[0c4bd7] | 1 | /////////////////////////////////////////////////////////////////////////////// |
---|
[341696] | 2 | version="$Id$"; |
---|
[fd3fb7] | 3 | category="Singularities"; |
---|
[c52356d] | 4 | |
---|
[0c4bd7] | 5 | info=" |
---|
[46af92] | 6 | LIBRARY: gaussman.lib Gauss-Manin System of Isolated Singularities |
---|
[0c4bd7] | 7 | |
---|
[61dadae] | 8 | AUTHOR: Mathias Schulze, email: mschulze@mathematik.uni-kl.de |
---|
[0c4bd7] | 9 | |
---|
[2d3b6e6] | 10 | OVERVIEW: A library for the Gauss-Manin system |
---|
| 11 | of an isolated hypersurface singularity |
---|
[cc3a04] | 12 | |
---|
[0c4bd7] | 13 | PROCEDURES: |
---|
[26a4bb] | 14 | gmsring(t,s); Gauss-Manin system of t with variable s |
---|
[91fc5e] | 15 | gmsnf(p,K); Gauss-Manin normal form of p |
---|
| 16 | gmscoeffs(p,K); Gauss-Manin basis representation of p |
---|
[779ee3] | 17 | bernstein(t); roots of the Bernstein polynomial of t |
---|
[91fc5e] | 18 | monodromy(t); Jordan data of complex monodromy of t |
---|
[275721f] | 19 | spectrum(t); singularity spectrum of t |
---|
[e480544] | 20 | sppairs(t); spectral pairs of t |
---|
[04c344] | 21 | vfilt(t); V-filtration of t on Brieskorn lattice |
---|
| 22 | vwfilt(t); weighted V-filtration of t on Brieskorn lattice |
---|
[5dfbc0] | 23 | tmatrix(t); C[[s]]-matrix of t on Brieskorn lattice |
---|
[64eab4] | 24 | endvfilt(V); endomorphism V-filtration on Jacobian algebra |
---|
[2d3b6e6] | 25 | sppnf(a,w[,m]); spectral pairs normal form of (a,w[,m]) |
---|
[d70bc7] | 26 | sppprint(spp); print spectral pairs spp |
---|
| 27 | spadd(sp1,sp2); sum of spectra sp1 and sp2 |
---|
| 28 | spsub(sp1,sp2); difference of spectra sp1 and sp2 |
---|
[275721f] | 29 | spmul(sp0,k); linear combination of spectra sp |
---|
| 30 | spissemicont(sp[,opt]); semicontinuity test of spectrum sp |
---|
| 31 | spsemicont(sp0,sp[,opt]); semicontinuous combinations of spectra sp0 in sp |
---|
[91fc5e] | 32 | spmilnor(sp); Milnor number of spectrum sp |
---|
[d70bc7] | 33 | spgeomgenus(sp); geometrical genus of spectrum sp |
---|
| 34 | spgamma(sp); gamma invariant of spectrum sp |
---|
[cc3a04] | 35 | |
---|
[b30162] | 36 | SEE ALSO: mondromy_lib, spectrum_lib |
---|
[8a87a6] | 37 | |
---|
[46af92] | 38 | KEYWORDS: singularities; Gauss-Manin system; Brieskorn lattice; |
---|
[d70bc7] | 39 | mixed Hodge structure; V-filtration; weight filtration |
---|
[3c4dcc] | 40 | Bernstein polynomial; monodromy; spectrum; spectral pairs; |
---|
[46af92] | 41 | good basis; |
---|
[0c4bd7] | 42 | "; |
---|
| 43 | |
---|
[e9124e] | 44 | LIB "linalg.lib"; |
---|
[0c4bd7] | 45 | |
---|
| 46 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 47 | |
---|
[442ed6] | 48 | static proc stdtrans(ideal I) |
---|
| 49 | { |
---|
[500122] | 50 | def R=basering; |
---|
[442ed6] | 51 | |
---|
[34a9eb1] | 52 | string os=ordstr(R); |
---|
| 53 | int j=find(os,",C"); |
---|
[442ed6] | 54 | if(j==0) |
---|
| 55 | { |
---|
[34a9eb1] | 56 | j=find(os,"C,"); |
---|
[442ed6] | 57 | } |
---|
| 58 | if(j==0) |
---|
| 59 | { |
---|
[34a9eb1] | 60 | j=find(os,",c"); |
---|
[442ed6] | 61 | } |
---|
| 62 | if(j==0) |
---|
| 63 | { |
---|
[34a9eb1] | 64 | j=find(os,"c,"); |
---|
[442ed6] | 65 | } |
---|
| 66 | if(j>0) |
---|
| 67 | { |
---|
[34a9eb1] | 68 | os[j..j+1]=" "; |
---|
[442ed6] | 69 | } |
---|
| 70 | |
---|
[34a9eb1] | 71 | execute("ring S="+charstr(R)+",(gmspoly,"+varstr(R)+"),(c,dp,"+os+");"); |
---|
[442ed6] | 72 | |
---|
[500122] | 73 | ideal I=homog(imap(R,I),gmspoly); |
---|
[442ed6] | 74 | module M=transpose(transpose(I)+freemodule(ncols(I))); |
---|
| 75 | M=std(M); |
---|
| 76 | |
---|
[500122] | 77 | setring(R); |
---|
[d341d0] | 78 | execute("map h=S,1,"+varstr(R)+";"); |
---|
[442ed6] | 79 | module M=h(M); |
---|
| 80 | |
---|
| 81 | for(int i=ncols(M);i>=1;i--) |
---|
| 82 | { |
---|
| 83 | for(j=ncols(M);j>=1;j--) |
---|
| 84 | { |
---|
| 85 | if(M[i][1]==0) |
---|
| 86 | { |
---|
| 87 | M[i]=0; |
---|
| 88 | } |
---|
| 89 | if(i!=j&&M[j][1]!=0) |
---|
| 90 | { |
---|
| 91 | if(lead(M[i][1])/lead(M[j][1])!=0) |
---|
| 92 | { |
---|
| 93 | M[i]=0; |
---|
| 94 | } |
---|
| 95 | } |
---|
| 96 | } |
---|
| 97 | } |
---|
| 98 | |
---|
| 99 | M=transpose(simplify(M,2)); |
---|
| 100 | I=M[1]; |
---|
| 101 | attrib(I,"isSB",1); |
---|
| 102 | M=M[2..ncols(M)]; |
---|
| 103 | module U=transpose(M); |
---|
| 104 | |
---|
| 105 | return(list(I,U)); |
---|
| 106 | } |
---|
| 107 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 108 | |
---|
[500122] | 109 | proc gmsring(poly t,string s) |
---|
[275721f] | 110 | "USAGE: gmsring(t,s); poly t, string s |
---|
| 111 | ASSUME: characteristic 0; local degree ordering; |
---|
[a8cc0a] | 112 | isolated critical point 0 of t |
---|
[e3f423] | 113 | RETURN: |
---|
| 114 | @format |
---|
[26a4bb] | 115 | ring G; Gauss-Manin system of t with variable s |
---|
[275721f] | 116 | poly gmspoly=t; |
---|
| 117 | ideal gmsjacob; Jacobian ideal of t |
---|
| 118 | ideal gmsstd; standard basis of Jacobian ideal |
---|
[04c344] | 119 | matrix gmsmatrix; matrix(gmsjacob)*gmsmatrix==matrix(gmsstd) |
---|
[275721f] | 120 | ideal gmsbasis; monomial vector space basis of Jacobian algebra |
---|
[9526639] | 121 | int gmsmaxdeg; maximal weight of variables |
---|
[e3f423] | 122 | @end format |
---|
[5dfbc0] | 123 | NOTE: gmsbasis is a C[[s]]-basis of H'' and [t,s]=s^2 |
---|
[46af92] | 124 | KEYWORDS: singularities; Gauss-Manin system; Brieskorn lattice |
