[f6e355] | 1 | version="$Id: equising.lib,v 1.9 2005-04-15 15:20:12 Singular Exp $"; |
---|
[fd3fb7] | 2 | category="Singularities"; |
---|
[e7cc147] | 3 | info=" |
---|
[558eb2] | 4 | LIBRARY: equising.lib Equisingularity Stratum of a Family of Plane Curves |
---|
[f6e355] | 5 | AUTHOR: Christoph Lossen, lossen@mathematik.uni-kl.de |
---|
| 6 | Andrea Mindnich, mindnich@mathematik.uni-kl.de |
---|
[4ac997] | 7 | |
---|
[f6e355] | 8 | MAIN PROCEDURES: |
---|
| 9 | tau_es(f); codim of mu-const stratum in semi-universal def. base |
---|
| 10 | esIdeal(f); (Wahl's) equisingularity ideal of f |
---|
| 11 | esStratum(F[,m,L]); equisingularity stratum of a family F |
---|
| 12 | isEquising(F[,m,L]); tests if a given deformation is equisingular |
---|
| 13 | |
---|
| 14 | AUXILIARY PROCEDURE: |
---|
| 15 | control_Matrix(M); computes list of blowing-up data |
---|
[5d9ec3] | 16 | "; |
---|
[e7cc147] | 17 | |
---|
[f6e355] | 18 | LIB "hnoether.lib"; |
---|
[e7cc147] | 19 | LIB "poly.lib"; |
---|
| 20 | LIB "elim.lib"; |
---|
[f6e355] | 21 | LIB "deform.lib"; |
---|
| 22 | LIB "sing.lib"; |
---|
| 23 | |
---|
| 24 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 25 | // |
---|
| 26 | // The following (static) procedures are used by esComputation |
---|
| 27 | // |
---|
| 28 | //////////////////////////////////////////////////////////////////////////////// |
---|
[e7cc147] | 29 | // COMPUTES a weight vector. x and y get weight 1 and all other |
---|
| 30 | // variables get weight 0. |
---|
| 31 | static proc xyVector() |
---|
| 32 | { |
---|
| 33 | intvec iv ; |
---|
| 34 | iv[nvars(basering)]=0 ; |
---|
| 35 | iv[rvar(x)] =1; |
---|
| 36 | iv[rvar(y)] =1; |
---|
| 37 | return (iv); |
---|
| 38 | } |
---|
[f6e355] | 39 | |
---|
| 40 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 41 | // exchanges the variables x and y in the polynomial f |
---|
[e7cc147] | 42 | static proc swapXY(poly f) |
---|
| 43 | { |
---|
| 44 | def r_base = basering; |
---|
| 45 | ideal MI = maxideal(1); |
---|
| 46 | MI[rvar(x)]=y; |
---|
| 47 | MI[rvar(y)]=x; |
---|
| 48 | map phi = r_base, MI; |
---|
| 49 | f=phi(f); |
---|
| 50 | return (f); |
---|
| 51 | } |
---|
[f6e355] | 52 | |
---|
| 53 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 54 | // computes m-jet w.r.t. the variables x,y (other variables weighted 0 |
---|
| 55 | static proc m_Jet(poly F,int m); |
---|
[e7cc147] | 56 | { |
---|
[f6e355] | 57 | intvec w=xyVector(); |
---|
| 58 | poly Fd=jet(F,m,w); |
---|
| 59 | return(Fd); |
---|
[e7cc147] | 60 | } |
---|
[f6e355] | 61 | |
---|
| 62 | |
---|
| 63 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 64 | // computes the 4 control matrices (input is multsequence(L)) |
---|
| 65 | proc control_Matrix(list M); |
---|
| 66 | "USAGE: control_Matrix(L); L list |
---|
| 67 | ASSUME: L is the output of multsequence(hnexpansion(f)). |
---|
| 68 | RETURN: list M of 4 intmat's: |
---|
| 69 | @format |
---|
| 70 | M[1] contains the multiplicities at the respective infinitely near points |
---|
| 71 | p[i,j] (i=step of blowup+1, j=branch) -- if branches j=k,...,k+m pass |
---|
| 72 | through the same p[i,j] then the multiplicity is stored in M[1][k,j], |
---|
| 73 | while M[1][k+1]=...=M[1][k+m]=0. |
---|
| 74 | M[2] contains the number of branches meeting at p[i,j] (again, the information |
---|
| 75 | is stored according to the above rule) |
---|
| 76 | M[3] contains the information about the splitting of M[1][i,j] with respect to |
---|
| 77 | different tangents of branches at p[i,j] (information is stored only for |
---|
| 78 | minimal j>=k corresponding to a new tangent direction). |
---|
| 79 | The entries are the sum of multiplicities of all branches with the |
---|
| 80 | respective tangent. |
---|
| 81 | M[4] contains the maximal sum of higher multiplicities for a branch passing |
---|
| 82 | through p[i,j] ( = degree Bound for blowing up) |
---|
| 83 | @end format |
---|
| 84 | NOTE: the branches are ordered in such a way that only consecutive branches |
---|
| 85 | can meet at an infinitely near point. @* |
---|
| 86 | the final rows of the matrices M[1],...,M[3] is (1,1,1,...,1), and |
---|
| 87 | correspond to infinitely near points such that the strict transforms |
---|
| 88 | of the branches are smooth and intersect the exceptional divisor |
---|
| 89 | transversally. |
---|
| 90 | SEE ALSO: multsequence |
---|
| 91 | EXAMPLE: example control_Matrix; shows an example |
---|
| 92 | " |
---|
[e7cc147] | 93 | { |
---|
[f6e355] | 94 | int i,j,k,dummy; |
---|
| 95 | |
---|
| 96 | dummy=0; |
---|
| 97 | for (j=1;j<=ncols(M[2]);j++) |
---|
| 98 | { |
---|
| 99 | dummy=dummy+M[1][nrows(M[1])-1,j]-M[1][nrows(M[1]),j]; |
---|
[e7cc147] | 100 | } |
---|
[f6e355] | 101 | intmat S[nrows(M[1])+dummy][ncols(M[1])]; |
---|
| 102 | intmat T[nrows(M[1])+dummy][ncols(M[1])]; |
---|
| 103 | intmat U[nrows(M[1])+dummy][ncols(M[1])]; |
---|
| 104 | intmat maxDeg[nrows(M[1])+dummy][ncols(M[1])]; |
---|
| 105 | |
---|
| 106 | for (i=1;i<=nrows(M[2]);i++) |
---|
[e7cc147] | 107 | { |
---|
[f6e355] | 108 | dummy=1; |
---|
| 109 | for (j=1;j<=ncols(M[2]);j++) |
---|
[e7cc147] | 110 | { |
---|
[f6e355] | 111 | for (k=dummy;k<dummy+M[2][i,j];k++) |
---|
| 112 | { |
---|
| 113 | T[i,dummy]=T[i,dummy]+1; |
---|
| 114 | S[i,dummy]=S[i,dummy]+M[1][i,k]; |
---|
| 115 | if (i>1) |
---|
| 116 | { |
---|
| 117 | U[i-1,dummy]=U[i-1,dummy]+M[1][i-1,k]; |
---|
| 118 | } |
---|
| 119 | } |
---|
| 120 | dummy=k; |
---|
[e7cc147] | 121 | } |
---|
| 122 | } |
---|
| 123 | |
---|
[f6e355] | 124 | // adding an extra row (in some cases needed to control ES-Stratum |
---|
| 125 | // computation) |
---|
| 126 | for (i=nrows(M[1]);i<=nrows(S);i++) |
---|
| 127 | { |
---|
| 128 | for (j=1;j<=ncols(M[2]);j++) |
---|
| 129 | { |
---|
| 130 | S[i,j]=1; |
---|
| 131 | T[i,j]=1; |
---|
| 132 | U[i,j]=1; |
---|
| 133 | } |
---|
| 134 | } |
---|
| 135 | |
---|
| 136 | // Computing the degree Bounds to be stored in M[4]: |
---|
| 137 | for (i=1;i<=nrows(S);i++) |
---|
[e7cc147] | 138 | { |
---|
[f6e355] | 139 | dummy=1; |
---|
| 140 | for (j=1;j<=ncols(S);j++) |
---|
| 141 | { |
---|
| 142 | for (k=dummy;k<dummy+T[i,j];k++) |
---|
| 143 | { |
---|
| 144 | maxDeg[i,k]=S[i,dummy]; // multiplicity at i-th blowup |
---|
| 145 | } |
---|
| 146 | dummy=k; |
---|
| 147 | } |
---|
[e7cc147] | 148 | } |
---|
[f6e355] | 149 | // adding up multiplicities |
---|
| 150 | for (i=nrows(S);i>=2;i--) |
---|
[e7cc147] | 151 | { |
---|
[f6e355] | 152 | for (j=1;j<=ncols(S);j++) |
---|
| 153 | { |
---|
| 154 | maxDeg[i-1,j]=maxDeg[i-1,j]+maxDeg[i,j]; |
---|
| 155 | } |
---|
[e7cc147] | 156 | } |
---|
[f6e355] | 157 | |
---|
| 158 | list L=S,T,U,maxDeg; |
---|
| 159 | return(L); |
---|
[e7cc147] | 160 | } |
---|
| 161 | |
---|
| 162 | |
---|
[f6e355] | 163 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 164 | // matrix of higher tangent directions: |
---|
| 165 | // returns list: 1) tangent directions |
---|
| 166 | // 2) swapping information (x <--> y) |
---|
| 167 | static proc inf_Tangents(list L,int s); // L aus hnexpansion, |
---|
| 168 | { |
---|
| 169 | int nv=nvars(basering); |
---|
| 170 | matrix M; |
---|
| 171 | matrix B[s][size(L)]; |
---|
| 172 | intvec V; |
---|
| 173 | intmat Mult=multsequence(L)[1]; |
---|
| 174 | |
---|
| 175 | int i,j,k,counter,e; |
---|
| 176 | for (k=1;k<=size(L);k++) |
---|
[e7cc147] | 177 | { |
---|
[f6e355] | 178 | V[k]=L[k][3]; // switch: 0 --> tangent 2nd parameter |
---|
| 179 | // 1 --> tangent 1st parameter |
---|
| 180 | e=0; |
---|
| 181 | M=L[k][1]; |
---|
| 182 | counter=1; |
---|
| 183 | B[counter,k]=M[1,1]; |
---|
| 184 | |
---|
| 185 | for (i=1;i<=nrows(M);i++) |
---|
[e7cc147] | 186 | { |
---|
[f6e355] | 187 | for (j=2;j<=ncols(M);j++) |
---|
| 188 | { |
---|
| 189 | counter=counter+1; |
---|
| 190 | if (M[i,j]==var(nv-1)) |
---|
| 191 | { |
---|
| 192 | if (i<>nrows(M)) |
---|
| 193 | { |
---|
| 194 | B[counter,k]=M[i,j]; |
---|
| 195 | j=ncols(M)+1; // goto new row of HNmatrix... |
---|
| 196 | if (counter<>s) |
---|
| 197 | { |
---|
| 198 | if (counter+1<=nrows(Mult)) |
---|
| 199 | { |
---|
| 200 | e=Mult[counter-1,k]-Mult[counter,k]-Mult[counter+1,k]; |
---|
| 201 | } |
---|
| 202 | else |
---|
| 203 | { |
---|
| 204 | e=Mult[counter-1,k]-Mult[counter,k]-1; |
---|
| 205 | } |
---|
| 206 | } |
---|
| 207 | } |
---|
| 208 | else |
---|
| 209 | { |
---|
| 210 | B[counter,k]=0; |
---|
| 211 | j=ncols(M)+1; // goto new row of HNmatrix... |
---|
| 212 | } |
---|
| 213 | } |
---|
| 214 | else |
---|
| 215 | { |
---|
| 216 | if (e<=0) |
---|
| 217 | { |
---|
| 218 | B[counter,k]=M[i,j]; |
---|
| 219 | } |
---|
| 220 | else // point is still proximate to an earlier point |
---|
| 221 | { |
---|
| 222 | B[counter,k]=y; // marking proximity (without swap....) |
---|
| 223 | if (counter+1<=nrows(Mult)) |
---|
| 224 | { |
---|
| 225 | e=e-Mult[counter+1,k]; |
---|
| 226 | } |
---|
| 227 | else |
---|
| 228 | { |
---|
| 229 | e=e-1; |
---|
| 230 | } |
---|
| 231 | } |
---|
| 232 | } |
---|
| 233 | |
---|
| 234 | if (counter==s) // given number of points determined |
---|
| 235 | { |
---|
| 236 | j=ncols(M)+1; |
---|
| 237 | i=nrows(M)+1; |
---|
| 238 | // leave procedure |
---|
| 239 | } |
---|
| 240 | } |
---|
[e7cc147] | 241 | } |
---|
[f6e355] | 242 | } |
---|
| 243 | L=B,V; |
---|
| 244 | return(L); |
---|
| 245 | } |
---|
[e7cc147] | 246 | |
---|
[f6e355] | 247 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 248 | // compute "good" upper bound for needed number of help variables |
---|
| 249 | // |
---|
| 250 | static proc Determine_no_b(intmat U,matrix B) |
---|
| 251 | // U is assumed to be 3rd output of control_Matrix |
---|
| 252 | // B is assumed to be 1st output of inf_Tangents |
---|
| 253 | { |
---|
| 254 | int nv=nvars(basering); |
---|
| 255 | int i,j,counter; |
---|
