 | pr=minAssPrimes(i); |
---|
| 2622 | } |
---|
| 2623 | if(n<nvars(basering)) |
---|
| 2624 | { |
---|
| 2625 | gnir="ring RS="+string(char(R))+",("+varstr(RH) |
---|
[24f458] | 2626 | +"),(dp("+string(n)+"),lp);"; |
---|
[a36e78] | 2627 | execute(gnir); |
---|
| 2628 | list pr=imap(RH,pr); |
---|
| 2629 | ideal K; |
---|
| 2630 | for(j=1;j<=size(pr);j++) |
---|
| 2631 | { |
---|
| 2632 | K=groebner(pr[j]); |
---|
| 2633 | K=K[2..size(K)]; |
---|
| 2634 | pr[j]=K; |
---|
| 2635 | } |
---|
| 2636 | setring R; |
---|
| 2637 | list pr=imap(RS,pr); |
---|
| 2638 | } |
---|
| 2639 | else |
---|
| 2640 | { |
---|
| 2641 | setring R; |
---|
| 2642 | list pr=imap(RH,pr); |
---|
| 2643 | } |
---|
| 2644 | list re; |
---|
| 2645 | if(w==2) |
---|
| 2646 | { |
---|
| 2647 | re=pr; |
---|
| 2648 | } |
---|
| 2649 | else |
---|
| 2650 | { |
---|
| 2651 | re=convList(pr); |
---|
| 2652 | } |
---|
| 2653 | return(re); |
---|
[24f458] | 2654 | } |
---|
[ab8937] | 2655 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 2656 | static proc prepare_absprimdec(list primary) |
---|
| 2657 | { |
---|
| 2658 | list resu,tempo; |
---|
| 2659 | string absotto; |
---|
[4173c7] | 2660 | resu[size(primary) div 2]=list(); |
---|
| 2661 | for(int ab=1;ab<=size(primary) div 2;ab++) |
---|
[ab8937] | 2662 | { |
---|
| 2663 | absotto= absFactorize(primary[2*ab][1],77); |
---|
| 2664 | tempo=primary[2*ab-1],primary[2*ab],absotto,string(var(nvars(basering))); |
---|
| 2665 | resu[ab]=tempo; |
---|
| 2666 | } |
---|
| 2667 | return(resu); |
---|
| 2668 | } |
---|
[67bd4c] | 2669 | /////////////////////////////////////////////////////////////////////////////// |
---|
[07c623] | 2670 | static proc decomp(ideal i,list #) |
---|
[7a7df90] | 2671 | "USAGE: decomp(i); i ideal (for primary decomposition) (resp. |
---|
| 2672 | decomp(i,1); (for the associated primes of dimension of i) ) |
---|
| 2673 | decomp(i,2); (for the minimal associated primes) ) |
---|
[6fa3af] | 2674 | decomp(i,3); (for the absolute primary decomposition) ) |
---|
[d6db1f2] | 2675 | RETURN: list = list of primary ideals and their associated primes |
---|
| 2676 | (at even positions in the list) |
---|
| 2677 | (resp. a list of the minimal associated primes) |
---|
[7b3971] | 2678 | NOTE: Algorithm of Gianni/Trager/Zacharias |
---|
[d6db1f2] | 2679 | EXAMPLE: example decomp; shows an example |
---|
[d2b2a7] | 2680 | " |
---|
[d6db1f2] | 2681 | { |
---|
[7cd077] | 2682 | intvec op,@vv; |
---|
[d6db1f2] | 2683 | def @P = basering; |
---|
[67bd4c] | 2684 | list primary,indep,ltras; |
---|
[d36f7f] | 2685 | intvec @vh,isat,@w; |
---|
[6fa3af] | 2686 | int @wr,@k,@n,@m,@n1,@n2,@n3,homo,seri,keepdi,abspri,ab,nn; |
---|
[d6db1f2] | 2687 | ideal peek=i; |
---|
| 2688 | ideal ser,tras; |
---|
[24f458] | 2689 | int isS=(attrib(i,"isSB")==1); |
---|
[18dd47] | 2690 | |
---|
[6fa3af] | 2691 | |
---|
[d6db1f2] | 2692 | if(size(#)>0) |
---|
| 2693 | { |
---|
[1d430ab] | 2694 | if((#[1]==1)||(#[1]==2)||(#[1]==3)) |
---|
| 2695 | { |
---|
| 2696 | @wr=#[1]; |
---|
| 2697 | if(@wr==3){abspri=1;@wr=0;} |
---|
| 2698 | if(size(#)>1) |
---|
[d6db1f2] | 2699 | { |
---|
[e801fe] | 2700 | seri=1; |
---|
[1d430ab] | 2701 | peek=#[2]; |
---|
| 2702 | ser=#[3]; |
---|
[d6db1f2] | 2703 | } |
---|
[1d430ab] | 2704 | } |
---|
| 2705 | else |
---|
| 2706 | { |
---|
| 2707 | seri=1; |
---|
| 2708 | peek=#[1]; |
---|
| 2709 | ser=#[2]; |
---|
| 2710 | } |
---|
[d6db1f2] | 2711 | } |
---|
[6fa3af] | 2712 | if(abspri) |
---|
| 2713 | { |
---|
[1d430ab] | 2714 | list absprimary,abskeep,absprimarytmp,abskeeptmp; |
---|
[6fa3af] | 2715 | } |
---|
[e801fe] | 2716 | homo=homog(i); |
---|
[d6db1f2] | 2717 | if(homo==1) |
---|
| 2718 | { |
---|
[e801fe] | 2719 | if(attrib(i,"isSB")!=1) |
---|
| 2720 | { |
---|
[17407e] | 2721 | //ltras=mstd(i); |
---|
| 2722 | tras=groebner(i); |
---|
| 2723 | ltras=tras,tras; |
---|
[e801fe] | 2724 | attrib(ltras[1],"isSB",1); |
---|
| 2725 | } |
---|
| 2726 | else |
---|
| 2727 | { |
---|
| 2728 | ltras=i,i; |
---|
[24f458] | 2729 | attrib(ltras[1],"isSB",1); |
---|
[e801fe] | 2730 | } |
---|
| 2731 | tras=ltras[1]; |
---|
[24f458] | 2732 | attrib(tras,"isSB",1); |
---|
[adde988] | 2733 | if((dim(tras)==0) && (!abspri)) |
---|
[e801fe] | 2734 | { |
---|
[1d430ab] | 2735 | primary[1]=ltras[2]; |
---|
| 2736 | primary[2]=maxideal(1); |
---|
| 2737 | if(@wr>0) |
---|
| 2738 | { |
---|
| 2739 | list l; |
---|
| 2740 | l[1]=maxideal(1); |
---|
| 2741 | l[2]=maxideal(1); |
---|
| 2742 | return(l); |
---|
| 2743 | } |
---|
| 2744 | return(primary); |
---|
| 2745 | } |
---|
| 2746 | for(@n=1;@n<=nvars(basering);@n++) |
---|
| 2747 | { |
---|
| 2748 | @w[@n]=ord(var(@n)); |
---|
| 2749 | } |
---|
| 2750 | intvec @hilb=hilb(tras,1,@w); |
---|
| 2751 | intvec keephilb=@hilb; |
---|
[a36e78] | 2752 | } |
---|
| 2753 | |
---|
| 2754 | //---------------------------------------------------------------- |
---|
[d6db1f2] | 2755 | //i is the zero-ideal |
---|
| 2756 | //---------------------------------------------------------------- |
---|
[18dd47] | 2757 | |
---|
[d6db1f2] | 2758 | if(size(i)==0) |
---|
| 2759 | { |
---|
[810a4af] | 2760 | primary=ideal(0),ideal(0); |
---|
[ab8937] | 2761 | if (abspri) { return(prepare_absprimdec(primary));} |
---|
[1d430ab] | 2762 | return(primary); |
---|
[d6db1f2] | 2763 | } |
---|
[18dd47] | 2764 | |
---|
[d6db1f2] | 2765 | //---------------------------------------------------------------- |
---|
| 2766 | //pass to the lexicographical ordering and compute a standardbasis |
---|
| 2767 | //---------------------------------------------------------------- |
---|
| 2768 | |
---|
[24f458] | 2769 | int lp=islp(); |
---|
| 2770 | |
---|
[2d2cad9] | 2771 | execute("ring gnir = ("+charstr(basering)+"),("+varstr(basering)+"),(C,lp);"); |
---|
[466f80] | 2772 | op=option(get); |
---|
[d6db1f2] | 2773 | option(redSB); |
---|
[e801fe] | 2774 | |
---|
[3939bc] | 2775 | ideal ser=fetch(@P,ser); |
---|
[18dd47] | 2776 | |
---|
[d6db1f2] | 2777 | if(homo==1) |
---|
| 2778 | { |
---|
[1d430ab] | 2779 | if(!lp) |
---|
| 2780 | { |
---|
| 2781 | ideal @j=std(fetch(@P,i),@hilb,@w); |
---|
| 2782 | } |
---|
| 2783 | else |
---|
| 2784 | { |
---|
| 2785 | ideal @j=fetch(@P,tras); |
---|
| 2786 | attrib(@j,"isSB",1); |
---|
| 2787 | } |
---|
[d6db1f2] | 2788 | } |
---|
| 2789 | else |
---|
| 2790 | { |
---|
[1d430ab] | 2791 | if(lp&&isS) |
---|
| 2792 | { |
---|
| 2793 | ideal @j=fetch(@P,i); |
---|
| 2794 | attrib(@j,"isSB",1); |
---|
| 2795 | } |
---|
| 2796 | else |
---|
| 2797 | { |
---|
| 2798 | ideal @j=groebner(fetch(@P,i)); |
---|
| 2799 | } |
---|
[d6db1f2] | 2800 | } |
---|
[02335e] | 2801 | option(set,op); |
---|
[e801fe] | 2802 | if(seri==1) |
---|
| 2803 | { |
---|
| 2804 | ideal peek=fetch(@P,peek); |
---|
| 2805 | attrib(peek,"isSB",1); |
---|
| 2806 | } |
---|
| 2807 | else |
---|
| 2808 | { |
---|
| 2809 | ideal peek=@j; |
---|
| 2810 | } |
---|
[6fa3af] | 2811 | if((size(ser)==0)&&(!abspri)) |
---|
[e801fe] | 2812 | { |
---|
| 2813 | ideal fried; |
---|
| 2814 | @n=size(@j); |
---|
| 2815 | for(@k=1;@k<=@n;@k++) |
---|
| 2816 | { |
---|
| 2817 | if(deg(lead(@j[@k]))==1) |
---|
| 2818 | { |
---|
| 2819 | fried[size(fried)+1]=@j[@k]; |
---|
| 2820 | @j[@k]=0; |
---|
| 2821 | } |
---|
| 2822 | } |
---|
[5674d5] | 2823 | if(size(fried)==nvars(basering)) |
---|
| 2824 | { |
---|
[1d430ab] | 2825 | setring @P; |
---|
| 2826 | primary[1]=i; |
---|
| 2827 | primary[2]=i; |
---|
[ab8937] | 2828 | if (abspri) { return(prepare_absprimdec(primary));} |
---|
[1d430ab] | 2829 | return(primary); |
---|
[5674d5] | 2830 | } |
---|
[e801fe] | 2831 | if(size(fried)>0) |
---|
| 2832 | { |
---|
[1d430ab] | 2833 | string newva; |
---|
| 2834 | string newma; |
---|
[b15849d] | 2835 | poly f; |
---|
[1d430ab] | 2836 | for(@k=1;@k<=nvars(basering);@k++) |
---|
| 2837 | { |
---|
| 2838 | @n1=0; |
---|
| 2839 | for(@n=1;@n<=size(fried);@n++) |
---|
| 2840 | { |
---|
| 2841 | if(leadmonom(fried[@n])==var(@k)) |
---|
[a36e78] | 2842 | { |
---|
[1d430ab] | 2843 | @n1=1; |
---|
| 2844 | break; |
---|
[a36e78] | 2845 | } |
---|
[1d430ab] | 2846 | } |
---|
| 2847 | if(@n1==0) |
---|
| 2848 | { |
---|
| 2849 | newva=newva+string(var(@k))+","; |
---|
| 2850 | newma=newma+string(var(@k))+","; |
---|
| 2851 | } |
---|
| 2852 | else |
---|
| 2853 | { |
---|
| 2854 | newma=newma+string(0)+","; |
---|
[b15849d] | 2855 | fried[@n]=fried[@n]/leadcoef(fried[@n]); |
---|
| 2856 | f=fried[@n]-lead(fried[@n]); |
---|
| 2857 | @j=subst(@j,var(@k),-f); |
---|
[1d430ab] | 2858 | } |
---|
| 2859 | } |
---|
| 2860 | newva[size(newva)]=")"; |
---|
| 2861 | newma[size(newma)]=";"; |
---|
| 2862 | execute("ring @deirf=("+charstr(gnir)+"),("+newva+",lp;"); |
---|
| 2863 | execute("map @kappa=gnir,"+newma); |
---|
| 2864 | ideal @j= @kappa(@j); |
---|
[b15849d] | 2865 | @j=std(@j); |
---|
| 2866 | |
---|
[1d430ab] | 2867 | list pr=decomp(@j); |
---|
| 2868 | setring gnir; |
---|
| 2869 | list pr=imap(@deirf,pr); |
---|
| 2870 | for(@k=1;@k<=size(pr);@k++) |
---|
| 2871 | { |
---|
| 2872 | @j=pr[@k]+fried; |
---|
| 2873 | pr[@k]=@j; |
---|
| 2874 | } |
---|
| 2875 | setring @P; |
---|
[810a4af] | 2876 | primary=imap(gnir,pr); |
---|
| 2877 | if (abspri) { return(prepare_absprimdec(primary));} |
---|
| 2878 | return(primary); |
---|
[e801fe] | 2879 | } |
---|
| 2880 | } |
---|
[d6db1f2] | 2881 | //---------------------------------------------------------------- |
---|
| 2882 | //j is the ring |
---|
| 2883 | //---------------------------------------------------------------- |
---|
| 2884 | |
---|
| 2885 | if (dim(@j)==-1) |
---|
| 2886 | { |
---|
[e801fe] | 2887 | setring @P; |
---|
[651953] | 2888 | primary=ideal(1),ideal(1); |
---|
[ab8937] | 2889 | if (abspri) { return(prepare_absprimdec(primary));} |
---|
[651953] | 2890 | return(primary); |
---|
[d6db1f2] | 2891 | } |
---|
[18dd47] | 2892 | |
---|
[d6db1f2] | 2893 | //---------------------------------------------------------------- |
---|
| 2894 | // the case of one variable |
---|
| 2895 | //---------------------------------------------------------------- |
---|
| 2896 | |
---|
| 2897 | if(nvars(basering)==1) |
---|
| 2898 | { |
---|
[1d430ab] | 2899 | list fac=factor(@j[1]); |
---|
| 2900 | list gprimary; |
---|
| 2901 | for(@k=1;@k<=size(fac[1]);@k++) |
---|
| 2902 | { |
---|
| 2903 | if(@wr==0) |
---|
| 2904 | { |
---|
| 2905 | gprimary[2*@k-1]=ideal(fac[1][@k]^fac[2][@k]); |
---|
| 2906 | gprimary[2*@k]=ideal(fac[1][@k]); |
---|
| 2907 | } |
---|
| 2908 | else |
---|
| 2909 | { |
---|
| 2910 | gprimary[2*@k-1]=ideal(fac[1][@k]); |
---|
| 2911 | gprimary[2*@k]=ideal(fac[1][@k]); |
---|
| 2912 | } |
---|
| 2913 | } |
---|
| 2914 | setring @P; |
---|
| 2915 | primary=fetch(gnir,gprimary); |
---|
[d6db1f2] | 2916 | |
---|
[6fa3af] | 2917 | //HIER |
---|
[ab8937] | 2918 | if (abspri) { return(prepare_absprimdec(primary));} |
---|
[1d430ab] | 2919 | return(primary); |
---|
[d6db1f2] | 2920 | } |
---|
[3939bc] | 2921 | |
---|
[d6db1f2] | 2922 | //------------------------------------------------------------------ |
---|
| 2923 | //the zero-dimensional case |
---|
| 2924 | //------------------------------------------------------------------ |
---|
| 2925 | if (dim(@j)==0) |
---|
| 2926 | { |
---|
[466f80] | 2927 | op=option(get); |
---|
[e801fe] | 2928 | option(redSB); |
---|
| 2929 | list gprimary= zero_decomp(@j,ser,@wr); |
---|
[6fa3af] | 2930 | |
---|
[e801fe] | 2931 | setring @P; |
---|
| 2932 | primary=fetch(gnir,gprimary); |
---|
[6fa3af] | 2933 | |
---|
[e801fe] | 2934 | if(size(ser)>0) |
---|
| 2935 | { |
---|
| 2936 | primary=cleanPrimary(primary); |
---|
| 2937 | } |
---|
[6fa3af] | 2938 | //HIER |
---|
| 2939 | if(abspri) |
---|
| 2940 | { |
---|
[1d430ab] | 2941 | setring gnir; |
---|
| 2942 | list primary=imap(@P,primary); |
---|
| 2943 | list resu,tempo; |
---|
| 2944 | string absotto; |
---|
| 2945 | map sigma,invsigma; |
---|
| 2946 | ideal II,jmap; |
---|
| 2947 | nn=nvars(basering); |
---|
[4173c7] | 2948 | for(ab=1;ab<=size(primary) div 2;ab++) |
---|
[1d430ab] | 2949 | { |
---|
| 2950 | II=primary[2*ab]; |
---|
| 2951 | attrib(II,"isSB",1); |
---|
| 2952 | if(deg(II[1])==vdim(II)) |
---|
| 2953 | { |
---|
[a36e78] | 2954 | absotto= absFactorize(primary[2*ab][1],77); |
---|
[1d430ab] | 2955 | tempo= |
---|
| 2956 | primary[2*ab-1],primary[2*ab],absotto,string(var(nvars(basering))); |
---|
| 2957 | } |
---|
| 2958 | else |
---|
| 2959 | { |
---|
| 2960 | invsigma=basering,maxideal(1); |
---|
| 2961 | jmap=randomLast(50); |
---|
| 2962 | sigma=basering,jmap; |
---|
| 2963 | jmap[nn]=2*var(nn)-jmap[nn]; |
---|
| 2964 | invsigma=basering,jmap; |
---|
| 2965 | II=groebner(sigma(II)); |
---|
| 2966 | absotto = absFactorize(II[1],77); |
---|
| 2967 | II=var(nn); |
---|
| 2968 | tempo= primary[2*ab-1],primary[2*ab],absotto,string(invsigma(II)); |
---|
| 2969 | } |
---|
| 2970 | resu[ab]=tempo; |
---|
| 2971 | } |
---|
| 2972 | primary=resu; |
---|
| 2973 | setring @P; |
---|
| 2974 | primary=imap(gnir,primary); |
---|
[6fa3af] | 2975 | } |
---|
[1e1ec4] | 2976 | option(set,op); |
---|
[e801fe] | 2977 | return(primary); |
---|
| 2978 | } |
---|
[d6db1f2] | 2979 | |
---|
| 2980 | poly @gs,@gh,@p; |
---|
| 2981 | string @va,quotring; |
---|
| 2982 | list quprimary,htprimary,collectprimary,lsau,lnew,allindep,restindep; |
---|
| 2983 | ideal @h; |
---|
| 2984 | int jdim=dim(@j); |
---|
| 2985 | list fett; |
---|
[e801fe] | 2986 | int lauf,di,newtest; |
---|
[67bd4c] | 2987 | //------------------------------------------------------------------ |
---|
| 2988 | //search for a maximal independent set indep,i.e. |
---|
| 2989 | //look for subring such that the intersection with the ideal is zero |
---|
| 2990 | //j intersected with K[var(indep[3]+1),...,var(nvar] is zero, |
---|
[9050ca] | 2991 | //indep[1] is the new varstring and indep[2] the string for block-ordering |
---|
[67bd4c] | 2992 | //------------------------------------------------------------------ |
---|
[d6db1f2] | 2993 | if(@wr!=1) |
---|
| 2994 | { |
---|
[1d430ab] | 2995 | allindep=independSet(@j); |
---|
| 2996 | for(@m=1;@m<=size(allindep);@m++) |
---|
| 2997 | { |
---|
| 2998 | if(allindep[@m][3]==jdim) |
---|
| 2999 | { |
---|
| 3000 | di++; |
---|
| 3001 | indep[di]=allindep[@m]; |
---|
| 3002 | } |
---|
| 3003 | else |
---|
| 3004 | { |
---|
| 3005 | lauf++; |
---|
| 3006 | restindep[lauf]=allindep[@m]; |
---|
| 3007 | } |
---|
| 3008 | } |
---|
| 3009 | } |
---|
| 3010 | else |
---|
| 3011 | { |
---|
| 3012 | indep=maxIndependSet(@j); |
---|
| 3013 | } |
---|
[3939bc] | 3014 | |
---|
[d6db1f2] | 3015 | ideal jkeep=@j; |
---|
| 3016 | if(ordstr(@P)[1]=="w") |
---|
| 3017 | { |
---|
[1d430ab] | 3018 | execute("ring @Phelp=("+charstr(gnir)+"),("+varstr(gnir)+"),("+ordstr(@P)+");"); |
---|
[d6db1f2] | 3019 | } |
---|
| 3020 | else |
---|
| 3021 | { |
---|
[1d430ab] | 3022 | execute( "ring @Phelp=("+charstr(gnir)+"),("+varstr(gnir)+"),(C,dp);"); |
---|
[e801fe] | 3023 | } |
---|
| 3024 | |
---|
| 3025 | if(homo==1) |
---|
| 3026 | { |
---|
| 3027 | if((ordstr(@P)[3]=="d")||(ordstr(@P)[1]=="d")||(ordstr(@P)[1]=="w") |
---|
| 3028 | ||(ordstr(@P)[3]=="w")) |
---|
| 3029 | { |
---|
| 3030 | ideal jwork=imap(@P,tras); |
---|
| 3031 | attrib(jwork,"isSB",1); |
---|
| 3032 | } |
---|
| 3033 | else |
---|
| 3034 | { |
---|
[2d2c8be] | 3035 | ideal jwork=std(imap(gnir,@j),@hilb,@w); |
---|
[e801fe] | 3036 | } |
---|
| 3037 | } |
---|
| 3038 | else |
---|
| 3039 | { |
---|
[9a384e] | 3040 | ideal jwork=groebner(imap(gnir,@j)); |
---|
[d6db1f2] | 3041 | } |
---|
[e801fe] | 3042 | list hquprimary; |
---|
[d6db1f2] | 3043 | poly @p,@q; |
---|
[e801fe] | 3044 | ideal @h,fac,ser; |
---|
[5c7562] | 3045 | ideal @Ptest=1; |
---|
[d6db1f2] | 3046 | di=dim(jwork); |
---|
[e801fe] | 3047 | keepdi=di; |
---|
[3939bc] | 3048 | |
---|
[d6db1f2] | 3049 | setring gnir; |
---|
| 3050 | for(@m=1;@m<=size(indep);@m++) |
---|
| 3051 | { |
---|
[1d430ab] | 3052 | isat=0; |
---|
| 3053 | @n2=0; |
---|
| 3054 | if((indep[@m][1]==varstr(basering))&&(@m==1)) |
---|
| 3055 | //this is the good case, nothing to do, just to have the same notations |
---|
| 3056 | //change the ring |
---|
| 3057 | { |
---|
| 3058 | execute("ring gnir1 = ("+charstr(basering)+"),("+varstr(basering)+"),(" |
---|
[a36e78] | 3059 | +ordstr(basering)+");"); |
---|
[1d430ab] | 3060 | ideal @j=fetch(gnir,@j); |
---|
| 3061 | attrib(@j,"isSB",1); |
---|
| 3062 | ideal ser=fetch(gnir,ser); |
---|
| 3063 | } |
---|
| 3064 | else |
---|
| 3065 | { |
---|
| 3066 | @va=string(maxideal(1)); |
---|
| 3067 | if(@m==1) |
---|
| 3068 | { |
---|
| 3069 | @j=fetch(@P,i); |
---|
| 3070 | } |
---|
| 3071 | execute("ring gnir1 = ("+charstr(basering)+"),("+indep[@m][1]+"),(" |
---|
[2d2cad9] | 3072 | +indep[@m][2]+");"); |
---|
[1d430ab] | 3073 | execute("map phi=gnir,"+@va+";"); |
---|
| 3074 | op=option(get); |
---|
| 3075 | option(redSB); |
---|
| 3076 | if(homo==1) |
---|
| 3077 | { |
---|
| 3078 | ideal @j=std(phi(@j),@hilb,@w); |
---|
| 3079 | } |
---|
| 3080 | else |
---|
| 3081 | { |
---|
| 3082 | ideal @j=groebner(phi(@j)); |
---|
| 3083 | } |
---|
| 3084 | ideal ser=phi(ser); |
---|
[3939bc] | 3085 | |
---|
[1d430ab] | 3086 | option(set,op); |
---|
| 3087 | } |
---|
| 3088 | if((deg(@j[1])==0)||(dim(@j)<jdim)) |
---|
| 3089 | { |
---|
| 3090 | setring gnir; |
---|
| 3091 | break; |
---|
| 3092 | } |
---|
| 3093 | for (lauf=1;lauf<=size(@j);lauf++) |
---|
| 3094 | { |
---|
| 3095 | fett[lauf]=size(@j[lauf]); |
---|
| 3096 | } |
---|
| 3097 | //------------------------------------------------------------------------ |
---|
| 3098 | //we have now the following situation: |
---|
| 3099 | //j intersected with K[var(nnp+1),..,var(nva)] is zero so we may pass |
---|
| 3100 | //to this quotientring, j is their still a standardbasis, the |
---|
| 3101 | //leading coefficients of the polynomials there (polynomials in |
---|
| 3102 | //K[var(nnp+1),..,var(nva)]) are collected in the list h, |
---|
| 3103 | //we need their ggt, gh, because of the following: let |
---|
| 3104 | //(j:gh^n)=(j:gh^infinity) then j*K(var(nnp+1),..,var(nva))[..the rest..] |
---|
| 3105 | //intersected with K[var(1),...,var(nva)] is (j:gh^n) |
---|
| 3106 | //on the other hand j=(j,gh^n) intersected with (j:gh^n) |
---|
| 3107 | |
---|
| 3108 | //------------------------------------------------------------------------ |
---|
| 3109 | |
---|
| 3110 | //arrangement for quotientring K(var(nnp+1),..,var(nva))[..the rest..] and |
---|
| 3111 | //map phi:K[var(1),...,var(nva)] --->K(var(nnpr+1),..,var(nva))[..rest..] |
---|
| 3112 | //------------------------------------------------------------------------ |
---|
| 3113 | |
---|
| 3114 | quotring=prepareQuotientring(nvars(basering)-indep[@m][3]); |
---|
| 3115 | |
---|
| 3116 | //--------------------------------------------------------------------- |
---|
| 3117 | //we pass to the quotientring K(var(nnp+1),..,var(nva))[..the rest..] |
---|
| 3118 | //--------------------------------------------------------------------- |
---|
| 3119 | |
---|
| 3120 | ideal @jj=lead(@j); //!! vorn vereinbaren |
---|
| 3121 | execute(quotring); |
---|
| 3122 | |
---|
| 3123 | ideal @jj=imap(gnir1,@jj); |
---|
| 3124 | @vv=clearSBNeu(@jj,fett); //!! vorn vereinbaren |
---|
| 3125 | setring gnir1; |
---|
| 3126 | @k=size(@j); |
---|
| 3127 | for (lauf=1;lauf<=@k;lauf++) |
---|
| 3128 | { |
---|
| 3129 | if(@vv[lauf]==1) |
---|
| 3130 | { |
---|
| 3131 | @j[lauf]=0; |
---|
| 3132 | } |
---|
| 3133 | } |
---|
| 3134 | @j=simplify(@j,2); |
---|
| 3135 | setring quring; |
---|
| 3136 | // @j considered in the quotientring |
---|
| 3137 | ideal @j=imap(gnir1,@j); |
---|
[70ab73] | 3138 | |
---|
[1d430ab] | 3139 | ideal ser=imap(gnir1,ser); |
---|
[70ab73] | 3140 | |
---|
[1d430ab] | 3141 | kill gnir1; |
---|
[70ab73] | 3142 | |
---|
[1d430ab] | 3143 | //j is a standardbasis in the quotientring but usually not minimal |
---|
| 3144 | //here it becomes minimal |
---|
[70ab73] | 3145 | |
---|
[1d430ab] | 3146 | attrib(@j,"isSB",1); |
---|
[70ab73] | 3147 | |
---|
[1d430ab] | 3148 | //we need later ggt(h[1],...)=gh for saturation |
---|
| 3149 | ideal @h; |
---|
| 3150 | if(deg(@j[1])>0) |
---|
| 3151 | { |
---|
| 3152 | for(@n=1;@n<=size(@j);@n++) |
---|
| 3153 | { |
---|
| 3154 | @h[@n]=leadcoef(@j[@n]); |
---|
| 3155 | } |
---|
| 3156 | //the primary decomposition of j*K(var(nnp+1),..,var(nva))[..the rest..] |
---|
| 3157 | op=option(get); |
---|
| 3158 | option(redSB); |
---|
[70ab73] | 3159 | |
---|
[1d430ab] | 3160 | list uprimary= zero_decomp(@j,ser,@wr); |
---|
[a36e78] | 3161 | //HIER |
---|
[1d430ab] | 3162 | if(abspri) |
---|
| 3163 | { |
---|
| 3164 | ideal II; |
---|
| 3165 | ideal jmap; |
---|
| 3166 | map sigma; |
---|
| 3167 | nn=nvars(basering); |
---|
| 3168 | map invsigma=basering,maxideal(1); |
---|
[4173c7] | 3169 | for(ab=1;ab<=size(uprimary) div 2;ab++) |
---|
[a36e78] | 3170 | { |
---|
[1d430ab] | 3171 | II=uprimary[2*ab]; |
---|
| 3172 | attrib(II,"isSB",1); |
---|
| 3173 | if(deg(II[1])!=vdim(II)) |
---|
| 3174 | { |
---|
| 3175 | jmap=randomLast(50); |
---|
| 3176 | sigma=basering,jmap; |
---|
| 3177 | jmap[nn]=2*var(nn)-jmap[nn]; |
---|
| 3178 | invsigma=basering,jmap; |
---|
| 3179 | II=groebner(sigma(II)); |
---|
| 3180 | } |
---|
| 3181 | absprimarytmp[ab]= absFactorize(II[1],77); |
---|
| 3182 | II=var(nn); |
---|
| 3183 | abskeeptmp[ab]=string(invsigma(II)); |
---|
| 3184 | invsigma=basering,maxideal(1); |
---|
[a36e78] | 3185 | } |
---|
[1d430ab] | 3186 | } |
---|
| 3187 | option(set,op); |
---|
| 3188 | } |
---|
| 3189 | else |
---|
| 3190 | { |
---|
| 3191 | list uprimary; |
---|
| 3192 | uprimary[1]=ideal(1); |
---|
| 3193 | uprimary[2]=ideal(1); |
---|
| 3194 | } |
---|
| 3195 | //we need the intersection of the ideals in the list quprimary with the |
---|
| 3196 | //polynomialring, i.e. let q=(f1,...,fr) in the quotientring such an ideal |
---|
| 3197 | //but fi polynomials, then the intersection of q with the polynomialring |
---|
| 3198 | //is the saturation of the ideal generated by f1,...,fr with respect to |
---|
| 3199 | //h which is the lcm of the leading coefficients of the fi considered in |
---|
| 3200 | //in the quotientring: this is coded in saturn |
---|
[f54c83] | 3201 | |
---|
[1d430ab] | 3202 | list saturn; |
---|
| 3203 | ideal hpl; |
---|
[d6db1f2] | 3204 | |
---|
[1d430ab] | 3205 | for(@n=1;@n<=size(uprimary);@n++) |
---|
| 3206 | { |
---|
| 3207 | uprimary[@n]=interred(uprimary[@n]); // temporary fix |
---|
| 3208 | hpl=0; |
---|
| 3209 | for(@n1=1;@n1<=size(uprimary[@n]);@n1++) |
---|
| 3210 | { |
---|
| 3211 | hpl=hpl,leadcoef(uprimary[@n][@n1]); |
---|
| 3212 | } |
---|
| 3213 | saturn[@n]=hpl; |
---|
| 3214 | } |
---|
[18dd47] | 3215 | |
---|
[1d430ab] | 3216 | //-------------------------------------------------------------------- |
---|
| 3217 | //we leave the quotientring K(var(nnp+1),..,var(nva))[..the rest..] |
---|
| 3218 | //back to the polynomialring |
---|
| 3219 | //--------------------------------------------------------------------- |
---|
| 3220 | setring gnir; |
---|
[d6db1f2] | 3221 | |
---|
[1d430ab] | 3222 | collectprimary=imap(quring,uprimary); |
---|
| 3223 | lsau=imap(quring,saturn); |
---|
| 3224 | @h=imap(quring,@h); |
---|
[d6db1f2] | 3225 | |
---|
[1d430ab] | 3226 | kill quring; |
---|
[7a7df90] | 3227 | |
---|
[1d430ab] | 3228 | @n2=size(quprimary); |
---|
| 3229 | @n3=@n2; |
---|
[a36e78] | 3230 | |
---|
[4173c7] | 3231 | for(@n1=1;@n1<=size(collectprimary) div 2;@n1++) |
---|
[1d430ab] | 3232 | { |
---|
| 3233 | if(deg(collectprimary[2*@n1][1])>0) |
---|
| 3234 | { |
---|
| 3235 | @n2++; |
---|
| 3236 | quprimary[@n2]=collectprimary[2*@n1-1]; |
---|
| 3237 | lnew[@n2]=lsau[2*@n1-1]; |
---|
| 3238 | @n2++; |
---|
| 3239 | lnew[@n2]=lsau[2*@n1]; |
---|
| 3240 | quprimary[@n2]=collectprimary[2*@n1]; |
---|
| 3241 | if(abspri) |
---|
[d6db1f2] | 3242 | { |
---|
[4173c7] | 3243 | absprimary[@n2 div 2]=absprimarytmp[@n1]; |
---|
| 3244 | abskeep[@n2 div 2]=abskeeptmp[@n1]; |
---|
[d6db1f2] | 3245 | } |
---|
[1d430ab] | 3246 | } |
---|
| 3247 | } |
---|
| 3248 | //here the intersection with the polynomialring |
---|
| 3249 | //mentioned above is really computed |
---|
[4173c7] | 3250 | for(@n=@n3 div 2+1;@n<=@n2 div 2;@n++) |
---|
[1d430ab] | 3251 | { |
---|
| 3252 | if(specialIdealsEqual(quprimary[2*@n-1],quprimary[2*@n])) |
---|
| 3253 | { |
---|
| 3254 | quprimary[2*@n-1]=sat2(quprimary[2*@n-1],lnew[2*@n-1])[1]; |
---|
| 3255 | quprimary[2*@n]=quprimary[2*@n-1]; |
---|
| 3256 | } |
---|
| 3257 | else |
---|
| 3258 | { |
---|
| 3259 | if(@wr==0) |
---|
[d6db1f2] | 3260 | { |
---|
[1d430ab] | 3261 | quprimary[2*@n-1]=sat2(quprimary[2*@n-1],lnew[2*@n-1])[1]; |
---|
[d6db1f2] | 3262 | } |
---|
[1d430ab] | 3263 | quprimary[2*@n]=sat2(quprimary[2*@n],lnew[2*@n])[1]; |
---|
| 3264 | } |
---|
| 3265 | } |
---|
[3939bc] | 3266 | |
---|
[1d430ab] | 3267 | if(size(@h)>0) |
---|
| 3268 | { |
---|
| 3269 | //--------------------------------------------------------------- |
---|
| 3270 | //we change to @Phelp to have the ordering dp for saturation |
---|
| 3271 | //--------------------------------------------------------------- |
---|
| 3272 | setring @Phelp; |
---|
| 3273 | @h=imap(gnir,@h); |
---|
| 3274 | if(@wr!=1) |
---|
| 3275 | { |
---|
| 3276 | if(defined(@LL)){kill @LL;} |
---|
| 3277 | list @LL=minSat(jwork,@h); |
---|
| 3278 | @Ptest=intersect(@Ptest,@LL[1]); |
---|
| 3279 | @q=@LL[2]; |
---|
| 3280 | } |
---|
| 3281 | else |
---|
| 3282 | { |
---|
| 3283 | fac=ideal(0); |
---|
| 3284 | for(lauf=1;lauf<=ncols(@h);lauf++) |
---|
[a36e78] | 3285 | { |
---|
[1d430ab] | 3286 | if(deg(@h[lauf])>0) |
---|
| 3287 | { |
---|
| 3288 | fac=fac+factorize(@h[lauf],1); |
---|
| 3289 | } |
---|
[a36e78] | 3290 | } |
---|
[1d430ab] | 3291 | fac=simplify(fac,6); |
---|
| 3292 | @q=1; |
---|
| 3293 | for(lauf=1;lauf<=size(fac);lauf++) |
---|
[a36e78] | 3294 | { |
---|
[1d430ab] | 3295 | @q=@q*fac[lauf]; |
---|
[a36e78] | 3296 | } |
---|
[1d430ab] | 3297 | } |
---|
| 3298 | jwork=std(jwork,@q); |
---|
| 3299 | keepdi=dim(jwork); |
---|
| 3300 | if(keepdi<di) |
---|
| 3301 | { |
---|
[d6db1f2] | 3302 | setring gnir; |
---|
| 3303 | @j=imap(@Phelp,jwork); |
---|
[1d430ab] | 3304 | break; |
---|
| 3305 | } |
---|
| 3306 | if(homo==1) |
---|
| 3307 | { |
---|
| 3308 | @hilb=hilb(jwork,1,@w); |
---|
| 3309 | } |
---|
| 3310 | |
---|
| 3311 | setring gnir; |
---|
| 3312 | @j=imap(@Phelp,jwork); |
---|
| 3313 | } |
---|
[d6db1f2] | 3314 | } |
---|
[7a7df90] | 3315 | |
---|
| 3316 | if((size(quprimary)==0)&&(@wr==1)) |
---|
[d6db1f2] | 3317 | { |
---|
[1d430ab] | 3318 | @j=ideal(1); |
---|
| 3319 | quprimary[1]=ideal(1); |
---|
| 3320 | quprimary[2]=ideal(1); |
---|
[d6db1f2] | 3321 | } |
---|
[e801fe] | 3322 | if((size(quprimary)==0)) |
---|
| 3323 | { |
---|
| 3324 | keepdi=di-1; |
---|
[17407e] | 3325 | quprimary[1]=ideal(1); |
---|
| 3326 | quprimary[2]=ideal(1); |
---|
[3939bc] | 3327 | } |
---|
[d6db1f2] | 3328 | //--------------------------------------------------------------- |
---|
| 3329 | //notice that j=sat(j,gh) intersected with (j,gh^n) |
---|
| 3330 | //we finished with sat(j,gh) and have to start with (j,gh^n) |
---|
| 3331 | //--------------------------------------------------------------- |
---|
| 3332 | if((deg(@j[1])!=0)&&(@wr!=1)) |
---|
| 3333 | { |
---|
[1d430ab] | 3334 | if(size(quprimary)>0) |
---|
| 3335 | { |
---|
| 3336 | setring @Phelp; |
---|
| 3337 | ser=imap(gnir,ser); |
---|
| 3338 | hquprimary=imap(gnir,quprimary); |
---|
| 3339 | if(@wr==0) |
---|
| 3340 | { |
---|
| 3341 | //HIER STATT DURCHSCHNITT SATURIEREN! |
---|
| 3342 | ideal htest=@Ptest; |
---|
| 3343 | } |
---|
| 3344 | else |
---|
| 3345 | { |
---|
| 3346 | ideal htest=hquprimary[2]; |
---|
| 3347 | |
---|
[4173c7] | 3348 | for (@n1=2;@n1<=size(hquprimary) div 2;@n1++) |
---|
[d6db1f2] | 3349 | { |
---|
[1d430ab] | 3350 | htest=intersect(htest,hquprimary[2*@n1]); |
---|
[d6db1f2] | 3351 | } |
---|
[1d430ab] | 3352 | } |
---|
[d6db1f2] | 3353 | |
---|
[1d430ab] | 3354 | if(size(ser)>0) |
---|
| 3355 | { |
---|
| 3356 | ser=intersect(htest,ser); |
---|
| 3357 | } |
---|
| 3358 | else |
---|
| 3359 | { |
---|
| 3360 | ser=htest; |
---|
| 3361 | } |
---|
| 3362 | setring gnir; |
---|
| 3363 | ser=imap(@Phelp,ser); |
---|
| 3364 | } |
---|
| 3365 | if(size(reduce(ser,peek,1))!=0) |
---|
| 3366 | { |
---|
| 3367 | for(@m=1;@m<=size(restindep);@m++) |
---|
| 3368 | { |
---|
| 3369 | // if(restindep[@m][3]>=keepdi) |
---|
| 3370 | // { |
---|
| 3371 | isat=0; |
---|
| 3372 | @n2=0; |
---|
[e801fe] | 3373 | |
---|
[1d430ab] | 3374 | if(restindep[@m][1]==varstr(basering)) |
---|
| 3375 | //the good case, nothing to do, just to have the same notations |
---|
| 3376 | //change the ring |
---|
[3939bc] | 3377 | { |
---|
[1d430ab] | 3378 | execute("ring gnir1 = ("+charstr(basering)+"),("+ |
---|
| 3379 | varstr(basering)+"),("+ordstr(basering)+");"); |
---|
| 3380 | ideal @j=fetch(gnir,jkeep); |
---|
| 3381 | attrib(@j,"isSB",1); |
---|
[d6db1f2] | 3382 | } |
---|
[a36e78] | 3383 | else |
---|
[d6db1f2] | 3384 | { |
---|
[1d430ab] | 3385 | @va=string(maxideal(1)); |
---|
| 3386 | execute("ring gnir1 = ("+charstr(basering)+"),("+ |
---|
[a36e78] | 3387 | restindep[@m][1]+"),(" +restindep[@m][2]+");"); |
---|
[1d430ab] | 3388 | execute("map phi=gnir,"+@va+";"); |
---|
| 3389 | op=option(get); |
---|
| 3390 | option(redSB); |
---|
| 3391 | if(homo==1) |
---|
| 3392 | { |
---|
| 3393 | ideal @j=std(phi(jkeep),keephilb,@w); |
---|
| 3394 | } |
---|
| 3395 | else |
---|
| 3396 | { |
---|
| 3397 | ideal @j=groebner(phi(jkeep)); |
---|
| 3398 | } |
---|
| 3399 | ideal ser=phi(ser); |
---|
| 3400 | option(set,op); |
---|
| 3401 | } |
---|
[a36e78] | 3402 | |
---|
[1d430ab] | 3403 | for (lauf=1;lauf<=size(@j);lauf++) |
---|
| 3404 | { |
---|
| 3405 | fett[lauf]=size(@j[lauf]); |
---|
| 3406 | } |
---|
| 3407 | //------------------------------------------------------------------ |
---|
| 3408 | //we have now the following situation: |
---|
| 3409 | //j intersected with K[var(nnp+1),..,var(nva)] is zero so we may |
---|
| 3410 | //pass to this quotientring, j is their still a standardbasis, the |
---|
| 3411 | //leading coefficients of the polynomials there (polynomials in |