---|
[275721f] | 125 | EXAMPLE: example gmsring; shows examples |
---|
[e3f423] | 126 | " |
---|
[04b295] | 127 | { |
---|
[500122] | 128 | def R=basering; |
---|
| 129 | if(charstr(R)!="0") |
---|
[04b295] | 130 | { |
---|
| 131 | ERROR("characteristic 0 expected"); |
---|
| 132 | } |
---|
[500122] | 133 | for(int i=nvars(R);i>=1;i--) |
---|
[04b295] | 134 | { |
---|
| 135 | if(var(i)>1) |
---|
| 136 | { |
---|
| 137 | ERROR("local ordering expected"); |
---|
| 138 | } |
---|
| 139 | } |
---|
| 140 | |
---|
| 141 | ideal dt=jacob(t); |
---|
[442ed6] | 142 | list l=stdtrans(dt); |
---|
| 143 | ideal g=l[1]; |
---|
[04b295] | 144 | if(vdim(g)<=0) |
---|
| 145 | { |
---|
| 146 | if(vdim(g)==0) |
---|
| 147 | { |
---|
| 148 | ERROR("singularity at 0 expected"); |
---|
| 149 | } |
---|
| 150 | else |
---|
| 151 | { |
---|
[a8cc0a] | 152 | ERROR("isolated critical point 0 expected"); |
---|
[04b295] | 153 | } |
---|
[e9124e] | 154 | } |
---|
[9526639] | 155 | matrix B=l[2]; |
---|
[04b295] | 156 | ideal m=kbase(g); |
---|
| 157 | |
---|
[9526639] | 158 | int gmsmaxdeg; |
---|
[500122] | 159 | for(i=nvars(R);i>=1;i--) |
---|
| 160 | { |
---|
[9526639] | 161 | if(deg(var(i))>gmsmaxdeg) |
---|
[500122] | 162 | { |
---|
[9526639] | 163 | gmsmaxdeg=deg(var(i)); |
---|
[500122] | 164 | } |
---|
| 165 | } |
---|
| 166 | |
---|
[34a9eb1] | 167 | string os=ordstr(R); |
---|
| 168 | int j=find(os,",C"); |
---|
| 169 | if(j==0) |
---|
| 170 | { |
---|
| 171 | j=find(os,"C,"); |
---|
| 172 | } |
---|
| 173 | if(j==0) |
---|
| 174 | { |
---|
| 175 | j=find(os,",c"); |
---|
| 176 | } |
---|
| 177 | if(j==0) |
---|
| 178 | { |
---|
| 179 | j=find(os,"c,"); |
---|
| 180 | } |
---|
| 181 | if(j>0) |
---|
| 182 | { |
---|
| 183 | os[j..j+1]=" "; |
---|
| 184 | } |
---|
| 185 | |
---|
[d341d0] | 186 | execute("ring G="+string(charstr(R))+",("+s+","+varstr(R)+"),(ws("+ |
---|
[9526639] | 187 | string(deg(highcorner(g))+2*gmsmaxdeg)+"),"+os+",c);"); |
---|
[04b295] | 188 | |
---|
[500122] | 189 | poly gmspoly=imap(R,t); |
---|
| 190 | ideal gmsjacob=imap(R,dt); |
---|
| 191 | ideal gmsstd=imap(R,g); |
---|
[9526639] | 192 | matrix gmsmatrix=imap(R,B); |
---|
[500122] | 193 | ideal gmsbasis=imap(R,m); |
---|
[04b295] | 194 | |
---|
[500122] | 195 | attrib(gmsstd,"isSB",1); |
---|
[9526639] | 196 | export gmspoly,gmsjacob,gmsstd,gmsmatrix,gmsbasis,gmsmaxdeg; |
---|
[04b295] | 197 | |
---|
[500122] | 198 | return(G); |
---|
[e3f423] | 199 | } |
---|
| 200 | example |
---|
| 201 | { "EXAMPLE:"; echo=2; |
---|
| 202 | ring R=0,(x,y),ds; |
---|
[86c1f0] | 203 | poly t=x5+x2y2+y5; |
---|
| 204 | def G=gmsring(t,"s"); |
---|
[e3f423] | 205 | setring(G); |
---|
[500122] | 206 | gmspoly; |
---|
| 207 | print(gmsjacob); |
---|
| 208 | print(gmsstd); |
---|
| 209 | print(gmsmatrix); |
---|
[86c1f0] | 210 | print(gmsbasis); |
---|
[9526639] | 211 | gmsmaxdeg; |
---|
[04b295] | 212 | } |
---|
| 213 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 214 | |
---|
[38c0dca] | 215 | proc gmsnf(ideal p,int K) |
---|
| 216 | "USAGE: gmsnf(p,K); poly p, int K |
---|
| 217 | ASSUME: basering returned by gmsring |
---|
[e3f423] | 218 | RETURN: |
---|
| 219 | @format |
---|
[e9124e] | 220 | list nf; |
---|
[74d9b7] | 221 | ideal nf[1]; projection of p to <gmsbasis>C[[s]] mod s^(K+1) |
---|
[91fc5e] | 222 | ideal nf[2]; p==nf[1]+nf[2] |
---|
[e3f423] | 223 | @end format |
---|
[91fc5e] | 224 | NOTE: computation can be continued by setting p=nf[2] |
---|
[46af92] | 225 | KEYWORDS: singularities; Gauss-Manin system; Brieskorn lattice |
---|
[e3f423] | 226 | EXAMPLE: example gmsnf; shows examples |
---|
| 227 | " |
---|
[04b295] | 228 | { |
---|
[e9124e] | 229 | if(system("with","gms")) |
---|
| 230 | { |
---|
| 231 | return(system("gmsnf",p,gmsstd,gmsmatrix,(K+1)*deg(var(1))-2*gmsmaxdeg,K)); |
---|
| 232 | } |
---|
| 233 | |
---|
| 234 | intvec v=1; |
---|
| 235 | v[nvars(basering)]=0; |
---|
| 236 | |
---|
| 237 | int k; |
---|
| 238 | ideal r,q; |
---|
| 239 | r[ncols(p)]=0; |
---|
| 240 | q[ncols(p)]=0; |
---|
| 241 | |
---|
| 242 | poly s; |
---|
| 243 | int i,j; |
---|
| 244 | for(k=ncols(p);k>=1;k--) |
---|
| 245 | { |
---|
| 246 | while(p[k]!=0&°(lead(p[k]),v)<=K) |
---|
| 247 | { |
---|
| 248 | i=1; |
---|
| 249 | s=lead(p[k])/lead(gmsstd[i]); |
---|
| 250 | while(i<ncols(gmsstd)&&s==0) |
---|
| 251 | { |
---|
| 252 | i++; |
---|
| 253 | s=lead(p[k])/lead(gmsstd[i]); |
---|
| 254 | } |
---|
| 255 | if(s!=0) |
---|
| 256 | { |
---|
| 257 | p[k]=p[k]-s*gmsstd[i]; |
---|
| 258 | for(j=1;j<=nrows(gmsmatrix);j++) |
---|
| 259 | { |
---|
| 260 | p[k]=p[k]+diff(s*gmsmatrix[j,i],var(j+1))*var(1); |
---|
| 261 | } |
---|
| 262 | } |
---|
| 263 | else |
---|
| 264 | { |
---|
| 265 | r[k]=r[k]+lead(p[k]); |
---|
| 266 | p[k]=p[k]-lead(p[k]); |
---|
| 267 | } |
---|
| 268 | while(deg(lead(p[k]))>(K+1)*deg(var(1))-2*gmsmaxdeg&& |
---|
| 269 | deg(lead(p[k]),v)<=K) |
---|
| 270 | { |
---|
| 271 | q[k]=q[k]+lead(p[k]); |
---|
| 272 | p[k]=p[k]-lead(p[k]); |
---|
| 273 | } |
---|
| 274 | } |
---|
| 275 | q[k]=q[k]+p[k]; |
---|
| 276 | } |
---|
| 277 | |
---|
| 278 | return(list(r,q)); |
---|
[e3f423] | 279 | } |
---|
| 280 | example |
---|
| 281 | { "EXAMPLE:"; echo=2; |
---|
| 282 | ring R=0,(x,y),ds; |
---|
[86c1f0] | 283 | poly t=x5+x2y2+y5; |
---|
| 284 | def G=gmsring(t,"s"); |
---|
[e3f423] | 285 | setring(G); |
---|
[500122] | 286 | list l0=gmsnf(gmspoly,0); |