| 256 | for (j=1;j<=ncols(U);j++) |
---|
| 257 | { |
---|
| 258 | for (i=1;i<=nrows(U);i++) |
---|
[e7cc147] | 259 | { |
---|
[f6e355] | 260 | if (U[i,j]>1) |
---|
| 261 | { |
---|
| 262 | if (B[i,j]<>var(nv-1) and B[i,j]<>var(nv)) |
---|
| 263 | { |
---|
| 264 | counter=counter+1; |
---|
| 265 | } |
---|
| 266 | } |
---|
| 267 | |
---|
[e7cc147] | 268 | } |
---|
| 269 | } |
---|
[f6e355] | 270 | counter=counter+ncols(U); |
---|
| 271 | return(counter); |
---|
[e7cc147] | 272 | } |
---|
| 273 | |
---|
[f6e355] | 274 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 275 | // compute number of infinitely near free points corresponding to non-zero |
---|
| 276 | // entries in control_Matrix[1] (except first row) |
---|
| 277 | // |
---|
| 278 | static proc no_freePoints(intmat Mult,matrix B) |
---|
| 279 | // Mult is assumed to be 1st output of control_Matrix |
---|
| 280 | // U is assumed to be 3rd output of control_Matrix |
---|
| 281 | // B is assumed to be 1st output of inf_Tangents |
---|
| 282 | { |
---|
| 283 | int i,j,k,counter; |
---|
| 284 | for (j=1;j<=ncols(Mult);j++) |
---|
[e7cc147] | 285 | { |
---|
[f6e355] | 286 | for (i=2;i<=nrows(Mult);i++) |
---|
| 287 | { |
---|
| 288 | if (Mult[i,j]>=1) |
---|
| 289 | { |
---|
| 290 | if (B[i-1,j]<>x and B[i-1,j]<>y) |
---|
| 291 | { |
---|
| 292 | counter=counter+1; |
---|
| 293 | } |
---|
| 294 | } |
---|
| 295 | } |
---|
[e7cc147] | 296 | } |
---|
[f6e355] | 297 | return(counter); |
---|
[e7cc147] | 298 | } |
---|
| 299 | |
---|
| 300 | |
---|
| 301 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 302 | // COMPUTES string(minpoly) and substitutes the parameter by newParName |
---|
| 303 | static proc makeMinPolyString (string newParName) |
---|
| 304 | { |
---|
| 305 | int i; |
---|
| 306 | string parName = parstr(basering); |
---|
| 307 | int parNameSize = size(parName); |
---|
| 308 | |
---|
| 309 | string oldMinPolyStr = string (minpoly); |
---|
| 310 | int minPolySize = size(oldMinPolyStr); |
---|
| 311 | |
---|
| 312 | string newMinPolyStr = ""; |
---|
| 313 | |
---|
| 314 | for (i=1;i <= minPolySize; i++) |
---|
| 315 | { |
---|
| 316 | if (oldMinPolyStr[i,parNameSize] == parName) |
---|
| 317 | { |
---|
| 318 | newMinPolyStr = newMinPolyStr + newParName; |
---|
| 319 | i = i + parNameSize-1; |
---|
| 320 | } |
---|
| 321 | else |
---|
| 322 | { |
---|
| 323 | newMinPolyStr = newMinPolyStr + oldMinPolyStr[i]; |
---|
| 324 | } |
---|
| 325 | } |
---|
| 326 | |
---|
| 327 | return(newMinPolyStr); |
---|
| 328 | } |
---|
| 329 | |
---|
| 330 | |
---|
| 331 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f6e355] | 332 | // |
---|
| 333 | // DEFINES: A new basering, "myRing", |
---|
| 334 | // with new names for the parameters and variables. |
---|
| 335 | // The new names for the parameters are a(1..k), |
---|
| 336 | // and t(1..s),x,y for the variables |
---|
| 337 | // The ring ordering is ordStr. |
---|
| 338 | // NOTE: This proc uses 'execute'. |
---|
| 339 | static proc createMyRing_new(poly p_F, string ordStr, |
---|
| 340 | string minPolyStr, int no_b) |
---|
[e7cc147] | 341 | { |
---|
[f6e355] | 342 | def r_old = basering; |
---|
[e7cc147] | 343 | |
---|
[f6e355] | 344 | int chara = char(basering); |
---|
| 345 | string charaStr; |
---|
| 346 | int i; |
---|
| 347 | string helpStr; |
---|
| 348 | int nDefParams = nvars(r_old)-2; |
---|
[e7cc147] | 349 | |
---|
[f6e355] | 350 | ideal qIdeal = ideal(basering); |
---|
[e7cc147] | 351 | |
---|
[f6e355] | 352 | if ((npars(basering)==0) and (minPolyStr=="")) |
---|
[e7cc147] | 353 | { |
---|
[f6e355] | 354 | helpStr = "ring myRing1 =" |
---|
| 355 | + string(chara)+ ", (t(1..nDefParams), x, y),("+ ordStr +");"; |
---|
| 356 | execute(helpStr); |
---|
[e7cc147] | 357 | } |
---|
[f6e355] | 358 | else |
---|
[e7cc147] | 359 | { |
---|
[f6e355] | 360 | charaStr = charstr(basering); |
---|
| 361 | if (charaStr == string(chara) + "," + parstr(basering) or minPolyStr<>"") |
---|
| 362 | { |
---|
| 363 | if (minPolyStr<>"") |
---|
| 364 | { |
---|
| 365 | helpStr = "ring myRing1 = |
---|
| 366 | (" + string(chara) + ",a), |
---|
| 367 | (t(1..nDefParams), x, y),(" + ordStr + ");"; |
---|
| 368 | execute(helpStr); |
---|
[e7cc147] | 369 | |
---|
[f6e355] | 370 | execute (minPolyStr); |
---|
| 371 | } |
---|
| 372 | else // no minpoly given |
---|
| 373 | { |
---|
| 374 | helpStr = "ring myRing1 = |
---|
| 375 | (" + string(chara) + ",a(1..npars(basering)) ), |
---|
| 376 | (t(1..nDefParams), x, y),(" + ordStr + ");"; |
---|
| 377 | execute(helpStr); |
---|
| 378 | } |
---|
| 379 | } |
---|
| 380 | else |
---|
[e7cc147] | 381 | { |
---|
[f6e355] | 382 | // ground field is of type (p^k,a).... |
---|
| 383 | i = find (charaStr,","); |
---|
| 384 | helpStr = "ring myRing1 = (" + charaStr[1,i] + "a), |
---|
| 385 | (t(1..nDefParams), x, y),(" + ordStr + ");"; |
---|
| 386 | execute (helpStr); |
---|
[e7cc147] | 387 | } |
---|
[f6e355] | 388 | } |
---|
[e7cc147] | 389 | |
---|
[f6e355] | 390 | ideal mIdeal = maxideal(1); |
---|
| 391 | ideal qIdeal = fetch(r_old, qIdeal); |
---|
| 392 | poly p_F = fetch(r_old, p_F); |
---|
| 393 | export p_F,mIdeal; |
---|
[e7cc147] | 394 | |
---|
[f6e355] | 395 | // Extension by no_b auxiliary variables |
---|
| 396 | if (no_b>0) |
---|
[e7cc147] | 397 | { |
---|
[f6e355] | 398 | if (npars(basering) == 0) |
---|
| 399 | { |
---|
| 400 | ordStr = "(dp("+string(no_b)+"),"+ordStr+")"; |
---|
| 401 | helpStr = "ring myRing =" |
---|
| 402 | + string(chara)+ ", (b(1..no_b), t(1..nDefParams), x, y)," |
---|
| 403 | + ordStr +";"; |
---|
| 404 | execute(helpStr); |
---|
| 405 | } |
---|
| 406 | else |
---|
| 407 | { |
---|
| 408 | charaStr = charstr(basering); |
---|
| 409 | if (charaStr == string(chara) + "," + parstr(basering)) |
---|
| 410 | { |
---|
| 411 | if (minpoly !=0) |
---|
| 412 | { |
---|
| 413 | ordStr = "(dp(" + string(no_b) + ")," + ordStr + ")"; |
---|
| 414 | minPolyStr = makeMinPolyString("a"); |
---|
| 415 | helpStr = "ring myRing = |
---|
| 416 | (" + string(chara) + ",a), |
---|
| 417 | (b(1..no_b), t(1..nDefParams), x, y)," + ordStr + ";"; |
---|
| 418 | execute(helpStr); |
---|
| 419 | |
---|
| 420 | helpStr = "minpoly =" + minPolyStr + ";"; |
---|
| 421 | execute (helpStr); |
---|
| 422 | } |
---|
| 423 | else // no minpoly given |
---|
| 424 | { |
---|
| 425 | ordStr = "(dp(" + string(no_b) + ")," + ordStr + ")"; |
---|
| 426 | helpStr = "ring myRing = |
---|
| 427 | (" + string(chara) + ",a(1..npars(basering)) ), |
---|
| 428 | (b(1..no_b), t(1..nDefParams), x, y)," + ordStr + ";"; |
---|
| 429 | execute(helpStr); |
---|
| 430 | } |
---|
| 431 | } |
---|
| 432 | else |
---|
| 433 | { |
---|
| 434 | i = find (charaStr,","); |
---|
| 435 | ordStr = "(dp(" + string(no_b) + ")," + ordStr + ")"; |
---|
| 436 | helpStr = "ring myRing = |
---|
| 437 | (" + charaStr[1,i] + "a), |
---|
| 438 | (b(1..no_b), t(1..nDefParams), x, y)," + ordStr + ";"; |
---|
| 439 | execute (helpStr); |
---|
| 440 | } |
---|
| 441 | } |
---|
| 442 | ideal qIdeal = imap(myRing1, qIdeal); |
---|
| 443 | |
---|
| 444 | if(qIdeal != 0) |
---|
| 445 | { |
---|
| 446 | def r_base = basering; |
---|
| 447 | setring r_base; |
---|
| 448 | kill myRing; |
---|
| 449 | qring myRing = std(qIdeal); |
---|
| 450 | } |
---|
[e7cc147] | 451 | |
---|
[f6e355] | 452 | poly p_F = imap(myRing1, p_F); |
---|
| 453 | ideal mIdeal = imap(myRing1, mIdeal); |
---|
| 454 | export p_F,mIdeal; |
---|
| 455 | kill myRing1; |
---|
[e7cc147] | 456 | } |
---|
[f6e355] | 457 | else |
---|
| 458 | { |
---|
| 459 | if(qIdeal != 0) |
---|
| 460 | { |
---|
| 461 | def r_base = basering; |
---|
| 462 | setring r_base; |
---|
| 463 | kill myRing1; |
---|
| 464 | qring myRing = std(qIdeal); |
---|
| 465 | poly p_F = imap(myRing1, p_F); |
---|
| 466 | ideal mIdeal = imap(myRing1, mIdeal); |
---|
| 467 | export p_F,mIdeal; |
---|
| 468 | } |
---|
| 469 | else |
---|
| 470 | { |
---|
| 471 | def myRing=myRing1; |
---|
| 472 | } |
---|
| 473 | kill myRing1; |
---|
[e7cc147] | 474 | } |
---|
[f6e355] | 475 | |
---|
| 476 | setring r_old; |
---|
| 477 | return(myRing); |
---|
[e7cc147] | 478 | } |
---|
[f6e355] | 479 | |
---|
| 480 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 481 | // returns list of coef, leadmonomial |
---|
| 482 | // |
---|
| 483 | static proc determine_coef (poly Fm) |
---|
[e7cc147] | 484 | { |
---|
[f6e355] | 485 | def r_base = basering; // is assumed to be the result of createMyRing |
---|
[e7cc147] | 486 | |
---|
| 487 | int chara = char(basering); |
---|
| 488 | string charaStr; |
---|
| 489 | int i; |
---|
| 490 | string minPolyStr = ""; |
---|
[f6e355] | 491 | string helpStr = ""; |
---|
[e7cc147] | 492 | |
---|
| 493 | if (npars(basering) == 0) |
---|
| 494 | { |
---|
[f6e355] | 495 | helpStr = "ring myRing1 =" |
---|
| 496 | + string(chara)+ ", (y,x),ds;"; |
---|
[e7cc147] | 497 | execute(helpStr); |
---|
| 498 | } |
---|
| 499 | else |
---|
| 500 | { |
---|
| 501 | charaStr = charstr(basering); |
---|
| 502 | if (charaStr == string(chara) + "," + parstr(basering)) |
---|
| 503 | { |
---|
| 504 | if (minpoly !=0) |
---|
| 505 | { |
---|
| 506 | minPolyStr = makeMinPolyString("a"); |
---|
[f6e355] | 507 | helpStr = "ring myRing1 = (" + string(chara) + ",a), (y,x),ds;"; |
---|
[e7cc147] | 508 | execute(helpStr); |
---|
| 509 | |
---|
| 510 | helpStr = "minpoly =" + minPolyStr + ";"; |
---|
| 511 | execute (helpStr); |
---|
| 512 | } |
---|
[f6e355] | 513 | else // no minpoly given |
---|
[e7cc147] | 514 | { |
---|
[f6e355] | 515 | helpStr = "ring myRing1 = |
---|
| 516 | (" + string(chara) + ",a(1..npars(basering)) ), (y,x),ds;"; |
---|
[e7cc147] | 517 | execute(helpStr); |
---|
| 518 | } |
---|
| 519 | } |
---|
| 520 | else |
---|
| 521 | { |
---|
| 522 | i = find (charaStr,","); |
---|
| 523 | |
---|
[f6e355] | 524 | helpStr = " ring myRing1 = (" + charaStr[1,i] + "a), (y,x),ds;"; |