---|
| 3412 | //K[var(nnp+1),..,var(nva)]) are collected in the list h, |
---|
| 3413 | //we need their ggt, gh, because of the following: |
---|
| 3414 | //let (j:gh^n)=(j:gh^infinity) then |
---|
| 3415 | //j*K(var(nnp+1),..,var(nva))[..the rest..] |
---|
| 3416 | //intersected with K[var(1),...,var(nva)] is (j:gh^n) |
---|
| 3417 | //on the other hand j=(j,gh^n) intersected with (j:gh^n) |
---|
| 3418 | |
---|
| 3419 | //------------------------------------------------------------------ |
---|
| 3420 | |
---|
| 3421 | //the arrangement for the quotientring |
---|
| 3422 | // K(var(nnp+1),..,var(nva))[..the rest..] |
---|
| 3423 | //and the map phi:K[var(1),...,var(nva)] ----> |
---|
| 3424 | //--->K(var(nnpr+1),..,var(nva))[..the rest..] |
---|
| 3425 | //------------------------------------------------------------------ |
---|
| 3426 | |
---|
| 3427 | quotring=prepareQuotientring(nvars(basering)-restindep[@m][3]); |
---|
| 3428 | |
---|
| 3429 | //------------------------------------------------------------------ |
---|
| 3430 | //we pass to the quotientring K(var(nnp+1),..,var(nva))[..rest..] |
---|
| 3431 | //------------------------------------------------------------------ |
---|
| 3432 | |
---|
| 3433 | execute(quotring); |
---|
| 3434 | |
---|
| 3435 | // @j considered in the quotientring |
---|
| 3436 | ideal @j=imap(gnir1,@j); |
---|
| 3437 | ideal ser=imap(gnir1,ser); |
---|
| 3438 | |
---|
| 3439 | kill gnir1; |
---|
| 3440 | |
---|
| 3441 | //j is a standardbasis in the quotientring but usually not minimal |
---|
| 3442 | //here it becomes minimal |
---|
| 3443 | @j=clearSB(@j,fett); |
---|
| 3444 | attrib(@j,"isSB",1); |
---|
[a36e78] | 3445 | |
---|
[1d430ab] | 3446 | //we need later ggt(h[1],...)=gh for saturation |
---|
| 3447 | ideal @h; |
---|
[a36e78] | 3448 | |
---|
[1d430ab] | 3449 | for(@n=1;@n<=size(@j);@n++) |
---|
| 3450 | { |
---|
| 3451 | @h[@n]=leadcoef(@j[@n]); |
---|
| 3452 | } |
---|
| 3453 | //the primary decomposition of j*K(var(nnp+1),..,var(nva))[..rest..] |
---|
[a36e78] | 3454 | |
---|
[1d430ab] | 3455 | op=option(get); |
---|
| 3456 | option(redSB); |
---|
| 3457 | list uprimary= zero_decomp(@j,ser,@wr); |
---|
[a36e78] | 3458 | //HIER |
---|
[1d430ab] | 3459 | if(abspri) |
---|
| 3460 | { |
---|
| 3461 | ideal II; |
---|
| 3462 | ideal jmap; |
---|
| 3463 | map sigma; |
---|
| 3464 | nn=nvars(basering); |
---|
| 3465 | map invsigma=basering,maxideal(1); |
---|
[4173c7] | 3466 | for(ab=1;ab<=size(uprimary) div 2;ab++) |
---|
[1d430ab] | 3467 | { |
---|
| 3468 | II=uprimary[2*ab]; |
---|
| 3469 | attrib(II,"isSB",1); |
---|
| 3470 | if(deg(II[1])!=vdim(II)) |
---|
| 3471 | { |
---|
| 3472 | jmap=randomLast(50); |
---|
| 3473 | sigma=basering,jmap; |
---|
| 3474 | jmap[nn]=2*var(nn)-jmap[nn]; |
---|
| 3475 | invsigma=basering,jmap; |
---|
| 3476 | II=groebner(sigma(II)); |
---|
| 3477 | } |
---|
| 3478 | absprimarytmp[ab]= absFactorize(II[1],77); |
---|
| 3479 | II=var(nn); |
---|
| 3480 | abskeeptmp[ab]=string(invsigma(II)); |
---|
| 3481 | invsigma=basering,maxideal(1); |
---|
| 3482 | } |
---|
| 3483 | } |
---|
| 3484 | option(set,op); |
---|
[a36e78] | 3485 | |
---|
[1d430ab] | 3486 | //we need the intersection of the ideals in the list quprimary with |
---|
| 3487 | //the polynomialring, i.e. let q=(f1,...,fr) in the quotientring |
---|
| 3488 | //such an ideal but fi polynomials, then the intersection of q with |
---|
| 3489 | //the polynomialring is the saturation of the ideal generated by |
---|
| 3490 | //f1,...,fr with respect toh which is the lcm of the leading |
---|
| 3491 | //coefficients of the fi considered in the quotientring: |
---|
| 3492 | //this is coded in saturn |
---|
[a36e78] | 3493 | |
---|
[1d430ab] | 3494 | list saturn; |
---|
| 3495 | ideal hpl; |
---|
[a36e78] | 3496 | |
---|
[1d430ab] | 3497 | for(@n=1;@n<=size(uprimary);@n++) |
---|
| 3498 | { |
---|
| 3499 | hpl=0; |
---|
| 3500 | for(@n1=1;@n1<=size(uprimary[@n]);@n1++) |
---|
| 3501 | { |
---|
| 3502 | hpl=hpl,leadcoef(uprimary[@n][@n1]); |
---|
| 3503 | } |
---|
| 3504 | saturn[@n]=hpl; |
---|
| 3505 | } |
---|
| 3506 | //------------------------------------------------------------------ |
---|
| 3507 | //we leave the quotientring K(var(nnp+1),..,var(nva))[..rest..] |
---|
| 3508 | //back to the polynomialring |
---|
| 3509 | //------------------------------------------------------------------ |
---|
| 3510 | setring gnir; |
---|
| 3511 | collectprimary=imap(quring,uprimary); |
---|
| 3512 | lsau=imap(quring,saturn); |
---|
| 3513 | @h=imap(quring,@h); |
---|
[a36e78] | 3514 | |
---|
[1d430ab] | 3515 | kill quring; |
---|
[a36e78] | 3516 | |
---|
[1d430ab] | 3517 | @n2=size(quprimary); |
---|
| 3518 | @n3=@n2; |
---|
[a36e78] | 3519 | |
---|
[4173c7] | 3520 | for(@n1=1;@n1<=size(collectprimary) div 2;@n1++) |
---|
[1d430ab] | 3521 | { |
---|
| 3522 | if(deg(collectprimary[2*@n1][1])>0) |
---|
| 3523 | { |
---|
| 3524 | @n2++; |
---|
| 3525 | quprimary[@n2]=collectprimary[2*@n1-1]; |
---|
| 3526 | lnew[@n2]=lsau[2*@n1-1]; |
---|
| 3527 | @n2++; |
---|
| 3528 | lnew[@n2]=lsau[2*@n1]; |
---|
| 3529 | quprimary[@n2]=collectprimary[2*@n1]; |
---|
| 3530 | if(abspri) |
---|
| 3531 | { |
---|
[4173c7] | 3532 | absprimary[@n2 div 2]=absprimarytmp[@n1]; |
---|
| 3533 | abskeep[@n2 div 2]=abskeeptmp[@n1]; |
---|
[1d430ab] | 3534 | } |
---|
| 3535 | } |
---|
| 3536 | } |
---|
[a36e78] | 3537 | |
---|
| 3538 | |
---|
[1d430ab] | 3539 | //here the intersection with the polynomialring |
---|
| 3540 | //mentioned above is really computed |
---|
[70ab73] | 3541 | |
---|
[4173c7] | 3542 | for(@n=@n3 div 2+1;@n<=@n2 div 2;@n++) |
---|
[6fa3af] | 3543 | { |
---|
[1d430ab] | 3544 | if(specialIdealsEqual(quprimary[2*@n-1],quprimary[2*@n])) |
---|
[6fa3af] | 3545 | { |
---|
[1d430ab] | 3546 | quprimary[2*@n-1]=sat2(quprimary[2*@n-1],lnew[2*@n-1])[1]; |
---|
| 3547 | quprimary[2*@n]=quprimary[2*@n-1]; |
---|
| 3548 | } |
---|
| 3549 | else |
---|
| 3550 | { |
---|
| 3551 | if(@wr==0) |
---|
| 3552 | { |
---|
| 3553 | quprimary[2*@n-1]=sat2(quprimary[2*@n-1],lnew[2*@n-1])[1]; |
---|
| 3554 | } |
---|
| 3555 | quprimary[2*@n]=sat2(quprimary[2*@n],lnew[2*@n])[1]; |
---|
[6fa3af] | 3556 | } |
---|
| 3557 | } |
---|
[1d430ab] | 3558 | if(@n2>=@n3+2) |
---|
[d6db1f2] | 3559 | { |
---|
[1d430ab] | 3560 | setring @Phelp; |
---|
| 3561 | ser=imap(gnir,ser); |
---|
| 3562 | hquprimary=imap(gnir,quprimary); |
---|
[4173c7] | 3563 | for(@n=@n3 div 2+1;@n<=@n2 div 2;@n++) |
---|
[1d430ab] | 3564 | { |
---|
| 3565 | if(@wr==0) |
---|
| 3566 | { |
---|
| 3567 | ser=intersect(ser,hquprimary[2*@n-1]); |
---|
| 3568 | } |
---|
| 3569 | else |
---|
| 3570 | { |
---|
| 3571 | ser=intersect(ser,hquprimary[2*@n]); |
---|
| 3572 | } |
---|
| 3573 | } |
---|
| 3574 | setring gnir; |
---|
| 3575 | ser=imap(@Phelp,ser); |
---|
[d6db1f2] | 3576 | } |
---|
[3939bc] | 3577 | |
---|
[1d430ab] | 3578 | // } |
---|
| 3579 | } |
---|
| 3580 | //HIER |
---|
[6fa3af] | 3581 | if(abspri) |
---|
| 3582 | { |
---|
| 3583 | list resu,tempo; |
---|
[4173c7] | 3584 | for(ab=1;ab<=size(quprimary) div 2;ab++) |
---|
[6fa3af] | 3585 | { |
---|
[1d430ab] | 3586 | if (deg(quprimary[2*ab][1])!=0) |
---|
| 3587 | { |
---|
| 3588 | tempo=quprimary[2*ab-1],quprimary[2*ab], |
---|
| 3589 | absprimary[ab],abskeep[ab]; |
---|
| 3590 | resu[ab]=tempo; |
---|
| 3591 | } |
---|
[6fa3af] | 3592 | } |
---|
| 3593 | quprimary=resu; |
---|
[1d430ab] | 3594 | @wr=3; |
---|
[70ab73] | 3595 | } |
---|
[1d430ab] | 3596 | if(size(reduce(ser,peek,1))!=0) |
---|
| 3597 | { |
---|
| 3598 | if(@wr>0) |
---|
| 3599 | { |
---|
| 3600 | htprimary=decomp(@j,@wr,peek,ser); |
---|
| 3601 | } |
---|
| 3602 | else |
---|
| 3603 | { |
---|
| 3604 | htprimary=decomp(@j,peek,ser); |
---|
| 3605 | } |
---|
| 3606 | // here we collect now both results primary(sat(j,gh)) |
---|
| 3607 | // and primary(j,gh^n) |
---|
| 3608 | @n=size(quprimary); |
---|
| 3609 | for (@k=1;@k<=size(htprimary);@k++) |
---|
| 3610 | { |
---|
| 3611 | quprimary[@n+@k]=htprimary[@k]; |
---|
| 3612 | } |
---|
| 3613 | } |
---|
| 3614 | } |
---|
| 3615 | } |
---|
| 3616 | else |
---|
| 3617 | { |
---|
| 3618 | if(abspri) |
---|
| 3619 | { |
---|
| 3620 | list resu,tempo; |
---|
[4173c7] | 3621 | for(ab=1;ab<=size(quprimary) div 2;ab++) |
---|
[1d430ab] | 3622 | { |
---|
| 3623 | tempo=quprimary[2*ab-1],quprimary[2*ab], |
---|
| 3624 | absprimary[ab],abskeep[ab]; |
---|
| 3625 | resu[ab]=tempo; |
---|
| 3626 | } |
---|
| 3627 | quprimary=resu; |
---|
| 3628 | } |
---|
| 3629 | } |
---|
[091424] | 3630 | //--------------------------------------------------------------------------- |
---|
[d6db1f2] | 3631 | //back to the ring we started with |
---|
| 3632 | //the final result: primary |
---|
[091424] | 3633 | //--------------------------------------------------------------------------- |
---|
[d6db1f2] | 3634 | setring @P; |
---|
| 3635 | primary=imap(gnir,quprimary); |
---|
[0ccdf4] | 3636 | if(!abspri) |
---|
| 3637 | { |
---|
[1d430ab] | 3638 | primary=cleanPrimary(primary); |
---|
[0ccdf4] | 3639 | } |
---|
[d92713] | 3640 | if (abspri && (typeof(primary[1][1])=="poly")) |
---|
| 3641 | { return(prepare_absprimdec(primary));} |
---|
[d6db1f2] | 3642 | return(primary); |
---|
| 3643 | } |
---|
[a36e78] | 3644 | |
---|
| 3645 | |
---|
[d6db1f2] | 3646 | example |
---|
| 3647 | { "EXAMPLE:"; echo = 2; |
---|
| 3648 | ring r = 32003,(x,y,z),lp; |
---|
| 3649 | poly p = z2+1; |
---|
| 3650 | poly q = z4+2; |
---|
| 3651 | ideal i = p^2*q^3,(y-z3)^3,(x-yz+z4)^4; |
---|
| 3652 | list pr= decomp(i); |
---|
| 3653 | pr; |
---|
[18dd47] | 3654 | testPrimary( pr, i); |
---|
[d6db1f2] | 3655 | } |
---|
[67bd4c] | 3656 | |
---|
| 3657 | /////////////////////////////////////////////////////////////////////////////// |
---|
[07c623] | 3658 | static proc powerCoeffs(poly f,int e) |
---|
[80654d] | 3659 | //computes a polynomial with the same monomials as f but coefficients |
---|
| 3660 | //the p^e th power of the coefficients of f |
---|
[67bd4c] | 3661 | { |
---|
[a36e78] | 3662 | int i; |
---|
| 3663 | poly g; |
---|
| 3664 | int ex=char(basering)^e; |
---|
| 3665 | for(i=1;i<=size(f);i++) |
---|
| 3666 | { |
---|
| 3667 | g=g+leadcoef(f[i])^ex*leadmonom(f[i]); |
---|
| 3668 | } |
---|
| 3669 | return(g); |
---|
[80654d] | 3670 | } |
---|
| 3671 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 3672 | |
---|
[fc5095] | 3673 | proc sep(poly f,int i, list #) |
---|
[80654d] | 3674 | "USAGE: input: a polynomial f depending on the i-th variable and optional |
---|
| 3675 | an integer k considering the polynomial f defined over Fp(t1,...,tm) |
---|
| 3676 | as polynomial over Fp(t(1)^(p^-k),...,t(m)^(p^-k)) |
---|
| 3677 | RETURN: the separabel part of f as polynomial in Fp(t1,...,tm) |
---|
[b9b906] | 3678 | and an integer k to indicate that f should be considerd |
---|
[80654d] | 3679 | as polynomial over Fp(t(1)^(p^-k),...,t(m)^(p^-k)) |
---|
| 3680 | EXAMPLE: example sep; shows an example |
---|
| 3681 | { |
---|
[a36e78] | 3682 | def R=basering; |
---|
| 3683 | int k; |
---|
| 3684 | if(size(#)>0){k=#[1];} |
---|
[fc5095] | 3685 | |
---|
[80654d] | 3686 | |
---|
[a36e78] | 3687 | poly h=gcd(f,diff(f,var(i))); |
---|
| 3688 | if((reduce(f,std(h))!=0)||(reduce(diff(f,var(i)),std(h))!=0)) |
---|
| 3689 | { |
---|
[cb980ab] | 3690 | ERROR("FEHLER IN GCD"); |
---|
[a36e78] | 3691 | } |
---|
| 3692 | poly g1=lift(h,f)[1][1]; // f/h |
---|
| 3693 | poly h1; |
---|
| 3694 | |
---|
| 3695 | while(h!=h1) |
---|
| 3696 | { |
---|
| 3697 | h1=h; |
---|
| 3698 | h=gcd(h,diff(h,var(i))); |
---|
| 3699 | } |
---|
[80654d] | 3700 | |
---|
[a36e78] | 3701 | if(deg(h1)==0){return(list(g1,k));} //in characteristic 0 we return here |
---|
[80654d] | 3702 | |
---|
[a36e78] | 3703 | k++; |
---|
[80654d] | 3704 | |
---|
[a36e78] | 3705 | ideal ma=maxideal(1); |
---|
| 3706 | ma[i]=var(i)^char(R); |
---|
| 3707 | map phi=R,ma; |
---|
| 3708 | ideal hh=h; //this is technical because preimage works only for ideals |
---|
[80654d] | 3709 | |
---|
[a36e78] | 3710 | poly u=preimage(R,phi,hh)[1]; //h=u(x(i)^p) |
---|
[80654d] | 3711 | |
---|
[a36e78] | 3712 | list g2=sep(u,i,k); //we consider u(t(1)^(p^-1),...,t(m)^(p^-1)) |
---|
| 3713 | g1=powerCoeffs(g1,g2[2]-k+1); //to have g1 over the same field as g2[1] |
---|
[80654d] | 3714 | |
---|
[a36e78] | 3715 | list g3=sep(g1*g2[1],i,g2[2]); |
---|
| 3716 | return(g3); |
---|
[80654d] | 3717 | } |
---|
| 3718 | example |
---|
| 3719 | { "EXAMPLE:"; echo = 2; |
---|
| 3720 | ring R=(5,t,s),(x,y,z),dp; |
---|
| 3721 | poly f=(x^25-t*x^5+t)*(x^3+s); |
---|
| 3722 | sep(f,1); |
---|
| 3723 | } |
---|
| 3724 | |
---|
| 3725 | /////////////////////////////////////////////////////////////////////////////// |
---|
[24f458] | 3726 | proc zeroRad(ideal I,list #) |
---|
[80654d] | 3727 | "USAGE: zeroRad(I) , I a zero-dimensional ideal |
---|
| 3728 | RETURN: the radical of I |
---|
| 3729 | NOTE: Algorithm of Kemper |
---|
| 3730 | EXAMPLE: example zeroRad; shows an example |
---|
| 3731 | { |
---|
[a36e78] | 3732 | if(homog(I)==1){return(maxideal(1));} |
---|
| 3733 | //I needs to be a reduced standard basis |
---|
| 3734 | def R=basering; |
---|
| 3735 | int m=npars(R); |
---|
| 3736 | int n=nvars(R); |
---|
| 3737 | int p=char(R); |
---|
| 3738 | int d=vdim(I); |
---|
| 3739 | int i,k; |
---|
| 3740 | list l; |
---|
| 3741 | if(((p==0)||(p>d))&&(d==deg(I[1]))) |
---|
| 3742 | { |
---|
| 3743 | intvec e=leadexp(I[1]); |
---|
| 3744 | for(i=1;i<=nvars(basering);i++) |
---|
| 3745 | { |
---|
| 3746 | if(e[i]!=0) break; |
---|
| 3747 | } |
---|
| 3748 | I[1]=sep(I[1],i)[1]; |
---|
| 3749 | return(interred(I)); |
---|
| 3750 | } |
---|
| 3751 | intvec op=option(get); |
---|
[80654d] | 3752 | |
---|
[a36e78] | 3753 | option(redSB); |
---|
| 3754 | ideal F=finduni(I);//F[i] generates I intersected with K[var(i)] |
---|
[25c431] | 3755 | |
---|
[a36e78] | 3756 | option(set,op); |
---|
| 3757 | if(size(#)>0){I=#[1];} |
---|
[80654d] | 3758 | |
---|
[a36e78] | 3759 | for(i=1;i<=n;i++) |
---|
| 3760 | { |
---|
| 3761 | l[i]=sep(F[i],i); |
---|
| 3762 | F[i]=l[i][1]; |
---|
| 3763 | if(l[i][2]>k){k=l[i][2];} //computation of the maximal k |
---|
| 3764 | } |
---|
[80654d] | 3765 | |
---|
[a90eb0] | 3766 | if((k==0)||(m==0)) //the separable case |
---|
| 3767 | { |
---|
| 3768 | intvec save=option(get);option(redSB); |
---|
| 3769 | I=interred(I+F);option(set,save);return(I); |
---|
| 3770 | } |
---|
[80371e] | 3771 | //I=simplify(I,1); |
---|
[80654d] | 3772 | |
---|
[a36e78] | 3773 | for(i=1;i<=n;i++) //consider all polynomials over |
---|
| 3774 | { //Fp(t(1)^(p^-k),...,t(m)^(p^-k)) |
---|
| 3775 | F[i]=powerCoeffs(F[i],k-l[i][2]); |
---|
| 3776 | } |
---|
[24f458] | 3777 | |
---|
[a36e78] | 3778 | string cR="ring @R="+string(p)+",("+parstr(R)+","+varstr(R)+"),dp;"; |
---|
| 3779 | execute(cR); |
---|
| 3780 | ideal F=imap(R,F); |
---|
[24f458] | 3781 | |
---|
[69b030f] | 3782 | string nR1="ring @S1="+string(p)+",("+varstr(R)+","+parstr(R)+",@y(1..m)),dp;"; |
---|
| 3783 | execute(nR1); |
---|
| 3784 | list lR=ringlist(@S1)[2]; |
---|
| 3785 | lR=lR[(size(lR)-m+1)..(size(lR))]; |
---|
| 3786 | |
---|
| 3787 | string nR="ring @S="+string(p)+",("+string(lR)+","+varstr(R)+","+parstr(R)+"),dp;"; |
---|
[a36e78] | 3788 | execute(nR); |
---|
[80654d] | 3789 | |
---|
[a36e78] | 3790 | ideal G=fetch(@R,F); //G[i](t(1)^(p^-k),...,t(m)^(p^-k),x(i))=sep(F[i]) |
---|
[24f458] | 3791 | |
---|
[a36e78] | 3792 | ideal I=imap(R,I); |
---|
| 3793 | ideal J=I+G; |
---|
| 3794 | poly el=1; |
---|
| 3795 | k=p^k; |
---|
| 3796 | for(i=1;i<=m;i++) |
---|
| 3797 | { |
---|
| 3798 | J=J,var(i)^k-var(m+n+i); |
---|
[957c6a] | 3799 | el=el*var(i); |
---|
[a36e78] | 3800 | } |
---|
[80654d] | 3801 | |
---|
[a36e78] | 3802 | J=eliminate(J,el); |
---|
| 3803 | setring R; |
---|
| 3804 | ideal J=imap(@S,J); |
---|
| 3805 | return(J); |
---|
[80654d] | 3806 | } |
---|
| 3807 | example |
---|
| 3808 | { "EXAMPLE:"; echo = 2; |
---|
| 3809 | ring R=(5,t),(x,y),dp; |
---|
| 3810 | ideal I=x^5-t,y^5-t; |
---|
[24f458] | 3811 | zeroRad(I); |
---|
[80654d] | 3812 | } |
---|
| 3813 | |
---|
[ebecf83] | 3814 | /////////////////////////////////////////////////////////////////////////////// |
---|
[67bd4c] | 3815 | |
---|
[07c623] | 3816 | proc radicalEHV(ideal i) |
---|
| 3817 | "USAGE: radicalEHV(i); i ideal. |
---|
| 3818 | RETURN: ideal, the radical of i. |
---|
[7a7df90] | 3819 | NOTE: Uses the algorithm of Eisenbud/Huneke/Vasconcelos, which |
---|
[50cbdc] | 3820 | reduces the computation to the complete intersection case, |
---|
[7a7df90] | 3821 | by taking, in the general case, a generic linear combination |
---|
| 3822 | of the input. |
---|
[07c623] | 3823 | Works only in characteristic 0 or p large. |
---|
| 3824 | EXAMPLE: example radicalEHV; shows an example |
---|
| 3825 | " |
---|
[67bd4c] | 3826 | { |
---|
[cb980ab] | 3827 | if(attrib(basering,"global")!=1) |
---|
[a36e78] | 3828 | { |
---|
[cb980ab] | 3829 | ERROR( |
---|
| 3830 | "// Not implemented for this ordering, please change to global ordering." |
---|
| 3831 | ); |
---|
[a36e78] | 3832 | } |
---|
[cb980ab] | 3833 | if((char(basering)<100)&&(char(basering)!=0)) |
---|
[3a2b8e] | 3834 | { |
---|
[cb980ab] | 3835 | "WARNING: The characteristic is too small, the result may be wrong"; |
---|
[3a2b8e] | 3836 | } |
---|
[a36e78] | 3837 | ideal J,I,I0,radI0,L,radI1,I2,radI2; |
---|
| 3838 | int l,n; |
---|
| 3839 | intvec op=option(get); |
---|
| 3840 | matrix M; |
---|
| 3841 | |
---|
| 3842 | option(redSB); |
---|
| 3843 | list m=mstd(i); |
---|
| 3844 | I=m[2]; |
---|
| 3845 | option(set,op); |
---|
| 3846 | |
---|
| 3847 | int cod=nvars(basering)-dim(m[1]); |
---|
| 3848 | //-------------------complete intersection case:---------------------- |
---|
| 3849 | if(cod==size(m[2])) |
---|
| 3850 | { |
---|
| 3851 | J=minor(jacob(I),cod); |
---|
| 3852 | return(quotient(I,J)); |
---|
| 3853 | } |
---|
| 3854 | //-----first codim elements of I are a complete intersection:--------- |
---|
| 3855 | for(l=1;l<=cod;l++) |
---|
| 3856 | { |
---|
| 3857 | I0[l]=I[l]; |
---|
| 3858 | } |
---|
| 3859 | n=dim(std(I0))+cod-nvars(basering); |
---|
| 3860 | //-----last codim elements of I are a complete intersection:---------- |
---|
| 3861 | if(n!=0) |
---|
| 3862 | { |
---|
| 3863 | for(l=1;l<=cod;l++) |
---|
| 3864 | { |
---|
| 3865 | I0[l]=I[size(I)-l+1]; |
---|
| 3866 | } |
---|
| 3867 | n=dim(std(I0))+cod-nvars(basering); |
---|
| 3868 | } |
---|
| 3869 | //-----taking a generic linear combination of the input:-------------- |
---|
| 3870 | if(n!=0) |
---|
| 3871 | { |
---|
| 3872 | M=transpose(sparsetriag(size(m[2]),cod,95,1)); |
---|
| 3873 | I0=ideal(M*transpose(I)); |
---|
| 3874 | n=dim(std(I0))+cod-nvars(basering); |
---|
| 3875 | } |
---|
| 3876 | //-----taking a more generic linear combination of the input:--------- |
---|
| 3877 | if(n!=0) |
---|
| 3878 | { |
---|
| 3879 | M=transpose(sparsetriag(size(m[2]),cod,0,100)); |
---|
| 3880 | I0=ideal(M*transpose(I)); |
---|
| 3881 | n=dim(std(I0))+cod-nvars(basering); |
---|
| 3882 | } |
---|
| 3883 | if(n==0) |
---|
| 3884 | { |
---|
| 3885 | J=minor(jacob(I0),cod); |
---|
| 3886 | radI0=quotient(I0,J); |
---|
| 3887 | L=quotient(radI0,I); |
---|
| 3888 | radI1=quotient(radI0,L); |
---|
[67bd4c] | 3889 | |
---|
[a36e78] | 3890 | if(size(reduce(radI1,m[1],1))==0) |
---|
| 3891 | { |
---|
| 3892 | return(I); |
---|
| 3893 | } |
---|
[70ab73] | 3894 | |
---|
[a36e78] | 3895 | I2=sat(I,radI1)[1]; |
---|
| 3896 | |
---|
| 3897 | if(deg(I2[1])<=0) |
---|
| 3898 | { |
---|
| 3899 | return(radI1); |
---|
| 3900 | } |
---|
| 3901 | return(intersect(radI1,radicalEHV(I2))); |
---|
| 3902 | } |
---|
| 3903 | //---------------------general case------------------------------------- |
---|
| 3904 | return(radical(I)); |
---|
[67bd4c] | 3905 | } |
---|
[07c623] | 3906 | example |
---|
| 3907 | { "EXAMPLE:"; echo = 2; |
---|
| 3908 | ring r = 0,(x,y,z),dp; |
---|
| 3909 | poly p = z2+1; |
---|
| 3910 | poly q = z3+2; |
---|
| 3911 | ideal i = p*q^2,y-z2; |
---|
| 3912 | ideal pr= radicalEHV(i); |
---|
| 3913 | pr; |
---|
| 3914 | } |
---|
| 3915 | |
---|
[ebecf83] | 3916 | /////////////////////////////////////////////////////////////////////////////// |
---|
[67bd4c] | 3917 | |
---|
[24f458] | 3918 | proc Ann(module M) |
---|
[76aca2] | 3919 | "USAGE: Ann(M); M module |
---|
| 3920 | RETURN: ideal, the annihilator of coker(M) |
---|
| 3921 | NOTE: The output is the ideal of all elements a of the basering R such that |
---|
| 3922 | a * R^m is contained in M (m=number of rows of M). |
---|
| 3923 | EXAMPLE: example Ann; shows an example |
---|
| 3924 | " |
---|
[67bd4c] | 3925 | { |
---|
| 3926 | M=prune(M); //to obtain a small embedding |
---|
[d950c5] | 3927 | ideal ann=quotient1(M,freemodule(nrows(M))); |
---|
[e801fe] | 3928 | return(ann); |
---|
[67bd4c] | 3929 | } |
---|
[76aca2] | 3930 | example |
---|
| 3931 | { "EXAMPLE:"; echo = 2; |
---|
| 3932 | ring r = 0,(x,y,z),lp; |
---|
| 3933 | module M = x2-y2,z3; |
---|
| 3934 | Ann(M); |
---|
| 3935 | M = [1,x2],[y,x]; |
---|
| 3936 | Ann(M); |
---|
| 3937 | qring Q=std(xy-1); |
---|
| 3938 | module M=imap(r,M); |
---|
| 3939 | Ann(M); |
---|
| 3940 | } |
---|
| 3941 | |
---|
[ebecf83] | 3942 | /////////////////////////////////////////////////////////////////////////////// |
---|
[67bd4c] | 3943 | |
---|
| 3944 | //computes the equidimensional part of the ideal i of codimension e |
---|
[07c623] | 3945 | static proc int_ass_primary_e(ideal i, int e) |
---|
[67bd4c] | 3946 | { |
---|
| 3947 | if(homog(i)!=1) |
---|
| 3948 | { |
---|
[a36e78] | 3949 | i=std(i); |
---|
[67bd4c] | 3950 | } |
---|
| 3951 | list re=sres(i,0); //the resolution |
---|
| 3952 | re=minres(re); //minimized resolution |
---|
| 3953 | ideal ann=AnnExt_R(e,re); |
---|
| 3954 | if(nvars(basering)-dim(std(ann))!=e) |
---|
| 3955 | { |
---|
| 3956 | return(ideal(1)); |
---|
| 3957 | } |
---|
| 3958 | return(ann); |
---|
[3939bc] | 3959 | } |
---|
| 3960 | |
---|
[ebecf83] | 3961 | /////////////////////////////////////////////////////////////////////////////// |
---|
[67bd4c] | 3962 | |
---|
| 3963 | //computes the annihilator of Ext^n(R/i,R) with given resolution re |
---|
| 3964 | //n is not necessarily the number of variables |
---|
[07c623] | 3965 | static proc AnnExt_R(int n,list re) |
---|
[67bd4c] | 3966 | { |
---|
| 3967 | if(n<nvars(basering)) |
---|
| 3968 | { |
---|
[a36e78] | 3969 | matrix f=transpose(re[n+1]); //Hom(_,R) |
---|
| 3970 | module k=nres(f,2)[2]; //the kernel |
---|
| 3971 | matrix g=transpose(re[n]); //the image of Hom(_,R) |
---|
[d950c5] | 3972 | |
---|
[a36e78] | 3973 | ideal ann=quotient1(g,k); //the anihilator |
---|
[67bd4c] | 3974 | } |
---|
| 3975 | else |
---|
| 3976 | { |
---|
[a36e78] | 3977 | ideal ann=Ann(transpose(re[n])); |
---|
[67bd4c] | 3978 | } |
---|
[3939bc] | 3979 | return(ann); |
---|
[e801fe] | 3980 | } |
---|
[ebecf83] | 3981 | /////////////////////////////////////////////////////////////////////////////// |
---|
[e801fe] | 3982 | |
---|
[07c623] | 3983 | static proc analyze(list pr) |
---|
[3939bc] | 3984 | { |
---|
[a36e78] | 3985 | int ii,jj; |
---|
[4173c7] | 3986 | for(ii=1;ii<=size(pr) div 2;ii++) |
---|
[a36e78] | 3987 | { |
---|
| 3988 | dim(std(pr[2*ii])); |
---|
| 3989 | idealsEqual(pr[2*ii-1],pr[2*ii]); |
---|
| 3990 | "==========================="; |
---|
| 3991 | } |
---|
[e801fe] | 3992 | |
---|
[4173c7] | 3993 | for(ii=size(pr) div 2;ii>1;ii--) |
---|
[a36e78] | 3994 | { |
---|
| 3995 | for(jj=1;jj<ii;jj++) |
---|
[e801fe] | 3996 | { |
---|
[a36e78] | 3997 | if(size(reduce(pr[2*jj],std(pr[2*ii],1)))==0) |
---|
| 3998 | { |
---|
| 3999 | "eingebette Komponente"; |
---|
| 4000 | jj; |
---|
| 4001 | ii; |
---|
| 4002 | } |
---|
[e801fe] | 4003 | } |
---|
[a36e78] | 4004 | } |
---|
[e801fe] | 4005 | } |
---|
| 4006 | |
---|
[ebecf83] | 4007 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 4008 | // |
---|
| 4009 | // Shimoyama-Yokoyama |
---|
| 4010 | // |
---|
| 4011 | /////////////////////////////////////////////////////////////////////////////// |
---|
[e801fe] | 4012 | |
---|
[07c623] | 4013 | static proc simplifyIdeal(ideal i) |
---|
[e801fe] | 4014 | { |
---|
| 4015 | def r=basering; |
---|
[3939bc] | 4016 | |
---|
[e801fe] | 4017 | int j,k; |
---|
| 4018 | map phi; |
---|
| 4019 | poly p; |
---|
[3939bc] | 4020 | |
---|
[e801fe] | 4021 | ideal iwork=i; |
---|
| 4022 | ideal imap1=maxideal(1); |
---|
| 4023 | ideal imap2=maxideal(1); |
---|
[3939bc] | 4024 | |
---|
[e801fe] | 4025 | |
---|
| 4026 | for(j=1;j<=nvars(basering);j++) |
---|
| 4027 | { |
---|
[a36e78] | 4028 | for(k=1;k<=size(i);k++) |
---|
[e801fe] | 4029 | { |
---|
| 4030 | if(deg(iwork[k]/var(j))==0) |
---|
| 4031 | { |
---|
| 4032 | p=-1/leadcoef(iwork[k]/var(j))*iwork[k]; |
---|
| 4033 | imap1[j]=p+2*var(j); |
---|
| 4034 | phi=r,imap1; |
---|
| 4035 | iwork=phi(iwork); |
---|
| 4036 | iwork=subst(iwork,var(j),0); |
---|
| 4037 | iwork[k]=var(j); |
---|
| 4038 | imap1=maxideal(1); |
---|
[3939bc] | 4039 | imap2[j]=-p; |
---|
[e801fe] | 4040 | break; |
---|
| 4041 | } |
---|
| 4042 | } |
---|
| 4043 | } |
---|
| 4044 | return(iwork,imap2); |
---|
| 4045 | } |
---|
| 4046 | |
---|
[3939bc] | 4047 | |
---|
[e801fe] | 4048 | /////////////////////////////////////////////////////// |
---|
| 4049 | // ini_mod |
---|
| 4050 | // input: a polynomial p |
---|
| 4051 | // output: the initial term of p as needed |
---|
| 4052 | // in the context of characteristic sets |
---|
| 4053 | ////////////////////////////////////////////////////// |
---|
| 4054 | |
---|
[07c623] | 4055 | static proc ini_mod(poly p) |
---|
[e801fe] | 4056 | { |
---|
| 4057 | if (p==0) |
---|
| 4058 | { |
---|
| 4059 | return(0); |
---|
| 4060 | } |
---|
| 4061 | int n; matrix m; |
---|
[70ab73] | 4062 | for( n=nvars(basering); n>0; n--) |
---|
[e801fe] | 4063 | { |
---|
| 4064 | m=coef(p,var(n)); |
---|
| 4065 | if(m[1,1]!=1) |
---|
| 4066 | { |
---|
| 4067 | p=m[2,1]; |
---|
| 4068 | break; |
---|
| 4069 | } |
---|
| 4070 | } |
---|
| 4071 | if(deg(p)==0) |
---|
| 4072 | { |
---|
| 4073 | p=0; |
---|
| 4074 | } |
---|
| 4075 | return(p); |
---|
| 4076 | } |
---|
| 4077 | /////////////////////////////////////////////////////// |
---|
| 4078 | // min_ass_prim_charsets |
---|
| 4079 | // input: generators of an ideal PS and an integer cho |
---|
| 4080 | // If cho=0, the given ordering of the variables is used. |
---|
| 4081 | // Otherwise, the system tries to find an "optimal ordering", |
---|
| 4082 | // which in some cases may considerably speed up the algorithm |
---|
| 4083 | // output: the minimal associated primes of PS |
---|
| 4084 | // algorithm: via characteriostic sets |
---|
| 4085 | ////////////////////////////////////////////////////// |
---|
| 4086 | |
---|
| 4087 | |
---|
[07c623] | 4088 | static proc min_ass_prim_charsets (ideal PS, int cho) |
---|
[e801fe] | 4089 | { |
---|
| 4090 | if((cho<0) and (cho>1)) |
---|
| 4091 | { |
---|
[a36e78] | 4092 | ERROR("<int> must be 0 or 1"); |
---|
[e801fe] | 4093 | } |
---|
[1e1ec4] | 4094 | intvec saveopt=option(get); |
---|
[70ab73] | 4095 | option(notWarnSB); |
---|
[1e1ec4] | 4096 | list L; |
---|
[e801fe] | 4097 | if(cho==0) |
---|
| 4098 | { |
---|
[1e1ec4] | 4099 | L=min_ass_prim_charsets0(PS); |
---|
[e801fe] | 4100 | } |
---|
| 4101 | else |
---|
| 4102 | { |
---|
[1e1ec4] | 4103 | L=min_ass_prim_charsets1(PS); |
---|
[e801fe] | 4104 | } |
---|
[1e1ec4] | 4105 | option(set,saveopt); |
---|
| 4106 | return(L); |
---|
[67bd4c] | 4107 | } |
---|
[e801fe] | 4108 | /////////////////////////////////////////////////////// |
---|
| 4109 | // min_ass_prim_charsets0 |
---|
| 4110 | // input: generators of an ideal PS |
---|
| 4111 | // output: the minimal associated primes of PS |
---|
| 4112 | // algorithm: via characteristic sets |
---|
| 4113 | // the given ordering of the variables is used |
---|
| 4114 | ////////////////////////////////////////////////////// |
---|
[67bd4c] | 4115 | |
---|
[e801fe] | 4116 | |
---|
[07c623] | 4117 | static proc min_ass_prim_charsets0 (ideal PS) |
---|
[e801fe] | 4118 | { |
---|
[466f80] | 4119 | intvec op; |
---|
[e801fe] | 4120 | matrix m=char_series(PS); // We compute an irreducible |
---|
| 4121 | // characteristic series |
---|
| 4122 | int i,j,k; |
---|
| 4123 | list PSI; |
---|
| 4124 | list PHI; // the ideals given by the characteristic series |
---|
| 4125 | for(i=nrows(m);i>=1; i--) |
---|
| 4126 | { |
---|
[70ab73] | 4127 | PHI[i]=ideal(m[i,1..ncols(m)]); |
---|
[e801fe] | 4128 | } |
---|
| 4129 | // We compute the radical of each ideal in PHI |
---|
| 4130 | ideal I,JS,II; |
---|
| 4131 | int sizeJS, sizeII; |
---|
| 4132 | for(i=size(PHI);i>=1; i--) |
---|
| 4133 | { |
---|
[70ab73] | 4134 | I=0; |
---|
| 4135 | for(j=size(PHI[i]);j>0;j--) |
---|
| 4136 | { |
---|
| 4137 | I=I+ini_mod(PHI[i][j]); |
---|
| 4138 | } |
---|
| 4139 | JS=std(PHI[i]); |
---|
[a36e78] | 4140 | sizeJS=size(JS); |
---|
| 4141 | for(j=size(I);j>0;j--) |
---|
[70ab73] | 4142 | { |
---|
| 4143 | II=0; |
---|
| 4144 | sizeII=0; |
---|
| 4145 | k=0; |
---|
| 4146 | while(k<=sizeII) // successive saturation |
---|
| 4147 | { |
---|
| 4148 | op=option(get); |
---|
| 4149 | option(returnSB); |
---|
| 4150 | II=quotient(JS,I[j]); |
---|
| 4151 | option(set,op); |
---|
[a36e78] | 4152 | sizeII=size(II); |
---|
[70ab73] | 4153 | if(sizeII==sizeJS) |
---|
| 4154 | { |
---|
| 4155 | for(k=1;k<=sizeII;k++) |
---|
| 4156 | { |
---|
| 4157 | if(leadexp(II[k])!=leadexp(JS[k])) break; |
---|
| 4158 | } |
---|
| 4159 | } |
---|
| 4160 | JS=II; |
---|
| 4161 | sizeJS=sizeII; |
---|
| 4162 | } |
---|
[e801fe] | 4163 | } |
---|
| 4164 | PSI=insert(PSI,JS); |
---|
| 4165 | } |
---|
| 4166 | int sizePSI=size(PSI); |
---|
| 4167 | // We eliminate redundant ideals |
---|
| 4168 | for(i=1;i<sizePSI;i++) |
---|
| 4169 | { |
---|
| 4170 | for(j=i+1;j<=sizePSI;j++) |
---|
| 4171 | { |
---|
| 4172 | if(size(PSI[i])!=0) |
---|
| 4173 | { |
---|
| 4174 | if(size(PSI[j])!=0) |
---|
| 4175 | { |
---|
| 4176 | if(size(NF(PSI[i],PSI[j],1))==0) |
---|
| 4177 | { |
---|
| 4178 | PSI[j]=ideal(0); |
---|
| 4179 | } |
---|
| 4180 | else |
---|
| 4181 | { |
---|