---|
[e3f423] | 287 | print(l0[1]); |
---|
[500122] | 288 | list l1=gmsnf(gmspoly,1); |
---|
[e3f423] | 289 | print(l1[1]); |
---|
| 290 | list l=gmsnf(l0[2],1); |
---|
| 291 | print(l[1]); |
---|
[04b295] | 292 | } |
---|
| 293 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 294 | |
---|
[38c0dca] | 295 | proc gmscoeffs(ideal p,int K) |
---|
| 296 | "USAGE: gmscoeffs(p,K); poly p, int K |
---|
| 297 | ASSUME: basering constructed by gmsring |
---|
[e3f423] | 298 | RETURN: |
---|
| 299 | @format |
---|
[e9124e] | 300 | list l; |
---|
[5dfbc0] | 301 | matrix l[1]; C[[s]]-basis representation of p mod s^(K+1) |
---|
[91fc5e] | 302 | ideal l[2]; p==matrix(gmsbasis)*l[1]+l[2] |
---|
[e3f423] | 303 | @end format |
---|
[91fc5e] | 304 | NOTE: computation can be continued by setting p=l[2] |
---|
[46af92] | 305 | KEYWORDS: singularities; Gauss-Manin system; Brieskorn lattice |
---|
[e3f423] | 306 | EXAMPLE: example gmscoeffs; shows examples |
---|
| 307 | " |
---|
[04b295] | 308 | { |
---|
[38c0dca] | 309 | list l=gmsnf(p,K); |
---|
[e3f423] | 310 | ideal r,q=l[1..2]; |
---|
[04b295] | 311 | poly v=1; |
---|
| 312 | for(int i=2;i<=nvars(basering);i++) |
---|
| 313 | { |
---|
| 314 | v=v*var(i); |
---|
| 315 | } |
---|
[500122] | 316 | matrix C=coeffs(r,gmsbasis,v); |
---|
[04b295] | 317 | return(C,q); |
---|
| 318 | } |
---|
[e3f423] | 319 | example |
---|
| 320 | { "EXAMPLE:"; echo=2; |
---|
| 321 | ring R=0,(x,y),ds; |
---|
[86c1f0] | 322 | poly t=x5+x2y2+y5; |
---|
| 323 | def G=gmsring(t,"s"); |
---|
[e3f423] | 324 | setring(G); |
---|
[500122] | 325 | list l0=gmscoeffs(gmspoly,0); |
---|
[e3f423] | 326 | print(l0[1]); |
---|
[500122] | 327 | list l1=gmscoeffs(gmspoly,1); |
---|
[e3f423] | 328 | print(l1[1]); |
---|
| 329 | list l=gmscoeffs(l0[2],1); |
---|
| 330 | print(l[1]); |
---|
| 331 | } |
---|
[04b295] | 332 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 333 | |
---|
[e9124e] | 334 | static proc nmin(ideal e) |
---|
[0c4bd7] | 335 | { |
---|
[0ff6b5] | 336 | int i; |
---|
| 337 | number m=number(e[1]); |
---|
| 338 | for(i=2;i<=ncols(e);i++) |
---|
[0c4bd7] | 339 | { |
---|
[0ff6b5] | 340 | if(number(e[i])<m) |
---|
[0c4bd7] | 341 | { |
---|
[0ff6b5] | 342 | m=number(e[i]); |
---|
[0c4bd7] | 343 | } |
---|
| 344 | } |
---|
[0ff6b5] | 345 | return(m); |
---|
[0c4bd7] | 346 | } |
---|
| 347 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 348 | |
---|
[e9124e] | 349 | static proc nmax(ideal e) |
---|
[0c4bd7] | 350 | { |
---|
[0ff6b5] | 351 | int i; |
---|
| 352 | number m=number(e[1]); |
---|
| 353 | for(i=2;i<=ncols(e);i++) |
---|
[8960ec] | 354 | { |
---|
[0ff6b5] | 355 | if(number(e[i])>m) |
---|
[8960ec] | 356 | { |
---|
[0ff6b5] | 357 | m=number(e[i]); |
---|
[0c4bd7] | 358 | } |
---|
| 359 | } |
---|
[0ff6b5] | 360 | return(m); |
---|
| 361 | } |
---|
| 362 | /////////////////////////////////////////////////////////////////////////////// |
---|
[8960ec] | 363 | |
---|
[2ca72f] | 364 | static proc saturate() |
---|
[0ff6b5] | 365 | { |
---|
[86c1f0] | 366 | int mu=ncols(gmsbasis); |
---|
[34a9eb1] | 367 | ideal r=gmspoly*gmsbasis; |
---|
| 368 | matrix A0[mu][mu],C; |
---|
[0c4bd7] | 369 | module H0; |
---|
[0ff6b5] | 370 | module H,H1=freemodule(mu),freemodule(mu); |
---|
[0c4bd7] | 371 | int k=-1; |
---|
[0ff6b5] | 372 | list l; |
---|
| 373 | |
---|
[7b55a0] | 374 | dbprint(printlevel-voice+2,"// compute saturation of H''"); |
---|
[4f364b] | 375 | while(size(reduce(H,std(H0*var(1))))>0) |
---|
[0c4bd7] | 376 | { |
---|
[0ff6b5] | 377 | dbprint(printlevel-voice+2,"// compute matrix A of t"); |
---|
[0c4bd7] | 378 | k++; |
---|
[1418c4] | 379 | dbprint(printlevel-voice+2,"// k="+string(k)); |
---|
[38c0dca] | 380 | l=gmscoeffs(r,k); |
---|
[34a9eb1] | 381 | C,r=l[1..2]; |
---|
[04b295] | 382 | A0=A0+C; |
---|
[12c3e5] | 383 | |
---|
[7b55a0] | 384 | dbprint(printlevel-voice+2,"// compute saturation step"); |
---|
[04b295] | 385 | H0=H; |
---|
[4f364b] | 386 | H1=jet(module(A0*H1+var(1)^2*diff(matrix(H1),var(1))),k); |
---|
| 387 | H=H*var(1)+H1; |
---|
[04b295] | 388 | } |
---|
| 389 | |
---|
[4f364b] | 390 | A0=A0-k*var(1); |
---|
[1418c4] | 391 | dbprint(printlevel-voice+2,"// compute basis of saturation of H''"); |
---|
[34a9eb1] | 392 | H=std(H0); |
---|
[0ff6b5] | 393 | |
---|
| 394 | dbprint(printlevel-voice+2,"// transform H'' to saturation of H''"); |
---|
[4f364b] | 395 | H0=division(freemodule(mu)*var(1)^k,H,k*deg(var(1)))[1]; |
---|
[0ff6b5] | 396 | |
---|
[61549b] | 397 | return(A0,r,H,H0,k); |
---|
[0ff6b5] | 398 | } |
---|
| 399 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 400 | |
---|
[e9124e] | 401 | static proc tjet(matrix A0,ideal r,module H,int k0,int K) |
---|
[0ff6b5] | 402 | { |
---|
[1418c4] | 403 | dbprint(printlevel-voice+2,"// compute matrix A of t"); |
---|
[61549b] | 404 | dbprint(printlevel-voice+2,"// k="+string(K+k0+1)); |
---|
[38c0dca] | 405 | list l=gmscoeffs(r,K+k0+1); |
---|
[0ff6b5] | 406 | matrix C; |
---|
[34a9eb1] | 407 | C,r=l[1..2]; |
---|
[04b295] | 408 | A0=A0+C; |
---|
[1418c4] | 409 | dbprint(printlevel-voice+2,"// transform A to saturation of H''"); |
---|
[4f364b] | 410 | matrix A=division(A0*H+var(1)^2*diff(matrix(H),var(1)),H, |
---|
| 411 | (K+1)*deg(var(1)))[1]/var(1); |
---|
[0ff6b5] | 412 | return(A,A0,r); |
---|
| 413 | } |
---|
| 414 | /////////////////////////////////////////////////////////////////////////////// |