---|
[e7cc147] | 525 | execute (helpStr); |
---|
| 526 | } |
---|
| 527 | } |
---|
[f6e355] | 528 | poly f=imap(r_base,Fm); |
---|
| 529 | poly g=leadmonom(f); |
---|
| 530 | setring r_base; |
---|
| 531 | poly g=imap(myRing1,g); |
---|
| 532 | kill myRing1; |
---|
| 533 | def M=coef(Fm,xy); |
---|
| 534 | |
---|
| 535 | for (i=1; i<=ncols(M); i++) |
---|
[e7cc147] | 536 | { |
---|
[f6e355] | 537 | if (M[1,i]==g) |
---|
| 538 | { |
---|
| 539 | poly h=M[2,i]; // determine coefficient of leading monomial (in K[t]) |
---|
| 540 | i=ncols(M)+1; |
---|
| 541 | } |
---|
[e7cc147] | 542 | } |
---|
[f6e355] | 543 | return(list(h,g)); |
---|
[e7cc147] | 544 | } |
---|
| 545 | |
---|
| 546 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f6e355] | 547 | // RETURNS: 1, if p_f = 0 or char(basering) divides the order of p_f |
---|
| 548 | // or p_f is not squarefree. |
---|
| 549 | // 0, otherwise |
---|
| 550 | static proc checkPoly (poly p_f) |
---|
[e7cc147] | 551 | { |
---|
[f6e355] | 552 | int i_print = printlevel - voice + 3; |
---|
| 553 | int i_ord; |
---|
[e7cc147] | 554 | |
---|
[f6e355] | 555 | if (p_f == 0) |
---|
[e7cc147] | 556 | { |
---|
[f6e355] | 557 | print("Input is a 'deformation' of the zero polynomial!"); |
---|
| 558 | return(1); |
---|
[e7cc147] | 559 | } |
---|
| 560 | |
---|
[f6e355] | 561 | i_ord = mindeg1(p_f); |
---|
[e7cc147] | 562 | |
---|
[f6e355] | 563 | if (number(i_ord) == 0) |
---|
[e7cc147] | 564 | { |
---|
[f6e355] | 565 | print("Characteristic of coefficient field " |
---|
| 566 | +"divides order of zero-fiber !"); |
---|
| 567 | return(1); |
---|
[e7cc147] | 568 | } |
---|
| 569 | |
---|
[f6e355] | 570 | if (squarefree(p_f) != p_f) |
---|
[e7cc147] | 571 | { |
---|
[f6e355] | 572 | print("Original polynomial (= zero-fiber) is not reduced!"); |
---|
| 573 | return(1); |
---|
[e7cc147] | 574 | } |
---|
| 575 | |
---|
[f6e355] | 576 | return(0); |
---|
[e7cc147] | 577 | } |
---|
| 578 | |
---|
[f6e355] | 579 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 580 | static proc make_ring_small(ideal J) |
---|
| 581 | // returns varstr for new ring, the map and the number of vars |
---|
| 582 | { |
---|
| 583 | attrib(J,"isSB",1); |
---|
| 584 | int counter=0; |
---|
| 585 | ideal newmap; |
---|
| 586 | string newvar=""; |
---|
| 587 | for (int i=1; i<=nvars(basering); i++) |
---|
[e7cc147] | 588 | { |
---|
[f6e355] | 589 | if (reduce(var(i),J)<>0) |
---|
| 590 | { |
---|
| 591 | newmap[i]=var(i); |
---|
| 592 | |
---|
| 593 | if (newvar=="") |
---|
| 594 | { |
---|
| 595 | newvar=newvar+string(var(i)); |
---|
| 596 | counter=counter+1; |
---|
| 597 | } |
---|
| 598 | else |
---|
| 599 | { |
---|
| 600 | newvar=newvar+","+string(var(i)); |
---|
| 601 | counter=counter+1; |
---|
| 602 | } |
---|
| 603 | } |
---|
| 604 | else |
---|
| 605 | { |
---|
| 606 | newmap[i]=0; |
---|
| 607 | } |
---|
[e7cc147] | 608 | } |
---|
[f6e355] | 609 | list L=newvar,newmap,counter; |
---|
| 610 | attrib(J,"isSB",0); |
---|
| 611 | return(L); |
---|
[e7cc147] | 612 | } |
---|
| 613 | |
---|
| 614 | /////////////////////////////////////////////////////////////////////////////// |
---|
[f6e355] | 615 | // The following procedure is called by esStratum (typ=0), resp. by |
---|
| 616 | // isEquising (typ=1) |
---|
| 617 | /////////////////////////////////////////////////////////////////////////////// |
---|
[e7cc147] | 618 | |
---|
[f6e355] | 619 | static proc esComputation (int typ, poly p_F, list #) |
---|
| 620 | { |
---|
| 621 | // Initialize variables |
---|
| 622 | int branch=1; |
---|
| 623 | int blowup=1; |
---|
| 624 | int auxVar=1; |
---|
| 625 | int nVars; |
---|
| 626 | |
---|
| 627 | intvec upper_bound, upper_bound_old, fertig, soll; |
---|
| 628 | list blowup_string; |
---|
| 629 | int i_print= printlevel-voice+2; |
---|
| 630 | |
---|
| 631 | int no_b, number_of_branches, swapped; |
---|
| 632 | int i,j,k,m, counter, dummy; |
---|
| 633 | string helpStr = ""; |
---|
| 634 | string ordStr = ""; |
---|
| 635 | string MinPolyStr = ""; |
---|
| 636 | |
---|
| 637 | if (nvars(basering)<=2) |
---|
[e7cc147] | 638 | { |
---|
[f6e355] | 639 | print("family is trivial (no deformation parameters)!"); |
---|
| 640 | if (typ==1) //isEquising |
---|
[e7cc147] | 641 | { |
---|
[f6e355] | 642 | return(1); |
---|
[e7cc147] | 643 | } |
---|
| 644 | else |
---|
| 645 | { |
---|
[f6e355] | 646 | return(list(ideal(0),0)); |
---|
[e7cc147] | 647 | } |
---|
| 648 | } |
---|
| 649 | |
---|
[f6e355] | 650 | if (size(#)>0) |
---|
[e7cc147] | 651 | { |
---|
[f6e355] | 652 | if (typeof(#[1])=="int") |
---|
[e7cc147] | 653 | { |
---|
[f6e355] | 654 | def artin_bd=#[1]; // compute modulo maxideal(artin_bd) |
---|
| 655 | if (artin_bd <= 1) |
---|
| 656 | { |
---|
| 657 | print("Do you really want to compute over Basering/maxideal(" |
---|
| 658 | +string(artin_bd)+") ?"); |
---|
| 659 | print("No computation performed !"); |
---|
| 660 | if (typ==1) //isEquising |
---|
| 661 | { |
---|
| 662 | return(1); |
---|
| 663 | } |
---|
| 664 | else |
---|
[e7cc147] | 665 | { |
---|
[f6e355] | 666 | return(list(ideal(0),int(1))); |
---|
[e7cc147] | 667 | } |
---|
| 668 | } |
---|
[f6e355] | 669 | if (size(#)>1) |
---|
| 670 | { |
---|
| 671 | if (typeof(#[2])=="list") |
---|
| 672 | { |
---|
| 673 | def @L=#[2]; // is assumed to be the Hamburger-Noether matrix |
---|
| 674 | } |
---|
| 675 | } |
---|
| 676 | } |
---|
| 677 | if (typeof(#[1])=="list") |
---|
| 678 | { |
---|
| 679 | def @L=#[1]; // is assumed to be the Hamburger-Noether matrix |
---|
| 680 | } |
---|
| 681 | } |
---|
| 682 | int ring_is_changed; |
---|
| 683 | def old_ring=basering; |
---|
| 684 | if(defined(@L)<=0) |
---|
| 685 | { |
---|
| 686 | // define a new ring without deformation-parameters and change to it: |
---|
| 687 | string str; |
---|
| 688 | string minpolyStr = string(minpoly); |
---|
| 689 | str = " ring HNERing = (" + charstr(basering) + "), (x,y), ls;"; |
---|
| 690 | execute (str); |
---|
| 691 | str = "minpoly ="+ minpolyStr+";"; |
---|
| 692 | execute(str); |
---|
| 693 | ring_is_changed=1; |
---|
| 694 | // Basering changed to HNERing (variables x,y, with ls ordering) |
---|
| 695 | |
---|
| 696 | k=nvars(old_ring); |
---|
| 697 | matrix Map_Phi[1][k]; |
---|
| 698 | Map_Phi[1,k-1]=x; |
---|
| 699 | Map_Phi[1,k]=y; |
---|
| 700 | map phi=old_ring,Map_Phi; |
---|
| 701 | poly f=phi(p_F); |
---|
| 702 | |
---|
| 703 | // Heuristics: if x,y are transversal parameters then computation of HNE |
---|
| 704 | // can be much faster when exchanging variables...! |
---|
| 705 | if (2*size(coeffs(f,x))<size(coeffs(f,y))) |
---|
| 706 | { |
---|
| 707 | swapped=1; |
---|
| 708 | f=swapXY(f); |
---|
| 709 | } |
---|
| 710 | |
---|
| 711 | int error=checkPoly(f); |
---|
| 712 | if (error) |
---|
| 713 | { |
---|
| 714 | setring old_ring; |
---|
| 715 | if (typ==1) //isEquising |
---|
| 716 | { |
---|
| 717 | print("Return value (=0) has no meaning!"); |
---|
| 718 | return(0); |
---|
| 719 | } |
---|
| 720 | else |
---|
| 721 | { |
---|
| 722 | return(list( ideal(0),error)); |
---|
| 723 | } |
---|
| 724 | } |
---|
| 725 | |
---|
| 726 | dbprint(i_print,"// "); |
---|
| 727 | dbprint(i_print,"// Compute HN expansion"); |
---|
| 728 | dbprint(i_print,"// ---------------------"); |
---|
| 729 | i=printlevel; |
---|
| 730 | printlevel=printlevel-5; |
---|
| 731 | list LLL=hnexpansion(f); |
---|
| 732 | |
---|
| 733 | if (size(LLL)==0) { // empty list returned by hnexpansion |
---|
| 734 | setring old_ring; |
---|
| 735 | print(i_print,"Unable to compute HN expansion !"); |
---|
| 736 | if (typ==1) //isEquising |
---|
| 737 | { |
---|
| 738 | print("Return value (=0) has no meaning!"); |
---|
| 739 | return(0); |
---|
| 740 | } |
---|
| 741 | else |
---|
| 742 | { |
---|
| 743 | return(list(ideal(0),int(1))); |
---|
| 744 | } |
---|
| 745 | return(0); |
---|
[e7cc147] | 746 | } |
---|
| 747 | else |
---|
| 748 | { |
---|
[f6e355] | 749 | if (typeof(LLL[1])=="ring") { |
---|
| 750 | def HNering = LLL[1]; |
---|
| 751 | setring HNering; |
---|
| 752 | def @L=hne; |
---|
| 753 | } |
---|
| 754 | else { |
---|
| 755 | def @L=LLL; |
---|
| 756 | } |
---|
[e7cc147] | 757 | } |
---|
[f6e355] | 758 | printlevel=i; |
---|
| 759 | dbprint(i_print,"// finished"); |
---|
| 760 | dbprint(i_print,"// "); |
---|
| 761 | } |
---|
| 762 | def HNEring=basering; |
---|
| 763 | list M=multsequence(@L); |
---|
| 764 | M=control_Matrix(M); // this returns the 4 control matrices |
---|
| 765 | def maxDeg=M[4]; |
---|
| 766 | |
---|
| 767 | list L1=inf_Tangents(@L,nrows(M[1])); |
---|
| 768 | matrix B=L1[1]; |
---|
| 769 | intvec V=L1[2]; |
---|
| 770 | kill L1; |
---|
| 771 | |
---|
| 772 | // if we have computed the HNE for f after swapping x and y, we have |
---|
| 773 | // to reinterprete the (swap) matrix V: |
---|
| 774 | if (swapped==1) |
---|
| 775 | { |
---|
| 776 | for (i=1;i<=size(V);i++) { V[i]=V[i]-1; } // turns 0 into -1, 1 into 0 |
---|
[e7cc147] | 777 | } |
---|
| 778 | |
---|
[f6e355] | 779 | // Determine maximal number of needed auxiliary parameters (free tangents): |
---|
| 780 | no_b=Determine_no_b(M[3],B); |
---|
[e7cc147] | 781 | |
---|
[f6e355] | 782 | // test whether HNexpansion needed field extension.... |
---|
| 783 | string minPolyStr = ""; |
---|
| 784 | if (minpoly !=0) |
---|
| 785 | { |
---|
| 786 | minPolyStr = makeMinPolyString("a"); |
---|
| 787 | minPolyStr = "minpoly =" + minPolyStr + ";"; |
---|
| 788 | } |
---|
[e7cc147] | 789 | |
---|
[f6e355] | 790 | setring old_ring; |
---|
[e7cc147] | 791 | |
---|
[f6e355] | 792 | def myRing=createMyRing_new(p_F,"dp",minPolyStr,no_b); |
---|
| 793 | setring myRing; // comes with mIdeal |
---|
| 794 | map hole=HNEring,mIdeal; |
---|
| 795 | // basering has changed to myRing, in particular, the "old" |
---|