| 4182 | if(size(NF(PSI[j],PSI[i],1))==0) |
---|
| 4183 | { |
---|
| 4184 | PSI[i]=ideal(0); |
---|
| 4185 | } |
---|
| 4186 | } |
---|
| 4187 | } |
---|
| 4188 | } |
---|
| 4189 | } |
---|
| 4190 | } |
---|
| 4191 | for(i=sizePSI;i>=1;i--) |
---|
| 4192 | { |
---|
| 4193 | if(size(PSI[i])==0) |
---|
| 4194 | { |
---|
| 4195 | PSI=delete(PSI,i); |
---|
| 4196 | } |
---|
| 4197 | } |
---|
| 4198 | return (PSI); |
---|
| 4199 | } |
---|
| 4200 | |
---|
| 4201 | /////////////////////////////////////////////////////// |
---|
| 4202 | // min_ass_prim_charsets1 |
---|
| 4203 | // input: generators of an ideal PS |
---|
| 4204 | // output: the minimal associated primes of PS |
---|
| 4205 | // algorithm: via characteristic sets |
---|
| 4206 | // input: generators of an ideal PS and an integer i |
---|
| 4207 | // The system tries to find an "optimal ordering" of |
---|
| 4208 | // the variables |
---|
| 4209 | ////////////////////////////////////////////////////// |
---|
| 4210 | |
---|
| 4211 | |
---|
[07c623] | 4212 | static proc min_ass_prim_charsets1 (ideal PS) |
---|
[e801fe] | 4213 | { |
---|
[466f80] | 4214 | intvec op; |
---|
[e801fe] | 4215 | def oldring=basering; |
---|
| 4216 | string n=system("neworder",PS); |
---|
[2d2cad9] | 4217 | execute("ring r=("+charstr(oldring)+"),("+n+"),dp;"); |
---|
[e801fe] | 4218 | ideal PS=imap(oldring,PS); |
---|
| 4219 | matrix m=char_series(PS); // We compute an irreducible |
---|
| 4220 | // characteristic series |
---|
| 4221 | int i,j,k; |
---|
| 4222 | ideal I; |
---|
| 4223 | list PSI; |
---|
| 4224 | list PHI; // the ideals given by the characteristic series |
---|
| 4225 | list ITPHI; // their initial terms |
---|
| 4226 | for(i=nrows(m);i>=1; i--) |
---|
| 4227 | { |
---|
[70ab73] | 4228 | PHI[i]=ideal(m[i,1..ncols(m)]); |
---|
| 4229 | I=0; |
---|
| 4230 | for(j=size(PHI[i]);j>0;j=j-1) |
---|
| 4231 | { |
---|
| 4232 | I=I,ini_mod(PHI[i][j]); |
---|
| 4233 | } |
---|
| 4234 | I=I[2..ncols(I)]; |
---|
| 4235 | ITPHI[i]=I; |
---|
[e801fe] | 4236 | } |
---|
| 4237 | setring oldring; |
---|
| 4238 | matrix m=imap(r,m); |
---|
| 4239 | list PHI=imap(r,PHI); |
---|
| 4240 | list ITPHI=imap(r,ITPHI); |
---|
| 4241 | // We compute the radical of each ideal in PHI |
---|
| 4242 | ideal I,JS,II; |
---|
| 4243 | int sizeJS, sizeII; |
---|
| 4244 | for(i=size(PHI);i>=1; i--) |
---|
| 4245 | { |
---|
[70ab73] | 4246 | I=0; |
---|
| 4247 | for(j=size(PHI[i]);j>0;j--) |
---|
| 4248 | { |
---|
| 4249 | I=I+ITPHI[i][j]; |
---|
| 4250 | } |
---|
| 4251 | JS=std(PHI[i]); |
---|
| 4252 | sizeJS=size(JS); |
---|
[a36e78] | 4253 | for(j=size(I);j>0;j--) |
---|
[70ab73] | 4254 | { |
---|
| 4255 | II=0; |
---|
| 4256 | sizeII=0; |
---|
| 4257 | k=0; |
---|
| 4258 | while(k<=sizeII) // successive iteration |
---|
| 4259 | { |
---|
| 4260 | op=option(get); |
---|
| 4261 | option(returnSB); |
---|
| 4262 | II=quotient(JS,I[j]); |
---|
| 4263 | option(set,op); |
---|
[e801fe] | 4264 | //std |
---|
[a36e78] | 4265 | // II=std(II); |
---|
| 4266 | sizeII=size(II); |
---|
[70ab73] | 4267 | if(sizeII==sizeJS) |
---|
| 4268 | { |
---|
| 4269 | for(k=1;k<=sizeII;k++) |
---|
| 4270 | { |
---|
| 4271 | if(leadexp(II[k])!=leadexp(JS[k])) break; |
---|
| 4272 | } |
---|
| 4273 | } |
---|
| 4274 | JS=II; |
---|
| 4275 | sizeJS=sizeII; |
---|
| 4276 | } |
---|
[e801fe] | 4277 | } |
---|
| 4278 | PSI=insert(PSI,JS); |
---|
| 4279 | } |
---|
| 4280 | int sizePSI=size(PSI); |
---|
| 4281 | // We eliminate redundant ideals |
---|
| 4282 | for(i=1;i<sizePSI;i++) |
---|
| 4283 | { |
---|
| 4284 | for(j=i+1;j<=sizePSI;j++) |
---|
| 4285 | { |
---|
| 4286 | if(size(PSI[i])!=0) |
---|
| 4287 | { |
---|
| 4288 | if(size(PSI[j])!=0) |
---|
| 4289 | { |
---|
| 4290 | if(size(NF(PSI[i],PSI[j],1))==0) |
---|
| 4291 | { |
---|
| 4292 | PSI[j]=ideal(0); |
---|
| 4293 | } |
---|
| 4294 | else |
---|
| 4295 | { |
---|
| 4296 | if(size(NF(PSI[j],PSI[i],1))==0) |
---|
| 4297 | { |
---|
| 4298 | PSI[i]=ideal(0); |
---|
| 4299 | } |
---|
| 4300 | } |
---|
| 4301 | } |
---|
| 4302 | } |
---|
| 4303 | } |
---|
| 4304 | } |
---|
| 4305 | for(i=sizePSI;i>=1;i--) |
---|
| 4306 | { |
---|
| 4307 | if(size(PSI[i])==0) |
---|
| 4308 | { |
---|
| 4309 | PSI=delete(PSI,i); |
---|
| 4310 | } |
---|
| 4311 | } |
---|
| 4312 | return (PSI); |
---|
| 4313 | } |
---|
| 4314 | |
---|
| 4315 | |
---|
| 4316 | ///////////////////////////////////////////////////// |
---|
| 4317 | // proc prim_dec |
---|
| 4318 | // input: generators of an ideal I and an integer choose |
---|
| 4319 | // If choose=0, min_ass_prim_charsets with the given |
---|
| 4320 | // ordering of the variables is used. |
---|
| 4321 | // If choose=1, min_ass_prim_charsets with the "optimized" |
---|
| 4322 | // ordering of the variables is used. |
---|
| 4323 | // If choose=2, minAssPrimes from primdec.lib is used |
---|
| 4324 | // If choose=3, minAssPrimes+factorizing Buchberger from primdec.lib is used |
---|
| 4325 | // output: a primary decomposition of I, i.e., a list |
---|
| 4326 | // of pairs consisting of a standard basis of a primary component |
---|
| 4327 | // of I and a standard basis of the corresponding associated prime. |
---|
| 4328 | // To compute the minimal associated primes of a given ideal |
---|
| 4329 | // min_ass_prim_l is called, i.e., the minimal associated primes |
---|
| 4330 | // are computed via characteristic sets. |
---|
| 4331 | // In the homogeneous case, the performance of the procedure |
---|
| 4332 | // will be improved if I is already given by a minimal set of |
---|
| 4333 | // generators. Apply minbase if necessary. |
---|
| 4334 | ////////////////////////////////////////////////////////// |
---|
| 4335 | |
---|
| 4336 | |
---|
[07c623] | 4337 | static proc prim_dec(ideal I, int choose) |
---|
[e801fe] | 4338 | { |
---|
[70ab73] | 4339 | if((choose<0) or (choose>3)) |
---|
| 4340 | { |
---|
[cb980ab] | 4341 | ERROR("ERROR: <int> must be 0 or 1 or 2 or 3"); |
---|
[70ab73] | 4342 | } |
---|
[e801fe] | 4343 | ideal H=1; // The intersection of the primary components |
---|
| 4344 | list U; // the leaves of the decomposition tree, i.e., |
---|
| 4345 | // pairs consisting of a primary component of I |
---|
| 4346 | // and the corresponding associated prime |
---|
| 4347 | list W; // the non-leaf vertices in the decomposition tree. |
---|
| 4348 | // every entry has 6 components: |
---|
| 4349 | // 1- the vertex itself , i.e., a standard bais of the |
---|
| 4350 | // given ideal I (type 1), or a standard basis of a |
---|
| 4351 | // pseudo-primary component arising from |
---|
| 4352 | // pseudo-primary decomposition (type 2), or a |
---|
| 4353 | // standard basis of a remaining component arising from |
---|
| 4354 | // pseudo-primary decomposition or extraction (type 3) |
---|
| 4355 | // 2- the type of the vertex as indicated above |
---|
| 4356 | // 3- the weighted_tree_depth of the vertex |
---|
| 4357 | // 4- the tester of the vertex |
---|
| 4358 | // 5- a standard basis of the associated prime |
---|
| 4359 | // of a vertex of type 2, or 0 otherwise |
---|
| 4360 | // 6- a list of pairs consisting of a standard |
---|
| 4361 | // basis of a minimal associated prime ideal |
---|
| 4362 | // of the father of the vertex and the |
---|
| 4363 | // irreducible factors of the "minimal |
---|
| 4364 | // divisor" of the seperator or extractor |
---|
| 4365 | // corresponding to the prime ideal |
---|
| 4366 | // as computed by the procedure minsat, |
---|
| 4367 | // if the vertex is of type 3, or |
---|
| 4368 | // the empty list otherwise |
---|
| 4369 | ideal SI=std(I); |
---|
[333b889] | 4370 | if(SI[1]==1) // primdecSY(ideal(1)) |
---|
| 4371 | { |
---|
| 4372 | return(list()); |
---|
| 4373 | } |
---|
[1e1ec4] | 4374 | intvec save=option(get); |
---|
| 4375 | option(notWarnSB); |
---|
[e801fe] | 4376 | int ncolsSI=ncols(SI); |
---|
| 4377 | int ncolsH=1; |
---|
| 4378 | W[1]=list(I,1,0,poly(1),ideal(0),list()); // The root of the tree |
---|
| 4379 | int weighted_tree_depth; |
---|
| 4380 | int i,j; |
---|
| 4381 | int check; |
---|
| 4382 | list V; // current vertex |
---|
| 4383 | list VV; // new vertex |
---|
| 4384 | list QQ; |
---|
| 4385 | list WI; |
---|
| 4386 | ideal Qi,SQ,SRest,fac; |
---|
| 4387 | poly tester; |
---|
| 4388 | |
---|
| 4389 | while(1) |
---|
| 4390 | { |
---|
| 4391 | i=1; |
---|
| 4392 | while(1) |
---|
| 4393 | { |
---|
| 4394 | while(i<=size(W)) // find vertex V of smallest weighted tree-depth |
---|
| 4395 | { |
---|
| 4396 | if (W[i][3]<=weighted_tree_depth) break; |
---|
| 4397 | i++; |
---|
| 4398 | } |
---|
| 4399 | if (i<=size(W)) break; |
---|
| 4400 | i=1; |
---|
| 4401 | weighted_tree_depth++; |
---|
| 4402 | } |
---|
| 4403 | V=W[i]; |
---|
| 4404 | W=delete(W,i); // delete V from W |
---|
| 4405 | |
---|
| 4406 | // now proceed by type of vertex V |
---|
| 4407 | |
---|
| 4408 | if (V[2]==2) // extraction needed |
---|
| 4409 | { |
---|
| 4410 | SQ,SRest,fac=extraction(V[1],V[5]); |
---|
| 4411 | // standard basis of primary component, |
---|
| 4412 | // standard basis of remaining component, |
---|
| 4413 | // irreducible factors of |
---|
| 4414 | // the "minimal divisor" of the extractor |
---|
| 4415 | // as computed by the procedure minsat, |
---|
| 4416 | check=0; |
---|
| 4417 | for(j=1;j<=ncolsH;j++) |
---|
| 4418 | { |
---|
| 4419 | if (NF(H[j],SQ,1)!=0) // Q is not redundant |
---|
| 4420 | { |
---|
| 4421 | check=1; |
---|
| 4422 | break; |
---|
| 4423 | } |
---|
| 4424 | } |
---|
| 4425 | if(check==1) // Q is not redundant |
---|
| 4426 | { |
---|
| 4427 | QQ=list(); |
---|
| 4428 | QQ[1]=list(SQ,V[5]); // primary component, associated prime, |
---|
| 4429 | // i.e., standard bases thereof |
---|
| 4430 | U=U+QQ; |
---|
[d950c5] | 4431 | H=intersect(H,SQ); |
---|
[e801fe] | 4432 | H=std(H); |
---|
| 4433 | ncolsH=ncols(H); |
---|
| 4434 | check=0; |
---|
| 4435 | if(ncolsH==ncolsSI) |
---|
| 4436 | { |
---|
| 4437 | for(j=1;j<=ncolsSI;j++) |
---|
| 4438 | { |
---|
| 4439 | if(leadexp(H[j])!=leadexp(SI[j])) |
---|
| 4440 | { |
---|
| 4441 | check=1; |
---|
| 4442 | break; |
---|
| 4443 | } |
---|
| 4444 | } |
---|
| 4445 | } |
---|
| 4446 | else |
---|
| 4447 | { |
---|
| 4448 | check=1; |
---|
| 4449 | } |
---|
| 4450 | if(check==0) // H==I => U is a primary decomposition |
---|
| 4451 | { |
---|
[552d26] | 4452 | option(set,save); |
---|
[e801fe] | 4453 | return(U); |
---|
| 4454 | } |
---|
| 4455 | } |
---|
| 4456 | if (SRest[1]!=1) // the remaining component is not |
---|
| 4457 | // the whole ring |
---|
| 4458 | { |
---|
| 4459 | if (rad_con(V[4],SRest)==0) // the new vertex is not the |
---|
| 4460 | // root of a redundant subtree |
---|
| 4461 | { |
---|
| 4462 | VV[1]=SRest; // remaining component |
---|
| 4463 | VV[2]=3; // pseudoprimdec_special |
---|
| 4464 | VV[3]=V[3]+1; // weighted depth |
---|
| 4465 | VV[4]=V[4]; // the tester did not change |
---|
| 4466 | VV[5]=ideal(0); |
---|
| 4467 | VV[6]=list(list(V[5],fac)); |
---|
| 4468 | W=insert(W,VV,size(W)); |
---|
| 4469 | } |
---|
| 4470 | } |
---|
| 4471 | } |
---|
| 4472 | else |
---|
| 4473 | { |
---|
| 4474 | if (V[2]==3) // pseudo_prim_dec_special is needed |
---|
| 4475 | { |
---|
| 4476 | QQ,SRest=pseudo_prim_dec_special_charsets(V[1],V[6],choose); |
---|
| 4477 | // QQ = quadruples: |
---|
| 4478 | // standard basis of pseudo-primary component, |
---|
| 4479 | // standard basis of corresponding prime, |
---|
| 4480 | // seperator, irreducible factors of |
---|
| 4481 | // the "minimal divisor" of the seperator |
---|
| 4482 | // as computed by the procedure minsat, |
---|
| 4483 | // SRest=standard basis of remaining component |
---|
| 4484 | } |
---|
| 4485 | else // V is the root, pseudo_prim_dec is needed |
---|
| 4486 | { |
---|
| 4487 | QQ,SRest=pseudo_prim_dec_charsets(I,SI,choose); |
---|
| 4488 | // QQ = quadruples: |
---|
| 4489 | // standard basis of pseudo-primary component, |
---|
| 4490 | // standard basis of corresponding prime, |
---|
| 4491 | // seperator, irreducible factors of |
---|
| 4492 | // the "minimal divisor" of the seperator |
---|
| 4493 | // as computed by the procedure minsat, |
---|
| 4494 | // SRest=standard basis of remaining component |
---|
| 4495 | } |
---|
[091424] | 4496 | //check |
---|
[e801fe] | 4497 | for(i=size(QQ);i>=1;i--) |
---|
| 4498 | //for(i=1;i<=size(QQ);i++) |
---|
| 4499 | { |
---|
| 4500 | tester=QQ[i][3]*V[4]; |
---|
| 4501 | Qi=QQ[i][2]; |
---|
| 4502 | if(NF(tester,Qi,1)!=0) // the new vertex is not the |
---|
| 4503 | // root of a redundant subtree |
---|
| 4504 | { |
---|
| 4505 | VV[1]=QQ[i][1]; |
---|
| 4506 | VV[2]=2; |
---|
| 4507 | VV[3]=V[3]+1; |
---|
| 4508 | VV[4]=tester; // the new tester as computed above |
---|
| 4509 | VV[5]=Qi; // QQ[i][2]; |
---|
| 4510 | VV[6]=list(); |
---|
| 4511 | W=insert(W,VV,size(W)); |
---|
| 4512 | } |
---|
| 4513 | } |
---|
| 4514 | if (SRest[1]!=1) // the remaining component is not |
---|
| 4515 | // the whole ring |
---|
| 4516 | { |
---|
| 4517 | if (rad_con(V[4],SRest)==0) // the vertex is not the root |
---|
| 4518 | // of a redundant subtree |
---|
| 4519 | { |
---|
| 4520 | VV[1]=SRest; |
---|
| 4521 | VV[2]=3; |
---|
| 4522 | VV[3]=V[3]+2; |
---|
| 4523 | VV[4]=V[4]; // the tester did not change |
---|
| 4524 | VV[5]=ideal(0); |
---|
| 4525 | WI=list(); |
---|
| 4526 | for(i=1;i<=size(QQ);i++) |
---|
| 4527 | { |
---|
| 4528 | WI=insert(WI,list(QQ[i][2],QQ[i][4])); |
---|
| 4529 | } |
---|
| 4530 | VV[6]=WI; |
---|
| 4531 | W=insert(W,VV,size(W)); |
---|
| 4532 | } |
---|
| 4533 | } |
---|
| 4534 | } |
---|
| 4535 | } |
---|
[1e1ec4] | 4536 | option(set,save); |
---|
[e801fe] | 4537 | } |
---|
| 4538 | |
---|
| 4539 | ////////////////////////////////////////////////////////////////////////// |
---|
| 4540 | // proc pseudo_prim_dec_charsets |
---|
| 4541 | // input: Generators of an arbitrary ideal I, a standard basis SI of I, |
---|
| 4542 | // and an integer choo |
---|
| 4543 | // If choo=0, min_ass_prim_charsets with the given |
---|
| 4544 | // ordering of the variables is used. |
---|
| 4545 | // If choo=1, min_ass_prim_charsets with the "optimized" |
---|
| 4546 | // ordering of the variables is used. |
---|
| 4547 | // If choo=2, minAssPrimes from primdec.lib is used |
---|
| 4548 | // If choo=3, minAssPrimes+factorizing Buchberger from primdec.lib is used |
---|
| 4549 | // output: a pseudo primary decomposition of I, i.e., a list |
---|
| 4550 | // of pseudo primary components together with a standard basis of the |
---|
| 4551 | // remaining component. Each pseudo primary component is |
---|
| 4552 | // represented by a quadrupel: A standard basis of the component, |
---|
| 4553 | // a standard basis of the corresponding associated prime, the |
---|
| 4554 | // seperator of the component, and the irreducible factors of the |
---|
| 4555 | // "minimal divisor" of the seperator as computed by the procedure minsat, |
---|
| 4556 | // calls proc pseudo_prim_dec_i |
---|
| 4557 | ////////////////////////////////////////////////////////////////////////// |
---|
| 4558 | |
---|
| 4559 | |
---|
[07c623] | 4560 | static proc pseudo_prim_dec_charsets (ideal I, ideal SI, int choo) |
---|
[e801fe] | 4561 | { |
---|
| 4562 | list L; // The list of minimal associated primes, |
---|
| 4563 | // each one given by a standard basis |
---|
| 4564 | if((choo==0) or (choo==1)) |
---|
[70ab73] | 4565 | { |
---|
| 4566 | L=min_ass_prim_charsets(I,choo); |
---|
| 4567 | } |
---|
| 4568 | else |
---|
| 4569 | { |
---|
| 4570 | if(choo==2) |
---|
[e801fe] | 4571 | { |
---|
[70ab73] | 4572 | L=minAssPrimes(I); |
---|
[e801fe] | 4573 | } |
---|
[70ab73] | 4574 | else |
---|
[e801fe] | 4575 | { |
---|
[70ab73] | 4576 | L=minAssPrimes(I,1); |
---|
[e801fe] | 4577 | } |
---|
[70ab73] | 4578 | for(int i=size(L);i>=1;i--) |
---|
| 4579 | { |
---|
| 4580 | L[i]=std(L[i]); |
---|
| 4581 | } |
---|
| 4582 | } |
---|
[e801fe] | 4583 | return (pseudo_prim_dec_i(SI,L)); |
---|
| 4584 | } |
---|
| 4585 | |
---|
| 4586 | //////////////////////////////////////////////////////////////// |
---|
| 4587 | // proc pseudo_prim_dec_special_charsets |
---|
| 4588 | // input: a standard basis of an ideal I whose radical is the |
---|
| 4589 | // intersection of the radicals of ideals generated by one prime ideal |
---|
| 4590 | // P_i together with one polynomial f_i, the list V6 must be the list of |
---|
| 4591 | // pairs (standard basis of P_i, irreducible factors of f_i), |
---|
| 4592 | // and an integer choo |
---|
| 4593 | // If choo=0, min_ass_prim_charsets with the given |
---|
| 4594 | // ordering of the variables is used. |
---|
| 4595 | // If choo=1, min_ass_prim_charsets with the "optimized" |
---|
| 4596 | // ordering of the variables is used. |
---|
| 4597 | // If choo=2, minAssPrimes from primdec.lib is used |
---|
| 4598 | // If choo=3, minAssPrimes+factorizing Buchberger from primdec.lib is used |
---|
| 4599 | // output: a pseudo primary decomposition of I, i.e., a list |
---|
| 4600 | // of pseudo primary components together with a standard basis of the |
---|
| 4601 | // remaining component. Each pseudo primary component is |
---|
| 4602 | // represented by a quadrupel: A standard basis of the component, |
---|
| 4603 | // a standard basis of the corresponding associated prime, the |
---|
| 4604 | // seperator of the component, and the irreducible factors of the |
---|
| 4605 | // "minimal divisor" of the seperator as computed by the procedure minsat, |
---|
| 4606 | // calls proc pseudo_prim_dec_i |
---|
| 4607 | //////////////////////////////////////////////////////////////// |
---|
| 4608 | |
---|
| 4609 | |
---|
[07c623] | 4610 | static proc pseudo_prim_dec_special_charsets (ideal SI,list V6, int choo) |
---|
[e801fe] | 4611 | { |
---|
| 4612 | int i,j,l; |
---|
| 4613 | list m; |
---|
| 4614 | list L; |
---|
| 4615 | int sizeL; |
---|
| 4616 | ideal P,SP; ideal fac; |
---|
| 4617 | int dimSP; |
---|
| 4618 | for(l=size(V6);l>=1;l--) // creates a list of associated primes |
---|
| 4619 | // of I, possibly redundant |
---|
| 4620 | { |
---|
| 4621 | P=V6[l][1]; |
---|
| 4622 | fac=V6[l][2]; |
---|
| 4623 | for(i=ncols(fac);i>=1;i--) |
---|
| 4624 | { |
---|
| 4625 | SP=P+fac[i]; |
---|
| 4626 | SP=std(SP); |
---|
| 4627 | if(SP[1]!=1) |
---|
| 4628 | { |
---|
| 4629 | if((choo==0) or (choo==1)) |
---|
| 4630 | { |
---|
| 4631 | m=min_ass_prim_charsets(SP,choo); // a list of SB |
---|
| 4632 | } |
---|
| 4633 | else |
---|
| 4634 | { |
---|
| 4635 | if(choo==2) |
---|
| 4636 | { |
---|
| 4637 | m=minAssPrimes(SP); |
---|
| 4638 | } |
---|
| 4639 | else |
---|
| 4640 | { |
---|
| 4641 | m=minAssPrimes(SP,1); |
---|
| 4642 | } |
---|
| 4643 | for(j=size(m);j>=1;j=j-1) |
---|
[a36e78] | 4644 | { |
---|
| 4645 | m[j]=std(m[j]); |
---|
| 4646 | } |
---|
[e801fe] | 4647 | } |
---|
[3939bc] | 4648 | dimSP=dim(SP); |
---|
[e801fe] | 4649 | for(j=size(m);j>=1; j--) |
---|
| 4650 | { |
---|
| 4651 | if(dim(m[j])==dimSP) |
---|
| 4652 | { |
---|
| 4653 | L=insert(L,m[j],size(L)); |
---|
| 4654 | } |
---|
| 4655 | } |
---|
| 4656 | } |
---|
| 4657 | } |
---|
| 4658 | } |
---|
| 4659 | sizeL=size(L); |
---|
| 4660 | for(i=1;i<sizeL;i++) // get rid of redundant primes |
---|
| 4661 | { |
---|
| 4662 | for(j=i+1;j<=sizeL;j++) |
---|
| 4663 | { |
---|
| 4664 | if(size(L[i])!=0) |
---|
| 4665 | { |
---|
| 4666 | if(size(L[j])!=0) |
---|
| 4667 | { |
---|
| 4668 | if(size(NF(L[i],L[j],1))==0) |
---|
| 4669 | { |
---|
| 4670 | L[j]=ideal(0); |
---|
| 4671 | } |
---|
| 4672 | else |
---|
| 4673 | { |
---|
| 4674 | if(size(NF(L[j],L[i],1))==0) |
---|
| 4675 | { |
---|
| 4676 | L[i]=ideal(0); |
---|
| 4677 | } |
---|
| 4678 | } |
---|
| 4679 | } |
---|
| 4680 | } |
---|
| 4681 | } |
---|
| 4682 | } |
---|
| 4683 | for(i=sizeL;i>=1;i--) |
---|
| 4684 | { |
---|
| 4685 | if(size(L[i])==0) |
---|
| 4686 | { |
---|
| 4687 | L=delete(L,i); |
---|
| 4688 | } |
---|
| 4689 | } |
---|
| 4690 | return (pseudo_prim_dec_i(SI,L)); |
---|
| 4691 | } |
---|
| 4692 | |
---|
| 4693 | |
---|
| 4694 | //////////////////////////////////////////////////////////////// |
---|
| 4695 | // proc pseudo_prim_dec_i |
---|
| 4696 | // input: A standard basis of an arbitrary ideal I, and standard bases |
---|
| 4697 | // of the minimal associated primes of I |
---|
| 4698 | // output: a pseudo primary decomposition of I, i.e., a list |
---|
| 4699 | // of pseudo primary components together with a standard basis of the |
---|
| 4700 | // remaining component. Each pseudo primary component is |
---|
| 4701 | // represented by a quadrupel: A standard basis of the component Q_i, |
---|
| 4702 | // a standard basis of the corresponding associated prime P_i, the |
---|
| 4703 | // seperator of the component, and the irreducible factors of the |
---|
| 4704 | // "minimal divisor" of the seperator as computed by the procedure minsat, |
---|
| 4705 | //////////////////////////////////////////////////////////////// |
---|
| 4706 | |
---|
| 4707 | |
---|
[07c623] | 4708 | static proc pseudo_prim_dec_i (ideal SI, list L) |
---|
[e801fe] | 4709 | { |
---|
| 4710 | list Q; |
---|
| 4711 | if (size(L)==1) // one minimal associated prime only |
---|
| 4712 | // the ideal is already pseudo primary |
---|
| 4713 | { |
---|
| 4714 | Q=SI,L[1],1; |
---|
| 4715 | list QQ; |
---|
| 4716 | QQ[1]=Q; |
---|
| 4717 | return (QQ,ideal(1)); |
---|
| 4718 | } |
---|
| 4719 | |
---|
| 4720 | poly f0,f,g; |
---|
| 4721 | ideal fac; |
---|
| 4722 | int i,j,k,l; |
---|
| 4723 | ideal SQi; |
---|
| 4724 | ideal I'=SI; |
---|
| 4725 | list QP; |
---|
| 4726 | int sizeL=size(L); |
---|
| 4727 | for(i=1;i<=sizeL;i++) |
---|
| 4728 | { |
---|
| 4729 | fac=0; |
---|
| 4730 | for(j=1;j<=sizeL;j++) // compute the seperator sep_i |
---|
| 4731 | // of the i-th component |
---|
| 4732 | { |
---|
| 4733 | if (i!=j) // search g not in L[i], but L[j] |
---|
| 4734 | { |
---|
| 4735 | for(k=1;k<=ncols(L[j]);k++) |
---|
| 4736 | { |
---|
| 4737 | if(NF(L[j][k],L[i],1)!=0) |
---|
| 4738 | { |
---|
| 4739 | break; |
---|
| 4740 | } |
---|
| 4741 | } |
---|
| 4742 | fac=fac+L[j][k]; |
---|
| 4743 | } |
---|
| 4744 | } |
---|
| 4745 | // delete superfluous polynomials |
---|
[7f38f4] | 4746 | fac=simplify(fac,8+2); |
---|
[e801fe] | 4747 | // saturation |
---|
| 4748 | SQi,f0,f,fac=minsat_ppd(SI,fac); |
---|
| 4749 | I'=I',f; |
---|
| 4750 | QP=SQi,L[i],f0,fac; |
---|
| 4751 | // the quadrupel: |
---|
| 4752 | // a standard basis of Q_i, |
---|
| 4753 | // a standard basis of P_i, |
---|
| 4754 | // sep_i, |
---|
| 4755 | // irreducible factors of |
---|
| 4756 | // the "minimal divisor" of the seperator |
---|
| 4757 | // as computed by the procedure minsat, |
---|
| 4758 | Q[i]=QP; |
---|
| 4759 | } |
---|
| 4760 | I'=std(I'); |
---|
| 4761 | return (Q, I'); |
---|
| 4762 | // I' = remaining component |
---|
| 4763 | } |
---|
| 4764 | |
---|
| 4765 | |
---|
| 4766 | //////////////////////////////////////////////////////////////// |
---|
| 4767 | // proc extraction |
---|
| 4768 | // input: A standard basis of a pseudo primary ideal I, and a standard |
---|
| 4769 | // basis of the unique minimal associated prime P of I |
---|
| 4770 | // output: an extraction of I, i.e., a standard basis of the primary |
---|
| 4771 | // component Q of I with associated prime P, a standard basis of the |
---|
| 4772 | // remaining component, and the irreducible factors of the |
---|
| 4773 | // "minimal divisor" of the extractor as computed by the procedure minsat |
---|
| 4774 | //////////////////////////////////////////////////////////////// |
---|
| 4775 | |
---|
| 4776 | |
---|
[07c623] | 4777 | static proc extraction (ideal SI, ideal SP) |
---|
[e801fe] | 4778 | { |
---|
[aa3811c] | 4779 | list indsets=indepSet(SP,0); |
---|
[e801fe] | 4780 | poly f; |
---|
| 4781 | if(size(indsets)!=0) //check, whether dim P != 0 |
---|
| 4782 | { |
---|
| 4783 | intvec v; // a maximal independent set of variables |
---|
| 4784 | // modulo P |
---|
| 4785 | string U; // the independent variables |
---|
| 4786 | string A; // the dependent variables |
---|
| 4787 | int j,k; |
---|
| 4788 | int a; // the size of A |
---|
| 4789 | int degf; |
---|
| 4790 | ideal g; |
---|
| 4791 | list polys; |
---|
| 4792 | int sizepolys; |
---|
| 4793 | list newpoly; |
---|
| 4794 | def R=basering; |
---|
| 4795 | //intvec hv=hilb(SI,1); |
---|
| 4796 | for (k=1;k<=size(indsets);k++) |
---|
| 4797 | { |
---|
| 4798 | v=indsets[k]; |
---|
| 4799 | for (j=1;j<=nvars(R);j++) |
---|
| 4800 | { |
---|
| 4801 | if (v[j]==1) |
---|
| 4802 | { |
---|
| 4803 | U=U+varstr(j)+","; |
---|
| 4804 | } |
---|
| 4805 | else |
---|
| 4806 | { |
---|
| 4807 | A=A+varstr(j)+","; |
---|
| 4808 | a++; |
---|
| 4809 | } |
---|
| 4810 | } |
---|
| 4811 | |
---|
| 4812 | U[size(U)]=")"; // we compute the extractor of I (w.r.t. U) |
---|
[24f458] | 4813 | execute("ring RAU=("+charstr(basering)+"),("+A+U+",(dp("+string(a)+"),dp);"); |
---|
[e801fe] | 4814 | ideal I=imap(R,SI); |
---|
| 4815 | //I=std(I,hv); // the standard basis in (R[U])[A] |
---|
| 4816 | I=std(I); // the standard basis in (R[U])[A] |
---|
| 4817 | A[size(A)]=")"; |
---|
[2d2cad9] | 4818 | execute("ring Rloc=("+charstr(basering)+","+U+",("+A+",dp;"); |
---|
[e801fe] | 4819 | ideal I=imap(RAU,I); |
---|
| 4820 | //"std in lokalisierung:"+newline,I; |
---|
| 4821 | ideal h; |
---|
| 4822 | for(j=ncols(I);j>=1;j--) |
---|
| 4823 | { |
---|
| 4824 | h[j]=leadcoef(I[j]); // consider I in (R(U))[A] |
---|
| 4825 | } |
---|
| 4826 | setring R; |
---|
| 4827 | g=imap(Rloc,h); |
---|
| 4828 | kill RAU,Rloc; |
---|
| 4829 | U=""; |
---|
| 4830 | A=""; |
---|
| 4831 | a=0; |
---|
| 4832 | f=lcm(g); |
---|
| 4833 | newpoly[1]=f; |
---|
| 4834 | polys=polys+newpoly; |
---|
| 4835 | newpoly=list(); |
---|
| 4836 | } |
---|
| 4837 | f=polys[1]; |
---|
| 4838 | degf=deg(f); |
---|
| 4839 | sizepolys=size(polys); |
---|
| 4840 | for (k=2;k<=sizepolys;k++) |
---|
| 4841 | { |
---|
| 4842 | if (deg(polys[k])<degf) |
---|
| 4843 | { |
---|
| 4844 | f=polys[k]; |
---|
[3939bc] | 4845 | degf=deg(f); |
---|
[e801fe] | 4846 | } |
---|
| 4847 | } |
---|
| 4848 | } |
---|
| 4849 | else |
---|
| 4850 | { |
---|
| 4851 | f=1; |
---|
| 4852 | } |
---|
| 4853 | poly f0,h0; ideal SQ; ideal fac; |
---|
| 4854 | if(f!=1) |
---|
| 4855 | { |
---|
| 4856 | SQ,f0,h0,fac=minsat(SI,f); |
---|
| 4857 | return(SQ,std(SI+h0),fac); |
---|
| 4858 | // the tripel |
---|
| 4859 | // a standard basis of Q, |
---|
| 4860 | // a standard basis of remaining component, |
---|
| 4861 | // irreducible factors of |
---|
| 4862 | // the "minimal divisor" of the extractor |
---|
| 4863 | // as computed by the procedure minsat |
---|
| 4864 | } |
---|
| 4865 | else |
---|
| 4866 | { |
---|
| 4867 | return(SI,ideal(1),ideal(1)); |
---|
| 4868 | } |
---|
| 4869 | } |
---|
| 4870 | |
---|
| 4871 | ///////////////////////////////////////////////////// |
---|
| 4872 | // proc minsat |
---|
| 4873 | // input: a standard basis of an ideal I and a polynomial p |
---|
| 4874 | // output: a standard basis IS of the saturation of I w.r. to p, |
---|
| 4875 | // the maximal squarefree factor f0 of p, |
---|
| 4876 | // the "minimal divisor" f of f0 such that the saturation of |
---|
| 4877 | // I w.r. to f equals the saturation of I w.r. to f0 (which is IS), |
---|
| 4878 | // the irreducible factors of f |
---|
| 4879 | ////////////////////////////////////////////////////////// |
---|
| 4880 | |
---|
| 4881 | |
---|
[07c623] | 4882 | static proc minsat(ideal SI, poly p) |
---|
[e801fe] | 4883 | { |
---|
| 4884 | ideal fac=factorize(p,1); //the irreducible factors of p |
---|
| 4885 | fac=sort(fac)[1]; |
---|
| 4886 | int i,k; |
---|
| 4887 | poly f0=1; |
---|
| 4888 | for(i=ncols(fac);i>=1;i--) |
---|
| 4889 | { |
---|
| 4890 | f0=f0*fac[i]; |
---|
| 4891 | } |
---|
| 4892 | poly f=1; |
---|
| 4893 | ideal iold; |
---|
| 4894 | list quotM; |
---|
| 4895 | quotM[1]=SI; |
---|
| 4896 | quotM[2]=fac; |
---|
| 4897 | quotM[3]=f0; |
---|
| 4898 | // we deal seperately with the first quotient; |
---|
| 4899 | // factors, which do not contribute to this one, |
---|
| 4900 | // are omitted |
---|
| 4901 | iold=quotM[1]; |
---|
| 4902 | quotM=minquot(quotM); |
---|
| 4903 | fac=quotM[2]; |
---|
| 4904 | if(quotM[3]==1) |
---|
[a36e78] | 4905 | { |
---|
| 4906 | return(quotM[1],f0,f,fac); |
---|
| 4907 | } |
---|
[e801fe] | 4908 | while(special_ideals_equal(iold,quotM[1])==0) |
---|
[a36e78] | 4909 | { |
---|
| 4910 | f=f*quotM[3]; |
---|
| 4911 | iold=quotM[1]; |
---|
| 4912 | quotM=minquot(quotM); |
---|
| 4913 | } |
---|
[e801fe] | 4914 | return(quotM[1],f0,f,fac); // the quadrupel ((I:p),f0,f, irr. factors of f) |
---|
| 4915 | } |
---|
| 4916 | |
---|
| 4917 | ///////////////////////////////////////////////////// |
---|
| 4918 | // proc minsat_ppd |
---|
| 4919 | // input: a standard basis of an ideal I and a polynomial p |
---|
| 4920 | // output: a standard basis IS of the saturation of I w.r. to p, |
---|
| 4921 | // the maximal squarefree factor f0 of p, |
---|
| 4922 | // the "minimal divisor" f of f0 such that the saturation of |
---|
| 4923 | // I w.r. to f equals the saturation of I w.r. to f0 (which is IS), |
---|
| 4924 | // the irreducible factors of f |
---|
| 4925 | ////////////////////////////////////////////////////////// |
---|
| 4926 | |
---|
| 4927 | |
---|
[07c623] | 4928 | static proc minsat_ppd(ideal SI, ideal fac) |
---|
[e801fe] | 4929 | { |
---|
| 4930 | fac=sort(fac)[1]; |
---|
| 4931 | int i,k; |
---|
| 4932 | poly f0=1; |
---|
| 4933 | for(i=ncols(fac);i>=1;i--) |
---|
| 4934 | { |
---|
| 4935 | f0=f0*fac[i]; |
---|
| 4936 | } |
---|
| 4937 | poly f=1; |
---|
| 4938 | ideal iold; |
---|
| 4939 | list quotM; |
---|
| 4940 | quotM[1]=SI; |
---|
| 4941 | quotM[2]=fac; |
---|
| 4942 | quotM[3]=f0; |
---|
| 4943 | // we deal seperately with the first quotient; |
---|
| 4944 | // factors, which do not contribute to this one, |
---|
| 4945 | // are omitted |
---|
| 4946 | iold=quotM[1]; |
---|
| 4947 | quotM=minquot(quotM); |
---|
| 4948 | fac=quotM[2]; |
---|
| 4949 | if(quotM[3]==1) |
---|
[a36e78] | 4950 | { |
---|
| 4951 | return(quotM[1],f0,f,fac); |
---|
| 4952 | } |
---|
[e801fe] | 4953 | while(special_ideals_equal(iold,quotM[1])==0) |
---|
| 4954 | { |