---|
[04b295] | 415 | |
---|
[e9124e] | 416 | static proc eigenval(matrix A0,ideal r,module H,int k0) |
---|
[0ff6b5] | 417 | { |
---|
[04b295] | 418 | dbprint(printlevel-voice+2, |
---|
[7b55a0] | 419 | "// compute eigenvalues e with multiplicities m of A1"); |
---|
[0ff6b5] | 420 | matrix A; |
---|
[e9124e] | 421 | A,A0,r=tjet(A0,r,H,k0,0); |
---|
[275721f] | 422 | list l=eigenvals(A); |
---|
[1418c4] | 423 | def e,m=l[1..2]; |
---|
| 424 | dbprint(printlevel-voice+2,"// e="+string(e)); |
---|
| 425 | dbprint(printlevel-voice+2,"// m="+string(m)); |
---|
[2ca72f] | 426 | return(e,m,A0,r); |
---|
[0ff6b5] | 427 | } |
---|
| 428 | /////////////////////////////////////////////////////////////////////////////// |
---|
[1418c4] | 429 | |
---|
[4f364b] | 430 | static proc transform(matrix A,matrix A0,ideal r,module H,module H0,ideal e, |
---|
| 431 | intvec m,int k0,int K,int opt) |
---|
[0ff6b5] | 432 | { |
---|
[61549b] | 433 | int mu=ncols(gmsbasis); |
---|
| 434 | |
---|
[e9124e] | 435 | number e0,e1=nmin(e),nmax(e); |
---|
[2ca72f] | 436 | |
---|
| 437 | int i,j,k; |
---|
| 438 | intvec d; |
---|
| 439 | d[ncols(e)]=0; |
---|
| 440 | if(opt) |
---|
| 441 | { |
---|
| 442 | dbprint(printlevel-voice+2, |
---|
[7b55a0] | 443 | "// compute rounded maximal differences d of e"); |
---|
[2ca72f] | 444 | for(i=1;i<=ncols(e);i++) |
---|
| 445 | { |
---|
| 446 | d[i]=int(e[i]-e0); |
---|
| 447 | } |
---|
| 448 | } |
---|
| 449 | else |
---|
| 450 | { |
---|
| 451 | dbprint(printlevel-voice+2, |
---|
| 452 | "// compute maximal integer differences d of e"); |
---|
| 453 | for(i=1;i<ncols(e);i++) |
---|
| 454 | { |
---|
[97403d] | 455 | for(j=i+1;j<=ncols(e);j++) |
---|
[2ca72f] | 456 | { |
---|
| 457 | k=int(e[i]-e[j]); |
---|
| 458 | if(number(e[i]-e[j])==k) |
---|
| 459 | { |
---|
| 460 | if(k>d[i]) |
---|
| 461 | { |
---|
| 462 | d[i]=k; |
---|
| 463 | } |
---|
| 464 | if(-k>d[j]) |
---|
| 465 | { |
---|
| 466 | d[j]=-k; |
---|
| 467 | } |
---|
| 468 | } |
---|
| 469 | } |
---|
| 470 | } |
---|
| 471 | } |
---|
| 472 | dbprint(printlevel-voice+2,"// d="+string(d)); |
---|
| 473 | |
---|
| 474 | for(i,k=1,0;i<=size(d);i++) |
---|
| 475 | { |
---|
| 476 | if(k<d[i]) |
---|
| 477 | { |
---|
| 478 | k=d[i]; |
---|
| 479 | } |
---|
| 480 | } |
---|
| 481 | |
---|
[7b55a0] | 482 | A,A0,r=tjet(A0,r,H,k0,K+k); |
---|
[4f364b] | 483 | module U0=var(1)^k0*freemodule(mu); |
---|
[0ff6b5] | 484 | |
---|
[2ca72f] | 485 | if(k>0) |
---|
[0c4bd7] | 486 | { |
---|
[2ca72f] | 487 | int i0,j0,i1,j1; |
---|
[0ff6b5] | 488 | module U,V; |
---|
[e480544] | 489 | list l; |
---|
[0c4bd7] | 490 | |
---|
[2ca72f] | 491 | while(k>0) |
---|
[0c4bd7] | 492 | { |
---|
[0ff6b5] | 493 | dbprint(printlevel-voice+2,"// transform to separate eigenvalues"); |
---|
[34a9eb1] | 494 | U=0; |
---|
[ccf8d9] | 495 | for(i=1;i<=ncols(e);i++) |
---|
[0c4bd7] | 496 | { |
---|
[ccf8d9] | 497 | U=U+syz(power(jet(A,0)-e[i],m[i])); |
---|
[34a9eb1] | 498 | } |
---|
[0ff6b5] | 499 | V=inverse(U); |
---|
| 500 | A=V*A*U; |
---|
| 501 | H0=V*H0; |
---|
[61549b] | 502 | U0=U0*U; |
---|
[34a9eb1] | 503 | |
---|
[2ca72f] | 504 | dbprint(printlevel-voice+2, |
---|
[7b55a0] | 505 | "// transform to reduce maximum of d by 1"); |
---|
[0ff6b5] | 506 | for(i0,i=1,1;i0<=ncols(e);i0++) |
---|
[34a9eb1] | 507 | { |
---|
[0ff6b5] | 508 | for(i1=1;i1<=m[i0];i1,i=i1+1,i+1) |
---|
[0c4bd7] | 509 | { |
---|
[0ff6b5] | 510 | for(j0,j=1,1;j0<=ncols(e);j0++) |
---|
[0c4bd7] | 511 | { |
---|
[0ff6b5] | 512 | for(j1=1;j1<=m[j0];j1,j=j1+1,j+1) |
---|
[34a9eb1] | 513 | { |
---|
[2ca72f] | 514 | if(d[i0]==0&&d[j0]>0) |
---|
[0ff6b5] | 515 | { |
---|
[4f364b] | 516 | A[i,j]=A[i,j]/var(1); |
---|
[0ff6b5] | 517 | } |
---|
[2ca72f] | 518 | if(d[i0]>0&&d[j0]==0) |
---|
[0ff6b5] | 519 | { |
---|
[4f364b] | 520 | A[i,j]=A[i,j]*var(1); |
---|
[0ff6b5] | 521 | } |
---|
[34a9eb1] | 522 | } |
---|
[3c4dcc] | 523 | } |
---|
[0c4bd7] | 524 | } |
---|
| 525 | } |
---|
[0ff6b5] | 526 | |
---|
| 527 | H0=transpose(H0); |
---|
| 528 | for(i0,i=1,1;i0<=ncols(e);i0++) |
---|
[6ab855] | 529 | { |
---|
[2ca72f] | 530 | if(d[i0]>0) |
---|
[34a9eb1] | 531 | { |
---|
[0ff6b5] | 532 | for(i1=1;i1<=m[i0];i1,i=i1+1,i+1) |
---|
| 533 | { |
---|
| 534 | A[i,i]=A[i,i]-1; |
---|
[4f364b] | 535 | H0[i]=H0[i]*var(1); |
---|
| 536 | U0[i]=U0[i]/var(1); |
---|
[0ff6b5] | 537 | } |
---|
| 538 | e[i0]=e[i0]-1; |
---|
[2ca72f] | 539 | d[i0]=d[i0]-1; |
---|
[34a9eb1] | 540 | } |
---|
[ccf8d9] | 541 | else |
---|
| 542 | { |
---|
| 543 | i=i+m[i0]; |
---|
| 544 | } |
---|
[6ab855] | 545 | } |
---|
[0ff6b5] | 546 | H0=transpose(H0); |
---|
[1418c4] | 547 | |
---|
[2d3b6e6] | 548 | l=sppnf(list(e,d,m)); |
---|
[2ca72f] | 549 | e,d,m=l[1..3]; |
---|
[e480544] | 550 | |
---|
[2ca72f] | 551 | k--; |
---|
[6ab855] | 552 | } |
---|
[34a9eb1] | 553 | |
---|
[0ff6b5] | 554 | A=jet(A,K); |
---|
[0c4bd7] | 555 | } |
---|
| 556 | |
---|
[61549b] | 557 | return(A,A0,r,H0,U0,e,m); |
---|
[0ff6b5] | 558 | } |
---|
| 559 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 560 | |
---|
[779ee3] | 561 | proc bernstein(poly t) |
---|
| 562 | "USAGE: bernstein(t); poly t |
---|
| 563 | ASSUME: characteristic 0; local degree ordering; |
---|
| 564 | isolated critical point 0 of t |
---|
| 565 | RETURN: ideal r; roots of the Bernstein polynomial b excluding the root -1 |
---|
| 566 | NOTE: the roots of b are negative rational numbers and -1 is a root of b |