| 796 | // variable names, e.g., A,B,C,z,y are replaced by t(1),t(2),t(3),x,y |
---|
[e7cc147] | 797 | |
---|
[f6e355] | 798 | // Initialize some variables: |
---|
| 799 | map phi; |
---|
| 800 | poly G, F_save; |
---|
| 801 | poly b_dummy; |
---|
| 802 | ideal J,Jnew,final_Map; |
---|
| 803 | number_of_branches=ncols(M[1]); |
---|
| 804 | for (i=1;i<=number_of_branches;i++) |
---|
| 805 | { |
---|
| 806 | poly F(i); |
---|
| 807 | ideal bl_Map(i); |
---|
[e7cc147] | 808 | } |
---|
[f6e355] | 809 | upper_bound[number_of_branches]=0; |
---|
| 810 | upper_bound[1]=number_of_branches; |
---|
| 811 | upper_bound_old=upper_bound; |
---|
| 812 | fertig[number_of_branches]=0; |
---|
| 813 | for (i=1;i<=number_of_branches;i++){ soll[i]=1; } |
---|
| 814 | |
---|
| 815 | // Hole: B = matrix of blowup points |
---|
| 816 | if (ring_is_changed==0) { matrix B=hole(B); } |
---|
| 817 | else { matrix B=imap(HNEring,B); } |
---|
| 818 | m=M[1][blowup,branch]; // multiplicity at 0 |
---|
| 819 | |
---|
| 820 | // now, we start by checking equimultiplicity along trivial section |
---|
| 821 | poly Fm=m_Jet(p_F,m-1); |
---|
| 822 | |
---|
| 823 | matrix coef_Mat = coef(Fm,xy); |
---|
| 824 | Jnew=coef_Mat[2,1..ncols(coef_Mat)]; |
---|
| 825 | J=J,Jnew; |
---|
| 826 | |
---|
| 827 | if (defined(artin_bd)) // the artin_bd-th power of the maxideal of |
---|
| 828 | // deformation parameters can be cutted off |
---|
[e7cc147] | 829 | { |
---|
[f6e355] | 830 | J=jet(J,artin_bd-1); |
---|
[e7cc147] | 831 | } |
---|
| 832 | |
---|
[f6e355] | 833 | J=interred(J); |
---|
[e7cc147] | 834 | |
---|
[f6e355] | 835 | // J=std(J); |
---|
[e7cc147] | 836 | |
---|
[f6e355] | 837 | if (typ==1) // isEquising |
---|
| 838 | { |
---|
| 839 | if(ideal(nselect(J,1,no_b))<>0) |
---|
| 840 | { |
---|
| 841 | setring old_ring; |
---|
| 842 | return(0); |
---|
| 843 | } |
---|
| 844 | } |
---|
| 845 | |
---|
| 846 | F(1)=p_F; |
---|
| 847 | |
---|
| 848 | // and reduce the remaining terms in F(1): |
---|
| 849 | bl_Map(1)=maxideal(1); |
---|
| 850 | |
---|
| 851 | attrib(J,"isSB",1); |
---|
| 852 | bl_Map(1)=reduce(bl_Map(1),J); |
---|
| 853 | attrib(J,"isSB",0); |
---|
| 854 | |
---|
| 855 | phi=myRing,bl_Map(1); |
---|
| 856 | F(1)=phi(F(1)); |
---|
| 857 | |
---|
| 858 | // simplify F(1) |
---|
| 859 | attrib(J,"isSB",1); |
---|
| 860 | F(1)=reduce(F(1),J); |
---|
| 861 | attrib(J,"isSB",0); |
---|
| 862 | |
---|
| 863 | // now we compute the m-jet: |
---|
| 864 | Fm=m_Jet(F(1),m); |
---|
| 865 | |
---|
| 866 | G=1; |
---|
| 867 | counter=branch; |
---|
| 868 | k=upper_bound[branch]; |
---|
| 869 | |
---|
| 870 | F_save=F(1); // is truncated differently in the following loop |
---|
| 871 | |
---|
| 872 | while(counter<=k) |
---|
| 873 | { |
---|
| 874 | F(counter)=m_Jet(F_save,maxDeg[blowup,counter]); |
---|
| 875 | if (V[counter]==0) // 2nd ring variable is tangent to this branch |
---|
| 876 | { |
---|
| 877 | G=G*(y-(b(auxVar)+B[blowup,counter])*x)^(M[3][blowup,counter]); |
---|
| 878 | } |
---|
| 879 | else // 1st ring variable is tangent to this branch |
---|
| 880 | { |
---|
| 881 | G=G*(x-(b(auxVar)+B[blowup,counter])*y)^(M[3][blowup,counter]); |
---|
| 882 | F(counter)=swapXY(F(counter)); |
---|
| 883 | } |
---|
| 884 | bl_Map(counter)=maxideal(1); |
---|
| 885 | bl_Map(counter)[nvars(basering)]=xy+(b(auxVar)+B[blowup,counter])*x; |
---|
| 886 | |
---|
| 887 | auxVar=auxVar+1; |
---|
| 888 | upper_bound[counter]=counter+M[2][blowup+1,counter]-1; |
---|
| 889 | counter=counter+M[2][blowup+1,counter]; |
---|
[e7cc147] | 890 | |
---|
[f6e355] | 891 | } |
---|
[e7cc147] | 892 | |
---|
[f6e355] | 893 | list LeadDataFm=determine_coef(Fm); |
---|
| 894 | def LeadDataG=coef(G,xy); |
---|
| 895 | |
---|
| 896 | for (i=1; i<=ncols(LeadDataG); i++) |
---|
[e7cc147] | 897 | { |
---|
[f6e355] | 898 | if (LeadDataG[1,i]==LeadDataFm[2]) |
---|
| 899 | { |
---|
| 900 | poly LeadG = LeadDataG[2,i]; // determine the coefficient of G |
---|
| 901 | i=ncols(LeadDataG)+1; |
---|
| 902 | } |
---|
[e7cc147] | 903 | } |
---|
[f6e355] | 904 | |
---|
| 905 | G=LeadDataFm[1]*G-LeadG*Fm; // leading terms in y should cancel... |
---|
[e7cc147] | 906 | |
---|
[f6e355] | 907 | coef_Mat = coef(G,xy); |
---|
| 908 | Jnew=coef_Mat[2,1..ncols(coef_Mat)]; |
---|
[e7cc147] | 909 | |
---|
[f6e355] | 910 | if (defined(artin_bd)) // the artin_bd-th power of the maxideal of |
---|
| 911 | // deformation parameters can be cutted off |
---|
| 912 | { |
---|
| 913 | Jnew=jet(Jnew,artin_bd-1); |
---|
| 914 | } |
---|
[e7cc147] | 915 | |
---|
[f6e355] | 916 | // simplification of Jnew |
---|
| 917 | Jnew=interred(Jnew); |
---|
[e7cc147] | 918 | |
---|
[f6e355] | 919 | J=J,Jnew; |
---|
| 920 | if (typ==1) // isEquising |
---|
| 921 | { |
---|
| 922 | if(ideal(nselect(J,1,no_b))<>0) |
---|
[e7cc147] | 923 | { |
---|
[f6e355] | 924 | setring old_ring; |
---|
| 925 | return(0); |
---|
| 926 | } |
---|
| 927 | } |
---|
[e7cc147] | 928 | |
---|
| 929 | |
---|
[f6e355] | 930 | while (fertig<>soll and blowup<nrows(M[3])) |
---|
| 931 | { |
---|
| 932 | upper_bound_old=upper_bound; |
---|
| 933 | dbprint(i_print,"// Blowup Step "+string(blowup)+" completed"); |
---|
| 934 | blowup=blowup+1; |
---|
| 935 | |
---|
| 936 | for (branch=1;branch<=number_of_branches;branch=branch+1) |
---|
[e7cc147] | 937 | { |
---|
[f6e355] | 938 | Jnew=0; |
---|
| 939 | |
---|
| 940 | // First we check if the branch still has to be considered: |
---|
| 941 | if (branch==upper_bound_old[branch] and fertig[branch]<>1) |
---|
[e7cc147] | 942 | { |
---|
[f6e355] | 943 | if (M[3][blowup-1,branch]==1 and |
---|
| 944 | ((B[blowup,branch]<>x and B[blowup,branch]<>y) |
---|
| 945 | or (blowup==nrows(M[3])) )) |
---|
| 946 | { |
---|
| 947 | fertig[branch]=1; |
---|
| 948 | dbprint(i_print,"// 1 branch finished"); |
---|
| 949 | } |
---|
[e7cc147] | 950 | } |
---|
[f6e355] | 951 | |
---|
| 952 | if (branch<=upper_bound_old[branch] and fertig[branch]<>1) |
---|
| 953 | { |
---|
| 954 | for (i=branch;i>=1;i--) |
---|
| 955 | { |
---|
| 956 | if (M[1][blowup-1,i]<>0) |
---|
| 957 | { |
---|
| 958 | m=M[1][blowup-1,i]; // multiplicity before blowup |
---|
| 959 | i=0; |
---|
| 960 | } |
---|
| 961 | } |
---|
| 962 | |
---|
| 963 | // we blow up the branch and take the strict transform: |
---|
| 964 | attrib(J,"isSB",1); |
---|
| 965 | bl_Map(branch)=reduce(bl_Map(branch),J); |
---|
| 966 | attrib(J,"isSB",0); |
---|
| 967 | |
---|
| 968 | phi=myRing,bl_Map(branch); |
---|
| 969 | F(branch)=phi(F(branch))/x^m; |
---|
| 970 | |
---|
| 971 | // simplify F |
---|
| 972 | attrib(Jnew,"isSB",1); |
---|
| 973 | |
---|
| 974 | F(branch)=reduce(F(branch),Jnew); |
---|
| 975 | attrib(Jnew,"isSB",0); |
---|
| 976 | |
---|
| 977 | m=M[1][blowup,branch]; // multiplicity after blowup |
---|
| 978 | Fm=m_Jet(F(branch),m); // homogeneous part of lowest degree |
---|
| 979 | |
---|
| 980 | |
---|
| 981 | // we check for Fm=F[k]*...*F[k+s] where |
---|
| 982 | // |
---|
| 983 | // F[j]=(y-b'(j)*x)^m(j), respectively F[j]=(-b'(j)*y+x)^m(j) |
---|
| 984 | // |
---|
| 985 | // according to the entries m(j)= M[3][blowup,j] and |
---|
| 986 | // b'(j) mod m_A = B[blowup,j] |
---|
| 987 | // computed from the HNE of the special fibre of the family: |
---|
| 988 | G=1; |
---|
| 989 | counter=branch; |
---|
| 990 | k=upper_bound[branch]; |
---|
| 991 | |
---|
| 992 | F_save=F(branch); |
---|
| 993 | |
---|
| 994 | while(counter<=k) |
---|
| 995 | { |
---|
| 996 | F(counter)=m_Jet(F_save,maxDeg[blowup,counter]); |
---|
| 997 | |
---|
| 998 | if (B[blowup,counter]<>x and B[blowup,counter]<>y) |
---|
| 999 | { |
---|
| 1000 | G=G*(y-(b(auxVar)+B[blowup,counter])*x)^(M[3][blowup,counter]); |
---|
| 1001 | bl_Map(counter)=maxideal(1); |
---|
| 1002 | bl_Map(counter)[nvars(basering)]= |
---|
| 1003 | xy+(b(auxVar)+B[blowup,counter])*x; |
---|
| 1004 | auxVar=auxVar+1; |
---|
| 1005 | } |
---|
| 1006 | else |
---|
| 1007 | { |
---|
| 1008 | if (B[blowup,counter]==x) |
---|
| 1009 | { |
---|
| 1010 | G=G*x^(M[3][blowup,counter]); // branch has tangent x !! |
---|
| 1011 | F(counter)=swapXY(F(counter)); // will turn x to y for blow up |
---|
| 1012 | bl_Map(counter)=maxideal(1); |
---|
| 1013 | bl_Map(counter)[nvars(basering)]=xy; |
---|
| 1014 | } |
---|
| 1015 | else |
---|
| 1016 | { |
---|
| 1017 | G=G*y^(M[3][blowup,counter]); // tangent has to be y |
---|
| 1018 | bl_Map(counter)=maxideal(1); |
---|
| 1019 | bl_Map(counter)[nvars(basering)]=xy; |
---|
| 1020 | } |
---|
| 1021 | } |
---|
| 1022 | upper_bound[counter]=counter+M[2][blowup+1,counter]-1; |
---|
| 1023 | counter=counter+M[2][blowup+1,counter]; |
---|
| 1024 | } |
---|
| 1025 | G=determine_coef(Fm)[1]*G-Fm; // leading terms in y should cancel |
---|
| 1026 | coef_Mat = coef(G,xy); |
---|
| 1027 | Jnew=coef_Mat[2,1..ncols(coef_Mat)]; |
---|
| 1028 | if (defined(artin_bd)) // the artin_bd-th power of the maxideal of |
---|
| 1029 | // deformation parameters can be cutted off |
---|
| 1030 | { |
---|
| 1031 | Jnew=jet(Jnew,artin_bd-1); |
---|
| 1032 | } |
---|
[e7cc147] | 1033 | |
---|
[f6e355] | 1034 | // simplification of J |
---|
| 1035 | Jnew=interred(Jnew); |
---|
[e7cc147] | 1036 | |
---|
[f6e355] | 1037 | J=J,Jnew; |
---|
| 1038 | if (typ==1) // isEquising |
---|
[e7cc147] | 1039 | { |
---|
[f6e355] | 1040 | if(ideal(nselect(J,1,no_b))<>0) |
---|
| 1041 | { |
---|
| 1042 | setring old_ring; |
---|
| 1043 | return(0); |
---|
| 1044 | } |
---|
[e7cc147] | 1045 | } |
---|
[f6e355] | 1046 | |
---|
[e7cc147] | 1047 | } |
---|
| 1048 | } |
---|
[f6e355] | 1049 | if (number_of_branches>=2) |
---|
[e7cc147] | 1050 | { |
---|
[f6e355] | 1051 | J=interred(J); |
---|
| 1052 | if (typ==1) // isEquising |
---|
[e7cc147] | 1053 | { |
---|
[f6e355] | 1054 | if(ideal(nselect(J,1,no_b))<>0) |
---|
| 1055 | { |
---|
| 1056 | setring old_ring; |
---|
| 1057 | return(0); |
---|