---|
| 4955 | f=f*quotM[3]; |
---|
| 4956 | iold=quotM[1]; |
---|
| 4957 | quotM=minquot(quotM); |
---|
| 4958 | k++; |
---|
| 4959 | } |
---|
[a36e78] | 4960 | return(quotM[1],f0,f,fac); // the quadrupel ((I:p),f0,f, irr. factors of f) |
---|
[e801fe] | 4961 | } |
---|
| 4962 | ///////////////////////////////////////////////////////////////// |
---|
| 4963 | // proc minquot |
---|
| 4964 | // input: a list with 3 components: a standard basis |
---|
| 4965 | // of an ideal I, a set of irreducible polynomials, and |
---|
| 4966 | // there product f0 |
---|
| 4967 | // output: a standard basis of the ideal (I:f0), the irreducible |
---|
| 4968 | // factors of the "minimal divisor" f of f0 with (I:f0) = (I:f), |
---|
| 4969 | // the "minimal divisor" f |
---|
| 4970 | ///////////////////////////////////////////////////////////////// |
---|
| 4971 | |
---|
[07c623] | 4972 | static proc minquot(list tsil) |
---|
[e801fe] | 4973 | { |
---|
[a36e78] | 4974 | intvec op; |
---|
| 4975 | int i,j,k,action; |
---|
| 4976 | ideal verg; |
---|
| 4977 | list l; |
---|
| 4978 | poly g; |
---|
| 4979 | ideal laedi=tsil[1]; |
---|
| 4980 | ideal fac=tsil[2]; |
---|
| 4981 | poly f=tsil[3]; |
---|
[e801fe] | 4982 | |
---|
| 4983 | //std |
---|
| 4984 | // ideal star=quotient(laedi,f); |
---|
| 4985 | // star=std(star); |
---|
[a36e78] | 4986 | op=option(get); |
---|
| 4987 | option(returnSB); |
---|
| 4988 | ideal star=quotient(laedi,f); |
---|
| 4989 | option(set,op); |
---|
| 4990 | if(special_ideals_equal(laedi,star)==1) |
---|
| 4991 | { |
---|
| 4992 | return(laedi,ideal(1),1); |
---|
| 4993 | } |
---|
| 4994 | action=1; |
---|
| 4995 | while(action==1) |
---|
| 4996 | { |
---|
| 4997 | if(size(fac)==1) |
---|
[e801fe] | 4998 | { |
---|
[a36e78] | 4999 | action=0; |
---|
| 5000 | break; |
---|
[e801fe] | 5001 | } |
---|
[a36e78] | 5002 | for(i=1;i<=size(fac);i++) |
---|
| 5003 | { |
---|
| 5004 | g=1; |
---|
| 5005 | for(j=1;j<=size(fac);j++) |
---|
| 5006 | { |
---|
| 5007 | if(i!=j) |
---|
| 5008 | { |
---|
| 5009 | g=g*fac[j]; |
---|
| 5010 | } |
---|
| 5011 | } |
---|
[e801fe] | 5012 | //std |
---|
| 5013 | // verg=quotient(laedi,g); |
---|
| 5014 | // verg=std(verg); |
---|
[a36e78] | 5015 | op=option(get); |
---|
| 5016 | option(returnSB); |
---|
| 5017 | verg=quotient(laedi,g); |
---|
| 5018 | option(set,op); |
---|
| 5019 | if(special_ideals_equal(verg,star)==1) |
---|
| 5020 | { |
---|
| 5021 | f=g; |
---|
| 5022 | fac[i]=0; |
---|
| 5023 | fac=simplify(fac,2); |
---|
| 5024 | break; |
---|
| 5025 | } |
---|
| 5026 | if(i==size(fac)) |
---|
| 5027 | { |
---|
| 5028 | action=0; |
---|
| 5029 | } |
---|
[70ab73] | 5030 | } |
---|
[a36e78] | 5031 | } |
---|
| 5032 | l=star,fac,f; |
---|
| 5033 | return(l); |
---|
[e801fe] | 5034 | } |
---|
| 5035 | ///////////////////////////////////////////////// |
---|
| 5036 | // proc special_ideals_equal |
---|
| 5037 | // input: standard bases of ideal k1 and k2 such that |
---|
| 5038 | // k1 is contained in k2, or k2 is contained ink1 |
---|
| 5039 | // output: 1, if k1 equals k2, 0 otherwise |
---|
| 5040 | ////////////////////////////////////////////////// |
---|
| 5041 | |
---|
[07c623] | 5042 | static proc special_ideals_equal( ideal k1, ideal k2) |
---|
[e801fe] | 5043 | { |
---|
[a36e78] | 5044 | int j; |
---|
| 5045 | if(size(k1)==size(k2)) |
---|
| 5046 | { |
---|
| 5047 | for(j=1;j<=size(k1);j++) |
---|
[e801fe] | 5048 | { |
---|
[a36e78] | 5049 | if(leadexp(k1[j])!=leadexp(k2[j])) |
---|
| 5050 | { |
---|
| 5051 | return(0); |
---|
| 5052 | } |
---|
[70ab73] | 5053 | } |
---|
[a36e78] | 5054 | return(1); |
---|
| 5055 | } |
---|
| 5056 | return(0); |
---|
[e801fe] | 5057 | } |
---|
[3939bc] | 5058 | |
---|
| 5059 | |
---|
[ebecf83] | 5060 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 5061 | |
---|
[07c623] | 5062 | static proc convList(list l) |
---|
[ebecf83] | 5063 | { |
---|
[a36e78] | 5064 | int i; |
---|
| 5065 | list re,he; |
---|
[4173c7] | 5066 | for(i=1;i<=size(l) div 2;i++) |
---|
[a36e78] | 5067 | { |
---|
| 5068 | he=l[2*i-1],l[2*i]; |
---|
| 5069 | re[i]=he; |
---|
| 5070 | } |
---|
| 5071 | return(re); |
---|
[ebecf83] | 5072 | } |
---|
| 5073 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 5074 | |
---|
[07c623] | 5075 | static proc reconvList(list l) |
---|
[ebecf83] | 5076 | { |
---|
[a36e78] | 5077 | int i; |
---|
| 5078 | list re; |
---|
| 5079 | for(i=1;i<=size(l);i++) |
---|
| 5080 | { |
---|
| 5081 | re[2*i-1]=l[i][1]; |
---|
| 5082 | re[2*i]=l[i][2]; |
---|
| 5083 | } |
---|
| 5084 | return(re); |
---|
[ebecf83] | 5085 | } |
---|
| 5086 | |
---|
| 5087 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 5088 | // |
---|
| 5089 | // The main procedures |
---|
| 5090 | // |
---|
| 5091 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 5092 | |
---|
[cb980ab] | 5093 | proc primdecGTZ(ideal i, list #) |
---|
[091424] | 5094 | "USAGE: primdecGTZ(i); i ideal |
---|
[07c623] | 5095 | RETURN: a list pr of primary ideals and their associated primes: |
---|
[367e88] | 5096 | @format |
---|
[7b3971] | 5097 | pr[i][1] the i-th primary component, |
---|
| 5098 | pr[i][2] the i-th prime component. |
---|
| 5099 | @end format |
---|
[cb980ab] | 5100 | NOTE: - Algorithm of Gianni/Trager/Zacharias. |
---|
| 5101 | - Designed for characteristic 0, works also in char k > 0, if it |
---|
| 5102 | terminates (may result in an infinite loop in small characteristic!) |
---|
[ea87a9] | 5103 | - For local orderings, the result is considered in the localization |
---|
[cb980ab] | 5104 | of the polynomial ring, not in the power series ring |
---|
[ea87a9] | 5105 | - For local and mixed orderings, the decomposition in the |
---|
[cb980ab] | 5106 | corresponding global ring is returned if the string 'global' |
---|
| 5107 | is specified as second argument |
---|
[ebecf83] | 5108 | EXAMPLE: example primdecGTZ; shows an example |
---|
| 5109 | " |
---|
| 5110 | { |
---|
[cb980ab] | 5111 | if(size(#)>0) |
---|
| 5112 | { |
---|
| 5113 | int keep_comp=1; |
---|
| 5114 | } |
---|
[a36e78] | 5115 | if(attrib(basering,"global")!=1) |
---|
| 5116 | { |
---|
[cb980ab] | 5117 | // algorithms only work in global case! |
---|
| 5118 | // pass to appropriate global ring |
---|
| 5119 | def r=basering; |
---|
[1e1ec4] | 5120 | def s=changeord(list(list("dp",1:nvars(basering)))); |
---|
[cb980ab] | 5121 | setring s; |
---|
| 5122 | ideal i=imap(r,i); |
---|
[ea87a9] | 5123 | // decompose and go back |
---|
[cb980ab] | 5124 | list li=primdecGTZ(i); |
---|
| 5125 | setring r; |
---|
| 5126 | def li=imap(s,li); |
---|
| 5127 | // clean up |
---|
| 5128 | if(!defined(keep_comp)) |
---|
| 5129 | { |
---|
| 5130 | for(int k=size(li);k>=1;k--) |
---|
| 5131 | { |
---|
| 5132 | if(mindeg(std(lead(li[k][2]))[1])==0) |
---|
[ea87a9] | 5133 | { |
---|
[cb980ab] | 5134 | // 1 contained in ideal, i.e. component does not meet origin in local ordering |
---|
| 5135 | li=delete(li,k); |
---|
| 5136 | } |
---|
| 5137 | } |
---|
| 5138 | } |
---|
| 5139 | return(li); |
---|
[a36e78] | 5140 | } |
---|
[cb980ab] | 5141 | |
---|
[a36e78] | 5142 | if(minpoly!=0) |
---|
| 5143 | { |
---|
| 5144 | return(algeDeco(i,0)); |
---|
| 5145 | ERROR( |
---|
[cb980ab] | 5146 | "// Not implemented yet for algebraic extensions.Simulate the ring extension by adding the minpoly to the ideal" |
---|
[a36e78] | 5147 | ); |
---|
| 5148 | } |
---|
[24f458] | 5149 | return(convList(decomp(i))); |
---|
[ebecf83] | 5150 | } |
---|
| 5151 | example |
---|
| 5152 | { "EXAMPLE:"; echo = 2; |
---|
[07c623] | 5153 | ring r = 0,(x,y,z),lp; |
---|
[ebecf83] | 5154 | poly p = z2+1; |
---|
[07c623] | 5155 | poly q = z3+2; |
---|
| 5156 | ideal i = p*q^2,y-z2; |
---|
[091424] | 5157 | list pr = primdecGTZ(i); |
---|
[ebecf83] | 5158 | pr; |
---|
| 5159 | } |
---|
| 5160 | /////////////////////////////////////////////////////////////////////////////// |
---|
[cb980ab] | 5161 | proc absPrimdecGTZ(ideal I, list #) |
---|
[6fa3af] | 5162 | "USAGE: absPrimdecGTZ(I); I ideal |
---|
| 5163 | ASSUME: Ground field has characteristic 0. |
---|
[cb980ab] | 5164 | RETURN: a ring containing two lists: @code{absolute_primes}, the absolute |
---|
| 5165 | prime components of I, and @code{primary_decomp}, the output of |
---|
| 5166 | @code{primdecGTZ(I)}. |
---|
[6fa3af] | 5167 | The list absolute_primes has to be interpreted as follows: |
---|
| 5168 | each entry describes a class of conjugated absolute primes, |
---|
| 5169 | @format |
---|
[326dba] | 5170 | absolute_primes[i][1] the absolute prime component, |
---|
[6fa3af] | 5171 | absolute_primes[i][2] the number of conjugates. |
---|
| 5172 | @end format |
---|
| 5173 | The first entry of @code{absolute_primes[i][1]} is the minimal |
---|
| 5174 | polynomial of a minimal finite field extension over which the |
---|
| 5175 | absolute prime component is defined. |
---|
[ea87a9] | 5176 | For local orderings, the result is considered in the localization |
---|
[cb980ab] | 5177 | of the polynomial ring, not in the power series ring. |
---|
[ea87a9] | 5178 | For local and mixed orderings, the decomposition in the |
---|
[cb980ab] | 5179 | corresponding global ring is returned if the string 'global' |
---|
| 5180 | is specified as second argument |
---|
[6fa3af] | 5181 | NOTE: Algorithm of Gianni/Trager/Zacharias combined with the |
---|
| 5182 | @code{absFactorize} command. |
---|
| 5183 | SEE ALSO: primdecGTZ; absFactorize |
---|
| 5184 | EXAMPLE: example absPrimdecGTZ; shows an example |
---|
| 5185 | " |
---|
| 5186 | { |
---|
[70ab73] | 5187 | if (char(basering) != 0) |
---|
| 5188 | { |
---|
[6fa3af] | 5189 | ERROR("primdec.lib::absPrimdecGTZ is only implemented for "+ |
---|
| 5190 | +"characteristic 0"); |
---|
| 5191 | } |
---|
| 5192 | |
---|
[cb980ab] | 5193 | if(size(#)>0) |
---|
| 5194 | { |
---|
| 5195 | int keep_comp=1; |
---|
| 5196 | } |
---|
| 5197 | |
---|
[70ab73] | 5198 | if(attrib(basering,"global")!=1) |
---|
| 5199 | { |
---|
[cb980ab] | 5200 | // algorithm automatically passes to the global case |
---|
| 5201 | // hence prepare to go back to an appropriate new ring |
---|
| 5202 | def r=basering; |
---|
| 5203 | ideal max_of_r=maxideal(1); |
---|
[1e1ec4] | 5204 | def s=changeord(list(list("dp",1:nvars(basering)))); |
---|
[cb980ab] | 5205 | setring s; |
---|
| 5206 | def I=imap(r,I); |
---|
| 5207 | def S=absPrimdecGTZ(I); |
---|
| 5208 | setring S; |
---|
| 5209 | ring r1=char(basering),var(nvars(r)+1),dp; |
---|
| 5210 | def rS=r+r1; |
---|
| 5211 | // move objects to appropriate ring and clean up |
---|
| 5212 | setring rS; |
---|
| 5213 | def max_of_r=imap(r,max_of_r); |
---|
| 5214 | attrib(max_of_r,"isSB",1); |
---|
| 5215 | def absolute_primes=imap(S,absolute_primes); |
---|
| 5216 | def primary_decomp=imap(S,primary_decomp); |
---|
| 5217 | if(!defined(keep_comp)) |
---|
| 5218 | { |
---|
| 5219 | ideal tempid; |
---|
| 5220 | for(int k=size(absolute_primes);k>=1;k--) |
---|
| 5221 | { |
---|
| 5222 | tempid=absolute_primes[k][1]; |
---|
| 5223 | tempid[1]=0; // ignore minimal polynomial |
---|
| 5224 | if(size(reduce(lead(tempid),max_of_r))!=0) |
---|
[ea87a9] | 5225 | { |
---|
[cb980ab] | 5226 | // 1 contained in ideal, i.e. component does not meet origin in local ordering |
---|
| 5227 | absolute_primes=delete(absolute_primes,k); |
---|
| 5228 | } |
---|
[ea87a9] | 5229 | } |
---|
[cb980ab] | 5230 | for(k=size(primary_decomp);k>=1;k--) |
---|
| 5231 | { |
---|
| 5232 | if(mindeg(std(lead(primary_decomp[k][2]))[1])==0) |
---|
[ea87a9] | 5233 | { |
---|
[cb980ab] | 5234 | // 1 contained in ideal, i.e. component does not meet origin in local ordering |
---|
| 5235 | primary_decomp=delete(primary_decomp,k); |
---|
| 5236 | } |
---|
| 5237 | } |
---|
| 5238 | kill tempid; |
---|
| 5239 | } |
---|
| 5240 | export(primary_decomp); |
---|
| 5241 | export(absolute_primes); |
---|
| 5242 | return(rS); |
---|
[70ab73] | 5243 | } |
---|
| 5244 | if(minpoly!=0) |
---|
| 5245 | { |
---|
| 5246 | //return(algeDeco(i,0)); |
---|
| 5247 | ERROR( |
---|
[cb980ab] | 5248 | "// Not implemented yet for algebraic extensions.Simulate the ring extension by adding the minpoly to the ideal" |
---|
[70ab73] | 5249 | ); |
---|
| 5250 | } |
---|
[6fa3af] | 5251 | def R=basering; |
---|
| 5252 | int n=nvars(R); |
---|
| 5253 | list L=decomp(I,3); |
---|
[4719f0] | 5254 | string newvar=L[1][3]; |
---|
[6fa3af] | 5255 | int k=find(newvar,",",find(newvar,",")+1); |
---|
| 5256 | newvar=newvar[k+1..size(newvar)]; |
---|
| 5257 | list lR=ringlist(R); |
---|
[1d430ab] | 5258 | int i,de,ii; |
---|
| 5259 | intvec vv=1:n; |
---|
| 5260 | //for(i=1;i<=n;i++){vv[i]=1;} |
---|
[6fa3af] | 5261 | |
---|
| 5262 | list orst; |
---|
| 5263 | orst[1]=list("dp",vv); |
---|
| 5264 | orst[2]=list("dp",intvec(1)); |
---|
| 5265 | orst[3]=list("C",0); |
---|
| 5266 | lR[3]=orst; |
---|
| 5267 | lR[2][n+1] = newvar; |
---|
| 5268 | def Rz = ring(lR); |
---|
| 5269 | setring Rz; |
---|
| 5270 | list L=imap(R,L); |
---|
| 5271 | list absolute_primes,primary_decomp; |
---|
| 5272 | ideal I,M,N,K; |
---|
| 5273 | M=maxideal(1); |
---|
| 5274 | N=maxideal(1); |
---|
| 5275 | poly p,q,f,g; |
---|
| 5276 | map phi,psi; |
---|
[1d430ab] | 5277 | string tvar; |
---|
[6fa3af] | 5278 | for(i=1;i<=size(L);i++) |
---|
| 5279 | { |
---|
[1d430ab] | 5280 | tvar=L[i][4]; |
---|
| 5281 | ii=find(tvar,"+"); |
---|
| 5282 | while(ii) |
---|
| 5283 | { |
---|
| 5284 | tvar=tvar[ii+1..size(tvar)]; |
---|
| 5285 | ii=find(tvar,"+"); |
---|
| 5286 | } |
---|
| 5287 | for(ii=1;ii<=nvars(basering);ii++) |
---|
| 5288 | { |
---|
| 5289 | if(tvar==string(var(ii))) break; |
---|
| 5290 | } |
---|
[6fa3af] | 5291 | I=L[i][2]; |
---|
| 5292 | execute("K="+L[i][3]+";"); |
---|
| 5293 | p=K[1]; |
---|
| 5294 | q=K[2]; |
---|
| 5295 | execute("f="+L[i][4]+";"); |
---|
[1d430ab] | 5296 | g=2*var(ii)-f; |
---|
| 5297 | M[ii]=f; |
---|
| 5298 | N[ii]=g; |
---|
[9d7c01] | 5299 | de=deg(p); |
---|
[1d430ab] | 5300 | psi=Rz,M; |
---|
| 5301 | phi=Rz,N; |
---|
[6fa3af] | 5302 | I=phi(I),p,q; |
---|
| 5303 | I=std(I); |
---|
[9d7c01] | 5304 | absolute_primes[i]=list(psi(I),de); |
---|
[6fa3af] | 5305 | primary_decomp[i]=list(L[i][1],L[i][2]); |
---|
| 5306 | } |
---|
| 5307 | export(primary_decomp); |
---|
| 5308 | export(absolute_primes); |
---|
| 5309 | setring R; |
---|
[cb980ab] | 5310 | dbprint( printlevel-voice+3," |
---|
| 5311 | // 'absPrimdecGTZ' created a ring, in which two lists absolute_primes (the |
---|
| 5312 | // absolute prime components) and primary_decomp (the primary and prime |
---|
| 5313 | // components over the current basering) are stored. |
---|
| 5314 | // To access the list of absolute prime components, type (if the name S was |
---|
| 5315 | // assigned to the return value): |
---|
| 5316 | setring S; absolute_primes; "); |
---|
| 5317 | |
---|
[6fa3af] | 5318 | return(Rz); |
---|
| 5319 | } |
---|
| 5320 | example |
---|
| 5321 | { "EXAMPLE:"; echo = 2; |
---|
| 5322 | ring r = 0,(x,y,z),lp; |
---|
| 5323 | poly p = z2+1; |
---|
| 5324 | poly q = z3+2; |
---|
| 5325 | ideal i = p*q^2,y-z2; |
---|
| 5326 | def S = absPrimdecGTZ(i); |
---|
| 5327 | setring S; |
---|
| 5328 | absolute_primes; |
---|
| 5329 | } |
---|
[1d430ab] | 5330 | |
---|
[6fa3af] | 5331 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 5332 | |
---|
[7b3971] | 5333 | proc primdecSY(ideal i, list #) |
---|
[7f7c25e] | 5334 | "USAGE: primdecSY(I, c); I ideal, c int (optional) |
---|
[07c623] | 5335 | RETURN: a list pr of primary ideals and their associated primes: |
---|
[367e88] | 5336 | @format |
---|
[7b3971] | 5337 | pr[i][1] the i-th primary component, |
---|
| 5338 | pr[i][2] the i-th prime component. |
---|
| 5339 | @end format |
---|
| 5340 | NOTE: Algorithm of Shimoyama/Yokoyama. |
---|
| 5341 | @format |
---|
| 5342 | if c=0, the given ordering of the variables is used, |
---|
[7f7c25e] | 5343 | if c=1, minAssChar tries to use an optimal ordering (default), |
---|
[7b3971] | 5344 | if c=2, minAssGTZ is used, |
---|
| 5345 | if c=3, minAssGTZ and facstd are used. |
---|
| 5346 | @end format |
---|
[ea87a9] | 5347 | For local orderings, the result is considered in the localization |
---|
[cb980ab] | 5348 | of the polynomial ring, not in the power series ring. |
---|
[ea87a9] | 5349 | For local and mixed orderings, the decomposition in the |
---|
[cb980ab] | 5350 | corresponding global ring is returned if the string 'global' |
---|
[ea87a9] | 5351 | is specified as third argument |
---|
[ebecf83] | 5352 | EXAMPLE: example primdecSY; shows an example |
---|
| 5353 | " |
---|
| 5354 | { |
---|
[cb980ab] | 5355 | if(size(#)>1) |
---|
| 5356 | { |
---|
| 5357 | int keep_comp=1; |
---|
| 5358 | } |
---|
[a36e78] | 5359 | if(attrib(basering,"global")!=1) |
---|
| 5360 | { |
---|
[cb980ab] | 5361 | // algorithms only work in global case! |
---|
| 5362 | // pass to appropriate global ring |
---|
| 5363 | def r=basering; |
---|
[1e1ec4] | 5364 | def s=changeord(list(list("dp",1:nvars(basering)))); |
---|
[cb980ab] | 5365 | setring s; |
---|
| 5366 | ideal i=imap(r,i); |
---|
[ea87a9] | 5367 | // decompose and go back |
---|
[cb980ab] | 5368 | list li=primdecSY(i); |
---|
| 5369 | setring r; |
---|
| 5370 | def li=imap(s,li); |
---|
| 5371 | // clean up |
---|
| 5372 | if(!defined(keep_comp)) |
---|
| 5373 | { |
---|
| 5374 | for(int k=size(li);k>=1;k--) |
---|
[ea87a9] | 5375 | { |
---|
[cb980ab] | 5376 | if(mindeg(std(lead(li[k][2]))[1])==0) |
---|
[ea87a9] | 5377 | { |
---|
[cb980ab] | 5378 | // 1 contained in ideal, i.e. component does not meet origin in local ordering |
---|
| 5379 | li=delete(li,k); |
---|
| 5380 | } |
---|
| 5381 | } |
---|
| 5382 | } |
---|
| 5383 | return(li); |
---|
[a36e78] | 5384 | } |
---|
| 5385 | i=simplify(i,2); |
---|
| 5386 | if ((i[1]==0)||(i[1]==1)) |
---|
| 5387 | { |
---|
| 5388 | list L=list(ideal(i[1]),ideal(i[1])); |
---|
| 5389 | return(list(L)); |
---|
| 5390 | } |
---|
[cb980ab] | 5391 | |
---|
[a36e78] | 5392 | if(minpoly!=0) |
---|
| 5393 | { |
---|
| 5394 | return(algeDeco(i,1)); |
---|
| 5395 | } |
---|
[cb980ab] | 5396 | if (size(#)!=0) |
---|
[a36e78] | 5397 | { return(prim_dec(i,#[1])); } |
---|
| 5398 | else |
---|
| 5399 | { return(prim_dec(i,1)); } |
---|
[ebecf83] | 5400 | } |
---|
| 5401 | example |
---|
| 5402 | { "EXAMPLE:"; echo = 2; |
---|
[07c623] | 5403 | ring r = 0,(x,y,z),lp; |
---|
[ebecf83] | 5404 | poly p = z2+1; |
---|
[07c623] | 5405 | poly q = z3+2; |
---|
| 5406 | ideal i = p*q^2,y-z2; |
---|
[091424] | 5407 | list pr = primdecSY(i); |
---|
[ebecf83] | 5408 | pr; |
---|
| 5409 | } |
---|
| 5410 | /////////////////////////////////////////////////////////////////////////////// |
---|
[25c431] | 5411 | proc minAssGTZ(ideal i,list #) |
---|
[7f7c25e] | 5412 | "USAGE: minAssGTZ(I[, l]); I ideal, l list (optional) |
---|
| 5413 | @* Optional parameters in list l (can be entered in any order): |
---|
| 5414 | @* 0, \"facstd\" -> uses facstd to first decompose the ideal (default) |
---|
| 5415 | @* 1, \"noFacstd\" -> does not use facstd |
---|
| 5416 | @* \"GTZ\" -> the original algorithm by Gianni, Trager and Zacharias is used |
---|
| 5417 | @* \"SL\" -> GTZ algorithm with modificiations by Laplagne is used (default) |
---|
| 5418 | |
---|
| 5419 | RETURN: a list, the minimal associated prime ideals of I. |
---|
[cb980ab] | 5420 | NOTE: - Designed for characteristic 0, works also in char k > 0 based |
---|
| 5421 | on an algorithm of Yokoyama |
---|
[ea87a9] | 5422 | - For local orderings, the result is considered in the localization |
---|
[cb980ab] | 5423 | of the polynomial ring, not in the power series ring |
---|
[ea87a9] | 5424 | - For local and mixed orderings, the decomposition in the |
---|
[cb980ab] | 5425 | corresponding global ring is returned if the string 'global' |
---|
| 5426 | is specified as second argument |
---|
[ebecf83] | 5427 | EXAMPLE: example minAssGTZ; shows an example |
---|
| 5428 | " |
---|
| 5429 | { |
---|
[cb980ab] | 5430 | if(size(#)>0) |
---|
| 5431 | { |
---|
| 5432 | int keep_comp=1; |
---|
| 5433 | } |
---|
| 5434 | |
---|
| 5435 | if(attrib(basering,"global")!=1) |
---|
| 5436 | { |
---|
| 5437 | // algorithms only work in global case! |
---|
| 5438 | // pass to appropriate global ring |
---|
| 5439 | def r=basering; |
---|
[1e1ec4] | 5440 | def s=changeord(list(list("dp",1:nvars(basering)))); |
---|
[cb980ab] | 5441 | setring s; |
---|
| 5442 | ideal i=imap(r,i); |
---|
[ea87a9] | 5443 | // decompose and go back |
---|
[cb980ab] | 5444 | list li=minAssGTZ(i); |
---|
| 5445 | setring r; |
---|
| 5446 | def li=imap(s,li); |
---|
| 5447 | // clean up |
---|
| 5448 | if(!defined(keep_comp)) |
---|
| 5449 | { |
---|
| 5450 | for(int k=size(li);k>=1;k--) |
---|
| 5451 | { |
---|
| 5452 | if(mindeg(std(lead(li[k]))[1])==0) |
---|
[ea87a9] | 5453 | { |
---|
[cb980ab] | 5454 | // 1 contained in ideal, i.e. component does not meet origin in local ordering |
---|
| 5455 | li=delete(li,k); |
---|
| 5456 | } |
---|
| 5457 | } |
---|
| 5458 | } |
---|
| 5459 | return(li); |
---|
| 5460 | } |
---|
| 5461 | |
---|
[70ab73] | 5462 | int j; |
---|
| 5463 | string algorithm; |
---|
| 5464 | string facstdOption; |
---|
| 5465 | int useFac; |
---|
[808a9f3] | 5466 | |
---|
[70ab73] | 5467 | // Set input parameters |
---|
| 5468 | algorithm = "SL"; // Default: SL algorithm |
---|
| 5469 | facstdOption = "facstd"; |
---|
| 5470 | if(size(#) > 0) |
---|
| 5471 | { |
---|
| 5472 | int valid; |
---|
| 5473 | for(j = 1; j <= size(#); j++) |
---|
| 5474 | { |
---|
| 5475 | valid = 0; |
---|
| 5476 | if((typeof(#[j]) == "int") or (typeof(#[j]) == "number")) |
---|
| 5477 | { |
---|
| 5478 | if (#[j] == 1) {facstdOption = "noFacstd"; valid = 1;} // If #[j] == 1, facstd is not used. |
---|
| 5479 | if (#[j] == 0) {facstdOption = "facstd"; valid = 1;} // If #[j] == 0, facstd is used. |
---|
| 5480 | } |
---|
| 5481 | if(typeof(#[j]) == "string") |
---|
| 5482 | { |
---|
| 5483 | if((#[j] == "GTZ") || (#[j] == "SL")) |
---|
| 5484 | { |
---|
| 5485 | algorithm = #[j]; |
---|
| 5486 | valid = 1; |
---|
| 5487 | } |
---|
| 5488 | if((#[j] == "noFacstd") || (#[j] == "facstd")) |
---|
| 5489 | { |
---|
| 5490 | facstdOption = #[j]; |
---|
| 5491 | valid = 1; |
---|
| 5492 | } |
---|
| 5493 | } |
---|
| 5494 | if(valid == 0) |
---|
| 5495 | { |
---|
| 5496 | dbprint(1, "Warning! The following input parameter was not recognized:", #[j]); |
---|
| 5497 | } |
---|
| 5498 | } |
---|
| 5499 | } |
---|
[3a2b8e] | 5500 | |
---|
[70ab73] | 5501 | if(minpoly!=0) |
---|
| 5502 | { |
---|
| 5503 | return(algeDeco(i,2)); |
---|
| 5504 | } |
---|
[808a9f3] | 5505 | |
---|
[70ab73] | 5506 | list result = minAssPrimes(i, facstdOption, algorithm); |
---|
| 5507 | return(result); |
---|
[ebecf83] | 5508 | } |
---|
| 5509 | example |
---|
| 5510 | { "EXAMPLE:"; echo = 2; |
---|
[07c623] | 5511 | ring r = 0,(x,y,z),dp; |
---|
[ebecf83] | 5512 | poly p = z2+1; |
---|
[07c623] | 5513 | poly q = z3+2; |
---|
| 5514 | ideal i = p*q^2,y-z2; |
---|
| 5515 | list pr = minAssGTZ(i); |
---|
[ebecf83] | 5516 | pr; |
---|
| 5517 | } |
---|
| 5518 | |
---|
| 5519 | /////////////////////////////////////////////////////////////////////////////// |
---|
[2d3c9b] | 5520 | proc minAssChar(ideal i, list #) |
---|
[7f7c25e] | 5521 | "USAGE: minAssChar(I[,c]); i ideal, c int (optional). |
---|
[7b3971] | 5522 | RETURN: list, the minimal associated prime ideals of i. |
---|
| 5523 | NOTE: If c=0, the given ordering of the variables is used. @* |
---|
[2d3c9b] | 5524 | Otherwise, the system tries to find an optimal ordering, |
---|
[7b3971] | 5525 | which in some cases may considerably speed up the algorithm. @* |
---|
[ea87a9] | 5526 | For local orderings, the result is considered in the localization |
---|
[cb980ab] | 5527 | of the polynomial ring, not in the power series ring |
---|
[ea87a9] | 5528 | For local and mixed orderings, the decomposition in the |
---|
[cb980ab] | 5529 | corresponding global ring is returned if the string 'global' |
---|
| 5530 | is specified as third argument |
---|
[9050ca] | 5531 | EXAMPLE: example minAssChar; shows an example |
---|
[22c0fc9] | 5532 | " |
---|
| 5533 | { |
---|
[cb980ab] | 5534 | if(size(#)>1) |
---|
| 5535 | { |
---|
| 5536 | int keep_comp=1; |
---|
| 5537 | } |
---|
[a36e78] | 5538 | if(attrib(basering,"global")!=1) |
---|
| 5539 | { |
---|
[cb980ab] | 5540 | // algorithms only work in global case! |
---|
| 5541 | // pass to appropriate global ring |
---|
| 5542 | def r=basering; |
---|
[1e1ec4] | 5543 | def s=changeord(list(list("dp",1:nvars(basering)))); |
---|
[cb980ab] | 5544 | setring s; |
---|
| 5545 | ideal i=imap(r,i); |
---|
[ea87a9] | 5546 | // decompose and go back |
---|
[cb980ab] | 5547 | list li=minAssChar(i); |
---|
| 5548 | setring r; |
---|
| 5549 | def li=imap(s,li); |
---|
| 5550 | // clean up |
---|
| 5551 | if(!defined(keep_comp)) |
---|
| 5552 | { |
---|
| 5553 | for(int k=size(li);k>=1;k--) |
---|
| 5554 | { |
---|
| 5555 | if(mindeg(std(lead(li[k]))[1])==0) |
---|
[ea87a9] | 5556 | { |
---|
[cb980ab] | 5557 | // 1 contained in ideal, i.e. component does not meet origin in local ordering |
---|
| 5558 | li=delete(li,k); |
---|
| 5559 | } |
---|
| 5560 | } |
---|
| 5561 | } |
---|
| 5562 | return(li); |
---|
[a36e78] | 5563 | } |
---|
[cb980ab] | 5564 | if (size(#)>0) |
---|
[a36e78] | 5565 | { return(min_ass_prim_charsets(i,#[1])); } |
---|
| 5566 | else |
---|
| 5567 | { return(min_ass_prim_charsets(i,1)); } |
---|
[22c0fc9] | 5568 | } |
---|
| 5569 | example |
---|
| 5570 | { "EXAMPLE:"; echo = 2; |
---|
[07c623] | 5571 | ring r = 0,(x,y,z),dp; |
---|
[22c0fc9] | 5572 | poly p = z2+1; |
---|
[07c623] | 5573 | poly q = z3+2; |
---|
| 5574 | ideal i = p*q^2,y-z2; |
---|
| 5575 | list pr = minAssChar(i); |
---|
[22c0fc9] | 5576 | pr; |
---|
| 5577 | } |
---|
[ebecf83] | 5578 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 5579 | proc equiRadical(ideal i) |
---|
[7f7c25e] | 5580 | "USAGE: equiRadical(I); I ideal |
---|
| 5581 | RETURN: ideal, intersection of associated primes of I of maximal dimension. |
---|
| 5582 | NOTE: A combination of the algorithms of Krick/Logar (with modifications by Laplagne) and Kemper is used. |
---|
[07c623] | 5583 | Works also in positive characteristic (Kempers algorithm). |
---|
[ebecf83] | 5584 | EXAMPLE: example equiRadical; shows an example |
---|
| 5585 | " |
---|
| 5586 | { |
---|
[cb980ab] | 5587 | if(attrib(basering,"global")!=1) |
---|
| 5588 | { |
---|
| 5589 | ERROR( |
---|
| 5590 | "// Not implemented for this ordering, please change to global ordering." |
---|
| 5591 | ); |
---|
| 5592 | } |
---|
[d88470] | 5593 | return(radical(i, 1)); |
---|
[ebecf83] | 5594 | } |
---|
| 5595 | example |
---|
| 5596 | { "EXAMPLE:"; echo = 2; |
---|
[07c623] | 5597 | ring r = 0,(x,y,z),dp; |
---|
[ebecf83] | 5598 | poly p = z2+1; |
---|
[07c623] | 5599 | poly q = z3+2; |
---|
| 5600 | ideal i = p*q^2,y-z2; |
---|
[ebecf83] | 5601 | ideal pr= equiRadical(i); |
---|
| 5602 | pr; |
---|
| 5603 | } |
---|
[fc5095] | 5604 | |
---|
[ebecf83] | 5605 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f0daaa2] | 5606 | proc radical(ideal i, list #) |
---|
[7f7c25e] | 5607 | "USAGE: radical(I[, l]); I ideal, l list (optional) |
---|
| 5608 | @* Optional parameters in list l (can be entered in any order): |
---|
| 5609 | @* 0, \"fullRad\" -> full radical is computed (default) |
---|
| 5610 | @* 1, \"equiRad\" -> equiRadical is computed |
---|
| 5611 | @* \"KL\" -> Krick/Logar algorithm is used |
---|
| 5612 | @* \"SL\" -> modifications by Laplagne are used (default) |
---|
| 5613 | @* \"facstd\" -> uses facstd to first decompose the ideal (default for non homogeneous ideals) |
---|
| 5614 | @* \"noFacstd\" -> does not use facstd (default for homogeneous ideals) |
---|
| 5615 | RETURN: ideal, the radical of I (or the equiradical if required in the input parameters) |
---|
| 5616 | NOTE: A combination of the algorithms of Krick/Logar (with modifications by Laplagne) and Kemper is used. |
---|
[07c623] | 5617 | Works also in positive characteristic (Kempers algorithm). |
---|
[ebecf83] | 5618 | EXAMPLE: example radical; shows an example |
---|
| 5619 | " |
---|
| 5620 | { |
---|
[cb980ab] | 5621 | dbprint(printlevel - voice, "Radical, version 2006.05.08"); |
---|
| 5622 | if(attrib(basering,"global")!=1) |
---|
| 5623 | { |
---|
| 5624 | // algorithms only work in global case! |
---|
| 5625 | // pass to appropriate global ring |
---|
| 5626 | def r=basering; |
---|
[1e1ec4] | 5627 | def s=changeord(list(list("dp",1:nvars(basering)))); |
---|
[cb980ab] | 5628 | setring s; |
---|
| 5629 | ideal i=imap(r,i); |
---|
[ea87a9] | 5630 | // compute radical and go back |
---|
[cb980ab] | 5631 | def j=radical(i); |
---|
| 5632 | setring r; |
---|
| 5633 | def j=imap(s,j); |
---|
| 5634 | return(j); |
---|
| 5635 | } |
---|
[d88470] | 5636 | if(size(i) == 0){return(ideal(0));} |
---|
| 5637 | int j; |
---|
| 5638 | def P0 = basering; |
---|
| 5639 | list Pl=ringlist(P0); |
---|
| 5640 | intvec dp_w; |
---|
| 5641 | for(j=nvars(P0);j>0;j--) {dp_w[j]=1;} |
---|
| 5642 | Pl[3]=list(list("dp",dp_w),list("C",0)); |
---|
| 5643 | def @P=ring(Pl); |
---|
| 5644 | setring @P; |
---|
| 5645 | ideal i=imap(P0,i); |
---|
| 5646 | |
---|
| 5647 | int il; |
---|
| 5648 | string algorithm; |
---|
| 5649 | int useFac; |
---|
| 5650 | |
---|
| 5651 | // Set input parameters |
---|
| 5652 | algorithm = "SL"; // Default: SL algorithm |
---|
| 5653 | il = 0; // Default: Full radical (not only equiRadical) |
---|
| 5654 | if (homog(i) == 1) |
---|
| 5655 | { // Default: facStd is used, except if the ideal is homogeneous. |
---|
[70ab73] | 5656 | useFac = 0; |
---|
| 5657 | } |
---|
| 5658 | else |
---|
| 5659 | { |
---|
| 5660 | useFac = 1; |
---|
[d88470] | 5661 | } |
---|
[70ab73] | 5662 | if(size(#) > 0) |
---|
| 5663 | { |
---|
[d88470] | 5664 | int valid; |
---|
| 5665 | for(j = 1; j <= size(#); j++) |
---|
| 5666 | { |
---|
| 5667 | valid = 0; |
---|
| 5668 | if((typeof(#[j]) == "int") or (typeof(#[j]) == "number")) |
---|
| 5669 | { |
---|
| 5670 | il = #[j]; // If il == 1, equiRadical is computed |
---|
| 5671 | valid = 1; |
---|
[7f7c25e] | 5672 | } |
---|
[70ab73] | 5673 | if(typeof(#[j]) == "string") |
---|
| 5674 | { |
---|