---|
[3c4dcc] | 567 | KEYWORDS: singularities; Gauss-Manin system; Brieskorn lattice; |
---|
[779ee3] | 568 | Bernstein polynomial |
---|
| 569 | EXAMPLE: example bernstein; shows examples |
---|
| 570 | " |
---|
| 571 | { |
---|
| 572 | def R=basering; |
---|
| 573 | int n=nvars(R)-1; |
---|
| 574 | def G=gmsring(t,"s"); |
---|
| 575 | setring(G); |
---|
| 576 | |
---|
| 577 | matrix A; |
---|
| 578 | module U0; |
---|
| 579 | ideal e; |
---|
| 580 | intvec m; |
---|
| 581 | |
---|
| 582 | def A0,r,H,H0,k0=saturate(); |
---|
[b5fcae] | 583 | A,A0,r=tjet(A0,r,H,k0,0); |
---|
| 584 | list l=minipoly(A); |
---|
| 585 | e,m=l[1..2]; |
---|
[779ee3] | 586 | |
---|
[b5fcae] | 587 | for(int i=1;i<=ncols(e);i++) |
---|
[744dd08] | 588 | { |
---|
[b5fcae] | 589 | e[i]=-e[i]; |
---|
| 590 | if(e[i]==-1) |
---|
[744dd08] | 591 | { |
---|
[b5fcae] | 592 | m[i]=m[i]+1; |
---|
[744dd08] | 593 | } |
---|
| 594 | } |
---|
| 595 | |
---|
[779ee3] | 596 | setring(R); |
---|
[b5fcae] | 597 | ideal e=imap(G,e); |
---|
[779ee3] | 598 | kill G,gmsmaxdeg; |
---|
| 599 | |
---|
[b5fcae] | 600 | return(list(e,m)); |
---|
[779ee3] | 601 | } |
---|
| 602 | example |
---|
| 603 | { "EXAMPLE:"; echo=2; |
---|
| 604 | ring R=0,(x,y),ds; |
---|
| 605 | poly t=x5+x2y2+y5; |
---|
| 606 | bernstein(t); |
---|
| 607 | } |
---|
| 608 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 609 | |
---|
[275721f] | 610 | proc monodromy(poly t) |
---|
[0ff6b5] | 611 | "USAGE: monodromy(t); poly t |
---|
[275721f] | 612 | ASSUME: characteristic 0; local degree ordering; |
---|
[a8cc0a] | 613 | isolated critical point 0 of t |
---|
[e9124e] | 614 | RETURN: |
---|
| 615 | @format |
---|
[26a4bb] | 616 | list l; Jordan data jordan(M) of monodromy matrix exp(-2*pi*i*M) |
---|
[3c4dcc] | 617 | ideal l[1]; |
---|
[e9124e] | 618 | number l[1][i]; eigenvalue of i-th Jordan block of M |
---|
[3c4dcc] | 619 | intvec l[2]; |
---|
[e9124e] | 620 | int l[2][i]; size of i-th Jordan block of M |
---|
[3c4dcc] | 621 | intvec l[3]; |
---|
[e9124e] | 622 | int l[3][i]; multiplicity of i-th Jordan block of M |
---|
| 623 | @end format |
---|
| 624 | SEE ALSO: mondromy_lib, linalg_lib |
---|
[46af92] | 625 | KEYWORDS: singularities; Gauss-Manin system; Brieskorn lattice; monodromy |
---|
[0ff6b5] | 626 | EXAMPLE: example monodromy; shows examples |
---|
| 627 | " |
---|
| 628 | { |
---|
| 629 | def R=basering; |
---|
| 630 | int n=nvars(R)-1; |
---|
| 631 | def G=gmsring(t,"s"); |
---|
| 632 | setring(G); |
---|
| 633 | |
---|
| 634 | matrix A; |
---|
[61549b] | 635 | module U0; |
---|
[0ff6b5] | 636 | ideal e; |
---|
| 637 | intvec m; |
---|
| 638 | |
---|
[2ca72f] | 639 | def A0,r,H,H0,k0=saturate(); |
---|
[e9124e] | 640 | e,m,A0,r=eigenval(A0,r,H,k0); |
---|
| 641 | A,A0,r,H0,U0,e,m=transform(A,A0,r,H,H0,e,m,k0,0,0); |
---|
[0ff6b5] | 642 | |
---|
[51534b6] | 643 | list l=jordan(A,e,m); |
---|
[500122] | 644 | setring(R); |
---|
[51534b6] | 645 | list l=imap(G,l); |
---|
[9526639] | 646 | kill G,gmsmaxdeg; |
---|
[51534b6] | 647 | |
---|
| 648 | return(l); |
---|
[0c4bd7] | 649 | } |
---|
| 650 | example |
---|
| 651 | { "EXAMPLE:"; echo=2; |
---|
| 652 | ring R=0,(x,y),ds; |
---|
[d70bc7] | 653 | poly t=x5+x2y2+y5; |
---|
| 654 | monodromy(t); |
---|
[0c4bd7] | 655 | } |
---|
| 656 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 657 | |
---|
[86c1f0] | 658 | proc spectrum(poly t) |
---|
| 659 | "USAGE: spectrum(t); poly t |
---|
[275721f] | 660 | ASSUME: characteristic 0; local degree ordering; |
---|
[a8cc0a] | 661 | isolated critical point 0 of t |
---|
[057c22e] | 662 | RETURN: |
---|
[7c7ca9] | 663 | @format |
---|
[275721f] | 664 | list sp; singularity spectrum of t |
---|
| 665 | ideal sp[1]; |
---|
| 666 | number sp[1][i]; i-th spectral number |
---|
| 667 | intvec sp[2]; |
---|
| 668 | int sp[2][i]; multiplicity of i-th spectral number |
---|
[86c1f0] | 669 | @end format |
---|
| 670 | SEE ALSO: spectrum_lib |
---|
[46af92] | 671 | KEYWORDS: singularities; Gauss-Manin system; Brieskorn lattice; |
---|
[d70bc7] | 672 | mixed Hodge structure; V-filtration; spectrum |
---|
[61549b] | 673 | EXAMPLE: example spectrum; shows examples |
---|
[86c1f0] | 674 | " |
---|
| 675 | { |
---|
[61549b] | 676 | list l=vwfilt(t); |
---|
[2d3b6e6] | 677 | return(spnf(list(l[1],l[3]))); |
---|
[86c1f0] | 678 | } |
---|
| 679 | example |
---|
| 680 | { "EXAMPLE:"; echo=2; |
---|
| 681 | ring R=0,(x,y),ds; |
---|
| 682 | poly t=x5+x2y2+y5; |
---|
| 683 | spprint(spectrum(t)); |
---|
| 684 | } |
---|
| 685 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 686 | |
---|
[61549b] | 687 | proc sppairs(poly t) |
---|
| 688 | "USAGE: sppairs(t); poly t |
---|
[275721f] | 689 | ASSUME: characteristic 0; local degree ordering; |
---|
[a8cc0a] | 690 | isolated critical point 0 of t |
---|
[61549b] | 691 | RETURN: |
---|
| 692 | @format |
---|
[275721f] | 693 | list spp; spectral pairs of t |
---|
| 694 | ideal spp[1]; |
---|
| 695 | number spp[1][i]; V-filtration index of i-th spectral pair |
---|
| 696 | intvec spp[2]; |
---|
| 697 | int spp[2][i]; weight filtration index of i-th spectral pair |
---|
[e9124e] | 698 | intvec spp[3]; |
---|
[275721f] | 699 | int spp[3][i]; multiplicity of i-th spectral pair |
---|
[61549b] | 700 | @end format |
---|
| 701 | SEE ALSO: spectrum_lib |
---|
[46af92] | 702 | KEYWORDS: singularities; Gauss-Manin system; Brieskorn lattice; |
---|