| 1058 | } |
---|
[e7cc147] | 1059 | } |
---|
| 1060 | } |
---|
[f6e355] | 1061 | } |
---|
| 1062 | |
---|
| 1063 | // computation for all equimultiple sections being trivial (I^s(f)) |
---|
| 1064 | ideal Jtriv=J; |
---|
| 1065 | for (i=1;i<=no_b; i++) |
---|
| 1066 | { |
---|
| 1067 | Jtriv=subst(Jtriv,b(i),0); |
---|
| 1068 | } |
---|
| 1069 | Jtriv=std(Jtriv); |
---|
[e7cc147] | 1070 | |
---|
[f6e355] | 1071 | dbprint(i_print,"// "); |
---|
| 1072 | dbprint(i_print,"// Elimination starts:"); |
---|
| 1073 | dbprint(i_print,"// -------------------"); |
---|
[e7cc147] | 1074 | |
---|
[f6e355] | 1075 | poly gg; |
---|
| 1076 | int b_left=no_b; |
---|
[e7cc147] | 1077 | |
---|
[f6e355] | 1078 | for (i=1;i<=no_b; i++) |
---|
[e7cc147] | 1079 | { |
---|
[f6e355] | 1080 | attrib(J,"isSB",1); |
---|
| 1081 | gg=reduce(b(i),J); |
---|
| 1082 | if (gg==0) |
---|
| 1083 | { |
---|
| 1084 | b_left = b_left-1; // another b(i) has to be 0 |
---|
| 1085 | } |
---|
| 1086 | J = subst(J, b(i), gg); |
---|
| 1087 | attrib(J,"isSB",0); |
---|
[e7cc147] | 1088 | } |
---|
[f6e355] | 1089 | J=simplify(J,10); |
---|
| 1090 | if (typ==1) // isEquising |
---|
[e7cc147] | 1091 | { |
---|
[f6e355] | 1092 | if(ideal(nselect(J,1,no_b))<>0) |
---|
| 1093 | { |
---|
| 1094 | setring old_ring; |
---|
| 1095 | return(0); |
---|
| 1096 | } |
---|
[e7cc147] | 1097 | } |
---|
| 1098 | |
---|
[f6e355] | 1099 | ideal J_no_b = nselect(J,1,no_b); |
---|
| 1100 | if (size(J) > size(J_no_b)) |
---|
[e7cc147] | 1101 | { |
---|
[f6e355] | 1102 | dbprint(i_print,"// std computation started"); |
---|
| 1103 | // some b(i) didn't appear in linear conditions and have to be eliminated |
---|
| 1104 | if (defined(artin_bd)) |
---|
| 1105 | { |
---|
| 1106 | // first we make the ring smaller (removing variables, which are |
---|
| 1107 | // forced to 0 by J |
---|
| 1108 | list LL=make_ring_small(J); |
---|
| 1109 | ideal Shortmap=LL[2]; |
---|
| 1110 | minPolyStr = ""; |
---|
| 1111 | if (minpoly !=0) |
---|
| 1112 | { |
---|
| 1113 | minPolyStr = "minpoly = "+string(minpoly); |
---|
| 1114 | } |
---|
| 1115 | ordStr = "dp(" + string(b_left) + "),dp"; |
---|
| 1116 | ideal qId = ideal(basering); |
---|
| 1117 | |
---|
| 1118 | helpStr = "ring Shortring = (" |
---|
| 1119 | + charstr(basering) + "),("+ LL[1] +") , ("+ ordStr +");"; |
---|
| 1120 | execute(helpStr); |
---|
| 1121 | execute(minPolyStr); |
---|
| 1122 | // ring has changed to "Shortring" |
---|
| 1123 | |
---|
| 1124 | ideal MM=maxideal(artin_bd); |
---|
| 1125 | MM=subst(MM,x,0); |
---|
| 1126 | MM=subst(MM,y,0); |
---|
| 1127 | MM=simplify(MM,2); |
---|
| 1128 | dbprint(i_print-1,"// maxideal("+string(artin_bd)+") has " |
---|
| 1129 | +string(size(MM))+" elements"); |
---|
| 1130 | dbprint(i_print-1,"//"); |
---|
| 1131 | |
---|
| 1132 | // we change to the qring mod m^artin_bd |
---|
| 1133 | // first, we have to check if we were in a qring when starting |
---|
| 1134 | ideal qId = imap(myRing, qId); |
---|
| 1135 | if (qId == 0) |
---|
| 1136 | { |
---|
| 1137 | attrib(MM,"isSB",1); |
---|
| 1138 | qring QQ=MM; |
---|
| 1139 | } |
---|
| 1140 | else |
---|
| 1141 | { |
---|
| 1142 | qId=qId,MM; |
---|
| 1143 | qring QQ = std(qId); |
---|
| 1144 | } |
---|
[e7cc147] | 1145 | |
---|
[f6e355] | 1146 | ideal Shortmap=imap(myRing,Shortmap); |
---|
| 1147 | map phiphi=myRing,Shortmap; |
---|
[e7cc147] | 1148 | |
---|
[f6e355] | 1149 | ideal J=phiphi(J); |
---|
| 1150 | J=std(J); |
---|
| 1151 | J=nselect(J,1,no_b); |
---|
[e7cc147] | 1152 | |
---|
[f6e355] | 1153 | setring myRing; |
---|
| 1154 | // back to "myRing" |
---|
[e7cc147] | 1155 | |
---|
[f6e355] | 1156 | J=nselect(J,1,no_b); |
---|
| 1157 | Jnew=imap(QQ,J); |
---|
[e7cc147] | 1158 | |
---|
[f6e355] | 1159 | J=J,Jnew; |
---|
| 1160 | J=interred(J); |
---|
| 1161 | } |
---|
| 1162 | else |
---|
| 1163 | { |
---|
| 1164 | J=std(J); |
---|
| 1165 | J=nselect(J,1,no_b); |
---|
| 1166 | } |
---|
[e7cc147] | 1167 | } |
---|
[f6e355] | 1168 | |
---|
| 1169 | dbprint(i_print,"// finished"); |
---|
| 1170 | dbprint(i_print,"// "); |
---|
| 1171 | |
---|
| 1172 | minPolyStr = ""; |
---|
| 1173 | if (minpoly !=0) |
---|
[e7cc147] | 1174 | { |
---|
[f6e355] | 1175 | minPolyStr = "minpoly = "+string(minpoly); |
---|
[e7cc147] | 1176 | } |
---|
| 1177 | |
---|
[f6e355] | 1178 | kill HNEring; |
---|
| 1179 | |
---|
| 1180 | if (typ==1) // isEquising |
---|
[e7cc147] | 1181 | { |
---|
[f6e355] | 1182 | if(J<>0) |
---|
| 1183 | { |
---|
| 1184 | setring old_ring; |
---|
| 1185 | return(0); |
---|
| 1186 | } |
---|
| 1187 | else |
---|
| 1188 | { |
---|
| 1189 | setring old_ring; |
---|
| 1190 | return(1); |
---|
| 1191 | } |
---|
[e7cc147] | 1192 | } |
---|
[f6e355] | 1193 | |
---|
| 1194 | setring old_ring; |
---|
| 1195 | // we are back in the original ring |
---|
[e7cc147] | 1196 | |
---|
[f6e355] | 1197 | if (npars(myRing)<>0) |
---|
[e7cc147] | 1198 | { |
---|
[f6e355] | 1199 | ideal qIdeal = ideal(basering); |
---|
| 1200 | helpStr = "ring ESSring = (" |
---|
| 1201 | + string(char(basering))+ "," + parstr(myRing) + |
---|
| 1202 | ") , ("+ varstr(basering)+") , ("+ ordstr(basering) +");"; |
---|
| 1203 | execute(helpStr); |
---|
| 1204 | execute(minPolyStr); |
---|
| 1205 | // basering has changed to ESSring |
---|
| 1206 | |
---|
| 1207 | ideal qIdeal = fetch(old_ring, qIdeal); |
---|
| 1208 | if(qIdeal != 0) |
---|
| 1209 | { |
---|
| 1210 | def r_base = basering; |
---|
| 1211 | kill ESSring; |
---|
| 1212 | qring ESSring = std(qIdeal); |
---|
| 1213 | } |
---|
| 1214 | kill qIdeal; |
---|
| 1215 | |
---|
| 1216 | ideal SSS; |
---|
| 1217 | for (int ii=1;ii<=nvars(basering);ii++) |
---|
| 1218 | { |
---|
| 1219 | SSS[ii+no_b]=var(ii); |
---|
| 1220 | } |
---|
| 1221 | map phi=myRing,SSS; // b(i) variables are mapped to zero |
---|
| 1222 | |
---|
| 1223 | ideal ES=phi(J); |
---|
| 1224 | ideal ES_all_triv=phi(Jtriv); |
---|
| 1225 | kill phi; |
---|
| 1226 | |
---|
| 1227 | if (defined(p_F)<=0) |
---|
| 1228 | { |
---|
| 1229 | poly p_F=fetch(old_ring,p_F); |
---|
| 1230 | export(p_F); |
---|
| 1231 | } |
---|
| 1232 | export(ES); |
---|
| 1233 | export(ES_all_triv); |
---|
| 1234 | setring old_ring; |
---|
| 1235 | dbprint(i_print+1," |
---|
| 1236 | // 'esStratum' created a list M of a ring and an integer. |
---|
| 1237 | // To access the ideal defining the equisingularity stratum, type: |
---|
| 1238 | def ESSring = M[1]; setring ESSring; ES; "); |
---|
| 1239 | |
---|
| 1240 | return(list(ESSring,0)); |
---|
[e7cc147] | 1241 | } |
---|
| 1242 | else |
---|
[f6e355] | 1243 | { |
---|
| 1244 | // no new ring definition necessary |
---|
| 1245 | ideal SSS; |
---|
| 1246 | for (int ii=1;ii<=nvars(basering);ii++) |
---|
| 1247 | { |
---|
| 1248 | SSS[ii+no_b]=var(ii); |
---|
| 1249 | } |
---|
| 1250 | map phi=myRing,SSS; // b(i) variables are mapped to zero |
---|
| 1251 | |
---|
| 1252 | ideal ES=phi(J); |
---|
| 1253 | ideal ES_all_triv=phi(Jtriv); |
---|
| 1254 | kill phi; |
---|
| 1255 | |
---|
| 1256 | setring old_ring; |
---|
| 1257 | dbprint(i_print,"// output of 'esStratum' is a list consisting of: |
---|
| 1258 | // _[1][1] = ideal defining the equisingularity stratum |
---|
| 1259 | // _[1][2] = ideal defining the part of the equisingularity stratum |
---|
| 1260 | // where all equimultiple sections are trivial |
---|
| 1261 | // _[2] = 0"); |
---|
| 1262 | return(list(list(ES,ES_all_triv),0)); |
---|
[e7cc147] | 1263 | } |
---|
[f6e355] | 1264 | |
---|
[e7cc147] | 1265 | } |
---|
| 1266 | |
---|
[f6e355] | 1267 | //////////////////////////////////////////////////////////////////////////////// |
---|
| 1268 | |
---|
| 1269 | proc tau_es (poly f,list #) |
---|
| 1270 | "USAGE: tau_es(f); f poly |
---|
| 1271 | ASSUME: f is a reduced bivariate polynomial, the basering has precisely |
---|
| 1272 | two variables, is local and no qring. |
---|
| 1273 | RETURN: int, the codimension of the mu-const stratum in the semi-universal |
---|
| 1274 | deformation base. |
---|
| 1275 | NOTE: printlevel>=1 displays additional information. |
---|
| 1276 | When called with any additional parameter, the computation of the |
---|
| 1277 | Milnor number is avoided (no check for NND). |
---|
| 1278 | SEE ALSO: esIdeal, tjurina, invariants |
---|
| 1279 | EXAMPLE: example tau_es; shows an example. |
---|
| 1280 | " |
---|
[e7cc147] | 1281 | { |
---|
[f6e355] | 1282 | int i,j,k,s; |
---|
| 1283 | int slope_x, slope_y, upper; |
---|
| 1284 | int i_print = printlevel - voice + 3; |
---|
| 1285 | string MinPolyStr; |
---|
[e7cc147] | 1286 | |
---|
[f6e355] | 1287 | // some checks first |
---|
| 1288 | if ( nvars(basering)<>2 ) |
---|
[e7cc147] | 1289 | { |
---|
[f6e355] | 1290 | print("// basering has not the correct number (two) of variables !"); |
---|
| 1291 | print("// computation stopped"); |
---|
| 1292 | return(0); |
---|
[e7cc147] | 1293 | } |
---|
[f6e355] | 1294 | if ( mult(std(1+var(1)+var(2))) <> 0) |
---|
| 1295 | { |
---|
| 1296 | print("// basering is not local !"); |
---|
| 1297 | print("// computation stopped"); |
---|
| 1298 | return(0); |
---|
| 1299 | } |
---|
[e7cc147] | 1300 | |
---|
[f6e355] | 1301 | if (mult(std(f))<=1) |
---|
[e7cc147] | 1302 | { |
---|
[f6e355] | 1303 | // f is rigid |
---|
| 1304 | return(0); |
---|
[e7cc147] | 1305 | } |
---|
| 1306 | |
---|
[f6e355] | 1307 | if ( deg(squarefree(f))!=deg(f) ) |
---|
[e7cc147] | 1308 | { |
---|
[f6e355] | 1309 | print("// input polynomial was not reduced"); |
---|
| 1310 | print("// try squarefree(f); first"); |
---|
| 1311 | return(0); |
---|
[e7cc147] | 1312 | } |
---|
| 1313 | |
---|
[f6e355] | 1314 | def old_ring=basering; |
---|
| 1315 | execute("ring @myRing=("+charstr(basering)+"),("+varstr(basering)+"),ds;"); |
---|
| 1316 | poly f=imap(old_ring,f); |
---|
[e7cc147] | 1317 | |
---|
[f6e355] | 1318 | ideal Jacobi_Id = jacob(f); |