| 5675 | if(#[j] == "KL") |
---|
| 5676 | { |
---|
[d88470] | 5677 | algorithm = "KL"; |
---|
| 5678 | valid = 1; |
---|
| 5679 | } |
---|
[70ab73] | 5680 | if(#[j] == "SL") |
---|
| 5681 | { |
---|
[d88470] | 5682 | algorithm = "SL"; |
---|
| 5683 | valid = 1; |
---|
| 5684 | } |
---|
[70ab73] | 5685 | if(#[j] == "noFacstd") |
---|
| 5686 | { |
---|
[d88470] | 5687 | useFac = 0; |
---|
[70ab73] | 5688 | valid = 1; |
---|
| 5689 | } |
---|
| 5690 | if(#[j] == "facstd") |
---|
| 5691 | { |
---|
[d88470] | 5692 | useFac = 1; |
---|
[70ab73] | 5693 | valid = 1; |
---|
| 5694 | } |
---|
| 5695 | if(#[j] == "equiRad") |
---|
| 5696 | { |
---|
[d88470] | 5697 | il = 1; |
---|
[70ab73] | 5698 | valid = 1; |
---|
| 5699 | } |
---|
| 5700 | if(#[j] == "fullRad") |
---|
| 5701 | { |
---|
[d88470] | 5702 | il = 0; |
---|
[70ab73] | 5703 | valid = 1; |
---|
| 5704 | } |
---|
[d88470] | 5705 | } |
---|
| 5706 | if(valid == 0) |
---|
| 5707 | { |
---|
| 5708 | dbprint(1, "Warning! The following input parameter was not recognized:", #[j]); |
---|
| 5709 | } |
---|
| 5710 | } |
---|
| 5711 | } |
---|
[f0daaa2] | 5712 | |
---|
[d88470] | 5713 | ideal rad = 1; |
---|
| 5714 | intvec op = option(get); |
---|
| 5715 | list qr = simplifyIdeal(i); |
---|
| 5716 | map phi = @P, qr[2]; |
---|
[0c33fb] | 5717 | |
---|
[d88470] | 5718 | option(redSB); |
---|
| 5719 | i = groebner(qr[1]); |
---|
| 5720 | option(set, op); |
---|
| 5721 | int di = dim(i); |
---|
[0c33fb] | 5722 | |
---|
[d88470] | 5723 | if(di == 0) |
---|
| 5724 | { |
---|
| 5725 | i = zeroRad(i, qr[1]); |
---|
[a90eb0] | 5726 | option(redSB); |
---|
[d88470] | 5727 | i=interred(phi(i)); |
---|
[a90eb0] | 5728 | option(set, op); |
---|
[d88470] | 5729 | setring(P0); |
---|
| 5730 | i=imap(@P,i); |
---|
| 5731 | return(i); |
---|
| 5732 | } |
---|
[0c33fb] | 5733 | |
---|
[d88470] | 5734 | option(redSB); |
---|
| 5735 | list pr; |
---|
| 5736 | if(useFac == 1) |
---|
| 5737 | { |
---|
| 5738 | pr = facstd(i); |
---|
[70ab73] | 5739 | } |
---|
| 5740 | else |
---|
| 5741 | { |
---|
[d88470] | 5742 | pr = i; |
---|
| 5743 | } |
---|
| 5744 | option(set, op); |
---|
| 5745 | int s = size(pr); |
---|
[70ab73] | 5746 | if(useFac == 1) |
---|
| 5747 | { |
---|
[d88470] | 5748 | dbprint(printlevel - voice, "Number of components returned by facstd: ", s); |
---|
| 5749 | } |
---|
| 5750 | for(j = 1; j <= s; j++) |
---|
| 5751 | { |
---|
| 5752 | attrib(pr[s + 1 - j], "isSB", 1); |
---|
| 5753 | if((size(reduce(rad, pr[s + 1 - j], 1)) != 0) && ((dim(pr[s + 1 - j]) == di) || !il)) |
---|
| 5754 | { |
---|
| 5755 | // SL Debug messages |
---|
| 5756 | dbprint(printlevel-voice, "We shall compute the radical of ", pr[s + 1 - j]); |
---|
| 5757 | dbprint(printlevel-voice, "The dimension is: ", dim(pr[s+1-j])); |
---|
[f0daaa2] | 5758 | |
---|
[d88470] | 5759 | if(algorithm == "KL") |
---|
[0266ac] | 5760 | { |
---|
[d88470] | 5761 | rad = intersect(rad, radicalKL(pr[s + 1 - j], rad, il)); |
---|
[7f7c25e] | 5762 | } |
---|
[70ab73] | 5763 | if(algorithm == "SL") |
---|
| 5764 | { |
---|
[d88470] | 5765 | rad = intersect(rad, radicalSL(pr[s + 1 - j], il)); |
---|
| 5766 | } |
---|
| 5767 | } |
---|
| 5768 | else |
---|
| 5769 | { |
---|
| 5770 | // SL Debug |
---|
| 5771 | dbprint(printlevel-voice, "The radical of this component is not needed."); |
---|
| 5772 | dbprint(printlevel-voice, "size(reduce(rad, pr[s + 1 - j], 1))", |
---|
| 5773 | size(reduce(rad, pr[s + 1 - j], 1))); |
---|
| 5774 | dbprint(printlevel-voice, "dim(pr[s + 1 - j])", dim(pr[s + 1 - j])); |
---|
| 5775 | dbprint(printlevel-voice, "il", il); |
---|
| 5776 | } |
---|
| 5777 | } |
---|
| 5778 | rad=interred(phi(rad)); |
---|
| 5779 | setring(P0); |
---|
| 5780 | i=imap(@P,rad); |
---|
| 5781 | return(i); |
---|
[1918008] | 5782 | } |
---|
[ebecf83] | 5783 | example |
---|
| 5784 | { "EXAMPLE:"; echo = 2; |
---|
[07c623] | 5785 | ring r = 0,(x,y,z),dp; |
---|
[ebecf83] | 5786 | poly p = z2+1; |
---|
[07c623] | 5787 | poly q = z3+2; |
---|
| 5788 | ideal i = p*q^2,y-z2; |
---|
[f0daaa2] | 5789 | ideal pr = radical(i); |
---|
[ebecf83] | 5790 | pr; |
---|
| 5791 | } |
---|
[f0daaa2] | 5792 | |
---|
| 5793 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 5794 | // |
---|
| 5795 | // Computes the radical of I using KL algorithm. |
---|
[7d56875] | 5796 | // The only difference with the previous implementation of KL algorithm is |
---|
[f0daaa2] | 5797 | // that now it uses block dp instead of lp ordering for the reduction to the |
---|
| 5798 | // zerodimensional case. |
---|
[f995aa] | 5799 | // The reduction step has been moved to the new routine radicalReduction, so that it can be |
---|
| 5800 | // used also by radicalSL procedure. |
---|
[f0daaa2] | 5801 | // |
---|
[70ab73] | 5802 | static proc radicalKL(ideal I, ideal ser, list #) |
---|
| 5803 | { |
---|
[f3c6e5] | 5804 | // ideal I The ideal for which the radical is computed |
---|
| 5805 | // ideal ser Used to reduce components already obtained |
---|
| 5806 | // list # If #[1] = 1, equiradical is computed. |
---|
[f0daaa2] | 5807 | |
---|
[70ab73] | 5808 | // I needs to be a Groebner basis. |
---|
| 5809 | if (attrib(I, "isSB") != 1) |
---|
| 5810 | { |
---|
| 5811 | I = groebner(I); |
---|
| 5812 | } |
---|
[f0daaa2] | 5813 | |
---|
[70ab73] | 5814 | ideal rad; // The radical |
---|
| 5815 | int allIndep = 1; // All max independent sets are used |
---|
[0266ac] | 5816 | |
---|
[70ab73] | 5817 | list result = radicalReduction(I, ser, allIndep, #); |
---|
| 5818 | int done = result[3]; |
---|
| 5819 | rad = result[1]; |
---|
| 5820 | if (done == 0) |
---|
| 5821 | { |
---|
| 5822 | rad = intersect(rad, radicalKL(result[2], ideal(1), #)); |
---|
| 5823 | } |
---|
| 5824 | return(rad); |
---|
| 5825 | } |
---|
[f0daaa2] | 5826 | |
---|
| 5827 | |
---|
| 5828 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 5829 | // |
---|
[f995aa] | 5830 | // Computes the radical of I via Laplagne algorithm, using zerodimensional radical in |
---|
[f0daaa2] | 5831 | // the zero dimensional case. |
---|
| 5832 | // For the reduction to the zerodimensional case, it uses the procedure |
---|
[f995aa] | 5833 | // radical, with some modifications to avoid the recursion. |
---|
[f0daaa2] | 5834 | // |
---|
[f995aa] | 5835 | static proc radicalSL(ideal I, list #) |
---|
[f0daaa2] | 5836 | // Input = I, ideal |
---|
| 5837 | // #, list. If #[1] = 1, then computes only the equiradical. |
---|
| 5838 | // Output = (P, primaryDec) where P = rad(I) and primaryDec is the list of the radicals |
---|
| 5839 | // obtained in intermediate steps. |
---|
| 5840 | { |
---|
[70ab73] | 5841 | ideal rad = 1; |
---|
| 5842 | ideal equiRad = 1; |
---|
| 5843 | list primes; |
---|
| 5844 | int k; // Counter |
---|
| 5845 | int il; // If il = 1, only the equiradical is required. |
---|
| 5846 | int iDim; // The dimension of I |
---|
| 5847 | int stop = 0; // Checks if the radical has been obtained |
---|
| 5848 | |
---|
| 5849 | if (attrib(I, "isSB") != 1) |
---|
| 5850 | { |
---|
| 5851 | I = groebner(I); |
---|
| 5852 | } |
---|
| 5853 | iDim = dim(I); |
---|
| 5854 | |
---|
| 5855 | // Checks if only equiradical is required |
---|
| 5856 | if (size(#) > 0) |
---|
| 5857 | { |
---|
| 5858 | il = #[1]; |
---|
| 5859 | } |
---|
| 5860 | |
---|
| 5861 | while(stop == 0) |
---|
| 5862 | { |
---|
| 5863 | dbprint (printlevel-voice, "// We call radLoopR to find new prime ideals."); |
---|
| 5864 | primes = radicalSLIteration(I, rad); // A list of primes or intersections of primes, not included in P |
---|
| 5865 | dbprint (printlevel - voice, "// Output of Iteration Step:"); |
---|
| 5866 | dbprint (printlevel - voice, primes); |
---|
| 5867 | if (size(primes) > 0) |
---|
| 5868 | { |
---|
| 5869 | dbprint (printlevel - voice, "// We intersect P with the ideal just obtained."); |
---|
| 5870 | for(k = 1; k <= size(primes); k++) |
---|
| 5871 | { |
---|
| 5872 | rad = intersect(rad, primes[k]); |
---|
| 5873 | if (il == 1) |
---|
| 5874 | { |
---|
| 5875 | if (attrib(primes[k], "isSB") != 1) |
---|
| 5876 | { |
---|
| 5877 | primes[k] = groebner(primes[k]); |
---|
| 5878 | } |
---|
| 5879 | if (iDim == dim(primes[k])) |
---|
| 5880 | { |
---|
| 5881 | equiRad = intersect(equiRad, primes[k]); |
---|
| 5882 | } |
---|
[7f7c25e] | 5883 | } |
---|
[70ab73] | 5884 | } |
---|
[7f7c25e] | 5885 | } |
---|
[70ab73] | 5886 | else |
---|
| 5887 | { |
---|
| 5888 | stop = 1; |
---|
[7f7c25e] | 5889 | } |
---|
[70ab73] | 5890 | } |
---|
| 5891 | if (il == 0) |
---|
| 5892 | { |
---|
| 5893 | return(rad); |
---|
| 5894 | } |
---|
| 5895 | else |
---|
| 5896 | { |
---|
| 5897 | return(equiRad); |
---|
| 5898 | } |
---|
| 5899 | } |
---|
[f0daaa2] | 5900 | |
---|
| 5901 | ////////////////////////////////////////////////////////////////////////// |
---|
| 5902 | // Based on radicalKL. |
---|
[f995aa] | 5903 | // It contains all of old version of proc radicalKL except the recursion call. |
---|
[a36e78] | 5904 | // |
---|
[f0daaa2] | 5905 | // Output: |
---|
| 5906 | // #1 -> output ideal, the part of the radical that has been computed |
---|
| 5907 | // #2 -> complementary ideal, the part of the ideal I whose radical remains to be computed |
---|
| 5908 | // = (I, h) in KL algorithm |
---|
| 5909 | // This is not used in the new algorithm. It is part of KL algorithm |
---|
| 5910 | // #3 -> done, 1: output = radical, there is no need to continue |
---|
| 5911 | // 0: radical = output \cap \sqrt{complementary ideal} |
---|
| 5912 | // This is not used in the new algorithm. It is part of KL algorithm |
---|
| 5913 | |
---|
[70ab73] | 5914 | static proc radicalReduction(ideal I, ideal ser, int allIndep, list #) |
---|
| 5915 | { |
---|
[6fd3a2] | 5916 | // allMaximal 1 -> Indicates that the reduction to the zerodim case |
---|
| 5917 | // must be done for all indep set of the leading terms ideal |
---|
| 5918 | // 0 -> Otherwise |
---|
| 5919 | // ideal ser Only for radicalKL. (Same as in radicalKL) |
---|
[7d56875] | 5920 | // list # Only for radicalKL (If #[1] = 1, |
---|
[6fd3a2] | 5921 | // only equiradical is required. |
---|
| 5922 | // It is used to set the value of done.) |
---|
[f0daaa2] | 5923 | |
---|
[70ab73] | 5924 | attrib(I, "isSB", 1); // I needs to be a reduced standard basis |
---|
| 5925 | list indep, fett; |
---|
| 5926 | intvec @w, @hilb, op; |
---|
| 5927 | int @wr, @n, @m, lauf, di; |
---|
| 5928 | ideal fac, @h, collectrad, lsau; |
---|
| 5929 | poly @q; |
---|
| 5930 | string @va, quotring; |
---|
| 5931 | |
---|
| 5932 | def @P = basering; |
---|
| 5933 | int jdim = dim(I); // Computes the dimension of I |
---|
| 5934 | int homo = homog(I); // Finds out if I is homogeneous |
---|
| 5935 | ideal rad = ideal(1); // The unit ideal |
---|
| 5936 | ideal te = ser; |
---|
| 5937 | if(size(#) > 0) |
---|
| 5938 | { |
---|
| 5939 | @wr = #[1]; |
---|
| 5940 | } |
---|
| 5941 | if(homo == 1) |
---|
| 5942 | { |
---|
| 5943 | for(@n = 1; @n <= nvars(basering); @n++) |
---|
| 5944 | { |
---|
| 5945 | @w[@n] = ord(var(@n)); |
---|
| 5946 | } |
---|
| 5947 | @hilb = hilb(I, 1, @w); |
---|
| 5948 | } |
---|
[f0daaa2] | 5949 | |
---|
[70ab73] | 5950 | // SL 2006.04.11 1 Debug messages |
---|
| 5951 | dbprint(printlevel-voice, "//Computes the radical of the ideal:", I); |
---|
| 5952 | // SL 2006.04.11 2 Debug messages |
---|
[f0daaa2] | 5953 | |
---|
| 5954 | //--------------------------------------------------------------------------- |
---|
| 5955 | //j is the ring |
---|
| 5956 | //--------------------------------------------------------------------------- |
---|
| 5957 | |
---|
[70ab73] | 5958 | if (jdim==-1) |
---|
| 5959 | { |
---|
| 5960 | return(ideal(1), ideal(1), 1); |
---|
| 5961 | } |
---|
[f0daaa2] | 5962 | |
---|
| 5963 | //--------------------------------------------------------------------------- |
---|
| 5964 | //the zero-dimensional case |
---|
| 5965 | //--------------------------------------------------------------------------- |
---|
| 5966 | |
---|
[70ab73] | 5967 | if (jdim==0) |
---|
| 5968 | { |
---|
| 5969 | return(zeroRad(I), ideal(1), 1); |
---|
| 5970 | } |
---|
[f0daaa2] | 5971 | |
---|
[70ab73] | 5972 | //------------------------------------------------------------------------- |
---|
| 5973 | //search for a maximal independent set indep,i.e. |
---|
| 5974 | //look for subring such that the intersection with the ideal is zero |
---|
| 5975 | //j intersected with K[var(indep[3]+1),...,var(nvar)] is zero, |
---|
| 5976 | //indep[1] is the new varstring, indep[2] the string for the block-ordering |
---|
| 5977 | //------------------------------------------------------------------------- |
---|
| 5978 | |
---|
[6fd3a2] | 5979 | // SL 2006-04-24 1 If allIndep = 0, then it only computes one maximal |
---|
| 5980 | // independent set. |
---|
| 5981 | // This looks better for the new algorithm but not for KL |
---|
| 5982 | // algorithm |
---|
[70ab73] | 5983 | list parameters = allIndep; |
---|
| 5984 | indep = newMaxIndependSetDp(I, parameters); |
---|
| 5985 | // SL 2006-04-24 2 |
---|
| 5986 | |
---|
| 5987 | for(@m = 1; @m <= size(indep); @m++) |
---|
| 5988 | { |
---|
| 5989 | if((indep[@m][1] == varstr(basering)) && (@m == 1)) |
---|
| 5990 | //this is the good case, nothing to do, just to have the same notations |
---|
| 5991 | //change the ring |
---|
| 5992 | { |
---|
| 5993 | execute("ring gnir1 = ("+charstr(basering)+"),("+varstr(basering)+"),(" |
---|
[f0daaa2] | 5994 | +ordstr(basering)+");"); |
---|
[70ab73] | 5995 | ideal @j = fetch(@P, I); |
---|
| 5996 | attrib(@j, "isSB", 1); |
---|
| 5997 | } |
---|
| 5998 | else |
---|
| 5999 | { |
---|
| 6000 | @va = string(maxideal(1)); |
---|
[f0daaa2] | 6001 | |
---|
[70ab73] | 6002 | execute("ring gnir1 = (" + charstr(basering) + "), (" + indep[@m][1] + "),(" |
---|
[f0daaa2] | 6003 | + indep[@m][2] + ");"); |
---|
[70ab73] | 6004 | execute("map phi = @P," + @va + ";"); |
---|
| 6005 | if(homo == 1) |
---|
| 6006 | { |
---|
| 6007 | ideal @j = std(phi(I), @hilb, @w); |
---|
[0266ac] | 6008 | } |
---|
| 6009 | else |
---|
| 6010 | { |
---|
[70ab73] | 6011 | ideal @j = groebner(phi(I)); |
---|
| 6012 | } |
---|
| 6013 | } |
---|
| 6014 | if((deg(@j[1]) == 0) || (dim(@j) < jdim)) |
---|
| 6015 | { |
---|
| 6016 | setring @P; |
---|
| 6017 | break; |
---|
| 6018 | } |
---|
| 6019 | for (lauf = 1; lauf <= size(@j); lauf++) |
---|
| 6020 | { |
---|
| 6021 | fett[lauf] = size(@j[lauf]); |
---|
| 6022 | } |
---|
| 6023 | //------------------------------------------------------------------------ |
---|
| 6024 | // We have now the following situation: |
---|
| 6025 | // j intersected with K[var(nnp+1),..,var(nva)] is zero so we may pass |
---|
| 6026 | // to this quotientring, j is there still a standardbasis, the |
---|
| 6027 | // leading coefficients of the polynomials there (polynomials in |
---|
| 6028 | // K[var(nnp+1),..,var(nva)]) are collected in the list h, |
---|
| 6029 | // we need their LCM, gh, because of the following: |
---|
| 6030 | // let (j:gh^n)=(j:gh^infinity) then j*K(var(nnp+1),..,var(nva))[..rest..] |
---|
| 6031 | // intersected with K[var(1),...,var(nva)] is (j:gh^n) |
---|
| 6032 | // on the other hand j = ((j, gh^n) intersected with (j : gh^n)) |
---|
| 6033 | |
---|
| 6034 | //------------------------------------------------------------------------ |
---|
| 6035 | // The arrangement for the quotientring K(var(nnp+1),..,var(nva))[..rest..] |
---|
| 6036 | // and the map phi:K[var(1),...,var(nva)] -----> |
---|
| 6037 | // K(var(nnpr+1),..,var(nva))[..the rest..] |
---|
| 6038 | //------------------------------------------------------------------------ |
---|
| 6039 | quotring = prepareQuotientRingDp(nvars(basering) - indep[@m][3]); |
---|
| 6040 | //------------------------------------------------------------------------ |
---|
| 6041 | // We pass to the quotientring K(var(nnp+1),..,var(nva))[..the rest..] |
---|
| 6042 | //------------------------------------------------------------------------ |
---|
| 6043 | |
---|
| 6044 | execute(quotring); |
---|
| 6045 | |
---|
| 6046 | // @j considered in the quotientring |
---|
| 6047 | ideal @j = imap(gnir1, @j); |
---|
| 6048 | |
---|
| 6049 | kill gnir1; |
---|
| 6050 | |
---|
| 6051 | // j is a standardbasis in the quotientring but usually not minimal |
---|
| 6052 | // here it becomes minimal |
---|
| 6053 | |
---|
| 6054 | @j = clearSB(@j, fett); |
---|
| 6055 | |
---|
| 6056 | // We need later LCM(h[1],...) = gh for saturation |
---|
| 6057 | ideal @h; |
---|
| 6058 | if(deg(@j[1]) > 0) |
---|
| 6059 | { |
---|
| 6060 | for(@n = 1; @n <= size(@j); @n++) |
---|
| 6061 | { |
---|
| 6062 | @h[@n] = leadcoef(@j[@n]); |
---|
[0266ac] | 6063 | } |
---|
[70ab73] | 6064 | op = option(get); |
---|
| 6065 | option(redSB); |
---|
[6fd3a2] | 6066 | @j = std(@j); //to obtain a reduced standardbasis |
---|
[70ab73] | 6067 | option(set, op); |
---|
| 6068 | |
---|
| 6069 | // SL 1 Debug messages |
---|
| 6070 | dbprint(printlevel - voice, "zero_rad", basering, @j, dim(groebner(@j))); |
---|
| 6071 | ideal zero_rad = zeroRad(@j); |
---|
| 6072 | dbprint(printlevel - voice, "zero_rad passed"); |
---|
| 6073 | // SL 2 |
---|
| 6074 | } |
---|
| 6075 | else |
---|
| 6076 | { |
---|
| 6077 | ideal zero_rad = ideal(1); |
---|
| 6078 | } |
---|
[f0daaa2] | 6079 | |
---|
[70ab73] | 6080 | // We need the intersection of the ideals in the list quprimary with the |
---|
| 6081 | // polynomialring, i.e. let q=(f1,...,fr) in the quotientring such an ideal |
---|
| 6082 | // but fi polynomials, then the intersection of q with the polynomialring |
---|
| 6083 | // is the saturation of the ideal generated by f1,...,fr with respect to |
---|
| 6084 | // h which is the lcm of the leading coefficients of the fi considered in |
---|
| 6085 | // the quotientring: this is coded in saturn |
---|
[f0daaa2] | 6086 | |
---|
[70ab73] | 6087 | zero_rad = std(zero_rad); |
---|
[f0daaa2] | 6088 | |
---|
[70ab73] | 6089 | ideal hpl; |
---|
[f0daaa2] | 6090 | |
---|
[70ab73] | 6091 | for(@n = 1; @n <= size(zero_rad); @n++) |
---|
| 6092 | { |
---|
| 6093 | hpl = hpl, leadcoef(zero_rad[@n]); |
---|
| 6094 | } |
---|
[f0daaa2] | 6095 | |
---|
[70ab73] | 6096 | //------------------------------------------------------------------------ |
---|
| 6097 | // We leave the quotientring K(var(nnp+1),..,var(nva))[..the rest..] |
---|
| 6098 | // back to the polynomialring |
---|
| 6099 | //------------------------------------------------------------------------ |
---|
| 6100 | setring @P; |
---|
[f0daaa2] | 6101 | |
---|
[70ab73] | 6102 | collectrad = imap(quring, zero_rad); |
---|
| 6103 | lsau = simplify(imap(quring, hpl), 2); |
---|
| 6104 | @h = imap(quring, @h); |
---|
[f0daaa2] | 6105 | |
---|
[70ab73] | 6106 | kill quring; |
---|
[f0daaa2] | 6107 | |
---|
[70ab73] | 6108 | // Here the intersection with the polynomialring |
---|
| 6109 | // mentioned above is really computed |
---|
[f0daaa2] | 6110 | |
---|
[70ab73] | 6111 | collectrad = sat2(collectrad, lsau)[1]; |
---|
| 6112 | if(deg(@h[1])>=0) |
---|
| 6113 | { |
---|
| 6114 | fac = ideal(0); |
---|
| 6115 | for(lauf = 1; lauf <= ncols(@h); lauf++) |
---|
[f0daaa2] | 6116 | { |
---|
[70ab73] | 6117 | if(deg(@h[lauf]) > 0) |
---|
| 6118 | { |
---|
| 6119 | fac = fac + factorize(@h[lauf], 1); |
---|
| 6120 | } |
---|
[f0daaa2] | 6121 | } |
---|
[70ab73] | 6122 | fac = simplify(fac, 6); |
---|
| 6123 | @q = 1; |
---|
| 6124 | for(lauf = 1; lauf <= size(fac); lauf++) |
---|
[f0daaa2] | 6125 | { |
---|
[70ab73] | 6126 | @q = @q * fac[lauf]; |
---|
[f0daaa2] | 6127 | } |
---|
[70ab73] | 6128 | op = option(get); |
---|
| 6129 | option(returnSB); |
---|
| 6130 | option(redSB); |
---|
| 6131 | I = quotient(I + ideal(@q), rad); |
---|
| 6132 | attrib(I, "isSB", 1); |
---|
| 6133 | option(set, op); |
---|
| 6134 | } |
---|
| 6135 | if((deg(rad[1]) > 0) && (deg(collectrad[1]) > 0)) |
---|
| 6136 | { |
---|
| 6137 | rad = intersect(rad, collectrad); |
---|
| 6138 | te = intersect(te, collectrad); |
---|
| 6139 | te = simplify(reduce(te, I, 1), 2); |
---|
| 6140 | } |
---|
| 6141 | else |
---|
| 6142 | { |
---|
| 6143 | if(deg(collectrad[1]) > 0) |
---|
[f0daaa2] | 6144 | { |
---|
[70ab73] | 6145 | rad = collectrad; |
---|
| 6146 | te = intersect(te, collectrad); |
---|
| 6147 | te = simplify(reduce(te, I, 1), 2); |
---|
[f0daaa2] | 6148 | } |
---|
[70ab73] | 6149 | } |
---|
[f0daaa2] | 6150 | |
---|
[70ab73] | 6151 | if((dim(I) < jdim)||(size(te) == 0)) |
---|
| 6152 | { |
---|
| 6153 | break; |
---|
| 6154 | } |
---|
| 6155 | if(homo==1) |
---|
| 6156 | { |
---|
| 6157 | @hilb = hilb(I, 1, @w); |
---|
| 6158 | } |
---|
| 6159 | } |
---|
[f0daaa2] | 6160 | |
---|
[70ab73] | 6161 | // SL 2006.04.11 1 Debug messages |
---|
| 6162 | dbprint (printlevel-voice, "// Part of the Radical already computed:", rad); |
---|
| 6163 | dbprint (printlevel-voice, "// Dimension:", dim(groebner(rad))); |
---|
| 6164 | // SL 2006.04.11 2 Debug messages |
---|
[f0daaa2] | 6165 | |
---|
[6fd3a2] | 6166 | // SL 2006.04.21 1 New variable "done". |
---|
| 6167 | // It tells if the radical is already computed or |
---|
| 6168 | // if it still has to be computed the radical of the new ideal I |
---|
[70ab73] | 6169 | int done; |
---|
| 6170 | if(((@wr == 1) && (dim(I)<jdim)) || (deg(I[1])==0) || (size(te) == 0)) |
---|
| 6171 | { |
---|
| 6172 | done = 1; |
---|
| 6173 | } |
---|
| 6174 | else |
---|
| 6175 | { |
---|
| 6176 | done = 0; |
---|
| 6177 | } |
---|
| 6178 | // SL 2006.04.21 2 |
---|
[f0daaa2] | 6179 | |
---|
[6fd3a2] | 6180 | // SL 2006.04.21 1 See details of the output at the beginning of this proc. |
---|
[70ab73] | 6181 | list result = rad, I, done; |
---|
| 6182 | return(result); |
---|
| 6183 | // SL 2006.04.21 2 |
---|
| 6184 | } |
---|
[f0daaa2] | 6185 | |
---|
| 6186 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 6187 | // Given an ideal I and an ideal P (intersection of some minimal prime ideals |
---|
| 6188 | // associated to I), it calculates the intersection of new minimal prime ideals |
---|
| 6189 | // associated to I which where not used to calculate P. |
---|
| 6190 | // This version uses ZD Radical in the zerodimensional case. |
---|
[f995aa] | 6191 | static proc radicalSLIteration (ideal I, ideal P); |
---|
[f0daaa2] | 6192 | // Input: I, ideal. The ideal from which new prime components will be obtained. |
---|
| 6193 | // P, ideal. Intersection of some prime ideals of I. |
---|
| 6194 | // Output: ideal. Intersection of some primes of I different from the ones in P. |
---|
| 6195 | { |
---|
[6fd3a2] | 6196 | int k = 1; // Counter |
---|
| 6197 | int good = 0; // Checks if an element of P is in rad(I) |
---|
[70ab73] | 6198 | |
---|
| 6199 | dbprint (printlevel-voice, "// We search for an element in P - sqrt(I)."); |
---|
| 6200 | while ((k <= size(P)) and (good == 0)) |
---|
| 6201 | { |
---|
| 6202 | dbprint (printlevel-voice, "// We try with:", P[k]); |
---|
| 6203 | good = 1 - rad_con(P[k], I); |
---|
| 6204 | k++; |
---|
| 6205 | } |
---|
| 6206 | k--; |
---|
| 6207 | if (good == 0) |
---|
| 6208 | { |
---|
| 6209 | dbprint (printlevel-voice, "// No element was found, P = sqrt(I)."); |
---|
| 6210 | list emptyList = list(); |
---|
| 6211 | return (emptyList); |
---|
| 6212 | } |
---|
| 6213 | dbprint(printlevel - voice, "// That one was good!"); |
---|
| 6214 | dbprint(printlevel - voice, "// We saturate I with respect to this element."); |
---|
| 6215 | if (P[k] != 1) |
---|
| 6216 | { |
---|
[6fd3a2] | 6217 | intvec oo=option(get); |
---|
| 6218 | option(redSB); |
---|
[70ab73] | 6219 | ideal J = sat(I, P[k])[1]; |
---|
[6fd3a2] | 6220 | option(set,oo); |
---|
| 6221 | |
---|
[a36e78] | 6222 | } |
---|
| 6223 | else |
---|
| 6224 | { |
---|
| 6225 | dbprint(printlevel - voice, "// The polynomial is 1, the saturation in not actually computed."); |
---|
| 6226 | ideal J = I; |
---|
| 6227 | } |
---|
[7f7c25e] | 6228 | |
---|
[a36e78] | 6229 | // We now call proc radicalNew; |
---|
| 6230 | dbprint(printlevel - voice, "// We do the reduction to the zerodimensional case, via radical."); |
---|
| 6231 | dbprint(printlevel - voice, "// The ideal is ", J); |
---|
| 6232 | dbprint(printlevel - voice, "// The dimension is ", dim(groebner(J))); |
---|
[7f7c25e] | 6233 | |
---|
[6fd3a2] | 6234 | int allMaximal = 0; // Compute the zerodim reduction for only one indep set. |
---|
| 6235 | ideal re = 1; // No reduction is need, |
---|
| 6236 | // there are not redundant components. |
---|
| 6237 | list emptyList = list(); // Look for primes of any dimension, |
---|
| 6238 | // not only of max dimension. |
---|
[a36e78] | 6239 | list result = radicalReduction(J, re, allMaximal, emptyList); |
---|
[f0daaa2] | 6240 | |
---|
[a36e78] | 6241 | return(result[1]); |
---|
[70ab73] | 6242 | } |
---|
[f0daaa2] | 6243 | |
---|
| 6244 | /////////////////////////////////////////////////////////////////////////////////// |
---|
| 6245 | // Based on maxIndependSet |
---|
| 6246 | // Added list # as parameter |
---|
| 6247 | // If the first element of # is 0, the output is only 1 max indep set. |
---|
| 6248 | // If no list is specified or #[1] = 1, the output is all the max indep set of the |
---|
| 6249 | // leading terms ideal. This is the original output of maxIndependSet |
---|
| 6250 | |
---|
| 6251 | // The ordering given in the output has been changed to block dp instead of lp. |
---|
| 6252 | |
---|
[f995aa] | 6253 | proc newMaxIndependSetDp(ideal j, list #) |
---|
[7f7c25e] | 6254 | "USAGE: newMaxIndependentSetDp(I); I ideal (returns all maximal independent sets of the corresponding leading terms ideal) |
---|
| 6255 | newMaxIndependentSetDp(I, 0); I ideal (returns only one maximal independent set) |
---|
[f0daaa2] | 6256 | RETURN: list = #1. new varstring with the maximal independent set at the end, |
---|
[f995aa] | 6257 | #2. ordstring with the corresponding dp block ordering, |
---|
[f0daaa2] | 6258 | #3. the number of independent variables |
---|
| 6259 | NOTE: |
---|
[f995aa] | 6260 | EXAMPLE: example newMaxIndependentSetDp; shows an example |
---|
[f0daaa2] | 6261 | " |
---|
| 6262 | { |
---|
[70ab73] | 6263 | int n, k, di; |
---|
| 6264 | list resu, hilf; |
---|
| 6265 | string var1, var2; |
---|
| 6266 | list v = indepSet(j, 0); |
---|
| 6267 | |
---|
| 6268 | // SL 2006.04.21 1 Lines modified to use only one independent Set |
---|
| 6269 | int allMaximal; |
---|
| 6270 | if (size(#) > 0) |
---|
| 6271 | { |
---|
| 6272 | allMaximal = #[1]; |
---|
[a36e78] | 6273 | } |
---|
| 6274 | else |
---|
| 6275 | { |
---|
| 6276 | allMaximal = 1; |
---|
| 6277 | } |
---|
[f0daaa2] | 6278 | |
---|
[a36e78] | 6279 | int nMax; |
---|
| 6280 | if (allMaximal == 1) |
---|
| 6281 | { |
---|
| 6282 | nMax = size(v); |
---|
| 6283 | } |
---|
| 6284 | else |
---|
| 6285 | { |
---|
| 6286 | nMax = 1; |
---|
| 6287 | } |
---|
[f0daaa2] | 6288 | |
---|
[a36e78] | 6289 | for(n = 1; n <= nMax; n++) |
---|
| 6290 | // SL 2006.04.21 2 |
---|
| 6291 | { |
---|
| 6292 | di = 0; |
---|
| 6293 | var1 = ""; |
---|
| 6294 | var2 = ""; |
---|
| 6295 | for(k = 1; k <= size(v[n]); k++) |
---|
| 6296 | { |
---|
| 6297 | if(v[n][k] != 0) |
---|
| 6298 | { |
---|
| 6299 | di++; |
---|
| 6300 | var2 = var2 + "var(" + string(k) + "), "; |
---|
| 6301 | } |
---|
| 6302 | else |
---|
| 6303 | { |
---|
| 6304 | var1 = var1 + "var(" + string(k) + "), "; |
---|
| 6305 | } |
---|
| 6306 | } |
---|
| 6307 | if(di > 0) |
---|
| 6308 | { |
---|
| 6309 | var1 = var1 + var2; |
---|
| 6310 | var1 = var1[1..size(var1) - 2]; // The "- 2" removes the trailer comma |
---|
| 6311 | hilf[1] = var1; |
---|
| 6312 | // SL 2006.21.04 1 The order is now block dp instead of lp |
---|
| 6313 | hilf[2] = "dp(" + string(nvars(basering) - di) + "), dp(" + string(di) + ")"; |
---|
| 6314 | // SL 2006.21.04 2 |
---|
| 6315 | hilf[3] = di; |
---|
| 6316 | resu[n] = hilf; |
---|
| 6317 | } |
---|
| 6318 | else |
---|
| 6319 | { |
---|
| 6320 | resu[n] = varstr(basering), ordstr(basering), 0; |
---|
| 6321 | } |
---|
| 6322 | } |
---|
| 6323 | return(resu); |
---|
[f0daaa2] | 6324 | } |
---|
| 6325 | example |
---|
| 6326 | { "EXAMPLE:"; echo = 2; |
---|
| 6327 | ring s1 = (0, x, y), (a, b, c, d, e, f, g), lp; |
---|
| 6328 | ideal i = ea - fbg, fa + be, ec - fdg, fc + de; |
---|
| 6329 | i = std(i); |
---|
[f995aa] | 6330 | list l = newMaxIndependSetDp(i); |
---|
[f0daaa2] | 6331 | l; |
---|
| 6332 | i = i, g; |
---|
[f995aa] | 6333 | l = newMaxIndependSetDp(i); |
---|
[f0daaa2] | 6334 | l; |
---|
| 6335 | |
---|
| 6336 | ring s = 0, (x, y, z), lp; |
---|
| 6337 | ideal i = z, yx; |
---|
[f995aa] | 6338 | list l = newMaxIndependSetDp(i); |
---|
[f0daaa2] | 6339 | l; |
---|
| 6340 | } |
---|
| 6341 | |
---|
| 6342 | |
---|
| 6343 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 6344 | // based on prepareQuotientring |
---|
| 6345 | // The order returned is now (C, dp) instead of (C, lp) |
---|
| 6346 | |
---|
[f995aa] | 6347 | static proc prepareQuotientRingDp (int nnp) |
---|
| 6348 | "USAGE: prepareQuotientRingDp(nnp); nnp int |
---|
[f0daaa2] | 6349 | RETURN: string = to define Kvar(nnp+1),...,var(nvars)[..rest ] |
---|
| 6350 | NOTE: |
---|
[f995aa] | 6351 | EXAMPLE: example prepareQuotientRingDp; shows an example |
---|
[f0daaa2] | 6352 | " |
---|
| 6353 | { |
---|
| 6354 | ideal @ih,@jh; |
---|
| 6355 | int npar=npars(basering); |
---|
| 6356 | int @n; |
---|
| 6357 | |
---|
| 6358 | string quotring= "ring quring = ("+charstr(basering); |
---|
| 6359 | for(@n = nnp + 1; @n <= nvars(basering); @n++) |
---|
| 6360 | { |
---|
[a36e78] | 6361 | quotring = quotring + ", var(" + string(@n) + ")"; |
---|
| 6362 | @ih = @ih + var(@n); |
---|
[f0daaa2] | 6363 | } |
---|
| 6364 | |
---|
| 6365 | quotring = quotring+"),(var(1)"; |
---|
| 6366 | @jh = @jh + var(1); |
---|
| 6367 | for(@n = 2; @n <= nnp; @n++) |
---|
| 6368 | { |
---|
| 6369 | quotring = quotring + ", var(" + string(@n) + ")"; |
---|
| 6370 | @jh = @jh + var(@n); |
---|
| 6371 | } |
---|
| 6372 | // SL 2006-04-21 1 The order returned is now (C, dp) instead of (C, lp) |
---|
| 6373 | quotring = quotring + "), (C, dp);"; |
---|
| 6374 | // SL 2006-04-21 2 |
---|
| 6375 | |
---|
| 6376 | return(quotring); |
---|
| 6377 | } |
---|
| 6378 | example |
---|