[d70bc7] | 703 | mixed Hodge structure; V-filtration; weight filtration; |
---|
| 704 | spectrum; spectral pairs |
---|
[61549b] | 705 | EXAMPLE: example sppairs; shows examples |
---|
| 706 | " |
---|
[e480544] | 707 | { |
---|
[61549b] | 708 | list l=vwfilt(t); |
---|
| 709 | return(list(l[1],l[2],l[3])); |
---|
| 710 | } |
---|
| 711 | example |
---|
| 712 | { "EXAMPLE:"; echo=2; |
---|
| 713 | ring R=0,(x,y),ds; |
---|
| 714 | poly t=x5+x2y2+y5; |
---|
| 715 | sppprint(sppairs(t)); |
---|
[e480544] | 716 | } |
---|
| 717 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 718 | |
---|
[61549b] | 719 | proc vfilt(poly t) |
---|
| 720 | "USAGE: vfilt(t); poly t |
---|
[275721f] | 721 | ASSUME: characteristic 0; local degree ordering; |
---|
[a8cc0a] | 722 | isolated critical point 0 of t |
---|
[275721f] | 723 | RETURN: |
---|
[86c1f0] | 724 | @format |
---|
[275721f] | 725 | list v; V-filtration on H''/s*H'' |
---|
| 726 | ideal v[1]; |
---|
[91fc5e] | 727 | number v[1][i]; V-filtration index of i-th spectral number |
---|
[e9124e] | 728 | intvec v[2]; |
---|
[91fc5e] | 729 | int v[2][i]; multiplicity of i-th spectral number |
---|
[e9124e] | 730 | list v[3]; |
---|
[04c344] | 731 | module v[3][i]; vector space of i-th graded part in terms of v[4] |
---|
[275721f] | 732 | ideal v[4]; monomial vector space basis of H''/s*H'' |
---|
| 733 | ideal v[5]; standard basis of Jacobian ideal |
---|
[86c1f0] | 734 | @end format |
---|
| 735 | SEE ALSO: spectrum_lib |
---|
[46af92] | 736 | KEYWORDS: singularities; Gauss-Manin system; Brieskorn lattice; |
---|
[61549b] | 737 | mixed Hodge structure; V-filtration; spectrum |
---|
| 738 | EXAMPLE: example vfilt; shows examples |
---|
| 739 | " |
---|
| 740 | { |
---|
| 741 | list l=vwfilt(t); |
---|
[2d3b6e6] | 742 | return(spnf(list(l[1],l[3],l[4]))+list(l[5],l[6])); |
---|
[61549b] | 743 | } |
---|
| 744 | example |
---|
| 745 | { "EXAMPLE:"; echo=2; |
---|
| 746 | ring R=0,(x,y),ds; |
---|
| 747 | poly t=x5+x2y2+y5; |
---|
| 748 | vfilt(t); |
---|
| 749 | } |
---|
| 750 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 751 | |
---|
| 752 | proc vwfilt(poly t) |
---|
| 753 | "USAGE: vwfilt(t); poly t |
---|
[275721f] | 754 | ASSUME: characteristic 0; local degree ordering; |
---|
[a8cc0a] | 755 | isolated critical point 0 of t |
---|
[61549b] | 756 | RETURN: |
---|
| 757 | @format |
---|
[275721f] | 758 | list vw; weighted V-filtration on H''/s*H'' |
---|
| 759 | ideal vw[1]; |
---|
| 760 | number vw[1][i]; V-filtration index of i-th spectral pair |
---|
| 761 | intvec vw[2]; |
---|
| 762 | int vw[2][i]; weight filtration index of i-th spectral pair |
---|
[e9124e] | 763 | intvec vw[3]; |
---|
[275721f] | 764 | int vw[3][i]; multiplicity of i-th spectral pair |
---|
[e9124e] | 765 | list vw[4]; |
---|
[04c344] | 766 | module vw[4][i]; vector space of i-th graded part in terms of vw[5] |
---|
[275721f] | 767 | ideal vw[5]; monomial vector space basis of H''/s*H'' |
---|
| 768 | ideal vw[6]; standard basis of Jacobian ideal |
---|
[61549b] | 769 | @end format |
---|
| 770 | SEE ALSO: spectrum_lib |
---|
[46af92] | 771 | KEYWORDS: singularities; Gauss-Manin system; Brieskorn lattice; |
---|
[61549b] | 772 | mixed Hodge structure; V-filtration; weight filtration; |
---|
[86c1f0] | 773 | spectrum; spectral pairs |
---|
[61549b] | 774 | EXAMPLE: example vwfilt; shows examples |
---|
[86c1f0] | 775 | " |
---|
| 776 | { |
---|
| 777 | def R=basering; |
---|
| 778 | int n=nvars(R)-1; |
---|
| 779 | def G=gmsring(t,"s"); |
---|
| 780 | setring(G); |
---|
| 781 | |
---|
| 782 | int mu=ncols(gmsbasis); |
---|
[0ff6b5] | 783 | matrix A; |
---|
[61549b] | 784 | module U0; |
---|
[0ff6b5] | 785 | ideal e; |
---|
| 786 | intvec m; |
---|
[86c1f0] | 787 | |
---|
[2ca72f] | 788 | def A0,r,H,H0,k0=saturate(); |
---|
[e9124e] | 789 | e,m,A0,r=eigenval(A0,r,H,k0); |
---|
| 790 | A,A0,r,H0,U0,e,m=transform(A,A0,r,H,H0,e,m,k0,0,1); |
---|
[86c1f0] | 791 | |
---|
[409dbae] | 792 | dbprint(printlevel-voice+2,"// compute weight filtration basis"); |
---|
[e480544] | 793 | list l=jordanbasis(A,e,m); |
---|
[0ff6b5] | 794 | def U,v=l[1..2]; |
---|
[61549b] | 795 | kill l; |
---|
[ccf8d9] | 796 | vector u0; |
---|
| 797 | int v0; |
---|
[61549b] | 798 | int i,j,k,l; |
---|
| 799 | for(k,l=1,1;l<=ncols(e);k,l=k+m[l],l+1) |
---|
[86c1f0] | 800 | { |
---|
[61549b] | 801 | for(i=k+m[l]-1;i>=k+1;i--) |
---|
[86c1f0] | 802 | { |
---|
[ccf8d9] | 803 | for(j=i-1;j>=k;j--) |
---|
[86c1f0] | 804 | { |
---|
[ccf8d9] | 805 | if(v[i]>v[j]) |
---|
| 806 | { |
---|
| 807 | v0=v[i];v[i]=v[j];v[j]=v0; |
---|
| 808 | u0=U[i];U[i]=U[j];U[j]=u0; |
---|
| 809 | } |
---|
[86c1f0] | 810 | } |
---|
| 811 | } |
---|
| 812 | } |
---|
| 813 | |
---|
| 814 | dbprint(printlevel-voice+2,"// transform to weight filtration basis"); |
---|
[ccf8d9] | 815 | matrix V=inverse(U); |
---|
[86c1f0] | 816 | A=V*A*U; |
---|
[4f364b] | 817 | dbprint(printlevel-voice+2,"// compute standard basis of H''"); |
---|
[86c1f0] | 818 | H0=std(V*H0); |
---|
[61549b] | 819 | U0=U0*U; |
---|
[86c1f0] | 820 | |
---|
| 821 | dbprint(printlevel-voice+2,"// compute spectral pairs"); |
---|
| 822 | ideal a; |
---|
| 823 | intvec w; |
---|
| 824 | for(i=1;i<=mu;i++) |
---|
| 825 | { |
---|
| 826 | j=leadexp(H0[i])[nvars(basering)+1]; |
---|
[4f364b] | 827 | a[i]=A[j,j]+ord(H0[i])/deg(var(1))-1; |
---|
[0ff6b5] | 828 | w[i]=v[j]+n; |
---|
[86c1f0] | 829 | } |
---|
[61549b] | 830 | kill v; |