---|
[e7cc147] | 1319 | |
---|
[f6e355] | 1320 | // check for A_k singularity |
---|
| 1321 | // ---------------------------------------- |
---|
| 1322 | if (mult(std(f))==2) |
---|
| 1323 | { |
---|
| 1324 | dbprint(i_print-1,"// "); |
---|
| 1325 | dbprint(i_print-1,"// polynomial defined A_k singularity"); |
---|
| 1326 | dbprint(i_print-1,"// "); |
---|
| 1327 | return( vdim(std(Jacobi_Id)) ); |
---|
| 1328 | } |
---|
[e7cc147] | 1329 | |
---|
[f6e355] | 1330 | // check for D_k singularity |
---|
| 1331 | // ---------------------------------------- |
---|
| 1332 | if (mult(std(f))==3 and size(factorize(jet(f,3))[1])>=3) |
---|
[e7cc147] | 1333 | { |
---|
[f6e355] | 1334 | dbprint(i_print,"// "); |
---|
| 1335 | dbprint(i_print,"// polynomial defined D_k singularity"); |
---|
| 1336 | dbprint(i_print,"// "); |
---|
| 1337 | ideal ES_Id = f, jacob(f); |
---|
| 1338 | return( vdim(std(Jacobi_Id))); |
---|
[e7cc147] | 1339 | } |
---|
| 1340 | |
---|
| 1341 | |
---|
[f6e355] | 1342 | if (size(#)==0) |
---|
[e7cc147] | 1343 | { |
---|
[f6e355] | 1344 | // check if Newton polygon non-degenerate |
---|
| 1345 | // ---------------------------------------- |
---|
| 1346 | Jacobi_Id=std(Jacobi_Id); |
---|
| 1347 | int mu = vdim(Jacobi_Id); |
---|
| 1348 | poly f_tilde=f+var(1)^mu+var(2)^mu; //to obtain convenient Newton-polygon |
---|
| 1349 | |
---|
| 1350 | list NP=newtonpoly(f_tilde); |
---|
| 1351 | dbprint(i_print-1,"// Newton polygon:"); |
---|
| 1352 | dbprint(i_print-1,NP); |
---|
| 1353 | dbprint(i_print-1,""); |
---|
| 1354 | |
---|
| 1355 | if(is_NND(f,mu,NP)) // f is Newton non-degenerate |
---|
| 1356 | { |
---|
| 1357 | upper=NP[1][2]; |
---|
| 1358 | ideal ES_Id= x^k*y^upper; |
---|
| 1359 | dbprint(i_print-1,"polynomial is Newton non-degenerate"); |
---|
| 1360 | dbprint(i_print-1,""); |
---|
| 1361 | k=0; |
---|
| 1362 | for (i=1;i<=size(NP)-1;i++) |
---|
| 1363 | { |
---|
| 1364 | slope_x=NP[i+1][1]-NP[i][1]; |
---|
| 1365 | slope_y=NP[i][2]-NP[i+1][2]; |
---|
| 1366 | for (k=NP[i][1]+1; k<=NP[i+1][1]; k++) |
---|
| 1367 | { |
---|
| 1368 | while ( slope_x*upper + slope_y*k >= |
---|
| 1369 | slope_x*NP[i][2] + slope_y*NP[i][1]) |
---|
| 1370 | { |
---|
| 1371 | upper=upper-1; |
---|
| 1372 | } |
---|
| 1373 | upper=upper+1; |
---|
| 1374 | ES_Id=ES_Id, x^k*y^upper; |
---|
| 1375 | } |
---|
| 1376 | } |
---|
| 1377 | ES_Id=std(ES_Id); |
---|
| 1378 | dbprint(i_print-2,"ideal of monomials above Newton bd. is generated by:"); |
---|
| 1379 | dbprint(i_print-2,ES_Id); |
---|
| 1380 | ideal ESfix_Id = ES_Id, f, maxideal(1)*jacob(f); |
---|
| 1381 | ES_Id = ES_Id, Jacobi_Id; |
---|
| 1382 | ES_Id = std(ES_Id); |
---|
| 1383 | dbprint(i_print-1,"// "); |
---|
| 1384 | dbprint(i_print-1,"// Equisingularity ideal is computed!"); |
---|
| 1385 | dbprint(i_print-1,""); |
---|
| 1386 | return(vdim(ES_Id)); |
---|
| 1387 | } |
---|
| 1388 | else |
---|
| 1389 | { |
---|
| 1390 | dbprint(i_print-1,"polynomial is Newton degenerate !"); |
---|
| 1391 | dbprint(i_print-1,""); |
---|
| 1392 | } |
---|
[e7cc147] | 1393 | } |
---|
| 1394 | |
---|
[f6e355] | 1395 | // for Newton degenerate polynomials, we compute the HN expansion, and |
---|
| 1396 | // count the number of free points ..... |
---|
| 1397 | |
---|
| 1398 | dbprint(i_print-1,"// "); |
---|
| 1399 | dbprint(i_print-1,"// Compute HN expansion"); |
---|
| 1400 | dbprint(i_print-1,"// ---------------------"); |
---|
| 1401 | i=printlevel; |
---|
| 1402 | printlevel=printlevel-5; |
---|
| 1403 | if (2*size(coeffs(f,x))<size(coeffs(f,y))) |
---|
| 1404 | { |
---|
| 1405 | f=swapXY(f); |
---|
| 1406 | } |
---|
| 1407 | list LLL=hnexpansion(f); |
---|
| 1408 | if (size(LLL)==0) { // empty list returned by hnexpansion |
---|
| 1409 | setring old_ring; |
---|
| 1410 | ERROR("Unable to compute HN expansion !"); |
---|
| 1411 | } |
---|
| 1412 | else |
---|
| 1413 | { |
---|
| 1414 | if (typeof(LLL[1])=="ring") { |
---|
| 1415 | def HNering = LLL[1]; |
---|
| 1416 | setring HNering; |
---|
| 1417 | def @L=hne; |
---|
| 1418 | } |
---|
| 1419 | else { |
---|
| 1420 | def @L=LLL; |
---|
| 1421 | } |
---|
| 1422 | } |
---|
| 1423 | def HNEring=basering; |
---|
| 1424 | |
---|
| 1425 | printlevel=i; |
---|
| 1426 | dbprint(i_print-1,"// finished"); |
---|
| 1427 | dbprint(i_print-1,"// "); |
---|
| 1428 | |
---|
| 1429 | list M=multsequence(@L); |
---|
| 1430 | M=control_Matrix(M); // this returns the 4 control matrices |
---|
| 1431 | intmat Mult=M[1]; |
---|
| 1432 | |
---|
| 1433 | list L1=inf_Tangents(@L,nrows(M[1])); |
---|
| 1434 | matrix B=L1[1]; |
---|
| 1435 | |
---|
| 1436 | // determine sum_i m_i(m_i+1)/2 (over inf. near points) |
---|
| 1437 | int conditions=0; |
---|
| 1438 | for (i=1;i<=nrows(Mult);i++) |
---|
| 1439 | { |
---|
| 1440 | for (j=1;j<=ncols(Mult);j++) |
---|
| 1441 | { |
---|
| 1442 | conditions=conditions+(Mult[i,j]*(Mult[i,j]+1)/2); |
---|
| 1443 | } |
---|
| 1444 | } |
---|
| 1445 | int freePts=no_freePoints(M[1],B); |
---|
| 1446 | int taues=conditions-freePts-2; |
---|
| 1447 | |
---|
| 1448 | setring old_ring; |
---|
| 1449 | return(taues); |
---|
[e7cc147] | 1450 | } |
---|
[f6e355] | 1451 | example |
---|
[e7cc147] | 1452 | { |
---|
[f6e355] | 1453 | "EXAMPLE:"; echo=2; |
---|
| 1454 | ring r=32003,(x,y),ds; |
---|
| 1455 | poly f=(x4-y4)^2-x10; |
---|
| 1456 | tau_es(f); |
---|
| 1457 | } |
---|
[e7cc147] | 1458 | |
---|
| 1459 | |
---|
[f6e355] | 1460 | //////////////////////////////////////////////////////////////////////////////// |
---|
[e7cc147] | 1461 | |
---|
[f6e355] | 1462 | proc esIdeal (poly f,list #) |
---|
| 1463 | "USAGE: esIdeal(f[,any]]); f poly |
---|
| 1464 | ASSUME: f is a reduced bivariate polynomial, the basering has precisely |
---|
| 1465 | two variables, is local and no qring, and the characteristic of |
---|
| 1466 | the ground field does not divide mult(f). |
---|
| 1467 | RETURN: if called with only one parameter: list of two ideals, |
---|
| 1468 | @format |
---|
| 1469 | _[1]: equisingularity ideal of f (in sense of Wahl), |
---|
| 1470 | _[2]: ideal of equisingularity with fixed position of the |
---|
| 1471 | singularity; |
---|
| 1472 | @end format |
---|
| 1473 | if called with more than one parameter: list of three ideals, |
---|
| 1474 | @format |
---|
| 1475 | _[1]: equisingularity ideal of f (in sense of Wahl) |
---|
| 1476 | _[2]: ideal of equisingularity with fixed position of the |
---|
| 1477 | singularity; |
---|
| 1478 | _[3]: ideal of all g such that the deformation defined by f+eg |
---|
| 1479 | (e^2=0) is isomorphic to an equisingular deformation |
---|
| 1480 | of V(f) with all equimultiple sections being trivial. |
---|
| 1481 | @end format |
---|
| 1482 | NOTE: if some of the above condition is not satisfied then return |
---|
| 1483 | value is list(0,0). |
---|
| 1484 | SEE ALSO: tau_es, esStratum |
---|
| 1485 | KEYWORDS: equisingularity ideal |
---|
| 1486 | EXAMPLE: example esIdeal; shows examples. |
---|
| 1487 | " |
---|
| 1488 | { |
---|
[e7cc147] | 1489 | |
---|
[f6e355] | 1490 | int typ; |
---|
| 1491 | if (size(#)>0) { typ=1; } // I^s is also computed |
---|
| 1492 | int i,k,s; |
---|
| 1493 | int slope_x, slope_y, upper; |
---|
| 1494 | int i_print = printlevel - voice + 3; |
---|
| 1495 | string MinPolyStr; |
---|
| 1496 | |
---|
| 1497 | // some checks first |
---|
| 1498 | if ( nvars(basering)<>2 ) |
---|
[e7cc147] | 1499 | { |
---|
[f6e355] | 1500 | print("// basering has not the correct number (two) of variables !"); |
---|
| 1501 | print("// computation stopped"); |
---|
| 1502 | return(list(0,0)); |
---|
[e7cc147] | 1503 | } |
---|
[f6e355] | 1504 | if ( mult(std(1+var(1)+var(2))) <> 0) |
---|
| 1505 | { |
---|
| 1506 | print("// basering is not local !"); |
---|
| 1507 | print("// computation stopped"); |
---|
| 1508 | return(list(0,0)); |
---|
| 1509 | } |
---|
[e7cc147] | 1510 | |
---|
[f6e355] | 1511 | if (mult(std(f))<=1) |
---|
[e7cc147] | 1512 | { |
---|
[f6e355] | 1513 | // f is rigid |
---|
| 1514 | if (typ==0) |
---|
| 1515 | { |
---|
| 1516 | return(list(ideal(1),ideal(1))); |
---|
| 1517 | } |
---|
| 1518 | else |
---|
| 1519 | { |
---|
| 1520 | return(list(ideal(1),ideal(1),ideal(1))); |
---|
| 1521 | } |
---|
[e7cc147] | 1522 | } |
---|
| 1523 | |
---|
[f6e355] | 1524 | if ( deg(squarefree(f))!=deg(f) ) |
---|
| 1525 | { |
---|
| 1526 | print("// input polynomial was not squarefree"); |
---|
| 1527 | print("// try squarefree(f); first"); |
---|
| 1528 | return(list(0,0)); |
---|
| 1529 | } |
---|
[e7cc147] | 1530 | |
---|
[f6e355] | 1531 | if (char(basering)<>0) |
---|
[e7cc147] | 1532 | { |
---|
[f6e355] | 1533 | if (mult(std(f)) mod char(basering)==0) |
---|
| 1534 | { |
---|
| 1535 | print("// characteristic of ground field divides " |
---|
| 1536 | + "multiplicity of polynomial !"); |
---|
| 1537 | print("// computation stopped"); |
---|
| 1538 | return(list(0,0)); |
---|
| 1539 | } |
---|
[e7cc147] | 1540 | } |
---|
[f6e355] | 1541 | |
---|
| 1542 | // check for A_k singularity |
---|
| 1543 | // ---------------------------------------- |
---|
| 1544 | if (mult(std(f))==2) |
---|
[e7cc147] | 1545 | { |
---|
[f6e355] | 1546 | dbprint(i_print,"// "); |
---|
| 1547 | dbprint(i_print,"// polynomial defined A_k singularity"); |
---|
| 1548 | dbprint(i_print,"// "); |
---|
| 1549 | ideal ES_Id = f, jacob(f); |
---|
| 1550 | ES_Id = interred(ES_Id); |
---|
| 1551 | ideal ESfix_Id = f, maxideal(1)*jacob(f); |
---|
| 1552 | ESfix_Id= interred(ESfix_Id); |
---|
| 1553 | if (typ==0) // only for computation of I^es and I^es_fix |
---|
[e7cc147] | 1554 | { |
---|
[f6e355] | 1555 | return( list(ES_Id,ESfix_Id) ); |
---|
[e7cc147] | 1556 | } |
---|
| 1557 | else |
---|
| 1558 | { |
---|
[f6e355] | 1559 | return( list(ES_Id,ESfix_Id,ES_Id) ); |
---|
[e7cc147] | 1560 | } |
---|
| 1561 | } |
---|
| 1562 | |
---|