| 6379 | { "EXAMPLE:"; echo = 2; |
---|
| 6380 | ring s1=(0,x),(a,b,c,d,e,f,g),lp; |
---|
| 6381 | def @Q=basering; |
---|
[f995aa] | 6382 | list l= prepareQuotientRingDp(3); |
---|
[f0daaa2] | 6383 | l; |
---|
| 6384 | execute(l[1]); |
---|
| 6385 | execute(l[2]); |
---|
| 6386 | basering; |
---|
| 6387 | phi; |
---|
| 6388 | setring @Q; |
---|
[a36e78] | 6389 | |
---|
[f0daaa2] | 6390 | } |
---|
| 6391 | |
---|
[ebecf83] | 6392 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 6393 | proc prepareAss(ideal i) |
---|
[7f7c25e] | 6394 | "USAGE: prepareAss(I); I ideal |
---|
| 6395 | RETURN: list, the radicals of the maximal dimensional components of I. |
---|
[7b3971] | 6396 | NOTE: Uses algorithm of Eisenbud/Huneke/Vasconcelos. |
---|
[ebecf83] | 6397 | EXAMPLE: example prepareAss; shows an example |
---|
| 6398 | " |
---|
| 6399 | { |
---|
[d88470] | 6400 | if(attrib(basering,"global")!=1) |
---|
[07c623] | 6401 | { |
---|
[cb980ab] | 6402 | ERROR( |
---|
| 6403 | "// Not implemented for this ordering, please change to global ordering." |
---|
| 6404 | ); |
---|
[07c623] | 6405 | } |
---|
[ebecf83] | 6406 | ideal j=std(i); |
---|
[d950c5] | 6407 | int cod=nvars(basering)-dim(j); |
---|
[ebecf83] | 6408 | int e; |
---|
| 6409 | list er; |
---|
| 6410 | ideal ann; |
---|
| 6411 | if(homog(i)==1) |
---|
| 6412 | { |
---|
[a36e78] | 6413 | list re=sres(j,0); //the resolution |
---|
| 6414 | re=minres(re); //minimized resolution |
---|
[ebecf83] | 6415 | } |
---|
| 6416 | else |
---|
| 6417 | { |
---|
[3939bc] | 6418 | list re=mres(i,0); |
---|
| 6419 | } |
---|
[ebecf83] | 6420 | for(e=cod;e<=nvars(basering);e++) |
---|
| 6421 | { |
---|
[a36e78] | 6422 | ann=AnnExt_R(e,re); |
---|
[d950c5] | 6423 | |
---|
[a36e78] | 6424 | if(nvars(basering)-dim(std(ann))==e) |
---|
| 6425 | { |
---|
| 6426 | er[size(er)+1]=equiRadical(ann); |
---|
| 6427 | } |
---|
[ebecf83] | 6428 | } |
---|
| 6429 | return(er); |
---|
[3939bc] | 6430 | } |
---|
[ebecf83] | 6431 | example |
---|
| 6432 | { "EXAMPLE:"; echo = 2; |
---|
[07c623] | 6433 | ring r = 0,(x,y,z),dp; |
---|
[ebecf83] | 6434 | poly p = z2+1; |
---|
[07c623] | 6435 | poly q = z3+2; |
---|
| 6436 | ideal i = p*q^2,y-z2; |
---|
| 6437 | list pr = prepareAss(i); |
---|
[ebecf83] | 6438 | pr; |
---|
| 6439 | } |
---|
[03f29c] | 6440 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 6441 | proc equidimMaxEHV(ideal i) |
---|
[7f7c25e] | 6442 | "USAGE: equidimMaxEHV(I); I ideal |
---|
| 6443 | RETURN: ideal, the equidimensional component (of maximal dimension) of I. |
---|
[07c623] | 6444 | NOTE: Uses algorithm of Eisenbud, Huneke and Vasconcelos. |
---|
[03f29c] | 6445 | EXAMPLE: example equidimMaxEHV; shows an example |
---|
| 6446 | " |
---|
| 6447 | { |
---|
[cb980ab] | 6448 | if(attrib(basering,"global")!=1) |
---|
| 6449 | { |
---|
| 6450 | ERROR( |
---|
| 6451 | "// Not implemented for this ordering, please change to global ordering." |
---|
| 6452 | ); |
---|
| 6453 | } |
---|
[0ad359] | 6454 | ideal j=groebner(i); |
---|
[03f29c] | 6455 | int cod=nvars(basering)-dim(j); |
---|
| 6456 | int e; |
---|
| 6457 | ideal ann; |
---|
| 6458 | if(homog(i)==1) |
---|
| 6459 | { |
---|
[a36e78] | 6460 | list re=sres(j,0); //the resolution |
---|
| 6461 | re=minres(re); //minimized resolution |
---|
[03f29c] | 6462 | } |
---|
| 6463 | else |
---|
| 6464 | { |
---|
| 6465 | list re=mres(i,0); |
---|
| 6466 | } |
---|
| 6467 | ann=AnnExt_R(cod,re); |
---|
| 6468 | return(ann); |
---|
| 6469 | } |
---|
| 6470 | example |
---|
| 6471 | { "EXAMPLE:"; echo = 2; |
---|
[07c623] | 6472 | ring r = 0,(x,y,z),dp; |
---|
[03f29c] | 6473 | ideal i=intersect(ideal(z),ideal(x,y),ideal(x2,z2),ideal(x5,y5,z5)); |
---|
| 6474 | equidimMaxEHV(i); |
---|
| 6475 | } |
---|
[ebecf83] | 6476 | |
---|
[838d37] | 6477 | proc testPrimary(list pr, ideal k) |
---|
[7b3971] | 6478 | "USAGE: testPrimary(pr,k); pr a list, k an ideal. |
---|
| 6479 | ASSUME: pr is the result of primdecGTZ(k) or primdecSY(k). |
---|
| 6480 | RETURN: int, 1 if the intersection of the ideals in pr is k, 0 if not |
---|
[091424] | 6481 | EXAMPLE: example testPrimary; shows an example |
---|
[ebecf83] | 6482 | " |
---|
| 6483 | { |
---|
[a36e78] | 6484 | int i; |
---|
| 6485 | pr=reconvList(pr); |
---|
| 6486 | ideal j=pr[1]; |
---|
[4173c7] | 6487 | for (i=2;i<=size(pr) div 2;i++) |
---|
[a36e78] | 6488 | { |
---|
| 6489 | j=intersect(j,pr[2*i-1]); |
---|
| 6490 | } |
---|
| 6491 | return(idealsEqual(j,k)); |
---|
[ebecf83] | 6492 | } |
---|
| 6493 | example |
---|
| 6494 | { "EXAMPLE:"; echo = 2; |
---|
| 6495 | ring r = 32003,(x,y,z),dp; |
---|
| 6496 | poly p = z2+1; |
---|
| 6497 | poly q = z4+2; |
---|
| 6498 | ideal i = p^2*q^3,(y-z3)^3,(x-yz+z4)^4; |
---|
[091424] | 6499 | list pr = primdecGTZ(i); |
---|
[ebecf83] | 6500 | testPrimary(pr,i); |
---|
| 6501 | } |
---|
| 6502 | |
---|
[55fcff] | 6503 | /////////////////////////////////////////////////////////////////////////////// |
---|
[7f24dd7] | 6504 | proc zerodec(ideal I) |
---|
| 6505 | "USAGE: zerodec(I); I ideal |
---|
[7b3971] | 6506 | ASSUME: I is zero-dimensional, the characteristic of the ground field is 0 |
---|
| 6507 | RETURN: list of primary ideals, the zero-dimensional decomposition of I |
---|
| 6508 | NOTE: The algorithm (of Monico), works well only for a small total number |
---|
[367e88] | 6509 | of solutions (@code{vdim(std(I))} should be < 100) and without |
---|
| 6510 | parameters. In practice, it works also in large characteristic p>0 |
---|
[7b3971] | 6511 | but may fail for small p. |
---|
| 6512 | @* If printlevel > 0 (default = 0) additional information is displayed. |
---|
[55fcff] | 6513 | EXAMPLE: example zerodec; shows an example |
---|
[7f24dd7] | 6514 | " |
---|
| 6515 | { |
---|
[d88470] | 6516 | if(attrib(basering,"global")!=1) |
---|
| 6517 | { |
---|
[cb980ab] | 6518 | ERROR( |
---|
| 6519 | "// Not implemented for this ordering, please change to global ordering." |
---|
| 6520 | ); |
---|
[d88470] | 6521 | } |
---|
[cb980ab] | 6522 | def R=basering; |
---|
[d88470] | 6523 | poly q; |
---|
| 6524 | int j,time; |
---|
| 6525 | matrix m; |
---|
| 6526 | list re; |
---|
| 6527 | poly va=var(1); |
---|
| 6528 | ideal J=groebner(I); |
---|
| 6529 | ideal ba=kbase(J); |
---|
| 6530 | int d=vdim(J); |
---|
| 6531 | dbprint(printlevel-voice+2,"// multiplicity of ideal : "+ string(d)); |
---|
[55fcff] | 6532 | //------ compute matrix of multiplication on R/I with generic element p ----- |
---|
[d88470] | 6533 | int e=nvars(basering); |
---|
| 6534 | poly p=randomLast(100)[e]+random(-50,50); //the generic element |
---|
| 6535 | matrix n[d][d]; |
---|
| 6536 | time = timer; |
---|
| 6537 | for(j=2;j<=e;j++) |
---|
| 6538 | { |
---|
| 6539 | va=va*var(j); |
---|
| 6540 | } |
---|
| 6541 | for(j=1;j<=d;j++) |
---|
| 6542 | { |
---|
| 6543 | q=reduce(p*ba[j],J); |
---|
| 6544 | m=coeffs(q,ba,va); |
---|
| 6545 | n[j,1..d]=m[1..d,1]; |
---|
| 6546 | } |
---|
| 6547 | dbprint(printlevel-voice+2, |
---|
| 6548 | "// time for computing multiplication matrix (with generic element) : "+ |
---|
| 6549 | string(timer-time)); |
---|
[55fcff] | 6550 | //---------------- compute characteristic polynomial of matrix -------------- |
---|
[d88470] | 6551 | execute("ring P1=("+charstr(R)+"),T,dp;"); |
---|
| 6552 | matrix n=imap(R,n); |
---|
| 6553 | time = timer; |
---|
| 6554 | poly charpol=det(n-T*freemodule(d)); |
---|
| 6555 | dbprint(printlevel-voice+2,"// time for computing char poly: "+ |
---|
| 6556 | string(timer-time)); |
---|
[55fcff] | 6557 | //------------------- factorize characteristic polynomial ------------------- |
---|
[b9b906] | 6558 | //check first if constant term of charpoly is != 0 (which is true for |
---|
[55fcff] | 6559 | //sufficiently generic element) |
---|
[d88470] | 6560 | if(charpol[size(charpol)]!=0) |
---|
| 6561 | { |
---|
| 6562 | time = timer; |
---|
| 6563 | list fac=factor(charpol); |
---|
| 6564 | testFactor(fac,charpol); |
---|
| 6565 | dbprint(printlevel-voice+2,"// time for factorizing char poly: "+ |
---|
| 6566 | string(timer-time)); |
---|
| 6567 | int f=size(fac[1]); |
---|
[55fcff] | 6568 | //--------------------------- the irreducible case -------------------------- |
---|
[d88470] | 6569 | if(f==1) |
---|
| 6570 | { |
---|
| 6571 | setring R; |
---|
| 6572 | re=I; |
---|
| 6573 | return(re); |
---|
| 6574 | } |
---|
[55fcff] | 6575 | //---------------------------- the reducible case --------------------------- |
---|
[b9b906] | 6576 | //if f_i are the irreducible factors of charpoly, mult=ri, then <I,g_i^ri> |
---|
| 6577 | //are the primary components where g_i = f_i(p). However, substituting p in |
---|
| 6578 | //f_i may result in a huge object although the final result may be small. |
---|
| 6579 | //Hence it is better to simultaneously reduce with I. For this we need a new |
---|
[55fcff] | 6580 | //ring. |
---|
[d88470] | 6581 | execute("ring P=("+charstr(R)+"),(T,"+varstr(R)+"),(dp(1),dp);"); |
---|
| 6582 | list rfac=imap(P1,fac); |
---|
| 6583 | intvec ov=option(get);; |
---|
| 6584 | option(redSB); |
---|
| 6585 | list re1; |
---|
| 6586 | ideal new = T-imap(R,p),imap(R,J); |
---|
| 6587 | attrib(new, "isSB",1); //we know that new is a standard basis |
---|
| 6588 | for(j=1;j<=f;j++) |
---|
| 6589 | { |
---|
| 6590 | re1[j]=reduce(rfac[1][j]^rfac[2][j],new); |
---|
| 6591 | } |
---|
| 6592 | setring R; |
---|
| 6593 | re = imap(P,re1); |
---|
| 6594 | for(j=1;j<=f;j++) |
---|
| 6595 | { |
---|
| 6596 | J=I,re[j]; |
---|
| 6597 | re[j]=interred(J); |
---|
| 6598 | } |
---|
| 6599 | option(set,ov); |
---|
| 6600 | return(re); |
---|
[7f24dd7] | 6601 | } |
---|
| 6602 | else |
---|
[55fcff] | 6603 | //------------------- choice of generic element failed ------------------- |
---|
[7f24dd7] | 6604 | { |
---|
[d88470] | 6605 | dbprint(printlevel-voice+2,"// try new generic element!"); |
---|
| 6606 | setring R; |
---|
| 6607 | return(zerodec(I)); |
---|
[7f24dd7] | 6608 | } |
---|
| 6609 | } |
---|
| 6610 | example |
---|
| 6611 | { "EXAMPLE:"; echo = 2; |
---|
[07c623] | 6612 | ring r = 0,(x,y),dp; |
---|
| 6613 | ideal i = x2-2,y2-2; |
---|
| 6614 | list pr = zerodec(i); |
---|
[7f24dd7] | 6615 | pr; |
---|
| 6616 | } |
---|
[808a9f3] | 6617 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 6618 | static proc newDecompStep(ideal i, list #) |
---|
| 6619 | "USAGE: newDecompStep(i); i ideal (for primary decomposition) |
---|
| 6620 | newDecompStep(i,1); (for the associated primes of dimension of i) |
---|
| 6621 | newDecompStep(i,2); (for the minimal associated primes) |
---|
[f995aa] | 6622 | newDecompStep(i,3); (for the absolute primary decomposition (not tested!)) |
---|
[808a9f3] | 6623 | "oneIndep"; (for using only one max indep set) |
---|
| 6624 | "intersect"; (returns alse the intersection of the components founded) |
---|
| 6625 | |
---|
| 6626 | RETURN: list = list of primary ideals and their associated primes |
---|
| 6627 | (at even positions in the list) |
---|
| 6628 | (resp. a list of the minimal associated primes) |
---|
| 6629 | NOTE: Algorithm of Gianni/Trager/Zacharias |
---|
| 6630 | EXAMPLE: example newDecompStep; shows an example |
---|
| 6631 | " |
---|
| 6632 | { |
---|
| 6633 | intvec op,@vv; |
---|
| 6634 | def @P = basering; |
---|
| 6635 | list primary,indep,ltras; |
---|
| 6636 | intvec @vh,isat,@w; |
---|
| 6637 | int @wr,@k,@n,@m,@n1,@n2,@n3,homo,seri,keepdi,abspri,ab,nn; |
---|
| 6638 | ideal peek=i; |
---|
| 6639 | ideal ser,tras; |
---|
| 6640 | list data; |
---|
| 6641 | list result; |
---|
| 6642 | intvec @hilb; |
---|
| 6643 | int isS=(attrib(i,"isSB")==1); |
---|
| 6644 | |
---|
| 6645 | // Debug |
---|
| 6646 | dbprint(printlevel - voice, "newDecompStep, v2.0"); |
---|
| 6647 | |
---|
| 6648 | string indepOption = "allIndep"; |
---|
| 6649 | string intersectOption = "noIntersect"; |
---|
| 6650 | |
---|
| 6651 | if(size(#)>0) |
---|
| 6652 | { |
---|
[70ab73] | 6653 | int count = 1; |
---|
| 6654 | if(typeof(#[count]) == "string") |
---|
| 6655 | { |
---|
| 6656 | if ((#[count] == "oneIndep") or (#[count] == "allIndep")) |
---|
| 6657 | { |
---|
| 6658 | indepOption = #[count]; |
---|
| 6659 | count++; |
---|
[7f7c25e] | 6660 | } |
---|
[70ab73] | 6661 | } |
---|
| 6662 | if(typeof(#[count]) == "string") |
---|
| 6663 | { |
---|
| 6664 | if ((#[count] == "intersect") or (#[count] == "noIntersect")) |
---|
[7f7c25e] | 6665 | { |
---|
[70ab73] | 6666 | intersectOption = #[count]; |
---|
| 6667 | count++; |
---|
[7f7c25e] | 6668 | } |
---|
[70ab73] | 6669 | } |
---|
| 6670 | if((typeof(#[count]) == "int") or (typeof(#[count]) == "number")) |
---|
| 6671 | { |
---|
| 6672 | if ((#[count]==1)||(#[count]==2)||(#[count]==3)) |
---|
[7f7c25e] | 6673 | { |
---|
[70ab73] | 6674 | @wr=#[count]; |
---|
| 6675 | if(@wr==3){abspri = 1; @wr = 0;} |
---|
| 6676 | count++; |
---|
[7f7c25e] | 6677 | } |
---|
[70ab73] | 6678 | } |
---|
| 6679 | if(size(#)>count) |
---|
| 6680 | { |
---|
| 6681 | seri=1; |
---|
| 6682 | peek=#[count + 1]; |
---|
| 6683 | ser=#[count + 2]; |
---|
| 6684 | } |
---|
| 6685 | } |
---|
| 6686 | if(abspri) |
---|
| 6687 | { |
---|
| 6688 | list absprimary,abskeep,absprimarytmp,abskeeptmp; |
---|
| 6689 | } |
---|
| 6690 | homo=homog(i); |
---|
| 6691 | if(homo==1) |
---|
| 6692 | { |
---|
| 6693 | if(attrib(i,"isSB")!=1) |
---|
| 6694 | { |
---|
| 6695 | //ltras=mstd(i); |
---|
| 6696 | tras=groebner(i); |
---|
| 6697 | ltras=tras,tras; |
---|
| 6698 | attrib(ltras[1],"isSB",1); |
---|
| 6699 | } |
---|
| 6700 | else |
---|
| 6701 | { |
---|
| 6702 | ltras=i,i; |
---|
| 6703 | attrib(ltras[1],"isSB",1); |
---|
| 6704 | } |
---|
| 6705 | tras = ltras[1]; |
---|
| 6706 | attrib(tras,"isSB",1); |
---|
| 6707 | if(dim(tras)==0) |
---|
| 6708 | { |
---|
| 6709 | primary[1]=ltras[2]; |
---|
| 6710 | primary[2]=maxideal(1); |
---|
| 6711 | if(@wr>0) |
---|
| 6712 | { |
---|
| 6713 | list l; |
---|
| 6714 | l[2]=maxideal(1); |
---|
| 6715 | l[1]=maxideal(1); |
---|
| 6716 | if (intersectOption == "intersect") |
---|
[808a9f3] | 6717 | { |
---|
[70ab73] | 6718 | return(list(l, maxideal(1))); |
---|
[808a9f3] | 6719 | } |
---|
[70ab73] | 6720 | else |
---|
| 6721 | { |
---|
| 6722 | return(l); |
---|
[7f7c25e] | 6723 | } |
---|
[70ab73] | 6724 | } |
---|
| 6725 | if (intersectOption == "intersect") |
---|
| 6726 | { |
---|
| 6727 | return(list(primary, primary[1])); |
---|
| 6728 | } |
---|
| 6729 | else |
---|
| 6730 | { |
---|
| 6731 | return(primary); |
---|
| 6732 | } |
---|
| 6733 | } |
---|
| 6734 | for(@n=1;@n<=nvars(basering);@n++) |
---|
| 6735 | { |
---|
| 6736 | @w[@n]=ord(var(@n)); |
---|
| 6737 | } |
---|
| 6738 | @hilb=hilb(tras,1,@w); |
---|
| 6739 | intvec keephilb=@hilb; |
---|
[808a9f3] | 6740 | } |
---|
| 6741 | |
---|
| 6742 | //---------------------------------------------------------------- |
---|
| 6743 | //i is the zero-ideal |
---|
| 6744 | //---------------------------------------------------------------- |
---|
| 6745 | |
---|
| 6746 | if(size(i)==0) |
---|
| 6747 | { |
---|
[7f7c25e] | 6748 | primary=i,i; |
---|
[70ab73] | 6749 | if (intersectOption == "intersect") |
---|
| 6750 | { |
---|
| 6751 | return(list(primary, i)); |
---|
| 6752 | } |
---|
| 6753 | else |
---|
| 6754 | { |
---|
| 6755 | return(primary); |
---|
[7f7c25e] | 6756 | } |
---|
[808a9f3] | 6757 | } |
---|
| 6758 | |
---|
| 6759 | //---------------------------------------------------------------- |
---|
| 6760 | //pass to the lexicographical ordering and compute a standardbasis |
---|
| 6761 | //---------------------------------------------------------------- |
---|
| 6762 | |
---|
| 6763 | int lp=islp(); |
---|
| 6764 | |
---|
| 6765 | execute("ring gnir = ("+charstr(basering)+"),("+varstr(basering)+"),(C,lp);"); |
---|
| 6766 | op=option(get); |
---|
| 6767 | option(redSB); |
---|
| 6768 | |
---|
| 6769 | ideal ser=fetch(@P,ser); |
---|
| 6770 | if(homo==1) |
---|
| 6771 | { |
---|
[70ab73] | 6772 | if(!lp) |
---|
| 6773 | { |
---|
| 6774 | ideal @j=std(fetch(@P,i),@hilb,@w); |
---|
| 6775 | } |
---|
| 6776 | else |
---|
| 6777 | { |
---|
| 6778 | ideal @j=fetch(@P,tras); |
---|
| 6779 | attrib(@j,"isSB",1); |
---|
| 6780 | } |
---|
[808a9f3] | 6781 | } |
---|
| 6782 | else |
---|
| 6783 | { |
---|
[70ab73] | 6784 | if(lp&&isS) |
---|
| 6785 | { |
---|
| 6786 | ideal @j=fetch(@P,i); |
---|
| 6787 | attrib(@j,"isSB",1); |
---|
| 6788 | } |
---|
| 6789 | else |
---|
| 6790 | { |
---|
| 6791 | ideal @j=groebner(fetch(@P,i)); |
---|
| 6792 | } |
---|
[808a9f3] | 6793 | } |
---|
| 6794 | option(set,op); |
---|
| 6795 | if(seri==1) |
---|
| 6796 | { |
---|
| 6797 | ideal peek=fetch(@P,peek); |
---|
| 6798 | attrib(peek,"isSB",1); |
---|
| 6799 | } |
---|
| 6800 | else |
---|
| 6801 | { |
---|
| 6802 | ideal peek=@j; |
---|
| 6803 | } |
---|
| 6804 | if((size(ser)==0)&&(!abspri)) |
---|
| 6805 | { |
---|
| 6806 | ideal fried; |
---|
| 6807 | @n=size(@j); |
---|
| 6808 | for(@k=1;@k<=@n;@k++) |
---|
| 6809 | { |
---|
| 6810 | if(deg(lead(@j[@k]))==1) |
---|
| 6811 | { |
---|
| 6812 | fried[size(fried)+1]=@j[@k]; |
---|
| 6813 | @j[@k]=0; |
---|
| 6814 | } |
---|
| 6815 | } |
---|
| 6816 | if(size(fried)==nvars(basering)) |
---|
| 6817 | { |
---|
[70ab73] | 6818 | setring @P; |
---|
| 6819 | primary[1]=i; |
---|
| 6820 | primary[2]=i; |
---|
| 6821 | if (intersectOption == "intersect") |
---|
| 6822 | { |
---|
| 6823 | return(list(primary, i)); |
---|
| 6824 | } |
---|
| 6825 | else |
---|
| 6826 | { |
---|
| 6827 | return(primary); |
---|
| 6828 | } |
---|
[808a9f3] | 6829 | } |
---|
| 6830 | if(size(fried)>0) |
---|
| 6831 | { |
---|
[70ab73] | 6832 | string newva; |
---|
| 6833 | string newma; |
---|
| 6834 | for(@k=1;@k<=nvars(basering);@k++) |
---|
| 6835 | { |
---|
| 6836 | @n1=0; |
---|
| 6837 | for(@n=1;@n<=size(fried);@n++) |
---|
| 6838 | { |
---|
| 6839 | if(leadmonom(fried[@n])==var(@k)) |
---|
[808a9f3] | 6840 | { |
---|
[70ab73] | 6841 | @n1=1; |
---|
| 6842 | break; |
---|
[808a9f3] | 6843 | } |
---|
[70ab73] | 6844 | } |
---|
| 6845 | if(@n1==0) |
---|
| 6846 | { |
---|
| 6847 | newva=newva+string(var(@k))+","; |
---|
| 6848 | newma=newma+string(var(@k))+","; |
---|
| 6849 | } |
---|
| 6850 | else |
---|
| 6851 | { |
---|
| 6852 | newma=newma+string(0)+","; |
---|
| 6853 | } |
---|
| 6854 | } |
---|
| 6855 | newva[size(newva)]=")"; |
---|
| 6856 | newma[size(newma)]=";"; |
---|
| 6857 | execute("ring @deirf=("+charstr(gnir)+"),("+newva+",lp;"); |
---|
| 6858 | execute("map @kappa=gnir,"+newma); |
---|
| 6859 | ideal @j= @kappa(@j); |
---|
| 6860 | @j=simplify(@j, 2); |
---|
| 6861 | attrib(@j,"isSB",1); |
---|
| 6862 | result = newDecompStep(@j, indepOption, intersectOption, @wr); |
---|
| 6863 | if (intersectOption == "intersect") |
---|
| 6864 | { |
---|
[a36e78] | 6865 | list pr = result[1]; |
---|
| 6866 | ideal intersection = result[2]; |
---|
[70ab73] | 6867 | } |
---|
| 6868 | else |
---|
| 6869 | { |
---|
| 6870 | list pr = result; |
---|
| 6871 | } |
---|
[808a9f3] | 6872 | |
---|
[70ab73] | 6873 | setring gnir; |
---|
| 6874 | list pr=imap(@deirf,pr); |
---|
| 6875 | for(@k=1;@k<=size(pr);@k++) |
---|
| 6876 | { |
---|
| 6877 | @j=pr[@k]+fried; |
---|
| 6878 | pr[@k]=@j; |
---|
| 6879 | } |
---|
| 6880 | if (intersectOption == "intersect") |
---|
| 6881 | { |
---|
| 6882 | ideal intersection = imap(@deirf, intersection); |
---|
| 6883 | @j = intersection + fried; |
---|
| 6884 | intersection = @j; |
---|
| 6885 | } |
---|
| 6886 | setring @P; |
---|
| 6887 | if (intersectOption == "intersect") |
---|
| 6888 | { |
---|
| 6889 | return(list(imap(gnir,pr), imap(gnir,intersection))); |
---|
| 6890 | } |
---|
| 6891 | else |
---|
| 6892 | { |
---|
| 6893 | return(imap(gnir,pr)); |
---|
| 6894 | } |
---|
[808a9f3] | 6895 | } |
---|
| 6896 | } |
---|
| 6897 | //---------------------------------------------------------------- |
---|
| 6898 | //j is the ring |
---|
| 6899 | //---------------------------------------------------------------- |
---|
| 6900 | |
---|
| 6901 | if (dim(@j)==-1) |
---|
| 6902 | { |
---|
| 6903 | setring @P; |
---|
| 6904 | primary=ideal(1),ideal(1); |
---|
[70ab73] | 6905 | if (intersectOption == "intersect") |
---|
| 6906 | { |
---|
[7f7c25e] | 6907 | return(list(primary, ideal(1))); |
---|
[70ab73] | 6908 | } |
---|
| 6909 | else |
---|
| 6910 | { |
---|
[7f7c25e] | 6911 | return(primary); |
---|
| 6912 | } |
---|
[808a9f3] | 6913 | } |
---|
| 6914 | |
---|
| 6915 | //---------------------------------------------------------------- |
---|
| 6916 | // the case of one variable |
---|
| 6917 | //---------------------------------------------------------------- |
---|
| 6918 | |
---|
| 6919 | if(nvars(basering)==1) |
---|
| 6920 | { |
---|
[70ab73] | 6921 | list fac=factor(@j[1]); |
---|
| 6922 | list gprimary; |
---|
| 6923 | poly generator; |
---|
| 6924 | ideal gIntersection; |
---|
| 6925 | for(@k=1;@k<=size(fac[1]);@k++) |
---|
| 6926 | { |
---|
| 6927 | if(@wr==0) |
---|
| 6928 | { |
---|
| 6929 | gprimary[2*@k-1]=ideal(fac[1][@k]^fac[2][@k]); |
---|
| 6930 | gprimary[2*@k]=ideal(fac[1][@k]); |
---|
| 6931 | } |
---|
| 6932 | else |
---|
| 6933 | { |
---|
| 6934 | gprimary[2*@k-1]=ideal(fac[1][@k]); |
---|
| 6935 | gprimary[2*@k]=ideal(fac[1][@k]); |
---|
| 6936 | } |
---|
| 6937 | if (intersectOption == "intersect") |
---|
| 6938 | { |
---|
| 6939 | generator = generator * fac[1][@k]; |
---|
| 6940 | } |
---|
| 6941 | } |
---|
| 6942 | if (intersectOption == "intersect") |
---|
| 6943 | { |
---|
| 6944 | gIntersection = generator; |
---|
| 6945 | } |
---|
| 6946 | setring @P; |
---|
| 6947 | primary=fetch(gnir,gprimary); |
---|
| 6948 | if (intersectOption == "intersect") |
---|
| 6949 | { |
---|
| 6950 | ideal intersection = fetch(gnir,gIntersection); |
---|
| 6951 | } |
---|
[808a9f3] | 6952 | |
---|
| 6953 | //HIER |
---|
| 6954 | if(abspri) |
---|
| 6955 | { |
---|
[70ab73] | 6956 | list resu,tempo; |
---|
| 6957 | string absotto; |
---|
[4173c7] | 6958 | for(ab=1;ab<=size(primary) div 2;ab++) |
---|
[70ab73] | 6959 | { |
---|
| 6960 | absotto= absFactorize(primary[2*ab][1],77); |
---|
| 6961 | tempo=primary[2*ab-1],primary[2*ab],absotto,string(var(nvars(basering))); |
---|
| 6962 | resu[ab]=tempo; |
---|
| 6963 | } |
---|
| 6964 | primary=resu; |
---|
| 6965 | intersection = 1; |
---|
| 6966 | for(ab=1;ab<=size(primary);ab++) |
---|
| 6967 | { |
---|
| 6968 | intersection = intersect(intersection, primary[ab][2]); |
---|
| 6969 | } |
---|
| 6970 | } |
---|
| 6971 | if (intersectOption == "intersect") |
---|
| 6972 | { |
---|
| 6973 | return(list(primary, intersection)); |
---|
| 6974 | } |
---|
| 6975 | else |
---|
| 6976 | { |
---|
| 6977 | return(primary); |
---|
| 6978 | } |
---|
[808a9f3] | 6979 | } |
---|
| 6980 | |
---|
| 6981 | //------------------------------------------------------------------ |
---|
| 6982 | //the zero-dimensional case |
---|
| 6983 | //------------------------------------------------------------------ |
---|
| 6984 | if (dim(@j)==0) |
---|
| 6985 | { |
---|
| 6986 | op=option(get); |
---|
| 6987 | option(redSB); |
---|
| 6988 | list gprimary= newZero_decomp(@j,ser,@wr); |
---|
| 6989 | |
---|
| 6990 | setring @P; |
---|
| 6991 | primary=fetch(gnir,gprimary); |
---|
| 6992 | |
---|
| 6993 | if(size(ser)>0) |
---|
| 6994 | { |
---|
| 6995 | primary=cleanPrimary(primary); |
---|
| 6996 | } |
---|
| 6997 | //HIER |
---|
| 6998 | if(abspri) |
---|
| 6999 | { |
---|
[70ab73] | 7000 | list resu,tempo; |
---|
| 7001 | string absotto; |
---|
[4173c7] | 7002 | for(ab=1;ab<=size(primary) div 2;ab++) |
---|
[70ab73] | 7003 | { |
---|
| 7004 | absotto= absFactorize(primary[2*ab][1],77); |
---|
| 7005 | tempo=primary[2*ab-1],primary[2*ab],absotto,string(var(nvars(basering))); |
---|
| 7006 | resu[ab]=tempo; |
---|
| 7007 | } |
---|
| 7008 | primary=resu; |
---|
[808a9f3] | 7009 | } |
---|
[70ab73] | 7010 | if (intersectOption == "intersect") |
---|
| 7011 | { |
---|
[7f7c25e] | 7012 | return(list(primary, fetch(gnir,@j))); |
---|
[70ab73] | 7013 | } |
---|
| 7014 | else |
---|
| 7015 | { |
---|
[7f7c25e] | 7016 | return(primary); |
---|
| 7017 | } |
---|
[808a9f3] | 7018 | } |
---|
| 7019 | |
---|
| 7020 | poly @gs,@gh,@p; |
---|
| 7021 | string @va,quotring; |
---|
| 7022 | list quprimary,htprimary,collectprimary,lsau,lnew,allindep,restindep; |
---|
| 7023 | ideal @h; |
---|
| 7024 | int jdim=dim(@j); |
---|
| 7025 | list fett; |
---|
| 7026 | int lauf,di,newtest; |
---|
| 7027 | //------------------------------------------------------------------ |
---|
| 7028 | //search for a maximal independent set indep,i.e. |
---|
| 7029 | //look for subring such that the intersection with the ideal is zero |
---|
| 7030 | //j intersected with K[var(indep[3]+1),...,var(nvar] is zero, |
---|
| 7031 | //indep[1] is the new varstring and indep[2] the string for block-ordering |
---|
| 7032 | //------------------------------------------------------------------ |
---|
| 7033 | if(@wr!=1) |
---|
| 7034 | { |
---|
[70ab73] | 7035 | allindep = newMaxIndependSetLp(@j, indepOption); |
---|
| 7036 | for(@m=1;@m<=size(allindep);@m++) |
---|
| 7037 | { |
---|
| 7038 | if(allindep[@m][3]==jdim) |
---|
| 7039 | { |
---|
| 7040 | di++; |
---|
| 7041 | indep[di]=allindep[@m]; |
---|
| 7042 | } |
---|
| 7043 | else |
---|
| 7044 | { |
---|
| 7045 | lauf++; |
---|
| 7046 | restindep[lauf]=allindep[@m]; |
---|
| 7047 | } |
---|
| 7048 | } |
---|
| 7049 | } |
---|
| 7050 | else |
---|
| 7051 | { |
---|
| 7052 | indep = newMaxIndependSetLp(@j, indepOption); |
---|
| 7053 | } |
---|
[808a9f3] | 7054 | |
---|
| 7055 | ideal jkeep=@j; |
---|
| 7056 | if(ordstr(@P)[1]=="w") |
---|
| 7057 | { |
---|
[70ab73] | 7058 | execute("ring @Phelp=("+charstr(gnir)+"),("+varstr(gnir)+"),("+ordstr(@P)+");"); |
---|
[808a9f3] | 7059 | } |
---|
| 7060 | else |
---|
| 7061 | { |
---|
[70ab73] | 7062 | execute( "ring @Phelp=("+charstr(gnir)+"),("+varstr(gnir)+"),(C,dp);"); |
---|
[808a9f3] | 7063 | } |
---|
| 7064 | |
---|
| 7065 | if(homo==1) |
---|
| 7066 | { |
---|
| 7067 | if((ordstr(@P)[3]=="d")||(ordstr(@P)[1]=="d")||(ordstr(@P)[1]=="w") |
---|
| 7068 | ||(ordstr(@P)[3]=="w")) |
---|
| 7069 | { |
---|
| 7070 | ideal jwork=imap(@P,tras); |
---|
| 7071 | attrib(jwork,"isSB",1); |
---|
| 7072 | } |
---|
| 7073 | else |
---|
| 7074 | { |
---|
| 7075 | ideal jwork=std(imap(gnir,@j),@hilb,@w); |
---|
| 7076 | } |
---|
| 7077 | } |
---|
| 7078 | else |
---|
| 7079 | { |
---|
| 7080 | ideal jwork=groebner(imap(gnir,@j)); |
---|
| 7081 | } |
---|
| 7082 | list hquprimary; |
---|
| 7083 | poly @p,@q; |
---|
| 7084 | ideal @h,fac,ser; |
---|
| 7085 | //Aenderung================ |
---|
| 7086 | ideal @Ptest=1; |
---|
| 7087 | //========================= |
---|
| 7088 | di=dim(jwork); |
---|
| 7089 | keepdi=di; |
---|
| 7090 | |
---|
| 7091 | ser = 1; |
---|
| 7092 | |
---|
| 7093 | setring gnir; |
---|
| 7094 | for(@m=1; @m<=size(indep); @m++) |
---|
| 7095 | { |
---|
| 7096 | data[1] = indep[@m]; |
---|
| 7097 | result = newReduction(@j, ser, @hilb, @w, jdim, abspri, @wr, data); |
---|
| 7098 | quprimary = quprimary + result[1]; |
---|
[70ab73] | 7099 | if(abspri) |
---|
| 7100 | { |
---|
[808a9f3] | 7101 | absprimary = absprimary + result[2]; |
---|
| 7102 | abskeep = abskeep + result[3]; |
---|
[7f7c25e] | 7103 | } |
---|
[808a9f3] | 7104 | @h = result[5]; |
---|
| 7105 | ser = result[4]; |
---|
[7f7c25e] | 7106 | if(size(@h)>0) |
---|
| 7107 | { |
---|
[70ab73] | 7108 | //--------------------------------------------------------------- |
---|
| 7109 | //we change to @Phelp to have the ordering dp for saturation |
---|
| 7110 | //--------------------------------------------------------------- |
---|
[808a9f3] | 7111 | |
---|
[70ab73] | 7112 | setring @Phelp; |
---|
| 7113 | @h=imap(gnir,@h); |
---|
[808a9f3] | 7114 | //Aenderung================================== |
---|
[70ab73] | 7115 | if(defined(@LL)){kill @LL;} |
---|
| 7116 | list @LL=minSat(jwork,@h); |
---|
| 7117 | @Ptest=intersect(@Ptest,@LL[1]); |
---|
| 7118 | ser = intersect(ser, @LL[1]); |
---|
[808a9f3] | 7119 | //=========================================== |
---|
| 7120 | |
---|
[70ab73] | 7121 | if(@wr!=1) |
---|
| 7122 | { |
---|
[808a9f3] | 7123 | //Aenderung================================== |
---|
[70ab73] | 7124 | @q=@LL[2]; |
---|
[808a9f3] | 7125 | //=========================================== |
---|
[70ab73] | 7126 | //@q=minSat(jwork,@h)[2]; |
---|
| 7127 | } |
---|
| 7128 | else |
---|
| 7129 | { |
---|
| 7130 | fac=ideal(0); |
---|
| 7131 | for(lauf=1;lauf<=ncols(@h);lauf++) |
---|
[808a9f3] | 7132 | { |
---|
[70ab73] | 7133 | if(deg(@h[lauf])>0) |
---|
| 7134 | { |
---|
| 7135 | fac=fac+factorize(@h[lauf],1); |
---|
| 7136 | } |
---|
[808a9f3] | 7137 | } |
---|
[70ab73] | 7138 | fac=simplify(fac,6); |
---|
| 7139 | @q=1; |
---|
| 7140 | for(lauf=1;lauf<=size(fac);lauf++) |
---|
[808a9f3] | 7141 | { |
---|
[70ab73] | 7142 | @q=@q*fac[lauf]; |
---|
[808a9f3] | 7143 | } |
---|
[70ab73] | 7144 | } |
---|
| 7145 | jwork = std(jwork,@q); |
---|
| 7146 | keepdi = dim(jwork); |
---|
| 7147 | if(keepdi < di) |
---|
| 7148 | { |
---|
[808a9f3] | 7149 | setring gnir; |
---|
| 7150 | @j = imap(@Phelp, jwork); |
---|
[70ab73] | 7151 | ser = imap(@Phelp, ser); |
---|
| 7152 | break; |
---|
| 7153 | } |
---|
| 7154 | if(homo == 1) |
---|
| 7155 | { |
---|
| 7156 | @hilb = hilb(jwork, 1, @w); |
---|
| 7157 | } |
---|
| 7158 | |
---|
| 7159 | setring gnir; |
---|
| 7160 | ser = imap(@Phelp, ser); |
---|
| 7161 | @j = imap(@Phelp, jwork); |
---|
| 7162 | } |
---|
[808a9f3] | 7163 | } |
---|
| 7164 | |
---|
| 7165 | if((size(quprimary)==0)&&(@wr==1)) |
---|
| 7166 | { |
---|
[a36e78] | 7167 | @j=ideal(1); |
---|
| 7168 | quprimary[1]=ideal(1); |
---|
| 7169 | quprimary[2]=ideal(1); |
---|
[808a9f3] | 7170 | } |
---|
| 7171 | if((size(quprimary)==0)) |
---|
| 7172 | { |
---|
| 7173 | keepdi = di - 1; |
---|
| 7174 | quprimary[1]=ideal(1); |
---|
| 7175 | quprimary[2]=ideal(1); |
---|
| 7176 | } |
---|
| 7177 | //--------------------------------------------------------------- |
---|
| 7178 | //notice that j=sat(j,gh) intersected with (j,gh^n) |
---|
| 7179 | //we finished with sat(j,gh) and have to start with (j,gh^n) |
---|
| 7180 | //--------------------------------------------------------------- |
---|