---|
[4f364b] | 831 | module v=simplify(jet(H*U0*H0,2*k0)/var(1)^(2*k0),1); |
---|
[0ff6b5] | 832 | |
---|
[51534b6] | 833 | kill l; |
---|
[2d3b6e6] | 834 | list l=sppnf(list(a,w,v))+list(gmsbasis,gmsstd); |
---|
[86c1f0] | 835 | setring(R); |
---|
[51534b6] | 836 | list l=imap(G,l); |
---|
[9526639] | 837 | kill G,gmsmaxdeg; |
---|
[51534b6] | 838 | attrib(l[5],"isSB",1); |
---|
| 839 | |
---|
| 840 | return(l); |
---|
[ccf8d9] | 841 | } |
---|
| 842 | example |
---|
| 843 | { "EXAMPLE:"; echo=2; |
---|
| 844 | ring R=0,(x,y),ds; |
---|
| 845 | poly t=x5+x2y2+y5; |
---|
[61549b] | 846 | vwfilt(t); |
---|
[ccf8d9] | 847 | } |
---|
| 848 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 849 | |
---|
[275721f] | 850 | static proc commutator(matrix A) |
---|
| 851 | { |
---|
| 852 | int n=ncols(A); |
---|
| 853 | int i,j,k; |
---|
| 854 | matrix C[n^2][n^2]; |
---|
| 855 | for(i=0;i<n;i++) |
---|
| 856 | { |
---|
| 857 | for(j=0;j<n;j++) |
---|
| 858 | { |
---|
| 859 | for(k=0;k<n;k++) |
---|
| 860 | { |
---|
| 861 | C[i*n+j+1,k*n+j+1]=C[i*n+j+1,k*n+j+1]+A[i+1,k+1]; |
---|
| 862 | C[i*n+j+1,i*n+k+1]=C[i*n+j+1,i*n+k+1]-A[k+1,j+1]; |
---|
| 863 | } |
---|
| 864 | } |
---|
| 865 | } |
---|
| 866 | return(C); |
---|
| 867 | } |
---|
| 868 | |
---|
| 869 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 870 | |
---|
| 871 | proc tmatrix(poly t,list #) |
---|
| 872 | "USAGE: tmatrix(t); poly t |
---|
| 873 | ASSUME: characteristic 0; local degree ordering; |
---|
[a8cc0a] | 874 | isolated critical point 0 of t |
---|
[91fc5e] | 875 | RETURN: |
---|
| 876 | @format |
---|
[4f364b] | 877 | list l=A0,A1,T,M; |
---|
| 878 | matrix A0,A1; t=A0+s*A1+s^2*(d/ds) on H'' w.r.t. C[[s]]-basis M*T |
---|
| 879 | module T; C-basis of C^mu |
---|
| 880 | ideal M; monomial C-basis of H''/sH'' |
---|
[91fc5e] | 881 | @end format |
---|
[46af92] | 882 | KEYWORDS: singularities; Gauss-Manin system; Brieskorn lattice; |
---|
| 883 | mixed Hodge structure; V-filtration; good basis |
---|
[275721f] | 884 | EXAMPLE: example tmatrix; shows examples |
---|
[61549b] | 885 | " |
---|
| 886 | { |
---|
| 887 | def R=basering; |
---|
| 888 | int n=nvars(R)-1; |
---|
| 889 | def G=gmsring(t,"s"); |
---|
| 890 | setring(G); |
---|
| 891 | |
---|
| 892 | int mu=ncols(gmsbasis); |
---|
| 893 | matrix A; |
---|
| 894 | module U0; |
---|
| 895 | ideal e; |
---|
| 896 | intvec m; |
---|
| 897 | |
---|
[2ca72f] | 898 | def A0,r,H,H0,k0=saturate(); |
---|
[e9124e] | 899 | e,m,A0,r=eigenval(A0,r,H,k0); |
---|
| 900 | int k1=int(nmax(e)-nmin(e)); |
---|
| 901 | A,A0,r,H0,U0,e,m=transform(A,A0,r,H,H0,e,m,k0,k0+k1,1); |
---|
[4f364b] | 902 | module T=H*U0; |
---|
[61549b] | 903 | |
---|
[51534b6] | 904 | ring S=0,s,(ds,c); |
---|
| 905 | matrix A=imap(G,A); |
---|
| 906 | module H0=imap(G,H0); |
---|
[4f364b] | 907 | module T=imap(G,T); |
---|
[51534b6] | 908 | ideal e=imap(G,e); |
---|
| 909 | |
---|
[61549b] | 910 | dbprint(printlevel-voice+2,"// transform to Jordan basis"); |
---|
| 911 | module U=jordanbasis(A,e,m)[1]; |
---|
| 912 | matrix V=inverse(U); |
---|
| 913 | A=V*A*U; |
---|
[51534b6] | 914 | module H=V*H0; |
---|
[4f364b] | 915 | T=T*U; |
---|
[61549b] | 916 | |
---|
| 917 | dbprint(printlevel-voice+2,"// compute splitting of V-filtration"); |
---|
| 918 | int i,j,k; |
---|
| 919 | U=freemodule(mu); |
---|
| 920 | V=matrix(0,mu,mu); |
---|
| 921 | matrix v[mu^2][1]; |
---|
[51534b6] | 922 | matrix A0=commutator(jet(A,0)); |
---|
[61549b] | 923 | for(k=1;k<=k0+k1;k++) |
---|
| 924 | { |
---|
| 925 | for(j=0;j<k;j++) |
---|
| 926 | { |
---|
[4f364b] | 927 | V=matrix(V)-(jet(A,k-j)/var(1)^(k-j))*(jet(U,j)/var(1)^j); |
---|
[61549b] | 928 | } |
---|
| 929 | v=V[1..mu,1..mu]; |
---|
| 930 | v=inverse(A0+k)*v; |
---|
| 931 | V=v[1..mu^2,1]; |
---|
[4f364b] | 932 | U=matrix(U)+var(1)^k*V; |
---|
[61549b] | 933 | } |
---|
[962b1f] | 934 | attrib(U,"isSB",1); |
---|
[61549b] | 935 | |
---|
| 936 | dbprint(printlevel-voice+2,"// transform to V-splitting basis"); |
---|
| 937 | A=jet(A,0); |
---|
[4f364b] | 938 | H=std(division(H,U,(k0+k1)*deg(var(1)))[1]); |
---|
| 939 | T=T*U; |
---|
[61549b] | 940 | |
---|
| 941 | dbprint(printlevel-voice+2,"// compute V-leading terms of H''"); |
---|
| 942 | int i0,j0; |
---|
| 943 | module H1=H; |
---|
| 944 | for(k=ncols(H1);k>=1;k--) |
---|
| 945 | { |
---|
| 946 | i0=leadexp(H1[k])[nvars(basering)+1]; |
---|
[4f364b] | 947 | j0=ord(H1[k]); |
---|
[61549b] | 948 | H0[k]=lead(H1[k]); |
---|
| 949 | H1[k]=H1[k]-lead(H1[k]); |
---|
| 950 | if(H1[k]!=0) |
---|
| 951 | { |
---|
| 952 | i=leadexp(H1[k])[nvars(basering)+1]; |
---|
[4f364b] | 953 | j=ord(H1[k]); |
---|
[61549b] | 954 | while(A[i,i]+j==A[i0,i0]+j0) |
---|
| 955 | { |
---|
| 956 | H0[k]=H0[k]+lead(H1[k]); |
---|
| 957 | H1[k]=H1[k]-lead(H1[k]); |
---|
| 958 | i=leadexp(H1[k])[nvars(basering)+1]; |
---|
[4f364b] | 959 | j=ord(H1[k]); |
---|
[61549b] | 960 | } |
---|
| 961 | } |
---|
| 962 | } |
---|
| 963 | H0=simplify(H0,1); |
---|
| 964 | |
---|
| 965 | dbprint(printlevel-voice+2,"// compute N"); |
---|
| 966 | matrix N=A; |
---|
| 967 | for(i=1;i<=ncols(N);i++) |
---|
| 968 | { |
---|
| 969 | N[i,i]=0; |
---|
| 970 | } |
---|
| 971 | |
---|
| 972 | dbprint(printlevel-voice+2,"// compute splitting of Hodge filtration"); |
---|
| 973 | U=0; |
---|
| 974 | module U1; |
---|
| 975 | module C; |
---|
| 976 | list F,I; |
---|
[08fff3] | 977 | module F0,I0,U0; |