[f6e355] | 1563 | // check for D_k singularity |
---|
| 1564 | // ---------------------------------------- |
---|
| 1565 | if (mult(std(f))==3 and size(factorize(jet(f,3))[1])>=3) |
---|
[e7cc147] | 1566 | { |
---|
[f6e355] | 1567 | dbprint(i_print,"// "); |
---|
| 1568 | dbprint(i_print,"// polynomial defined D_k singularity"); |
---|
| 1569 | dbprint(i_print,"// "); |
---|
| 1570 | ideal ES_Id = f, jacob(f); |
---|
| 1571 | ES_Id = interred(ES_Id); |
---|
| 1572 | ideal ESfix_Id = f, maxideal(1)*jacob(f); |
---|
| 1573 | ESfix_Id= interred(ESfix_Id); |
---|
| 1574 | if (typ==0) // only for computation of I^es and I^es_fix |
---|
| 1575 | { |
---|
| 1576 | return( list(ES_Id,ESfix_Id) ); |
---|
| 1577 | } |
---|
| 1578 | else |
---|
| 1579 | { |
---|
| 1580 | return( list(ES_Id,ESfix_Id,ES_Id) ); |
---|
| 1581 | } |
---|
[e7cc147] | 1582 | } |
---|
| 1583 | |
---|
[f6e355] | 1584 | // check if Newton polygon non-degenerate |
---|
| 1585 | // ---------------------------------------- |
---|
| 1586 | int mu = milnor(f); |
---|
| 1587 | poly f_tilde=f+var(1)^mu+var(2)^mu; //to obtain a convenient Newton-polygon |
---|
| 1588 | |
---|
| 1589 | list NP=newtonpoly(f_tilde); |
---|
| 1590 | dbprint(i_print-1,"// Newton polygon:"); |
---|
| 1591 | dbprint(i_print-1,NP); |
---|
| 1592 | dbprint(i_print-1,""); |
---|
| 1593 | |
---|
| 1594 | if(is_NND(f,mu,NP)) // f is Newton non-degenerate |
---|
[e7cc147] | 1595 | { |
---|
[f6e355] | 1596 | upper=NP[1][2]; |
---|
| 1597 | ideal ES_Id= x^k*y^upper; |
---|
| 1598 | dbprint(i_print,"polynomial is Newton non-degenerate"); |
---|
| 1599 | dbprint(i_print,""); |
---|
| 1600 | k=0; |
---|
| 1601 | for (i=1;i<=size(NP)-1;i++) |
---|
[e7cc147] | 1602 | { |
---|
[f6e355] | 1603 | slope_x=NP[i+1][1]-NP[i][1]; |
---|
| 1604 | slope_y=NP[i][2]-NP[i+1][2]; |
---|
| 1605 | for (k=NP[i][1]+1; k<=NP[i+1][1]; k++) |
---|
| 1606 | { |
---|
| 1607 | while ( slope_x*upper + slope_y*k >= |
---|
| 1608 | slope_x*NP[i][2] + slope_y*NP[i][1]) |
---|
| 1609 | { |
---|
| 1610 | upper=upper-1; |
---|
| 1611 | } |
---|
| 1612 | upper=upper+1; |
---|
| 1613 | ES_Id=ES_Id, x^k*y^upper; |
---|
| 1614 | } |
---|
| 1615 | } |
---|
| 1616 | ES_Id=std(ES_Id); |
---|
| 1617 | dbprint(i_print-1,"ideal of monomials above Newton bd. is generated by:"); |
---|
| 1618 | dbprint(i_print-1,ES_Id); |
---|
| 1619 | ideal ESfix_Id = ES_Id, f, maxideal(1)*jacob(f); |
---|
| 1620 | ES_Id = ES_Id, f, jacob(f); |
---|
| 1621 | dbprint(i_print,"// "); |
---|
| 1622 | dbprint(i_print,"// equisingularity ideal is computed!"); |
---|
| 1623 | if (typ==0) |
---|
| 1624 | { |
---|
| 1625 | return(list(ES_Id,ESfix_Id)); |
---|
| 1626 | } |
---|
| 1627 | else |
---|
| 1628 | { |
---|
| 1629 | return(list(ES_Id,ESfix_Id,ES_Id)); |
---|
[e7cc147] | 1630 | } |
---|
| 1631 | } |
---|
[f6e355] | 1632 | else |
---|
[e7cc147] | 1633 | { |
---|
[f6e355] | 1634 | dbprint(i_print,"polynomial is Newton degenerate !"); |
---|
| 1635 | dbprint(i_print,""); |
---|
[e7cc147] | 1636 | } |
---|
[f6e355] | 1637 | |
---|
| 1638 | def old_ring=basering; |
---|
| 1639 | |
---|
| 1640 | dbprint(i_print,"// "); |
---|
| 1641 | dbprint(i_print,"// versal deformation with triv. section"); |
---|
| 1642 | dbprint(i_print,"// ====================================="); |
---|
| 1643 | dbprint(i_print,"// "); |
---|
| 1644 | |
---|
| 1645 | ideal JJ=maxideal(1)*jacob(f); |
---|
| 1646 | ideal kbase_versal=kbase(std(JJ)); |
---|
| 1647 | s=size(kbase_versal); |
---|
| 1648 | string ring_versal="ring @Px = ("+charstr(basering)+"),(t(1.."+string(s)+")," |
---|
| 1649 | +varstr(basering)+"),(ds("+string(s)+")," |
---|
| 1650 | +ordstr(basering)+");"; |
---|
| 1651 | MinPolyStr = string(minpoly); |
---|
| 1652 | |
---|
| 1653 | execute(ring_versal); |
---|
| 1654 | if (MinPolyStr<>"0") |
---|
| 1655 | { |
---|
| 1656 | MinPolyStr = "minpoly="+MinPolyStr; |
---|
| 1657 | execute(MinPolyStr); |
---|
| 1658 | } |
---|
| 1659 | // basering has changed to @Px |
---|
[e7cc147] | 1660 | |
---|
[f6e355] | 1661 | poly F=imap(old_ring,f); |
---|
| 1662 | ideal kbase_versal=imap(old_ring,kbase_versal); |
---|
| 1663 | for (i=1; i<=s; i++) |
---|
[e7cc147] | 1664 | { |
---|
[f6e355] | 1665 | F=F+var(i)*kbase_versal[i]; |
---|
[e7cc147] | 1666 | } |
---|
[f6e355] | 1667 | dbprint(i_print-1,F); |
---|
| 1668 | dbprint(i_print-1,""); |
---|
| 1669 | |
---|
| 1670 | |
---|
| 1671 | ideal ES_Id,ES_Id_all_triv; |
---|
| 1672 | poly Ftriv=F; |
---|
| 1673 | |
---|
| 1674 | dbprint(i_print,"// "); |
---|
| 1675 | dbprint(i_print,"// Compute equisingularity Stratum over Spec(C[t]/t^2)"); |
---|
| 1676 | dbprint(i_print,"// ==================================================="); |
---|
| 1677 | dbprint(i_print,"// "); |
---|
| 1678 | list M=esStratum(F,2); |
---|
| 1679 | dbprint(i_print,"// finished"); |
---|
| 1680 | dbprint(i_print,"// "); |
---|
| 1681 | |
---|
| 1682 | if (M[2]==1) // error occured during esStratum computation |
---|
| 1683 | { |
---|
| 1684 | print("Some error has occured during the computation"); |
---|
| 1685 | return(list(0,0)); |
---|
| 1686 | } |
---|
| 1687 | |
---|
| 1688 | if ( typeof(M[1])=="list" ) |
---|
| 1689 | { |
---|
| 1690 | int defpars = nvars(basering)-2; |
---|
| 1691 | poly Fred,Ftrivred; |
---|
| 1692 | poly g; |
---|
| 1693 | F=reduce(F,std(M[1][1])); |
---|
| 1694 | Ftriv=reduce(Ftriv,std(M[1][2])); |
---|
| 1695 | |
---|
| 1696 | for (i=1; i<=defpars; i++) |
---|
| 1697 | { |
---|
| 1698 | Fred=reduce(F,std(var(i))); |
---|
| 1699 | Ftrivred=reduce(Ftriv,std(var(i))); |
---|
[e7cc147] | 1700 | |
---|
[f6e355] | 1701 | g=subst(F-Fred,var(i),1); |
---|
| 1702 | ES_Id=ES_Id, g; |
---|
| 1703 | F=Fred; |
---|
[e7cc147] | 1704 | |
---|
[f6e355] | 1705 | g=subst(Ftriv-Ftrivred,var(i),1); |
---|
| 1706 | ES_Id_all_triv=ES_Id_all_triv, g; |
---|
| 1707 | Ftriv=Ftrivred; |
---|
| 1708 | } |
---|
[e7cc147] | 1709 | |
---|
[f6e355] | 1710 | setring old_ring; |
---|
| 1711 | // back to original ring |
---|
[e7cc147] | 1712 | |
---|
[f6e355] | 1713 | ideal ES_Id = imap(@Px,ES_Id); |
---|
| 1714 | ES_Id = interred(ES_Id); |
---|
[e7cc147] | 1715 | |
---|
[f6e355] | 1716 | ideal ES_Id_all_triv = imap(@Px,ES_Id_all_triv); |
---|
| 1717 | ES_Id_all_triv = interred(ES_Id_all_triv); |
---|
[e7cc147] | 1718 | |
---|
[f6e355] | 1719 | ideal ESfix_Id = ES_Id, f, maxideal(1)*jacob(f); |
---|
| 1720 | ES_Id = ES_Id, f, jacob(f); |
---|
| 1721 | ES_Id_all_triv = ES_Id_all_triv, f, jacob(f); |
---|
[e7cc147] | 1722 | |
---|
[f6e355] | 1723 | if (typ==0) |
---|
[e7cc147] | 1724 | { |
---|
[f6e355] | 1725 | return(list(ES_Id,ESfix_Id)); |
---|
[e7cc147] | 1726 | } |
---|
[f6e355] | 1727 | else |
---|
[e7cc147] | 1728 | { |
---|
[f6e355] | 1729 | return(list(ES_Id,ESfix_Id,ES_Id_all_triv)); |
---|
[e7cc147] | 1730 | } |
---|
[f6e355] | 1731 | } |
---|
| 1732 | else |
---|
| 1733 | { |
---|
| 1734 | def AuxRing=M[1]; |
---|
| 1735 | |
---|
| 1736 | dbprint(i_print,"// "); |
---|
| 1737 | dbprint(i_print,"// change ring to ESSring"); |
---|
| 1738 | |
---|
| 1739 | setring AuxRing; // contains p_F, ES |
---|
| 1740 | |
---|
| 1741 | int defpars = nvars(basering)-2; |
---|
| 1742 | poly Fred,Fredtriv; |
---|
| 1743 | poly g; |
---|
| 1744 | ideal ES_Id,ES_Id_all_triv; |
---|
| 1745 | |
---|
| 1746 | poly p_Ftriv=p_F |
---|
| 1747 | |
---|
| 1748 | p_F=reduce(p_F,std(ES)); |
---|
| 1749 | p_Ftriv=reduce(p_Ftriv,std(ES_all_triv)); |
---|
| 1750 | for (i=1; i<=defpars; i++) |
---|
[e7cc147] | 1751 | { |
---|
[f6e355] | 1752 | Fred=reduce(p_F,std(var(i))); |
---|
| 1753 | Fredtriv=reduce(p_Ftriv,std(var(i))); |
---|
[e7cc147] | 1754 | |
---|
[f6e355] | 1755 | g=subst(p_F-Fred,var(i),1); |
---|
| 1756 | ES_Id=ES_Id, g; |
---|
| 1757 | p_F=Fred; |
---|
| 1758 | |
---|
| 1759 | g=subst(p_Ftriv-Fredtriv,var(i),1); |
---|
| 1760 | ES_Id_all_triv=ES_Id_all_triv, g; |
---|
| 1761 | p_Ftriv=Fredtriv; |
---|
[e7cc147] | 1762 | |
---|
| 1763 | } |
---|
| 1764 | |
---|
[f6e355] | 1765 | dbprint(i_print,"// "); |
---|
| 1766 | dbprint(i_print,"// back to the original ring"); |
---|
[e7cc147] | 1767 | |
---|
[f6e355] | 1768 | setring old_ring; |
---|
| 1769 | // back to original ring |
---|
[e7cc147] | 1770 | |
---|
[f6e355] | 1771 | ideal ES_Id = imap(AuxRing,ES_Id); |
---|
| 1772 | ES_Id = interred(ES_Id); |
---|
[e7cc147] | 1773 | |
---|
[f6e355] | 1774 | ideal ES_Id_all_triv = imap(AuxRing,ES_Id_all_triv); |
---|
| 1775 | ES_Id_all_triv = interred(ES_Id_all_triv); |
---|
[e7cc147] | 1776 | |
---|
[f6e355] | 1777 | kill @Px; |
---|
| 1778 | kill AuxRing; |
---|
[e7cc147] | 1779 | |
---|
[f6e355] | 1780 | ideal ESfix_Id = ES_Id, f, maxideal(1)*jacob(f); |
---|
| 1781 | ES_Id = ES_Id, f, jacob(f); |
---|
| 1782 | ES_Id_all_triv = ES_Id_all_triv, f, jacob(f); |
---|
| 1783 | dbprint(i_print,"// "); |
---|
| 1784 | dbprint(i_print,"// equisingularity ideal is computed!"); |
---|
| 1785 | if (typ==0) |
---|
| 1786 | { |
---|
| 1787 | return(list(ES_Id,ESfix_Id)); |
---|
| 1788 | } |
---|
| 1789 | else |
---|
| 1790 | { |
---|
| 1791 | return(list(ES_Id,ESfix_Id,ES_Id_all_triv)); |
---|
[e7cc147] | 1792 | } |
---|
| 1793 | } |
---|
| 1794 | } |
---|
[f6e355] | 1795 | example |
---|
| 1796 | { |
---|
| 1797 | "EXAMPLE:"; echo=2; |
---|
| 1798 | ring r=0,(x,y),ds; |
---|
| 1799 | poly f=x7+y7+(x-y)^2*x2y2; |
---|
| 1800 | list K=esIdeal(f); |
---|
| 1801 | option(redSB); |
---|
| 1802 | // Wahl's equisingularity ideal: |
---|
| 1803 | std(K[1]); |
---|
| 1804 | |
---|
| 1805 | ring rr=0,(x,y),ds; |
---|
| 1806 | poly f=x4+4x3y+6x2y2+4xy3+y4+2x2y15+4xy16+2y17+xy23+y24+y30+y31; |
---|
| 1807 | list K=esIdeal(f); |
---|
| 1808 | vdim(std(K[1])); |
---|
| 1809 | // the latter should be equal to: |
---|
| 1810 | tau_es(f); |
---|
| 1811 | } |
---|
| 1812 | |
---|
[e7cc147] | 1813 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1814 | |
---|