| 7181 | if((deg(@j[1])!=0)&&(@wr!=1)) |
---|
| 7182 | { |
---|
[a36e78] | 7183 | if(size(quprimary)>0) |
---|
| 7184 | { |
---|
| 7185 | setring @Phelp; |
---|
| 7186 | ser=imap(gnir,ser); |
---|
[808a9f3] | 7187 | |
---|
[a36e78] | 7188 | hquprimary=imap(gnir,quprimary); |
---|
| 7189 | if(@wr==0) |
---|
| 7190 | { |
---|
[808a9f3] | 7191 | //Aenderung==================================================== |
---|
| 7192 | //HIER STATT DURCHSCHNITT SATURIEREN! |
---|
[a36e78] | 7193 | ideal htest=@Ptest; |
---|
[808a9f3] | 7194 | /* |
---|
[a36e78] | 7195 | ideal htest=hquprimary[1]; |
---|
[4173c7] | 7196 | for (@n1=2;@n1<=size(hquprimary) div 2;@n1++) |
---|
[a36e78] | 7197 | { |
---|
| 7198 | htest=intersect(htest,hquprimary[2*@n1-1]); |
---|
| 7199 | } |
---|
[808a9f3] | 7200 | */ |
---|
| 7201 | //============================================================= |
---|
| 7202 | } |
---|
| 7203 | else |
---|
| 7204 | { |
---|
[a36e78] | 7205 | ideal htest=hquprimary[2]; |
---|
[808a9f3] | 7206 | |
---|
[4173c7] | 7207 | for (@n1=2;@n1<=size(hquprimary) div 2;@n1++) |
---|
[a36e78] | 7208 | { |
---|
| 7209 | htest=intersect(htest,hquprimary[2*@n1]); |
---|
| 7210 | } |
---|
[808a9f3] | 7211 | } |
---|
[70ab73] | 7212 | |
---|
[a36e78] | 7213 | if(size(ser)>0) |
---|
[808a9f3] | 7214 | { |
---|
[a36e78] | 7215 | ser=intersect(htest,ser); |
---|
[808a9f3] | 7216 | } |
---|
[a36e78] | 7217 | else |
---|
[808a9f3] | 7218 | { |
---|
[a36e78] | 7219 | ser=htest; |
---|
[70ab73] | 7220 | } |
---|
| 7221 | setring gnir; |
---|
[a36e78] | 7222 | ser=imap(@Phelp,ser); |
---|
| 7223 | } |
---|
| 7224 | if(size(reduce(ser,peek,1))!=0) |
---|
| 7225 | { |
---|
| 7226 | for(@m=1;@m<=size(restindep);@m++) |
---|
| 7227 | { |
---|
| 7228 | // if(restindep[@m][3]>=keepdi) |
---|
| 7229 | // { |
---|
| 7230 | isat=0; |
---|
| 7231 | @n2=0; |
---|
| 7232 | |
---|
| 7233 | if(restindep[@m][1]==varstr(basering)) |
---|
| 7234 | //the good case, nothing to do, just to have the same notations |
---|
| 7235 | //change the ring |
---|
| 7236 | { |
---|
| 7237 | execute("ring gnir1 = ("+charstr(basering)+"),("+ |
---|
| 7238 | varstr(basering)+"),("+ordstr(basering)+");"); |
---|
| 7239 | ideal @j=fetch(gnir,jkeep); |
---|
| 7240 | attrib(@j,"isSB",1); |
---|
| 7241 | } |
---|
| 7242 | else |
---|
| 7243 | { |
---|
| 7244 | @va=string(maxideal(1)); |
---|
| 7245 | execute("ring gnir1 = ("+charstr(basering)+"),("+ |
---|
| 7246 | restindep[@m][1]+"),(" +restindep[@m][2]+");"); |
---|
| 7247 | execute("map phi=gnir,"+@va+";"); |
---|
| 7248 | op=option(get); |
---|
| 7249 | option(redSB); |
---|
| 7250 | if(homo==1) |
---|
| 7251 | { |
---|
| 7252 | ideal @j=std(phi(jkeep),keephilb,@w); |
---|
| 7253 | } |
---|
| 7254 | else |
---|
| 7255 | { |
---|
| 7256 | ideal @j=groebner(phi(jkeep)); |
---|
| 7257 | } |
---|
| 7258 | ideal ser=phi(ser); |
---|
| 7259 | option(set,op); |
---|
| 7260 | } |
---|
| 7261 | |
---|
| 7262 | for (lauf=1;lauf<=size(@j);lauf++) |
---|
| 7263 | { |
---|
| 7264 | fett[lauf]=size(@j[lauf]); |
---|
| 7265 | } |
---|
| 7266 | //------------------------------------------------------------------ |
---|
| 7267 | //we have now the following situation: |
---|
| 7268 | //j intersected with K[var(nnp+1),..,var(nva)] is zero so we may |
---|
| 7269 | //pass to this quotientring, j is their still a standardbasis, the |
---|
| 7270 | //leading coefficients of the polynomials there (polynomials in |
---|
| 7271 | //K[var(nnp+1),..,var(nva)]) are collected in the list h, |
---|
| 7272 | //we need their ggt, gh, because of the following: |
---|
| 7273 | //let (j:gh^n)=(j:gh^infinity) then |
---|
| 7274 | //j*K(var(nnp+1),..,var(nva))[..the rest..] |
---|
| 7275 | //intersected with K[var(1),...,var(nva)] is (j:gh^n) |
---|
| 7276 | //on the other hand j=(j,gh^n) intersected with (j:gh^n) |
---|
| 7277 | |
---|
| 7278 | //------------------------------------------------------------------ |
---|
| 7279 | |
---|
| 7280 | //the arrangement for the quotientring |
---|
| 7281 | // K(var(nnp+1),..,var(nva))[..the rest..] |
---|
| 7282 | //and the map phi:K[var(1),...,var(nva)] ----> |
---|
| 7283 | //--->K(var(nnpr+1),..,var(nva))[..the rest..] |
---|
| 7284 | //------------------------------------------------------------------ |
---|
| 7285 | |
---|
| 7286 | quotring=prepareQuotientring(nvars(basering)-restindep[@m][3]); |
---|
| 7287 | |
---|
| 7288 | //------------------------------------------------------------------ |
---|
| 7289 | //we pass to the quotientring K(var(nnp+1),..,var(nva))[..rest..] |
---|
| 7290 | //------------------------------------------------------------------ |
---|
| 7291 | |
---|
| 7292 | execute(quotring); |
---|
| 7293 | |
---|
| 7294 | // @j considered in the quotientring |
---|
| 7295 | ideal @j=imap(gnir1,@j); |
---|
| 7296 | ideal ser=imap(gnir1,ser); |
---|
| 7297 | |
---|
| 7298 | kill gnir1; |
---|
| 7299 | |
---|
| 7300 | //j is a standardbasis in the quotientring but usually not minimal |
---|
| 7301 | //here it becomes minimal |
---|
| 7302 | @j=clearSB(@j,fett); |
---|
| 7303 | attrib(@j,"isSB",1); |
---|
| 7304 | |
---|
| 7305 | //we need later ggt(h[1],...)=gh for saturation |
---|
| 7306 | ideal @h; |
---|
| 7307 | |
---|
| 7308 | for(@n=1;@n<=size(@j);@n++) |
---|
| 7309 | { |
---|
| 7310 | @h[@n]=leadcoef(@j[@n]); |
---|
| 7311 | } |
---|
| 7312 | //the primary decomposition of j*K(var(nnp+1),..,var(nva))[..rest..] |
---|
| 7313 | |
---|
| 7314 | op=option(get); |
---|
| 7315 | option(redSB); |
---|
| 7316 | list uprimary= newZero_decomp(@j,ser,@wr); |
---|
| 7317 | //HIER |
---|
| 7318 | if(abspri) |
---|
| 7319 | { |
---|
| 7320 | ideal II; |
---|
| 7321 | ideal jmap; |
---|
| 7322 | map sigma; |
---|
| 7323 | nn=nvars(basering); |
---|
| 7324 | map invsigma=basering,maxideal(1); |
---|
[4173c7] | 7325 | for(ab=1;ab<=size(uprimary) div 2;ab++) |
---|
[a36e78] | 7326 | { |
---|
| 7327 | II=uprimary[2*ab]; |
---|
| 7328 | attrib(II,"isSB",1); |
---|
| 7329 | if(deg(II[1])!=vdim(II)) |
---|
| 7330 | { |
---|
| 7331 | jmap=randomLast(50); |
---|
| 7332 | sigma=basering,jmap; |
---|
| 7333 | jmap[nn]=2*var(nn)-jmap[nn]; |
---|
| 7334 | invsigma=basering,jmap; |
---|
| 7335 | II=groebner(sigma(II)); |
---|
| 7336 | } |
---|
| 7337 | absprimarytmp[ab]= absFactorize(II[1],77); |
---|
| 7338 | II=var(nn); |
---|
| 7339 | abskeeptmp[ab]=string(invsigma(II)); |
---|
| 7340 | invsigma=basering,maxideal(1); |
---|
| 7341 | } |
---|
| 7342 | } |
---|
| 7343 | option(set,op); |
---|
| 7344 | |
---|
| 7345 | //we need the intersection of the ideals in the list quprimary with |
---|
| 7346 | //the polynomialring, i.e. let q=(f1,...,fr) in the quotientring |
---|
| 7347 | //such an ideal but fi polynomials, then the intersection of q with |
---|
| 7348 | //the polynomialring is the saturation of the ideal generated by |
---|
| 7349 | //f1,...,fr with respect toh which is the lcm of the leading |
---|
| 7350 | //coefficients of the fi considered in the quotientring: |
---|
| 7351 | //this is coded in saturn |
---|
| 7352 | |
---|
| 7353 | list saturn; |
---|
| 7354 | ideal hpl; |
---|
| 7355 | |
---|
| 7356 | for(@n=1;@n<=size(uprimary);@n++) |
---|
| 7357 | { |
---|
| 7358 | hpl=0; |
---|
| 7359 | for(@n1=1;@n1<=size(uprimary[@n]);@n1++) |
---|
| 7360 | { |
---|
| 7361 | hpl=hpl,leadcoef(uprimary[@n][@n1]); |
---|
| 7362 | } |
---|
| 7363 | saturn[@n]=hpl; |
---|
| 7364 | } |
---|
| 7365 | //------------------------------------------------------------------ |
---|
| 7366 | //we leave the quotientring K(var(nnp+1),..,var(nva))[..rest..] |
---|
| 7367 | //back to the polynomialring |
---|
| 7368 | //------------------------------------------------------------------ |
---|
| 7369 | setring gnir; |
---|
| 7370 | collectprimary=imap(quring,uprimary); |
---|
| 7371 | lsau=imap(quring,saturn); |
---|
| 7372 | @h=imap(quring,@h); |
---|
| 7373 | |
---|
| 7374 | kill quring; |
---|
| 7375 | |
---|
| 7376 | |
---|
| 7377 | @n2=size(quprimary); |
---|
[808a9f3] | 7378 | //================NEU========================================= |
---|
[a36e78] | 7379 | if(deg(quprimary[1][1])<=0){ @n2=0; } |
---|
[808a9f3] | 7380 | //============================================================ |
---|
| 7381 | |
---|
[a36e78] | 7382 | @n3=@n2; |
---|
| 7383 | |
---|
[4173c7] | 7384 | for(@n1=1;@n1<=size(collectprimary) div 2;@n1++) |
---|
[a36e78] | 7385 | { |
---|
| 7386 | if(deg(collectprimary[2*@n1][1])>0) |
---|
| 7387 | { |
---|
| 7388 | @n2++; |
---|
| 7389 | quprimary[@n2]=collectprimary[2*@n1-1]; |
---|
| 7390 | lnew[@n2]=lsau[2*@n1-1]; |
---|
| 7391 | @n2++; |
---|
| 7392 | lnew[@n2]=lsau[2*@n1]; |
---|
| 7393 | quprimary[@n2]=collectprimary[2*@n1]; |
---|
| 7394 | if(abspri) |
---|
| 7395 | { |
---|
[4173c7] | 7396 | absprimary[@n2 div 2]=absprimarytmp[@n1]; |
---|
| 7397 | abskeep[@n2 div 2]=abskeeptmp[@n1]; |
---|
[a36e78] | 7398 | } |
---|
| 7399 | } |
---|
| 7400 | } |
---|
| 7401 | |
---|
| 7402 | |
---|
| 7403 | //here the intersection with the polynomialring |
---|
| 7404 | //mentioned above is really computed |
---|
| 7405 | |
---|
[4173c7] | 7406 | for(@n=@n3 div 2+1;@n<=@n2 div 2;@n++) |
---|
[a36e78] | 7407 | { |
---|
| 7408 | if(specialIdealsEqual(quprimary[2*@n-1],quprimary[2*@n])) |
---|
| 7409 | { |
---|
| 7410 | quprimary[2*@n-1]=sat2(quprimary[2*@n-1],lnew[2*@n-1])[1]; |
---|
| 7411 | quprimary[2*@n]=quprimary[2*@n-1]; |
---|
| 7412 | } |
---|
| 7413 | else |
---|
| 7414 | { |
---|
| 7415 | if(@wr==0) |
---|
| 7416 | { |
---|
| 7417 | quprimary[2*@n-1]=sat2(quprimary[2*@n-1],lnew[2*@n-1])[1]; |
---|
| 7418 | } |
---|
| 7419 | quprimary[2*@n]=sat2(quprimary[2*@n],lnew[2*@n])[1]; |
---|
| 7420 | } |
---|
| 7421 | } |
---|
| 7422 | if(@n2>=@n3+2) |
---|
| 7423 | { |
---|
| 7424 | setring @Phelp; |
---|
| 7425 | ser=imap(gnir,ser); |
---|
| 7426 | hquprimary=imap(gnir,quprimary); |
---|
[4173c7] | 7427 | for(@n=@n3 div 2+1;@n<=@n2 div 2;@n++) |
---|
[a36e78] | 7428 | { |
---|
| 7429 | if(@wr==0) |
---|
| 7430 | { |
---|
| 7431 | ser=intersect(ser,hquprimary[2*@n-1]); |
---|
| 7432 | } |
---|
| 7433 | else |
---|
| 7434 | { |
---|
| 7435 | ser=intersect(ser,hquprimary[2*@n]); |
---|
| 7436 | } |
---|
| 7437 | } |
---|
| 7438 | setring gnir; |
---|
| 7439 | ser=imap(@Phelp,ser); |
---|
| 7440 | } |
---|
[808a9f3] | 7441 | |
---|
[a36e78] | 7442 | // } |
---|
[808a9f3] | 7443 | } |
---|
[a36e78] | 7444 | //HIER |
---|
| 7445 | if(abspri) |
---|
[808a9f3] | 7446 | { |
---|
[a36e78] | 7447 | list resu,tempo; |
---|
[4173c7] | 7448 | for(ab=1;ab<=size(quprimary) div 2;ab++) |
---|
[70ab73] | 7449 | { |
---|
[a36e78] | 7450 | if (deg(quprimary[2*ab][1])!=0) |
---|
| 7451 | { |
---|
| 7452 | tempo=quprimary[2*ab-1],quprimary[2*ab], |
---|
| 7453 | absprimary[ab],abskeep[ab]; |
---|
| 7454 | resu[ab]=tempo; |
---|
| 7455 | } |
---|
[808a9f3] | 7456 | } |
---|
[a36e78] | 7457 | quprimary=resu; |
---|
| 7458 | @wr=3; |
---|
[808a9f3] | 7459 | } |
---|
[a36e78] | 7460 | if(size(reduce(ser,peek,1))!=0) |
---|
[808a9f3] | 7461 | { |
---|
[a36e78] | 7462 | if(@wr>0) |
---|
| 7463 | { |
---|
| 7464 | // The following line was dropped to avoid the recursion step: |
---|
| 7465 | //htprimary=newDecompStep(@j,@wr,peek,ser); |
---|
| 7466 | htprimary = list(); |
---|
| 7467 | } |
---|
| 7468 | else |
---|
| 7469 | { |
---|
| 7470 | // The following line was dropped to avoid the recursion step: |
---|
| 7471 | //htprimary=newDecompStep(@j,peek,ser); |
---|
| 7472 | htprimary = list(); |
---|
| 7473 | } |
---|
| 7474 | // here we collect now both results primary(sat(j,gh)) |
---|
| 7475 | // and primary(j,gh^n) |
---|
| 7476 | @n=size(quprimary); |
---|
| 7477 | if (deg(quprimary[1][1])<=0) { @n=0; } |
---|
| 7478 | for (@k=1;@k<=size(htprimary);@k++) |
---|
| 7479 | { |
---|
| 7480 | quprimary[@n+@k]=htprimary[@k]; |
---|
| 7481 | } |
---|
[808a9f3] | 7482 | } |
---|
[a36e78] | 7483 | } |
---|
| 7484 | } |
---|
| 7485 | else |
---|
| 7486 | { |
---|
[808a9f3] | 7487 | if(abspri) |
---|
| 7488 | { |
---|
| 7489 | list resu,tempo; |
---|
[4173c7] | 7490 | for(ab=1;ab<=size(quprimary) div 2;ab++) |
---|
[808a9f3] | 7491 | { |
---|
[a36e78] | 7492 | tempo=quprimary[2*ab-1],quprimary[2*ab], |
---|
| 7493 | absprimary[ab],abskeep[ab]; |
---|
| 7494 | resu[ab]=tempo; |
---|
[808a9f3] | 7495 | } |
---|
| 7496 | quprimary=resu; |
---|
[70ab73] | 7497 | } |
---|
[a36e78] | 7498 | } |
---|
[808a9f3] | 7499 | //--------------------------------------------------------------------------- |
---|
| 7500 | //back to the ring we started with |
---|
| 7501 | //the final result: primary |
---|
| 7502 | //--------------------------------------------------------------------------- |
---|
| 7503 | |
---|
| 7504 | setring @P; |
---|
| 7505 | primary=imap(gnir,quprimary); |
---|
| 7506 | |
---|
[70ab73] | 7507 | if (intersectOption == "intersect") |
---|
| 7508 | { |
---|
[a36e78] | 7509 | return(list(primary, imap(gnir, ser))); |
---|
[70ab73] | 7510 | } |
---|
| 7511 | else |
---|
| 7512 | { |
---|
| 7513 | return(primary); |
---|
| 7514 | } |
---|
[808a9f3] | 7515 | } |
---|
| 7516 | example |
---|
| 7517 | { "EXAMPLE:"; echo = 2; |
---|
| 7518 | ring r = 32003,(x,y,z),lp; |
---|
| 7519 | poly p = z2+1; |
---|
| 7520 | poly q = z4+2; |
---|
| 7521 | ideal i = p^2*q^3,(y-z3)^3,(x-yz+z4)^4; |
---|
| 7522 | list pr= newDecompStep(i); |
---|
| 7523 | pr; |
---|
| 7524 | testPrimary( pr, i); |
---|
| 7525 | } |
---|
| 7526 | |
---|
[7f7c25e] | 7527 | // This was part of proc decomp. |
---|
| 7528 | // In proc newDecompStep, used for the computation of the minimal associated primes, |
---|
| 7529 | // this part was separated as a soubrutine to make the code more clear. |
---|
| 7530 | // Also, since the reduction is performed twice in proc newDecompStep, it should use both times this routine. |
---|
| 7531 | // This is not yet implemented, since the reduction is not exactly the same and some changes should be made. |
---|
| 7532 | static proc newReduction(ideal @j, ideal ser, intvec @hilb, intvec @w, int jdim, int abspri, int @wr, list data) |
---|
[808a9f3] | 7533 | { |
---|
[a36e78] | 7534 | string @va; |
---|
| 7535 | string quotring; |
---|
| 7536 | intvec op; |
---|
| 7537 | intvec @vv; |
---|
| 7538 | def gnir = basering; |
---|
| 7539 | ideal isat=0; |
---|
| 7540 | int @n; |
---|
| 7541 | int @n1 = 0; |
---|
| 7542 | int @n2 = 0; |
---|
| 7543 | int @n3 = 0; |
---|
| 7544 | int homo = homog(@j); |
---|
| 7545 | int lauf; |
---|
| 7546 | int @k; |
---|
| 7547 | list fett; |
---|
| 7548 | int keepdi; |
---|
| 7549 | list collectprimary; |
---|
| 7550 | list lsau; |
---|
| 7551 | list lnew; |
---|
| 7552 | ideal @h; |
---|
| 7553 | |
---|
| 7554 | list indepInfo = data[1]; |
---|
| 7555 | list quprimary = list(); |
---|
| 7556 | |
---|
| 7557 | //if(abspri) |
---|
| 7558 | //{ |
---|
[808a9f3] | 7559 | int ab; |
---|
| 7560 | list absprimarytmp,abskeeptmp; |
---|
| 7561 | list absprimary, abskeep; |
---|
[a36e78] | 7562 | //} |
---|
| 7563 | // Debug |
---|
| 7564 | dbprint(printlevel - voice, "newReduction, v2.0"); |
---|
[808a9f3] | 7565 | |
---|
[a36e78] | 7566 | if((indepInfo[1]==varstr(basering))) // &&(@m==1) |
---|
| 7567 | //this is the good case, nothing to do, just to have the same notations |
---|
| 7568 | //change the ring |
---|
| 7569 | { |
---|
| 7570 | execute("ring gnir1 = ("+charstr(basering)+"),("+varstr(basering)+"),(" |
---|
[808a9f3] | 7571 | +ordstr(basering)+");"); |
---|
[a36e78] | 7572 | ideal @j = fetch(gnir, @j); |
---|
| 7573 | attrib(@j,"isSB",1); |
---|
| 7574 | ideal ser = fetch(gnir, ser); |
---|
| 7575 | } |
---|
| 7576 | else |
---|
| 7577 | { |
---|
| 7578 | @va=string(maxideal(1)); |
---|
[808a9f3] | 7579 | //Aenderung============== |
---|
[a36e78] | 7580 | //if(@m==1) |
---|
| 7581 | //{ |
---|
| 7582 | // @j=fetch(@P,i); |
---|
| 7583 | //} |
---|
[808a9f3] | 7584 | //======================= |
---|
[a36e78] | 7585 | execute("ring gnir1 = ("+charstr(basering)+"),("+indepInfo[1]+"),(" |
---|
[808a9f3] | 7586 | +indepInfo[2]+");"); |
---|
[a36e78] | 7587 | execute("map phi=gnir,"+@va+";"); |
---|
| 7588 | op=option(get); |
---|
| 7589 | option(redSB); |
---|
| 7590 | if(homo==1) |
---|
| 7591 | { |
---|
| 7592 | ideal @j=std(phi(@j),@hilb,@w); |
---|
| 7593 | } |
---|
| 7594 | else |
---|
| 7595 | { |
---|
| 7596 | ideal @j=groebner(phi(@j)); |
---|
| 7597 | } |
---|
| 7598 | ideal ser=phi(ser); |
---|
[808a9f3] | 7599 | |
---|
[a36e78] | 7600 | option(set,op); |
---|
| 7601 | } |
---|
| 7602 | if((deg(@j[1])==0)||(dim(@j)<jdim)) |
---|
| 7603 | { |
---|
| 7604 | setring gnir; |
---|
| 7605 | break; |
---|
| 7606 | } |
---|
| 7607 | for (lauf=1;lauf<=size(@j);lauf++) |
---|
| 7608 | { |
---|
| 7609 | fett[lauf]=size(@j[lauf]); |
---|
| 7610 | } |
---|
| 7611 | //------------------------------------------------------------------------ |
---|
| 7612 | //we have now the following situation: |
---|
| 7613 | //j intersected with K[var(nnp+1),..,var(nva)] is zero so we may pass |
---|
| 7614 | //to this quotientring, j is their still a standardbasis, the |
---|
| 7615 | //leading coefficients of the polynomials there (polynomials in |
---|
| 7616 | //K[var(nnp+1),..,var(nva)]) are collected in the list h, |
---|
| 7617 | //we need their ggt, gh, because of the following: let |
---|
| 7618 | //(j:gh^n)=(j:gh^infinity) then j*K(var(nnp+1),..,var(nva))[..the rest..] |
---|
| 7619 | //intersected with K[var(1),...,var(nva)] is (j:gh^n) |
---|
| 7620 | //on the other hand j=(j,gh^n) intersected with (j:gh^n) |
---|
| 7621 | |
---|
| 7622 | //------------------------------------------------------------------------ |
---|
| 7623 | |
---|
| 7624 | //arrangement for quotientring K(var(nnp+1),..,var(nva))[..the rest..] and |
---|
| 7625 | //map phi:K[var(1),...,var(nva)] --->K(var(nnpr+1),..,var(nva))[..rest..] |
---|
| 7626 | //------------------------------------------------------------------------ |
---|
| 7627 | |
---|
| 7628 | quotring=prepareQuotientring(nvars(basering)-indepInfo[3]); |
---|
| 7629 | |
---|
| 7630 | //--------------------------------------------------------------------- |
---|
| 7631 | //we pass to the quotientring K(var(nnp+1),..,var(nva))[..the rest..] |
---|
| 7632 | //--------------------------------------------------------------------- |
---|
| 7633 | |
---|
| 7634 | ideal @jj=lead(@j); //!! vorn vereinbaren |
---|
| 7635 | execute(quotring); |
---|
| 7636 | |
---|
| 7637 | ideal @jj=imap(gnir1,@jj); |
---|
| 7638 | @vv=clearSBNeu(@jj,fett); //!! vorn vereinbaren |
---|
| 7639 | setring gnir1; |
---|
| 7640 | @k=size(@j); |
---|
| 7641 | for (lauf=1;lauf<=@k;lauf++) |
---|
| 7642 | { |
---|
| 7643 | if(@vv[lauf]==1) |
---|
| 7644 | { |
---|
| 7645 | @j[lauf]=0; |
---|
| 7646 | } |
---|
| 7647 | } |
---|
| 7648 | @j=simplify(@j,2); |
---|
| 7649 | setring quring; |
---|
| 7650 | // @j considered in the quotientring |
---|
| 7651 | ideal @j=imap(gnir1,@j); |
---|
[808a9f3] | 7652 | |
---|
[a36e78] | 7653 | ideal ser=imap(gnir1,ser); |
---|
[808a9f3] | 7654 | |
---|
[a36e78] | 7655 | kill gnir1; |
---|
[808a9f3] | 7656 | |
---|
[a36e78] | 7657 | //j is a standardbasis in the quotientring but usually not minimal |
---|
| 7658 | //here it becomes minimal |
---|
[808a9f3] | 7659 | |
---|
[a36e78] | 7660 | attrib(@j,"isSB",1); |
---|
[808a9f3] | 7661 | |
---|
[a36e78] | 7662 | //we need later ggt(h[1],...)=gh for saturation |
---|
| 7663 | ideal @h; |
---|
| 7664 | if(deg(@j[1])>0) |
---|
| 7665 | { |
---|
| 7666 | for(@n=1;@n<=size(@j);@n++) |
---|
| 7667 | { |
---|
| 7668 | @h[@n]=leadcoef(@j[@n]); |
---|
| 7669 | } |
---|
| 7670 | //the primary decomposition of j*K(var(nnp+1),..,var(nva))[..the rest..] |
---|
| 7671 | op=option(get); |
---|
| 7672 | option(redSB); |
---|
[808a9f3] | 7673 | |
---|
[a36e78] | 7674 | int zeroMinAss = @wr; |
---|
| 7675 | if (@wr == 2) {zeroMinAss = 1;} |
---|
| 7676 | list uprimary= newZero_decomp(@j, ser, zeroMinAss); |
---|
[808a9f3] | 7677 | |
---|
[a36e78] | 7678 | //HIER |
---|
| 7679 | if(abspri) |
---|
| 7680 | { |
---|
| 7681 | ideal II; |
---|
| 7682 | ideal jmap; |
---|
| 7683 | map sigma; |
---|
| 7684 | nn=nvars(basering); |
---|
| 7685 | map invsigma=basering,maxideal(1); |
---|
[4173c7] | 7686 | for(ab=1;ab<=size(uprimary) div 2;ab++) |
---|
[a36e78] | 7687 | { |
---|
| 7688 | II=uprimary[2*ab]; |
---|
| 7689 | attrib(II,"isSB",1); |
---|
| 7690 | if(deg(II[1])!=vdim(II)) |
---|
| 7691 | { |
---|
| 7692 | jmap=randomLast(50); |
---|
| 7693 | sigma=basering,jmap; |
---|
| 7694 | jmap[nn]=2*var(nn)-jmap[nn]; |
---|
| 7695 | invsigma=basering,jmap; |
---|
| 7696 | II=groebner(sigma(II)); |
---|
| 7697 | } |
---|
| 7698 | absprimarytmp[ab]= absFactorize(II[1],77); |
---|
| 7699 | II=var(nn); |
---|
| 7700 | abskeeptmp[ab]=string(invsigma(II)); |
---|
| 7701 | invsigma=basering,maxideal(1); |
---|
| 7702 | } |
---|
| 7703 | } |
---|
| 7704 | option(set,op); |
---|
| 7705 | } |
---|
| 7706 | else |
---|
| 7707 | { |
---|
| 7708 | list uprimary; |
---|
| 7709 | uprimary[1]=ideal(1); |
---|
| 7710 | uprimary[2]=ideal(1); |
---|
| 7711 | } |
---|
| 7712 | //we need the intersection of the ideals in the list quprimary with the |
---|
| 7713 | //polynomialring, i.e. let q=(f1,...,fr) in the quotientring such an ideal |
---|
| 7714 | //but fi polynomials, then the intersection of q with the polynomialring |
---|
| 7715 | //is the saturation of the ideal generated by f1,...,fr with respect to |
---|
| 7716 | //h which is the lcm of the leading coefficients of the fi considered in |
---|
| 7717 | //in the quotientring: this is coded in saturn |
---|
[70ab73] | 7718 | |
---|
[a36e78] | 7719 | list saturn; |
---|
| 7720 | ideal hpl; |
---|
[70ab73] | 7721 | |
---|
[a36e78] | 7722 | for(@n=1;@n<=size(uprimary);@n++) |
---|
| 7723 | { |
---|
| 7724 | uprimary[@n]=interred(uprimary[@n]); // temporary fix |
---|
| 7725 | hpl=0; |
---|
| 7726 | for(@n1=1;@n1<=size(uprimary[@n]);@n1++) |
---|
| 7727 | { |
---|
| 7728 | hpl=hpl,leadcoef(uprimary[@n][@n1]); |
---|
| 7729 | } |
---|
| 7730 | saturn[@n]=hpl; |
---|
| 7731 | } |
---|
[808a9f3] | 7732 | |
---|
[a36e78] | 7733 | //-------------------------------------------------------------------- |
---|
| 7734 | //we leave the quotientring K(var(nnp+1),..,var(nva))[..the rest..] |
---|
| 7735 | //back to the polynomialring |
---|
| 7736 | //--------------------------------------------------------------------- |
---|
| 7737 | setring gnir; |
---|
[808a9f3] | 7738 | |
---|
[a36e78] | 7739 | collectprimary=imap(quring,uprimary); |
---|
| 7740 | lsau=imap(quring,saturn); |
---|
| 7741 | @h=imap(quring,@h); |
---|
[808a9f3] | 7742 | |
---|
[a36e78] | 7743 | kill quring; |
---|
[808a9f3] | 7744 | |
---|
[a36e78] | 7745 | @n2=size(quprimary); |
---|
| 7746 | @n3=@n2; |
---|
[808a9f3] | 7747 | |
---|
[4173c7] | 7748 | for(@n1=1;@n1<=size(collectprimary) div 2;@n1++) |
---|
[a36e78] | 7749 | { |
---|
| 7750 | if(deg(collectprimary[2*@n1][1])>0) |
---|
| 7751 | { |
---|
| 7752 | @n2++; |
---|
| 7753 | quprimary[@n2]=collectprimary[2*@n1-1]; |
---|
| 7754 | lnew[@n2]=lsau[2*@n1-1]; |
---|
| 7755 | @n2++; |
---|
| 7756 | lnew[@n2]=lsau[2*@n1]; |
---|
| 7757 | quprimary[@n2]=collectprimary[2*@n1]; |
---|
| 7758 | if(abspri) |
---|
| 7759 | { |
---|
[4173c7] | 7760 | absprimary[@n2 div 2]=absprimarytmp[@n1]; |
---|
| 7761 | abskeep[@n2 div 2]=abskeeptmp[@n1]; |
---|
[a36e78] | 7762 | } |
---|
| 7763 | } |
---|
| 7764 | } |
---|
[808a9f3] | 7765 | |
---|
[a36e78] | 7766 | //here the intersection with the polynomialring |
---|
| 7767 | //mentioned above is really computed |
---|
[4173c7] | 7768 | for(@n=@n3 div 2+1;@n<=@n2 div 2;@n++) |
---|
[a36e78] | 7769 | { |
---|
| 7770 | if(specialIdealsEqual(quprimary[2*@n-1],quprimary[2*@n])) |
---|
| 7771 | { |
---|
| 7772 | quprimary[2*@n-1]=sat2(quprimary[2*@n-1],lnew[2*@n-1])[1]; |
---|
| 7773 | quprimary[2*@n]=quprimary[2*@n-1]; |
---|
| 7774 | } |
---|
| 7775 | else |
---|
| 7776 | { |
---|
| 7777 | if(@wr==0) |
---|
| 7778 | { |
---|
| 7779 | quprimary[2*@n-1]=sat2(quprimary[2*@n-1],lnew[2*@n-1])[1]; |
---|
| 7780 | } |
---|
| 7781 | quprimary[2*@n]=sat2(quprimary[2*@n],lnew[2*@n])[1]; |
---|
| 7782 | } |
---|
| 7783 | } |
---|
[808a9f3] | 7784 | |
---|
[a36e78] | 7785 | return(quprimary, absprimary, abskeep, ser, @h); |
---|
| 7786 | } |
---|
[808a9f3] | 7787 | |
---|
| 7788 | |
---|
[a36e78] | 7789 | //////////////////////////////////////////////////////////////////////////// |
---|
[808a9f3] | 7790 | |
---|
| 7791 | |
---|
| 7792 | |
---|
| 7793 | |
---|
| 7794 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 7795 | // Based on minAssGTZ |
---|
| 7796 | |
---|
[f995aa] | 7797 | proc minAss(ideal i,list #) |
---|
[7f7c25e] | 7798 | "USAGE: minAss(I[, l]); i ideal, l list (optional) of parameters, same as minAssGTZ |
---|
| 7799 | RETURN: a list, the minimal associated prime ideals of I. |
---|
[808a9f3] | 7800 | NOTE: Designed for characteristic 0, works also in char k > 0 based |
---|
| 7801 | on an algorithm of Yokoyama |
---|
[f995aa] | 7802 | EXAMPLE: example minAss; shows an example |
---|
[808a9f3] | 7803 | " |
---|
| 7804 | { |
---|
[70ab73] | 7805 | return(minAssGTZ(i,#)); |
---|
[808a9f3] | 7806 | } |
---|
| 7807 | example |
---|
| 7808 | { "EXAMPLE:"; echo = 2; |
---|
| 7809 | ring r = 0, (x, y, z), dp; |
---|
| 7810 | poly p = z2 + 1; |
---|
| 7811 | poly q = z3 + 2; |
---|
| 7812 | ideal i = p * q^2, y - z2; |
---|
[f995aa] | 7813 | list pr = minAss(i); |
---|
[808a9f3] | 7814 | pr; |
---|
| 7815 | } |
---|
| 7816 | |
---|
| 7817 | |
---|
| 7818 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 7819 | // |
---|
[f995aa] | 7820 | // Computes the minimal associated primes of I via Laplagne algorithm, |
---|
[808a9f3] | 7821 | // using primary decomposition in the zero dimensional case. |
---|
| 7822 | // For reduction to the zerodimensional case, it uses the procedure |
---|
[f995aa] | 7823 | // decomp, with some modifications to avoid the recursion. |
---|
[808a9f3] | 7824 | // |
---|
| 7825 | |
---|
[f995aa] | 7826 | static proc minAssSL(ideal I) |
---|
[808a9f3] | 7827 | // Input = I, ideal |
---|
| 7828 | // Output = primaryDec where primaryDec is the list of the minimal |
---|
| 7829 | // associated primes and the primary components corresponding to these primes. |
---|
| 7830 | { |
---|
| 7831 | ideal P = 1; |
---|
| 7832 | list pd = list(); |
---|
| 7833 | int k; |
---|
| 7834 | int stop = 0; |
---|
| 7835 | list primaryDec = list(); |
---|
| 7836 | |
---|
[70ab73] | 7837 | while (stop == 0) |
---|
| 7838 | { |
---|
[808a9f3] | 7839 | // Debug |
---|
[f995aa] | 7840 | dbprint(printlevel - voice, "// We call minAssSLIteration to find new prime ideals!"); |
---|
| 7841 | pd = minAssSLIteration(I, P); |
---|
[808a9f3] | 7842 | // Debug |
---|
[f995aa] | 7843 | dbprint(printlevel - voice, "// Output of minAssSLIteration:"); |
---|
[808a9f3] | 7844 | dbprint(printlevel - voice, pd); |
---|
[70ab73] | 7845 | if (size(pd[1]) > 0) |
---|
| 7846 | { |
---|
[808a9f3] | 7847 | primaryDec = primaryDec + pd[1]; |
---|
| 7848 | // Debug |
---|
| 7849 | dbprint(printlevel - voice, "// We intersect the prime ideals obtained."); |
---|
| 7850 | P = intersect(P, pd[2]); |
---|
| 7851 | // Debug |
---|
| 7852 | dbprint(printlevel - voice, "// Intersection finished."); |
---|
[70ab73] | 7853 | } |
---|
| 7854 | else |
---|
| 7855 | { |
---|
[f3c6e5] | 7856 | stop = 1; |
---|
[7f7c25e] | 7857 | } |
---|
| 7858 | } |
---|
[f3c6e5] | 7859 | |
---|
[808a9f3] | 7860 | // Returns only the primary components, not the radical. |
---|
| 7861 | return(primaryDec); |
---|
[f3c6e5] | 7862 | } |
---|
[808a9f3] | 7863 | |
---|
| 7864 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 7865 | // Given an ideal I and an ideal P (intersection of some minimal prime ideals |
---|
| 7866 | // associated to I), it calculates new minimal prime ideals associated to I |
---|
| 7867 | // which were not used to calculate P. |
---|
| 7868 | // This version uses Primary Decomposition in the zerodimensional case. |
---|
[f995aa] | 7869 | static proc minAssSLIteration(ideal I, ideal P); |
---|
[808a9f3] | 7870 | { |
---|
| 7871 | int k = 1; |
---|
| 7872 | int good = 0; |
---|
| 7873 | list primaryDec = list(); |
---|
| 7874 | // Debug |
---|
| 7875 | dbprint (printlevel-voice, "// We search for an element in P - sqrt(I)."); |
---|
[70ab73] | 7876 | while ((k <= size(P)) and (good == 0)) |
---|
| 7877 | { |
---|
[808a9f3] | 7878 | good = 1 - rad_con(P[k], I); |
---|
| 7879 | k++; |
---|
[7f7c25e] | 7880 | } |
---|
[808a9f3] | 7881 | k--; |
---|
[70ab73] | 7882 | if (good == 0) |
---|
| 7883 | { |
---|
[808a9f3] | 7884 | // Debug |
---|
| 7885 | dbprint (printlevel - voice, "// No element was found, P = sqrt(I)."); |
---|
| 7886 | return (list(primaryDec, ideal(0))); |
---|
[7f7c25e] | 7887 | } |
---|
[808a9f3] | 7888 | // Debug |
---|
| 7889 | dbprint (printlevel - voice, "// We found h = ", P[k]); |
---|
| 7890 | dbprint (printlevel - voice, "// We calculate the saturation of I with respect to the element just founded."); |
---|
| 7891 | ideal J = sat(I, P[k])[1]; |
---|
| 7892 | |
---|
| 7893 | // Uses decomp from primdec, modified to avoid the recursion. |
---|
| 7894 | // Debug |
---|
| 7895 | dbprint(printlevel - voice, "// We do the reduction to the zerodimensional case, via decomp."); |
---|
| 7896 | |
---|
| 7897 | primaryDec = newDecompStep(J, "oneIndep", "intersect", 2); |
---|
| 7898 | // Debug |
---|
[4173c7] | 7899 | dbprint(printlevel - voice, "// Proc decomp has found", size(primaryDec) div 2, "new primary components."); |
---|
[808a9f3] | 7900 | |
---|
| 7901 | return(primaryDec); |
---|
| 7902 | } |
---|
| 7903 | |
---|
| 7904 | |
---|
| 7905 | |
---|
| 7906 | /////////////////////////////////////////////////////////////////////////////////// |
---|
| 7907 | // Based on maxIndependSet |
---|
| 7908 | // Added list # as parameter |
---|
| 7909 | // If the first element of # is 0, the output is only 1 max indep set. |
---|
| 7910 | // If no list is specified or #[1] = 1, the output is all the max indep set of the |
---|
| 7911 | // leading terms ideal. This is the original output of maxIndependSet |
---|
| 7912 | |
---|
| 7913 | proc newMaxIndependSetLp(ideal j, list #) |
---|