[f6e355] | 1815 | proc esStratum (poly p_F, list #) |
---|
| 1816 | "USAGE: esStratum(F[,m,L]); F poly, m int, L list |
---|
| 1817 | ASSUME: F defines a deformation of a reduced bivariate polynomial f |
---|
[7b3971] | 1818 | and the characteristic of the basering does not divide mult(f). @* |
---|
[f6e355] | 1819 | If nv is the number of variables of the basering, then the first |
---|
| 1820 | nv-2 variables are the deformation parameters. @* |
---|
[e7cc147] | 1821 | If the basering is a qring, ideal(basering) must only depend |
---|
| 1822 | on the deformation parameters. |
---|
[f6e355] | 1823 | COMPUTE: equations for the stratum of equisingular deformations with |
---|
| 1824 | fixed (trivial) section. |
---|
| 1825 | RETURN: list l: either consisting of a list and an integer, where |
---|
[50cbdc] | 1826 | @format |
---|
[f6e355] | 1827 | l[1][1]=ideal defining the equisingularity stratum |
---|
| 1828 | l[1][2]=ideal defining the part of the equisingularity stratum where all |
---|
| 1829 | equimultiple sections through the non-nodes of the reduced total |
---|
| 1830 | transform are trivial sections |
---|
| 1831 | l[2]=1 if some error has occured, l[2]=0 otherwise; |
---|
[7b3971] | 1832 | @end format |
---|
[f6e355] | 1833 | or consisting of a ring and an integer, where |
---|
| 1834 | @format |
---|
| 1835 | l[1]=ESSring is a ring extension of basering containing the ideal ES |
---|
| 1836 | (describing the ES-stratum), the ideal ES_all_triv (describing the |
---|
| 1837 | part with trival equimultiple sections) and the poly p_F=F, |
---|
| 1838 | l[2]=1 if some error has occured, l[2]=0 otherwise. |
---|
| 1839 | @end format |
---|
| 1840 | NOTE: L is supposed to be the output of hnexpansion (with the given ordering |
---|
| 1841 | of the variables appearing in f). @* |
---|
| 1842 | If m is given, the ES Stratum over A/maxideal(m) is computed. @* |
---|
| 1843 | This procedure uses @code{execute} or calls a procedure using |
---|
[7b3971] | 1844 | @code{execute}. |
---|
[f6e355] | 1845 | printlevel>=2 displays additional information. |
---|
| 1846 | SEE ALSO: esIdeal, isEquising |
---|
| 1847 | KEYWORDS: equisingularity stratum |
---|
| 1848 | EXAMPLE: example esStratum; shows examples. |
---|
[e7cc147] | 1849 | " |
---|
| 1850 | { |
---|
[f6e355] | 1851 | list l=esComputation (0,p_F,#); |
---|
| 1852 | return(l); |
---|
| 1853 | } |
---|
| 1854 | example |
---|
| 1855 | { |
---|
| 1856 | "EXAMPLE:"; echo=2; |
---|
| 1857 | int p=printlevel; |
---|
| 1858 | printlevel=1; |
---|
| 1859 | ring r = 0,(a,b,c,d,e,f,g,x,y),ds; |
---|
| 1860 | poly F = (x2+2xy+y2+x5)+ax+by+cx2+dxy+ey2+fx3+gx4; |
---|
| 1861 | list M = esStratum(F); |
---|
| 1862 | M[1][1]; |
---|
| 1863 | |
---|
| 1864 | printlevel=3; // displays additional information |
---|
| 1865 | esStratum(F,2) ; // ES-stratum over Q[a,b,c,d,e,f,g] / <a,b,c,d,e,f,g>^2 |
---|
[e7cc147] | 1866 | |
---|
[f6e355] | 1867 | ideal I = f-fa,e+b; |
---|
| 1868 | qring q = std(I); |
---|
| 1869 | poly F = imap(r,F); |
---|
| 1870 | esStratum(F); |
---|
| 1871 | printlevel=p; |
---|
| 1872 | } |
---|
[e7cc147] | 1873 | |
---|
[f6e355] | 1874 | /////////////////////////////////////////////////////////////////////////////// |
---|
[e7cc147] | 1875 | |
---|
[f6e355] | 1876 | proc isEquising (poly p_F, list #) |
---|
| 1877 | "USAGE: isEquising(F[,m,L]); F poly, m int, L list |
---|
| 1878 | ASSUME: F defines a deformation of a reduced bivariate polynomial f |
---|
| 1879 | and the characteristic of the basering does not divide mult(f). @* |
---|
| 1880 | If nv is the number of variables of the basering, then the first |
---|
| 1881 | nv-2 variables are the deformation parameters. @* |
---|
| 1882 | If the basering is a qring, ideal(basering) must only depend |
---|
| 1883 | on the deformation parameters. |
---|
| 1884 | COMPUTE: tests if the given family is equisingular along the trivial |
---|
| 1885 | section. |
---|
| 1886 | RETURN: int: 1 if the family is equisingular, 0 otherwise. |
---|
| 1887 | NOTE: L is supposed to be the output of hnexpansion (with the given ordering |
---|
| 1888 | of the variables appearing in f). @* |
---|
| 1889 | If m is given, the family is considered over A/maxideal(m). @* |
---|
| 1890 | This procedure uses @code{execute} or calls a procedure using |
---|
| 1891 | @code{execute}. |
---|
| 1892 | printlevel>=2 displays additional information. |
---|
| 1893 | EXAMPLE: example isEquising; shows examples. |
---|
| 1894 | " |
---|
| 1895 | { |
---|
| 1896 | int check=esComputation (1,p_F,#); |
---|
| 1897 | return(check); |
---|
[e7cc147] | 1898 | } |
---|
| 1899 | example |
---|
| 1900 | { |
---|
| 1901 | "EXAMPLE:"; echo=2; |
---|
[f6e355] | 1902 | ring r = 0,(a,b,x,y),ds; |
---|
[7b3971] | 1903 | poly F = (x2+2xy+y2+x5)+ay3+bx5; |
---|
[e7cc147] | 1904 | isEquising(F); |
---|
[7b3971] | 1905 | ideal I = ideal(a); |
---|
[e7cc147] | 1906 | qring q = std(I); |
---|
| 1907 | poly F = imap(r,F); |
---|
[7b3971] | 1908 | isEquising(F); |
---|
[f6e355] | 1909 | |
---|
| 1910 | ring rr=0,(A,B,C,x,y),ls; |
---|
| 1911 | poly f=x7+y7+(x-y)^2*x2y2; |
---|
| 1912 | poly F=f+A*y*diff(f,x)+B*x*diff(f,x); |
---|
| 1913 | isEquising(F); |
---|
| 1914 | isEquising(F,2); // computation over Q[a,b] / <a,b>^2 |
---|
[e7cc147] | 1915 | } |
---|
[f6e355] | 1916 | |
---|
| 1917 | |
---|
| 1918 | |
---|
| 1919 | /* Examples: |
---|
| 1920 | |
---|
| 1921 | LIB "equising.lib"; |
---|
| 1922 | ring r = 0,(x,y),ds; |
---|
| 1923 | poly p1 = y^2+x^3; |
---|
| 1924 | poly p2 = p1^2+x5y; |
---|
| 1925 | poly p3 = p2^2+x^10*p1; |
---|
| 1926 | poly p=p3^2+x^20*p2; |
---|
| 1927 | p; |
---|
| 1928 | versal(p); |
---|
| 1929 | setring Px; |
---|
| 1930 | poly F=Fs[1,1]; |
---|
| 1931 | int t=timer; |
---|
| 1932 | list M=esStratum(F); |
---|
| 1933 | timer-t; //-> 3 |
---|
| 1934 | |
---|
| 1935 | LIB "equising.lib"; |
---|
| 1936 | option(prot); |
---|
| 1937 | printlevel=2; |
---|
| 1938 | ring r=0,(x,y),ds; |
---|
| 1939 | poly f=(x-yx+y2)^2-(y+x)^31; |
---|
| 1940 | versal(f); |
---|
| 1941 | setring Px; |
---|
| 1942 | poly F=Fs[1,1]; |
---|
| 1943 | int t=timer; |
---|
| 1944 | list M=esStratum(F); |
---|
| 1945 | timer-t; //-> 233 |
---|
| 1946 | |
---|
| 1947 | |
---|
| 1948 | LIB "equising.lib"; |
---|
| 1949 | printlevel=2; |
---|
| 1950 | option(prot); |
---|
| 1951 | timer=1; |
---|
| 1952 | ring r=32003,(x,y),ds; |
---|
| 1953 | poly f=(x4-y4)^2-x10; |
---|
| 1954 | versal(f); |
---|
| 1955 | setring Px; |
---|
| 1956 | poly F=Fs[1,1]; |
---|
| 1957 | int t=timer; |
---|
| 1958 | list M=esStratum(F,3); |
---|
| 1959 | timer-t; //-> 8 |
---|
| 1960 | |
---|
| 1961 | LIB "equising.lib"; |
---|
| 1962 | printlevel=2; |
---|
| 1963 | timer=1; |
---|
| 1964 | ring rr=0,(x,y),ls; |
---|
| 1965 | poly f=x7+y7+(x-y)^2*x2y2; |
---|
| 1966 | list K=esIdeal(f); |
---|
| 1967 | // tau_es |
---|
| 1968 | vdim(std(K[1])); //-> 22 |
---|
| 1969 | // tau_es_fix |
---|
| 1970 | vdim(std(K[2])); //-> 24 |
---|
| 1971 | |
---|
| 1972 | |
---|
| 1973 | LIB "equising.lib"; |
---|
| 1974 | printlevel=2; |
---|
| 1975 | timer=1; |
---|
| 1976 | ring rr=0,(x,y),ls; |
---|
| 1977 | poly f=x7+y7+(x-y)^2*x2y2+x2y4; // Newton non-deg. |
---|
| 1978 | list K=esIdeal(f); |
---|
| 1979 | // tau_es |
---|
| 1980 | vdim(std(K[1])); //-> 21 |
---|
| 1981 | // tau_es_fix |
---|
| 1982 | vdim(std(K[2])); //-> 23 |
---|
| 1983 | |
---|
| 1984 | LIB "equising.lib"; |
---|
| 1985 | ring r=0,(w,v),ds; |
---|
| 1986 | poly f=w2-v199; |
---|
| 1987 | list L=hnexpansion(f); |
---|
| 1988 | versal(f); |
---|
| 1989 | setring Px; |
---|
| 1990 | list L=imap(r,L); |
---|
| 1991 | poly F=Fs[1,1]; |
---|
| 1992 | list M=esStratum(F,2,L); |
---|
| 1993 | |
---|
| 1994 | LIB "equising.lib"; |
---|
| 1995 | printlevel=2; |
---|
| 1996 | timer=1; |
---|
| 1997 | ring rr=0,(A,B,C,x,y),ls; |
---|
| 1998 | poly f=x7+y7+(x-y)^2*x2y2; |
---|
| 1999 | poly F=f+A*y*diff(f,x)+B*x*diff(f,x)+C*diff(f,y); |
---|
| 2000 | list M=esStratum(F,6); |
---|
| 2001 | std(M[1]); // standard basis of equisingularity ideal |
---|
| 2002 | |
---|
| 2003 | LIB "equising.lib"; |
---|
| 2004 | printlevel=2; |
---|
| 2005 | timer=1; |
---|
| 2006 | ring rr=0,(x,y),ls; |
---|
| 2007 | poly f=x20+y7+(x-y)^2*x2y2+x2y4; // Newton non-degenerate |
---|
| 2008 | list K=esIdeal(f); |
---|
| 2009 | |
---|
| 2010 | ring rr=0,(x,y),ls; |
---|
| 2011 | poly f=x6y-3x4y4-x4y5+3x2y7-x4y6+2x2y8-y10+2x2y9-y11+x2y10-y12-y13; |
---|
| 2012 | list K=esIdeal(f); |
---|
| 2013 | versal(f); |
---|
| 2014 | setring Px; |
---|
| 2015 | poly F=Fs[1,1]; |
---|
| 2016 | list M=esStratum(F,2); |
---|
| 2017 | |
---|
| 2018 | LIB "equising.lib"; |
---|
| 2019 | printlevel=2; |
---|
| 2020 | ring rr=0,(x,y),ls; |
---|
| 2021 | poly f=x6y-3x4y4-x4y5+3x2y7-x4y6+2x2y8-y10+2x2y9-y11+x2y10-y12-y13; |
---|
| 2022 | list K=esIdeal(f); |
---|
| 2023 | vdim(std(K[1])); //-> 51 |
---|
| 2024 | tau_es(f); //-> 51 |
---|
| 2025 | |
---|
| 2026 | printlevel=3; |
---|
| 2027 | f=f*(y-x2)*(y2-x3)*(x-y5); |
---|
| 2028 | int t=timer; |
---|
| 2029 | list L=esIdeal(f); |
---|
| 2030 | vdim(std(L[1])); //-> 99 |
---|
| 2031 | timer-t; //-> 42 |
---|
| 2032 | t=timer; |
---|
| 2033 | tau_es(f); //-> 99 |
---|
| 2034 | timer-t; //-> 23 |
---|
| 2035 | |
---|
| 2036 | |
---|
| 2037 | LIB "equising.lib"; |
---|
| 2038 | printlevel=3; |
---|
| 2039 | ring rr=0,(x,y),ds; |
---|
| 2040 | poly f=x4+4x3y+6x2y2+4xy3+y4+2x2y15+4xy16+2y17+xy23+y24+y30+y31; |
---|
| 2041 | list K=esIdeal(f); |
---|
| 2042 | vdim(std(K[1])); //-> 68 |
---|
| 2043 | tau_es(f); //-> 68 |
---|
| 2044 | |
---|
| 2045 | versal(f); |
---|
| 2046 | setring Px; |
---|
| 2047 | poly F=Fs[1,1]; |
---|
| 2048 | int t=timer; |
---|
| 2049 | list M=esStratum(F); |
---|
| 2050 | timer-t; //-> 0 |
---|
| 2051 | |
---|
| 2052 | |
---|
| 2053 | |
---|
[558eb2] | 2054 | */ |
---|