[f995aa] | 7914 | "USAGE: newMaxIndependentSetLp(i); i ideal (returns all maximal independent sets of the corresponding leading terms ideal) |
---|
| 7915 | newMaxIndependentSetLp(i, 0); i ideal (returns only one maximal independent set) |
---|
[808a9f3] | 7916 | RETURN: list = #1. new varstring with the maximal independent set at the end, |
---|
[f995aa] | 7917 | #2. ordstring with the lp ordering, |
---|
[808a9f3] | 7918 | #3. the number of independent variables |
---|
| 7919 | NOTE: |
---|
[f995aa] | 7920 | EXAMPLE: example newMaxIndependentSetLp; shows an example |
---|
[808a9f3] | 7921 | " |
---|
| 7922 | { |
---|
[70ab73] | 7923 | int n, k, di; |
---|
| 7924 | list resu, hilf; |
---|
| 7925 | string var1, var2; |
---|
| 7926 | list v = indepSet(j, 0); |
---|
[808a9f3] | 7927 | |
---|
[70ab73] | 7928 | // SL 2006.04.21 1 Lines modified to use only one independent Set |
---|
| 7929 | string indepOption; |
---|
| 7930 | if (size(#) > 0) |
---|
| 7931 | { |
---|
| 7932 | indepOption = #[1]; |
---|
| 7933 | } |
---|
| 7934 | else |
---|
| 7935 | { |
---|
| 7936 | indepOption = "allIndep"; |
---|
| 7937 | } |
---|
[808a9f3] | 7938 | |
---|
[70ab73] | 7939 | int nMax; |
---|
| 7940 | if (indepOption == "allIndep") |
---|
| 7941 | { |
---|
| 7942 | nMax = size(v); |
---|
| 7943 | } |
---|
| 7944 | else |
---|
| 7945 | { |
---|
| 7946 | nMax = 1; |
---|
| 7947 | } |
---|
| 7948 | |
---|
| 7949 | for(n = 1; n <= nMax; n++) |
---|
| 7950 | // SL 2006.04.21 2 |
---|
| 7951 | { |
---|
| 7952 | di = 0; |
---|
| 7953 | var1 = ""; |
---|
| 7954 | var2 = ""; |
---|
| 7955 | for(k = 1; k <= size(v[n]); k++) |
---|
| 7956 | { |
---|
| 7957 | if(v[n][k] != 0) |
---|
| 7958 | { |
---|
| 7959 | di++; |
---|
| 7960 | var2 = var2 + "var(" + string(k) + "), "; |
---|
[808a9f3] | 7961 | } |
---|
| 7962 | else |
---|
| 7963 | { |
---|
[70ab73] | 7964 | var1 = var1 + "var(" + string(k) + "), "; |
---|
[808a9f3] | 7965 | } |
---|
[70ab73] | 7966 | } |
---|
| 7967 | if(di > 0) |
---|
| 7968 | { |
---|
| 7969 | var1 = var1 + var2; |
---|
| 7970 | var1 = var1[1..size(var1) - 2]; // The "- 2" removes the trailer comma |
---|
| 7971 | hilf[1] = var1; |
---|
| 7972 | // SL 2006.21.04 1 The order is now block dp instead of lp |
---|
| 7973 | //hilf[2] = "dp(" + string(nvars(basering) - di) + "), dp(" + string(di) + ")"; |
---|
| 7974 | // SL 2006.21.04 2 |
---|
| 7975 | // For decomp, lp ordering is needed. Nothing is changed. |
---|
| 7976 | hilf[2] = "lp"; |
---|
| 7977 | hilf[3] = di; |
---|
| 7978 | resu[n] = hilf; |
---|
| 7979 | } |
---|
| 7980 | else |
---|
| 7981 | { |
---|
| 7982 | resu[n] = varstr(basering), ordstr(basering), 0; |
---|
| 7983 | } |
---|
| 7984 | } |
---|
| 7985 | return(resu); |
---|
[808a9f3] | 7986 | } |
---|
| 7987 | example |
---|
| 7988 | { "EXAMPLE:"; echo = 2; |
---|
| 7989 | ring s1 = (0, x, y), (a, b, c, d, e, f, g), lp; |
---|
| 7990 | ideal i = ea - fbg, fa + be, ec - fdg, fc + de; |
---|
| 7991 | i = std(i); |
---|
| 7992 | list l = newMaxIndependSetLp(i); |
---|
| 7993 | l; |
---|
| 7994 | i = i, g; |
---|
| 7995 | l = newMaxIndependSetLp(i); |
---|
| 7996 | l; |
---|
| 7997 | |
---|
| 7998 | ring s = 0, (x, y, z), lp; |
---|
| 7999 | ideal i = z, yx; |
---|
| 8000 | list l = newMaxIndependSetLp(i); |
---|
| 8001 | l; |
---|
| 8002 | } |
---|
| 8003 | |
---|
| 8004 | |
---|
| 8005 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 8006 | |
---|
| 8007 | proc newZero_decomp (ideal j, ideal ser, int @wr, list #) |
---|
| 8008 | "USAGE: newZero_decomp(j,ser,@wr); j,ser ideals, @wr=0 or 1 |
---|
[7f7c25e] | 8009 | (@wr=0 for primary decomposition, @wr=1 for computation of associated |
---|
[808a9f3] | 8010 | primes) |
---|
| 8011 | if #[1] = "nest", then #[2] indicates the nest level (number of recursive calls) |
---|
| 8012 | When the nest level is high it indicates that the computation is difficult, |
---|
| 8013 | and different methods are applied. |
---|
| 8014 | RETURN: list = list of primary ideals and their radicals (at even positions |
---|
| 8015 | in the list) if the input is zero-dimensional and a standardbases |
---|
| 8016 | with respect to lex-ordering |
---|
| 8017 | If ser!=(0) and ser is contained in j or if j is not zero-dimen- |
---|
| 8018 | sional then ideal(1),ideal(1) is returned |
---|
| 8019 | NOTE: Algorithm of Gianni/Trager/Zacharias |
---|
| 8020 | EXAMPLE: example newZero_decomp; shows an example |
---|
| 8021 | " |
---|
| 8022 | { |
---|
| 8023 | def @P = basering; |
---|
| 8024 | int uytrewq; |
---|
| 8025 | int nva = nvars(basering); |
---|
| 8026 | int @k,@s,@n,@k1,zz; |
---|
| 8027 | list primary,lres0,lres1,act,@lh,@wh; |
---|
| 8028 | map phi,psi,phi1,psi1; |
---|
| 8029 | ideal jmap,jmap1,jmap2,helpprim,@qh,@qht,ser1; |
---|
| 8030 | intvec @vh,@hilb; |
---|
| 8031 | string @ri; |
---|
| 8032 | poly @f; |
---|
| 8033 | |
---|
| 8034 | // Debug |
---|
| 8035 | dbprint(printlevel - voice, "proc newZero_decomp"); |
---|
| 8036 | |
---|
| 8037 | if (dim(j)>0) |
---|
| 8038 | { |
---|
[70ab73] | 8039 | primary[1]=ideal(1); |
---|
| 8040 | primary[2]=ideal(1); |
---|
| 8041 | return(primary); |
---|
[808a9f3] | 8042 | } |
---|
| 8043 | j=interred(j); |
---|
| 8044 | |
---|
| 8045 | attrib(j,"isSB",1); |
---|
| 8046 | |
---|
| 8047 | int nestLevel = 0; |
---|
[70ab73] | 8048 | if (size(#) > 0) |
---|
| 8049 | { |
---|
| 8050 | if (typeof(#[1]) == "string") |
---|
| 8051 | { |
---|
| 8052 | if (#[1] == "nest") |
---|
| 8053 | { |
---|
[808a9f3] | 8054 | nestLevel = #[2]; |
---|
[7f7c25e] | 8055 | } |
---|
[808a9f3] | 8056 | # = list(); |
---|
[7f7c25e] | 8057 | } |
---|
| 8058 | } |
---|
[808a9f3] | 8059 | |
---|
| 8060 | if(vdim(j)==deg(j[1])) |
---|
| 8061 | { |
---|
[70ab73] | 8062 | act=factor(j[1]); |
---|
| 8063 | for(@k=1;@k<=size(act[1]);@k++) |
---|
| 8064 | { |
---|
| 8065 | @qh=j; |
---|
| 8066 | if(@wr==0) |
---|
| 8067 | { |
---|
| 8068 | @qh[1]=act[1][@k]^act[2][@k]; |
---|
| 8069 | } |
---|
| 8070 | else |
---|
| 8071 | { |
---|
| 8072 | @qh[1]=act[1][@k]; |
---|
| 8073 | } |
---|
| 8074 | primary[2*@k-1]=interred(@qh); |
---|
| 8075 | @qh=j; |
---|
| 8076 | @qh[1]=act[1][@k]; |
---|
| 8077 | primary[2*@k]=interred(@qh); |
---|
| 8078 | attrib( primary[2*@k-1],"isSB",1); |
---|
[808a9f3] | 8079 | |
---|
[70ab73] | 8080 | if((size(ser)>0)&&(size(reduce(ser,primary[2*@k-1],1))==0)) |
---|
| 8081 | { |
---|
| 8082 | primary[2*@k-1]=ideal(1); |
---|
| 8083 | primary[2*@k]=ideal(1); |
---|
| 8084 | } |
---|
| 8085 | } |
---|
| 8086 | return(primary); |
---|
[808a9f3] | 8087 | } |
---|
| 8088 | |
---|
| 8089 | if(homog(j)==1) |
---|
| 8090 | { |
---|
[70ab73] | 8091 | primary[1]=j; |
---|
| 8092 | if((size(ser)>0)&&(size(reduce(ser,j,1))==0)) |
---|
| 8093 | { |
---|
| 8094 | primary[1]=ideal(1); |
---|
| 8095 | primary[2]=ideal(1); |
---|
| 8096 | return(primary); |
---|
| 8097 | } |
---|
| 8098 | if(dim(j)==-1) |
---|
| 8099 | { |
---|
| 8100 | primary[1]=ideal(1); |
---|
| 8101 | primary[2]=ideal(1); |
---|
| 8102 | } |
---|
| 8103 | else |
---|
| 8104 | { |
---|
| 8105 | primary[2]=maxideal(1); |
---|
| 8106 | } |
---|
| 8107 | return(primary); |
---|
[808a9f3] | 8108 | } |
---|
| 8109 | |
---|
| 8110 | //the first element in the standardbase is factorized |
---|
| 8111 | if(deg(j[1])>0) |
---|
| 8112 | { |
---|
| 8113 | act=factor(j[1]); |
---|
| 8114 | testFactor(act,j[1]); |
---|
| 8115 | } |
---|
| 8116 | else |
---|
| 8117 | { |
---|
[70ab73] | 8118 | primary[1]=ideal(1); |
---|
| 8119 | primary[2]=ideal(1); |
---|
| 8120 | return(primary); |
---|
[808a9f3] | 8121 | } |
---|
| 8122 | |
---|
| 8123 | //with the factors new ideals (hopefully the primary decomposition) |
---|
| 8124 | //are created |
---|
| 8125 | if(size(act[1])>1) |
---|
| 8126 | { |
---|
[70ab73] | 8127 | if(size(#)>1) |
---|
| 8128 | { |
---|
| 8129 | primary[1]=ideal(1); |
---|
| 8130 | primary[2]=ideal(1); |
---|
| 8131 | primary[3]=ideal(1); |
---|
| 8132 | primary[4]=ideal(1); |
---|
| 8133 | return(primary); |
---|
| 8134 | } |
---|
| 8135 | for(@k=1;@k<=size(act[1]);@k++) |
---|
| 8136 | { |
---|
| 8137 | if(@wr==0) |
---|
| 8138 | { |
---|
| 8139 | primary[2*@k-1]=std(j,act[1][@k]^act[2][@k]); |
---|
| 8140 | } |
---|
| 8141 | else |
---|
| 8142 | { |
---|
| 8143 | primary[2*@k-1]=std(j,act[1][@k]); |
---|
| 8144 | } |
---|
| 8145 | if((act[2][@k]==1)&&(vdim(primary[2*@k-1])==deg(act[1][@k]))) |
---|
| 8146 | { |
---|
| 8147 | primary[2*@k] = primary[2*@k-1]; |
---|
| 8148 | } |
---|
| 8149 | else |
---|
| 8150 | { |
---|
| 8151 | primary[2*@k] = primaryTest(primary[2*@k-1],act[1][@k]); |
---|
| 8152 | } |
---|
| 8153 | } |
---|
[808a9f3] | 8154 | } |
---|
| 8155 | else |
---|
| 8156 | { |
---|
[70ab73] | 8157 | primary[1]=j; |
---|
| 8158 | if((size(#)>0)&&(act[2][1]>1)) |
---|
| 8159 | { |
---|
| 8160 | act[2]=1; |
---|
| 8161 | primary[1]=std(primary[1],act[1][1]); |
---|
| 8162 | } |
---|
| 8163 | if(@wr!=0) |
---|
| 8164 | { |
---|
| 8165 | primary[1]=std(j,act[1][1]); |
---|
| 8166 | } |
---|
| 8167 | if((act[2][1]==1)&&(vdim(primary[1])==deg(act[1][1]))) |
---|
| 8168 | { |
---|
| 8169 | primary[2]=primary[1]; |
---|
| 8170 | } |
---|
| 8171 | else |
---|
| 8172 | { |
---|
| 8173 | primary[2]=primaryTest(primary[1],act[1][1]); |
---|
| 8174 | } |
---|
[808a9f3] | 8175 | } |
---|
| 8176 | |
---|
| 8177 | if(size(#)==0) |
---|
| 8178 | { |
---|
[70ab73] | 8179 | primary=splitPrimary(primary,ser,@wr,act); |
---|
[808a9f3] | 8180 | } |
---|
| 8181 | |
---|
| 8182 | if((voice>=6)&&(char(basering)<=181)) |
---|
| 8183 | { |
---|
[70ab73] | 8184 | primary=splitCharp(primary); |
---|
[808a9f3] | 8185 | } |
---|
| 8186 | |
---|
| 8187 | if((@wr==2)&&(npars(basering)>0)&&(voice>=6)&&(char(basering)>0)) |
---|
| 8188 | { |
---|
| 8189 | //the prime decomposition of Yokoyama in characteristic p |
---|
[70ab73] | 8190 | list ke,ek; |
---|
| 8191 | @k=0; |
---|
[4173c7] | 8192 | while(@k<size(primary) div 2) |
---|
[70ab73] | 8193 | { |
---|
| 8194 | @k++; |
---|
| 8195 | if(size(primary[2*@k])==0) |
---|
| 8196 | { |
---|
| 8197 | ek=insepDecomp(primary[2*@k-1]); |
---|
| 8198 | primary=delete(primary,2*@k); |
---|
| 8199 | primary=delete(primary,2*@k-1); |
---|
| 8200 | @k--; |
---|
| 8201 | } |
---|
| 8202 | ke=ke+ek; |
---|
| 8203 | } |
---|
| 8204 | for(@k=1;@k<=size(ke);@k++) |
---|
| 8205 | { |
---|
| 8206 | primary[size(primary)+1]=ke[@k]; |
---|
| 8207 | primary[size(primary)+1]=ke[@k]; |
---|
| 8208 | } |
---|
[808a9f3] | 8209 | } |
---|
| 8210 | |
---|
[7f7c25e] | 8211 | if(nestLevel > 1){primary=extF(primary);} |
---|
[808a9f3] | 8212 | |
---|
| 8213 | //test whether all ideals in the decomposition are primary and |
---|
| 8214 | //in general position |
---|
| 8215 | //if not after a random coordinate transformation of the last |
---|
| 8216 | //variable the corresponding ideal is decomposed again. |
---|
| 8217 | if((npars(basering)>0)&&(nestLevel > 1)) |
---|
| 8218 | { |
---|
[70ab73] | 8219 | poly randp; |
---|
| 8220 | for(zz=1;zz<nvars(basering);zz++) |
---|
| 8221 | { |
---|
| 8222 | randp=randp |
---|
[808a9f3] | 8223 | +(random(0,5)*par(1)^2+random(0,5)*par(1)+random(0,5))*var(zz); |
---|
[70ab73] | 8224 | } |
---|
| 8225 | randp=randp+var(nvars(basering)); |
---|
[808a9f3] | 8226 | } |
---|
| 8227 | @k=0; |
---|
[4173c7] | 8228 | while(@k<(size(primary) div 2)) |
---|
[808a9f3] | 8229 | { |
---|
| 8230 | @k++; |
---|
| 8231 | if (size(primary[2*@k])==0) |
---|
| 8232 | { |
---|
[70ab73] | 8233 | for(zz=1;zz<size(primary[2*@k-1])-1;zz++) |
---|
| 8234 | { |
---|
| 8235 | attrib(primary[2*@k-1],"isSB",1); |
---|
| 8236 | if(vdim(primary[2*@k-1])==deg(primary[2*@k-1][zz])) |
---|
| 8237 | { |
---|
| 8238 | primary[2*@k]=primary[2*@k-1]; |
---|
| 8239 | } |
---|
| 8240 | } |
---|
[808a9f3] | 8241 | } |
---|
| 8242 | } |
---|
| 8243 | |
---|
| 8244 | @k=0; |
---|
| 8245 | ideal keep; |
---|
[4173c7] | 8246 | while(@k<(size(primary) div 2)) |
---|
[808a9f3] | 8247 | { |
---|
| 8248 | @k++; |
---|
| 8249 | if (size(primary[2*@k])==0) |
---|
| 8250 | { |
---|
[70ab73] | 8251 | jmap=randomLast(100); |
---|
| 8252 | jmap1=maxideal(1); |
---|
| 8253 | jmap2=maxideal(1); |
---|
| 8254 | @qht=primary[2*@k-1]; |
---|
| 8255 | if((npars(basering)>0)&&(nestLevel > 1)) |
---|
| 8256 | { |
---|
| 8257 | jmap[size(jmap)]=randp; |
---|
| 8258 | } |
---|
[808a9f3] | 8259 | |
---|
[70ab73] | 8260 | for(@n=2;@n<=size(primary[2*@k-1]);@n++) |
---|
| 8261 | { |
---|
| 8262 | if(deg(lead(primary[2*@k-1][@n]))==1) |
---|
| 8263 | { |
---|
| 8264 | for(zz=1;zz<=nva;zz++) |
---|
[808a9f3] | 8265 | { |
---|
[70ab73] | 8266 | if(lead(primary[2*@k-1][@n])/var(zz)!=0) |
---|
| 8267 | { |
---|
| 8268 | jmap1[zz]=-1/leadcoef(primary[2*@k-1][@n])*primary[2*@k-1][@n] |
---|
[a36e78] | 8269 | +2/leadcoef(primary[2*@k-1][@n])*lead(primary[2*@k-1][@n]); |
---|
[70ab73] | 8270 | jmap2[zz]=primary[2*@k-1][@n]; |
---|
| 8271 | @qht[@n]=var(zz); |
---|
| 8272 | } |
---|
[808a9f3] | 8273 | } |
---|
[70ab73] | 8274 | jmap[nva]=subst(jmap[nva],lead(primary[2*@k-1][@n]),0); |
---|
| 8275 | } |
---|
| 8276 | } |
---|
| 8277 | if(size(subst(jmap[nva],var(1),0)-var(nva))!=0) |
---|
| 8278 | { |
---|
| 8279 | // jmap[nva]=subst(jmap[nva],var(1),0); |
---|
| 8280 | //hier geaendert +untersuchen!!!!!!!!!!!!!! |
---|
| 8281 | } |
---|
| 8282 | phi1=@P,jmap1; |
---|
| 8283 | phi=@P,jmap; |
---|
| 8284 | for(@n=1;@n<=nva;@n++) |
---|
| 8285 | { |
---|
| 8286 | jmap[@n]=-(jmap[@n]-2*var(@n)); |
---|
| 8287 | } |
---|
| 8288 | psi=@P,jmap; |
---|
| 8289 | psi1=@P,jmap2; |
---|
| 8290 | @qh=phi(@qht); |
---|
[808a9f3] | 8291 | |
---|
| 8292 | //=================== the new part ============================ |
---|
| 8293 | |
---|
[8992ed] | 8294 | if (npars(basering)>1) { @qh=groebner(@qh,"par2var"); } |
---|
| 8295 | else { @qh=groebner(@qh); } |
---|
[808a9f3] | 8296 | |
---|
| 8297 | //============================================================= |
---|
| 8298 | // if(npars(@P)>0) |
---|
| 8299 | // { |
---|
| 8300 | // @ri= "ring @Phelp =" |
---|
| 8301 | // +string(char(@P))+", |
---|
| 8302 | // ("+varstr(@P)+","+parstr(@P)+",@t),(C,dp);"; |
---|
| 8303 | // } |
---|
| 8304 | // else |
---|
| 8305 | // { |
---|
| 8306 | // @ri= "ring @Phelp =" |
---|
| 8307 | // +string(char(@P))+",("+varstr(@P)+",@t),(C,dp);"; |
---|
| 8308 | // } |
---|
| 8309 | // execute(@ri); |
---|
| 8310 | // ideal @qh=homog(imap(@P,@qht),@t); |
---|
| 8311 | // |
---|
| 8312 | // ideal @qh1=std(@qh); |
---|
| 8313 | // @hilb=hilb(@qh1,1); |
---|
| 8314 | // @ri= "ring @Phelp1 =" |
---|
| 8315 | // +string(char(@P))+",("+varstr(@Phelp)+"),(C,lp);"; |
---|
| 8316 | // execute(@ri); |
---|
| 8317 | // ideal @qh=homog(imap(@P,@qh),@t); |
---|
| 8318 | // kill @Phelp; |
---|
| 8319 | // @qh=std(@qh,@hilb); |
---|
| 8320 | // @qh=subst(@qh,@t,1); |
---|
| 8321 | // setring @P; |
---|
| 8322 | // @qh=imap(@Phelp1,@qh); |
---|
| 8323 | // kill @Phelp1; |
---|
| 8324 | // @qh=clearSB(@qh); |
---|
| 8325 | // attrib(@qh,"isSB",1); |
---|
| 8326 | //============================================================= |
---|
| 8327 | |
---|
[70ab73] | 8328 | ser1=phi1(ser); |
---|
| 8329 | @lh=newZero_decomp (@qh,phi(ser1),@wr, list("nest", nestLevel + 1)); |
---|
[808a9f3] | 8330 | |
---|
[70ab73] | 8331 | kill lres0; |
---|
| 8332 | list lres0; |
---|
| 8333 | if(size(@lh)==2) |
---|
| 8334 | { |
---|
| 8335 | helpprim=@lh[2]; |
---|
| 8336 | lres0[1]=primary[2*@k-1]; |
---|
| 8337 | ser1=psi(helpprim); |
---|
| 8338 | lres0[2]=psi1(ser1); |
---|
| 8339 | if(size(reduce(lres0[2],lres0[1],1))==0) |
---|
| 8340 | { |
---|
| 8341 | primary[2*@k]=primary[2*@k-1]; |
---|
| 8342 | continue; |
---|
| 8343 | } |
---|
| 8344 | } |
---|
| 8345 | else |
---|
| 8346 | { |
---|
| 8347 | lres1=psi(@lh); |
---|
| 8348 | lres0=psi1(lres1); |
---|
| 8349 | } |
---|
[808a9f3] | 8350 | |
---|
| 8351 | //=================== the new part ============================ |
---|
| 8352 | |
---|
[70ab73] | 8353 | primary=delete(primary,2*@k-1); |
---|
| 8354 | primary=delete(primary,2*@k-1); |
---|
| 8355 | @k--; |
---|
| 8356 | if(size(lres0)==2) |
---|
| 8357 | { |
---|
[8992ed] | 8358 | if (npars(basering)>1) { lres0[2]=groebner(lres0[2],"par2var"); } |
---|
| 8359 | else { lres0[2]=groebner(lres0[2]); } |
---|
[70ab73] | 8360 | } |
---|
| 8361 | else |
---|
| 8362 | { |
---|
[4173c7] | 8363 | for(@n=1;@n<=size(lres0) div 2;@n++) |
---|
[70ab73] | 8364 | { |
---|
| 8365 | if(specialIdealsEqual(lres0[2*@n-1],lres0[2*@n])==1) |
---|
[808a9f3] | 8366 | { |
---|
[a36e78] | 8367 | lres0[2*@n-1]=groebner(lres0[2*@n-1]); |
---|
[70ab73] | 8368 | lres0[2*@n]=lres0[2*@n-1]; |
---|
| 8369 | attrib(lres0[2*@n],"isSB",1); |
---|
[808a9f3] | 8370 | } |
---|
[70ab73] | 8371 | else |
---|
| 8372 | { |
---|
[a36e78] | 8373 | lres0[2*@n-1]=groebner(lres0[2*@n-1]); |
---|
| 8374 | lres0[2*@n]=groebner(lres0[2*@n]); |
---|
[70ab73] | 8375 | } |
---|
| 8376 | } |
---|
| 8377 | } |
---|
| 8378 | primary=primary+lres0; |
---|
[808a9f3] | 8379 | |
---|
| 8380 | //============================================================= |
---|
| 8381 | // if(npars(@P)>0) |
---|
| 8382 | // { |
---|
| 8383 | // @ri= "ring @Phelp =" |
---|
| 8384 | // +string(char(@P))+", |
---|
| 8385 | // ("+varstr(@P)+","+parstr(@P)+",@t),(C,dp);"; |
---|
| 8386 | // } |
---|
| 8387 | // else |
---|
| 8388 | // { |
---|
| 8389 | // @ri= "ring @Phelp =" |
---|
| 8390 | // +string(char(@P))+",("+varstr(@P)+",@t),(C,dp);"; |
---|
| 8391 | // } |
---|
| 8392 | // execute(@ri); |
---|
| 8393 | // list @lvec; |
---|
| 8394 | // list @lr=imap(@P,lres0); |
---|
| 8395 | // ideal @lr1; |
---|
| 8396 | // |
---|
| 8397 | // if(size(@lr)==2) |
---|
| 8398 | // { |
---|
| 8399 | // @lr[2]=homog(@lr[2],@t); |
---|
| 8400 | // @lr1=std(@lr[2]); |
---|
| 8401 | // @lvec[2]=hilb(@lr1,1); |
---|
| 8402 | // } |
---|
| 8403 | // else |
---|
| 8404 | // { |
---|
[4173c7] | 8405 | // for(@n=1;@n<=size(@lr) div 2;@n++) |
---|
[808a9f3] | 8406 | // { |
---|
| 8407 | // if(specialIdealsEqual(@lr[2*@n-1],@lr[2*@n])==1) |
---|
| 8408 | // { |
---|
| 8409 | // @lr[2*@n-1]=homog(@lr[2*@n-1],@t); |
---|
| 8410 | // @lr1=std(@lr[2*@n-1]); |
---|
| 8411 | // @lvec[2*@n-1]=hilb(@lr1,1); |
---|
| 8412 | // @lvec[2*@n]=@lvec[2*@n-1]; |
---|
| 8413 | // } |
---|
| 8414 | // else |
---|
| 8415 | // { |
---|
| 8416 | // @lr[2*@n-1]=homog(@lr[2*@n-1],@t); |
---|
| 8417 | // @lr1=std(@lr[2*@n-1]); |
---|
| 8418 | // @lvec[2*@n-1]=hilb(@lr1,1); |
---|
| 8419 | // @lr[2*@n]=homog(@lr[2*@n],@t); |
---|
| 8420 | // @lr1=std(@lr[2*@n]); |
---|
| 8421 | // @lvec[2*@n]=hilb(@lr1,1); |
---|
| 8422 | // |
---|
| 8423 | // } |
---|
| 8424 | // } |
---|
| 8425 | // } |
---|
| 8426 | // @ri= "ring @Phelp1 =" |
---|
| 8427 | // +string(char(@P))+",("+varstr(@Phelp)+"),(C,lp);"; |
---|
| 8428 | // execute(@ri); |
---|
| 8429 | // list @lr=imap(@Phelp,@lr); |
---|
| 8430 | // |
---|
| 8431 | // kill @Phelp; |
---|
| 8432 | // if(size(@lr)==2) |
---|
| 8433 | // { |
---|
| 8434 | // @lr[2]=std(@lr[2],@lvec[2]); |
---|
| 8435 | // @lr[2]=subst(@lr[2],@t,1); |
---|
| 8436 | // |
---|
| 8437 | // } |
---|
| 8438 | // else |
---|
| 8439 | // { |
---|
[4173c7] | 8440 | // for(@n=1;@n<=size(@lr) div 2;@n++) |
---|
[808a9f3] | 8441 | // { |
---|
| 8442 | // if(specialIdealsEqual(@lr[2*@n-1],@lr[2*@n])==1) |
---|
| 8443 | // { |
---|
| 8444 | // @lr[2*@n-1]=std(@lr[2*@n-1],@lvec[2*@n-1]); |
---|
| 8445 | // @lr[2*@n-1]=subst(@lr[2*@n-1],@t,1); |
---|
| 8446 | // @lr[2*@n]=@lr[2*@n-1]; |
---|
| 8447 | // attrib(@lr[2*@n],"isSB",1); |
---|
| 8448 | // } |
---|
| 8449 | // else |
---|
| 8450 | // { |
---|
| 8451 | // @lr[2*@n-1]=std(@lr[2*@n-1],@lvec[2*@n-1]); |
---|
| 8452 | // @lr[2*@n-1]=subst(@lr[2*@n-1],@t,1); |
---|
| 8453 | // @lr[2*@n]=std(@lr[2*@n],@lvec[2*@n]); |
---|
| 8454 | // @lr[2*@n]=subst(@lr[2*@n],@t,1); |
---|
| 8455 | // } |
---|
| 8456 | // } |
---|
| 8457 | // } |
---|
| 8458 | // kill @lvec; |
---|
| 8459 | // setring @P; |
---|
| 8460 | // lres0=imap(@Phelp1,@lr); |
---|
| 8461 | // kill @Phelp1; |
---|
| 8462 | // for(@n=1;@n<=size(lres0);@n++) |
---|
| 8463 | // { |
---|
| 8464 | // lres0[@n]=clearSB(lres0[@n]); |
---|
| 8465 | // attrib(lres0[@n],"isSB",1); |
---|
| 8466 | // } |
---|
| 8467 | // |
---|
| 8468 | // primary[2*@k-1]=lres0[1]; |
---|
| 8469 | // primary[2*@k]=lres0[2]; |
---|
[4173c7] | 8470 | // @s=size(primary) div 2; |
---|
| 8471 | // for(@n=1;@n<=size(lres0) div 2-1;@n++) |
---|
[808a9f3] | 8472 | // { |
---|
| 8473 | // primary[2*@s+2*@n-1]=lres0[2*@n+1]; |
---|
| 8474 | // primary[2*@s+2*@n]=lres0[2*@n+2]; |
---|
| 8475 | // } |
---|
| 8476 | // @k--; |
---|
| 8477 | //============================================================= |
---|
[70ab73] | 8478 | } |
---|
[808a9f3] | 8479 | } |
---|
| 8480 | return(primary); |
---|
| 8481 | } |
---|
| 8482 | example |
---|
| 8483 | { "EXAMPLE:"; echo = 2; |
---|
| 8484 | ring r = 0,(x,y,z),lp; |
---|
| 8485 | poly p = z2+1; |
---|
| 8486 | poly q = z4+2; |
---|
| 8487 | ideal i = p^2*q^3,(y-z3)^3,(x-yz+z4)^4; |
---|
| 8488 | i=std(i); |
---|
| 8489 | list pr= newZero_decomp(i,ideal(0),0); |
---|
| 8490 | pr; |
---|
| 8491 | } |
---|
| 8492 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 8493 | |
---|
[55fcff] | 8494 | //////////////////////////////////////////////////////////////////////////// |
---|
| 8495 | /* |
---|
| 8496 | //Beispiele Wenk-Dipl (in ~/Texfiles/Diplom/Wenk/Examples/) |
---|
| 8497 | //Zeiten: Singular/Singular/Singular -r123456789 -v :wilde13 (PentiumPro200) |
---|
| 8498 | //Singular for HPUX-9 version 1-3-8 (2000060214) Jun 2 2000 15:31:26 |
---|
| 8499 | //(wilde13) |
---|
| 8500 | |
---|
| 8501 | //1. vdim=20, 3 Komponenten |
---|
| 8502 | //zerodec-time:2(1) (matrix:1 charpoly:0 factor:1) |
---|
| 8503 | //primdecGTZ-time: 1(0) |
---|
| 8504 | ring rs= 0,(a,b,c),dp; |
---|
| 8505 | poly f1= a^2*b*c + a*b^2*c + a*b*c^2 + a*b*c + a*b + a*c + b*c; |
---|
| 8506 | poly f2= a^2*b^2*c + a*b^2*c^2 + a^2*b*c + a*b*c + b*c + a + c; |
---|
| 8507 | poly f3= a^2*b^2*c^2 + a^2*b^2*c + a*b^2*c + a*b*c + a*c + c + 1; |
---|
| 8508 | ideal gls=f1,f2,f3; |
---|
| 8509 | int time=timer; |
---|
| 8510 | printlevel =1; |
---|
| 8511 | time=timer; list pr1=zerodec(gls); timer-time;size(pr1); |
---|
| 8512 | time=timer; list pr =primdecGTZ(gls); timer-time;size(pr); |
---|
[07c623] | 8513 | time=timer; ideal ra =radical(gls); timer-time;size(pr); |
---|
[55fcff] | 8514 | |
---|
| 8515 | //2.cyclic5 vdim=70, 20 Komponenten |
---|
| 8516 | //zerodec-time:36(28) (matrix:1(0) charpoly:18(19) factor:17(9) |
---|
| 8517 | //primdecGTZ-time: 28(5) |
---|
[b9b906] | 8518 | //radical : 0 |
---|
[55fcff] | 8519 | ring rs= 0,(a,b,c,d,e),dp; |
---|
| 8520 | poly f0= a + b + c + d + e + 1; |
---|
| 8521 | poly f1= a + b + c + d + e; |
---|
| 8522 | poly f2= a*b + b*c + c*d + a*e + d*e; |
---|
| 8523 | poly f3= a*b*c + b*c*d + a*b*e + a*d*e + c*d*e; |
---|
| 8524 | poly f4= a*b*c*d + a*b*c*e + a*b*d*e + a*c*d*e + b*c*d*e; |
---|
| 8525 | poly f5= a*b*c*d*e - 1; |
---|
| 8526 | ideal gls= f1,f2,f3,f4,f5; |
---|
| 8527 | |
---|
| 8528 | //3. random vdim=40, 1 Komponente |
---|
| 8529 | //zerodec-time:126(304) (matrix:1 charpoly:115(298) factor:10(5)) |
---|
[b9b906] | 8530 | //primdecGTZ-time:17 (11) |
---|
[55fcff] | 8531 | ring rs=0,(x,y,z),dp; |
---|
| 8532 | poly f1=2*x^2 + 4*x + 3*y^2 + 7*x*z + 9*y*z + 5*z^2; |
---|
| 8533 | poly f2=7*x^3 + 8*x*y + 12*y^2 + 18*x*z + 3*y^4*z + 10*z^3 + 12; |
---|
| 8534 | poly f3=3*x^4 + 1*x*y*z + 6*y^3 + 3*x*z^2 + 2*y*z^2 + 4*z^2 + 5; |
---|
| 8535 | ideal gls=f1,f2,f3; |
---|
| 8536 | |
---|
| 8537 | //4. introduction into resultants, sturmfels, vdim=28, 1 Komponente |
---|
| 8538 | //zerodec-time:4 (matrix:0 charpoly:0 factor:4) |
---|
[b9b906] | 8539 | //primdecGTZ-time:1 |
---|
[55fcff] | 8540 | ring rs=0,(x,y),dp; |
---|
| 8541 | poly f1= x4+y4-1; |
---|
| 8542 | poly f2= x5y2-4x3y3+x2y5-1; |
---|
| 8543 | ideal gls=f1,f2; |
---|
| 8544 | |
---|
| 8545 | //5. 3 quadratic equations with random coeffs, vdim=8, 1 Komponente |
---|
| 8546 | //zerodec-time:0(0) (matrix:0 charpoly:0 factor:0) |
---|
[b9b906] | 8547 | //primdecGTZ-time:1(0) |
---|
[55fcff] | 8548 | ring rs=0,(x,y,z),dp; |
---|
| 8549 | poly f1=2*x^2 + 4*x*y + 3*y^2 + 7*x*z + 9*y*z + 5*z^2 + 2; |
---|
| 8550 | poly f2=7*x^2 + 8*x*y + 12*y^2 + 18*x*z + 3*y*z + 10*z^2 + 12; |
---|
| 8551 | poly f3=3*x^2 + 1*x*y + 6*y^2 + 3*x*z + 2*y*z + 4*z^2 + 5; |
---|
| 8552 | ideal gls=f1,f2,f3; |
---|
| 8553 | |
---|
| 8554 | //6. 3 polys vdim=24, 1 Komponente |
---|
| 8555 | // run("ex14",2); |
---|
| 8556 | //zerodec-time:5(4) (matrix:0 charpoly:3(3) factor:2(1)) |
---|
| 8557 | //primdecGTZ-time:4 (2) |
---|
| 8558 | ring rs=0,(x1,x2,x3,x4),dp; |
---|
| 8559 | poly f1=16*x1^2 + 3*x2^2 + 5*x3^4 - 1 - 4*x4 + x4^3; |
---|
| 8560 | poly f2=5*x1^3 + 3*x2^2 + 4*x3^2*x4 + 2*x1*x4 - 1 + x4 + 4*x1 + x2 + x3 + x4; |
---|
| 8561 | poly f3=-4*x1^2 + x2^2 + x3^2 - 3 + x4^2 + 4*x1 + x2 + x3 + x4; |
---|
| 8562 | poly f4=-4*x1 + x2 + x3 + x4; |
---|
| 8563 | ideal gls=f1,f2,f3,f4; |
---|
| 8564 | |
---|
| 8565 | //7. ex43, PoSSo, caprasse vdim=56, 16 Komponenten |
---|
[b9b906] | 8566 | //zerodec-time:23(15) (matrix:0 charpoly:16(13) factor:3(2)) |
---|
[55fcff] | 8567 | //primdecGTZ-time:3 (2) |
---|
| 8568 | ring rs= 0,(y,z,x,t),dp; |
---|
| 8569 | ideal gls=y^2*z+2*y*x*t-z-2*x, |
---|
| 8570 | 4*y^2*z*x-z*x^3+2*y^3*t+4*y*x^2*t-10*y^2+4*z*x+4*x^2-10*y*t+2, |
---|
| 8571 | 2*y*z*t+x*t^2-2*z-x, |
---|
| 8572 | -z^3*x+4*y*z^2*t+4*z*x*t^2+2*y*t^3+4*z^2+4*z*x-10*y*t-10*t^2+2; |
---|
| 8573 | |
---|
| 8574 | //8. Arnborg-System, n=6 (II), vdim=156, 90 Komponenten |
---|
| 8575 | //zerodec-time (char32003):127(45)(matrix:2(0) charpoly:106(37) factor:16(7)) |
---|
| 8576 | //primdecGTZ-time(char32003) :81 (18) |
---|
| 8577 | //ring rs= 0,(a,b,c,d,x,f),dp; |
---|
| 8578 | ring rs= 32003,(a,b,c,d,x,f),dp; |
---|
[b9b906] | 8579 | ideal gls=a+b+c+d+x+f, ab+bc+cd+dx+xf+af, abc+bcd+cdx+d*xf+axf+abf, |
---|
| 8580 | abcd+bcdx+cd*xf+ad*xf+abxf+abcf, abcdx+bcd*xf+acd*xf+abd*xf+abcxf+abcdf, |
---|
[55fcff] | 8581 | abcd*xf-1; |
---|
| 8582 | |
---|
| 8583 | //9. ex42, PoSSo, Methan6_1, vdim=27, 2 Komponenten |
---|
| 8584 | //zerodec-time:610 (matrix:10 charpoly:557 factor:26) |
---|
| 8585 | //primdecGTZ-time: 118 |
---|
| 8586 | //zerodec-time(char32003):2 |
---|
| 8587 | //primdecGTZ-time(char32003):4 |
---|
| 8588 | //ring rs= 0,(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10),dp; |
---|
| 8589 | ring rs= 32003,(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10),dp; |
---|
| 8590 | ideal gls=64*x2*x7-10*x1*x8+10*x7*x9+11*x7*x10-320000*x1, |
---|
| 8591 | -32*x2*x7-5*x2*x8-5*x2*x10+160000*x1-5000*x2, |
---|
| 8592 | -x3*x8+x6*x8+x9*x10+210*x6+1300000, |
---|
| 8593 | -x4*x8+700000, |
---|
| 8594 | x10^2-2*x5, |
---|
| 8595 | -x6*x8+x7*x9-210*x6, |
---|
| 8596 | -64*x2*x7-10*x7*x9-11*x7*x10+320000*x1-16*x7+7000000, |
---|
| 8597 | -10*x1*x8-10*x2*x8-10*x3*x8-10*x4*x8-10*x6*x8+10*x2*x10+11*x7*x10 |
---|
| 8598 | +20000*x2+14*x5, |
---|
| 8599 | x4*x8-x7*x9-x9*x10-410*x9, |
---|
| 8600 | 10*x2*x8+10*x3*x8+10*x6*x8+10*x7*x9-10*x2*x10-11*x7*x10-10*x9*x10 |
---|
| 8601 | -10*x10^2+1400*x6-4200*x10; |
---|
| 8602 | |
---|
| 8603 | //10. ex33, walk-s7, Diplomarbeit von Tim, vdim=114 |
---|
| 8604 | //zerfaellt in unterschiedlich viele Komponenten in versch. Charkteristiken: |
---|
| 8605 | //char32003:30, char0:3(2xdeg1,1xdeg112!), char181:4(2xdeg1,1xdeg28,1xdeg84) |
---|
| 8606 | //char 0: zerodec-time:10075 (ca 3h) (matrix:3 charpoly:9367, factor:680 |
---|
| 8607 | // + 24 sec fuer Normalform (anstatt einsetzen), total [29623k]) |
---|
| 8608 | // primdecGTZ-time: 214 |
---|
| 8609 | //char 32003:zerodec-time:197(68) (matrix:2(1) charpoly:173(60) factor:15(6)) |
---|
| 8610 | // primdecGTZ-time:14 (5) |
---|
| 8611 | //char 181:zerodec-time:(87) (matrix:(1) charpoly:(58) factor:(25)) |
---|
| 8612 | // primdecGTZ-time:(2) |
---|
| 8613 | //in char181 stimmen Ergebnisse von zerodec und primdecGTZ ueberein (gecheckt) |
---|
| 8614 | |
---|
| 8615 | //ring rs= 0,(a,b,c,d,e,f,g),dp; |
---|
| 8616 | ring rs= 32003,(a,b,c,d,e,f,g),dp; |
---|
| 8617 | poly f1= 2gb + 2fc + 2ed + a2 + a; |
---|
| 8618 | poly f2= 2gc + 2fd + e2 + 2ba + b; |
---|
| 8619 | poly f3= 2gd + 2fe + 2ca + c + b2; |
---|
| 8620 | poly f4= 2ge + f2 + 2da + d + 2cb; |
---|
| 8621 | poly f5= 2fg + 2ea + e + 2db + c2; |
---|
| 8622 | poly f6= g2 + 2fa + f + 2eb + 2dc; |
---|
| 8623 | poly f7= 2ga + g + 2fb + 2ec + d2; |
---|
| 8624 | ideal gls= f1,f2,f3,f4,f5,f6,f7; |
---|
| 8625 | |
---|
| 8626 | ~/Singular/Singular/Singular -r123456789 -v |
---|
| 8627 | LIB"./primdec.lib"; |
---|
| 8628 | timer=1; |
---|
| 8629 | int time=timer; |
---|
| 8630 | printlevel =1; |
---|
| 8631 | option(prot,mem); |
---|
| 8632 | time=timer; list pr1=zerodec(gls); timer-time; |
---|
| 8633 | |
---|
| 8634 | time=timer; list pr =primdecGTZ(gls); timer-time; |
---|
| 8635 | time=timer; list pr =primdecSY(gls); timer-time; |
---|
[07c623] | 8636 | time=timer; ideal ra =radical(gls); timer-time;size(pr); |
---|
[24f458] | 8637 | LIB"all.lib"; |
---|
| 8638 | |
---|
| 8639 | ring R=0,(a,b,c,d,e,f),dp; |
---|
| 8640 | ideal I=cyclic(6); |
---|
| 8641 | minAssGTZ(I); |
---|
| 8642 | |
---|
| 8643 | |
---|
| 8644 | ring S=(2,a,b),(x,y),lp; |
---|
| 8645 | ideal I=x8-b,y4+a; |
---|
| 8646 | minAssGTZ(I); |
---|
| 8647 | |
---|
| 8648 | ring S1=2,(x,y,a,b),lp; |
---|
| 8649 | ideal I=x8-b,y4+a; |
---|
| 8650 | minAssGTZ(I); |
---|
| 8651 | |
---|
| 8652 | |
---|
| 8653 | ring S2=(2,z),(x,y),dp; |
---|
| 8654 | minpoly=z2+z+1; |
---|
| 8655 | ideal I=y3+y+1,x4+x+1; |
---|
| 8656 | primdecGTZ(I); |
---|
| 8657 | minAssGTZ(I); |
---|
| 8658 | |
---|
| 8659 | ring S3=2,(x,y,z),dp; |
---|
| 8660 | ideal I=y3+y+1,x4+x+1,z2+z+1; |
---|
| 8661 | primdecGTZ(I); |
---|
| 8662 | minAssGTZ(I); |
---|
| 8663 | |
---|
| 8664 | |
---|
| 8665 | ring R1=2,(x,y,z),lp; |
---|
| 8666 | ideal I=y6+y5+y3+y2+1,x4+x+1,z2+z+1; |
---|
| 8667 | primdecGTZ(I); |
---|
| 8668 | minAssGTZ(I); |
---|
| 8669 | |
---|
| 8670 | |
---|
| 8671 | ring R2=(2,z),(x,y),lp; |
---|
| 8672 | minpoly=z3+z+1; |
---|
| 8673 | ideal I=y2+y+(z2+z+1),x4+x+1; |
---|
| 8674 | primdecGTZ(I); |
---|
| 8675 | minAssGTZ(I); |
---|
| 8676 | |
---|
[55fcff] | 8677 | */ |
---|