---|
[61549b] | 978 | for(i0,j0=1,1;i0<=ncols(e);i0++) |
---|
| 979 | { |
---|
| 980 | C=matrix(0,mu,1); |
---|
| 981 | for(j=m[i0];j>=1;j,j0=j-1,j0+1) |
---|
| 982 | { |
---|
| 983 | C=C+gen(j0); |
---|
| 984 | } |
---|
| 985 | F0=intersect(C,H0); |
---|
[275721f] | 986 | |
---|
[61549b] | 987 | F=list(); |
---|
| 988 | j=0; |
---|
| 989 | while(size(F0)>0) |
---|
| 990 | { |
---|
| 991 | j++; |
---|
| 992 | F[j]=matrix(0,mu,1); |
---|
| 993 | if(size(jet(F0,0))>0) |
---|
| 994 | { |
---|
| 995 | for(i=ncols(F0);i>=1;i--) |
---|
| 996 | { |
---|
| 997 | if(ord(F0[i])==0) |
---|
| 998 | { |
---|
| 999 | F[j]=F[j]+F0[i]; |
---|
| 1000 | } |
---|
| 1001 | } |
---|
| 1002 | } |
---|
| 1003 | for(i=ncols(F0);i>=1;i--) |
---|
| 1004 | { |
---|
[4f364b] | 1005 | F0[i]=F0[i]/var(1); |
---|
[61549b] | 1006 | } |
---|
| 1007 | } |
---|
| 1008 | |
---|
| 1009 | I=list(); |
---|
| 1010 | I0=module(); |
---|
[08fff3] | 1011 | U0=std(module()); |
---|
[61549b] | 1012 | for(i=size(F);i>=1;i--) |
---|
| 1013 | { |
---|
| 1014 | I[i]=module(); |
---|
| 1015 | } |
---|
| 1016 | for(i=1;i<=size(F);i++) |
---|
| 1017 | { |
---|
| 1018 | I0=reduce(F[i],U0); |
---|
| 1019 | j=i; |
---|
| 1020 | while(size(I0)>0) |
---|
| 1021 | { |
---|
| 1022 | U0=std(U0+I0); |
---|
| 1023 | I[j]=I[j]+I0; |
---|
[e5dcf2e] | 1024 | I0=reduce(N*matrix(I0,nrows(N),ncols(N)),U0); |
---|
[61549b] | 1025 | j++; |
---|
| 1026 | } |
---|
| 1027 | } |
---|
| 1028 | |
---|
| 1029 | for(i=1;i<=size(I);i++) |
---|
| 1030 | { |
---|
| 1031 | U=U+I[i]; |
---|
| 1032 | } |
---|
| 1033 | } |
---|
| 1034 | |
---|
| 1035 | dbprint(printlevel-voice+2,"// transform to Hodge splitting basis"); |
---|
| 1036 | V=inverse(U); |
---|
| 1037 | A=V*A*U; |
---|
| 1038 | H=V*H; |
---|
[4f364b] | 1039 | T=T*U; |
---|
[61549b] | 1040 | |
---|
| 1041 | dbprint(printlevel-voice+2,"// compute reduced standard basis of H''"); |
---|
[51534b6] | 1042 | degBound=k0+k1+2; |
---|
[17f79e9] | 1043 | option(redSB); |
---|
[61549b] | 1044 | H=std(H); |
---|
[17f79e9] | 1045 | option(noredSB); |
---|
[61549b] | 1046 | degBound=0; |
---|
| 1047 | H=simplify(jet(H,k0+k1),1); |
---|
[962b1f] | 1048 | attrib(H,"isSB",1); |
---|
[61549b] | 1049 | dbprint(printlevel-voice+2,"// compute matrix A0+sA1 of t"); |
---|
[4f364b] | 1050 | A=division(var(1)*A*H+var(1)^2*diff(matrix(H),var(1)),H,deg(var(1)))[1]; |
---|
[d70bc7] | 1051 | A0=jet(A,0); |
---|
[4f364b] | 1052 | A=jet(A,1)/var(1); |
---|
| 1053 | T=jet(T*H,2*k0)/var(1)^(2*k0); |
---|
[61549b] | 1054 | |
---|
| 1055 | setring(R); |
---|
[51534b6] | 1056 | matrix A0=imap(S,A0); |
---|
| 1057 | matrix A1=imap(S,A); |
---|
[4f364b] | 1058 | module T=imap(S,T); |
---|
| 1059 | ideal M=imap(G,gmsbasis); |
---|
| 1060 | kill G,gmsmaxdeg; |
---|
[51534b6] | 1061 | kill S; |
---|
[4f364b] | 1062 | return(list(A0,A1,T,M)); |
---|
[61549b] | 1063 | } |
---|
| 1064 | example |
---|
| 1065 | { "EXAMPLE:"; echo=2; |
---|
| 1066 | ring R=0,(x,y),ds; |
---|
| 1067 | poly t=x5+x2y2+y5; |
---|
[275721f] | 1068 | list A=tmatrix(t); |
---|
[61549b] | 1069 | print(A[1]); |
---|
| 1070 | print(A[2]); |
---|
[4f364b] | 1071 | print(A[3]); |
---|
| 1072 | print(A[4]); |
---|
[61549b] | 1073 | } |
---|
| 1074 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1075 | |
---|
[275721f] | 1076 | proc endvfilt(list v) |
---|
| 1077 | "USAGE: endvfilt(v); list v |
---|
| 1078 | ASSUME: v returned by vfilt |
---|
[057c22e] | 1079 | RETURN: |
---|
[7c7ca9] | 1080 | @format |
---|
[04c344] | 1081 | list ev; V-filtration on Jacobian algebra |
---|
[275721f] | 1082 | ideal ev[1]; |
---|
[26a4bb] | 1083 | number ev[1][i]; i-th V-filtration index |
---|
[e9124e] | 1084 | intvec ev[2]; |
---|
[26a4bb] | 1085 | int ev[2][i]; i-th multiplicity |
---|
[e9124e] | 1086 | list ev[3]; |
---|
[04c344] | 1087 | module ev[3][i]; vector space of i-th graded part in terms of ev[4] |
---|
[275721f] | 1088 | ideal ev[4]; monomial vector space basis of Jacobian algebra |
---|
| 1089 | ideal ev[5]; standard basis of Jacobian ideal |
---|
[c52356d] | 1090 | @end format |
---|
[46af92] | 1091 | KEYWORDS: singularities; Gauss-Manin system; Brieskorn lattice; |
---|
[d70bc7] | 1092 | mixed Hodge structure; V-filtration; endomorphism filtration |
---|
[275721f] | 1093 | EXAMPLE: example endvfilt; shows examples |
---|
[0c4bd7] | 1094 | " |
---|
| 1095 | { |
---|
[275721f] | 1096 | def a,d,V,m,g=v[1..5]; |
---|
[0049b4] | 1097 | attrib(g,"isSB",1); |
---|
[0c4bd7] | 1098 | int mu=ncols(m); |
---|
| 1099 | |
---|
[275721f] | 1100 | module V0=V[1]; |
---|
| 1101 | for(int i=2;i<=size(V);i++) |
---|
[0c4bd7] | 1102 | { |
---|
[275721f] | 1103 | V0=V0,V[i]; |
---|
[0c4bd7] | 1104 | } |
---|
[d08457] | 1105 | |
---|
[1418c4] | 1106 | dbprint(printlevel-voice+2,"// compute multiplication in Jacobian algebra"); |
---|
[0c4bd7] | 1107 | list M; |
---|
[409dbae] | 1108 | module U=freemodule(ncols(m)); |
---|
[0c4bd7] | 1109 | for(i=ncols(m);i>=1;i--) |
---|
| 1110 | { |
---|
[215349c] | 1111 | M[i]=division(coeffs(reduce(m[i]*m,g,U),m)*V0,V0)[1]; |
---|
[0c4bd7] | 1112 | } |
---|
| 1113 | |
---|
[8960ec] | 1114 | int j,k,i0,j0,i1,j1; |
---|
[8c4269a] | 1115 | number b0=number(a[1]-a[ncols(a)]); |
---|
| 1116 | number b1,b2; |
---|
[0c4bd7] | 1117 | matrix M0; |
---|
| 1118 | module L; |
---|
| 1119 | list v0=freemodule(ncols(m)); |
---|
[8c4269a] | 1120 | ideal a0=b0; |
---|
[a25a6a] | 1121 | list l; |
---|
[0c4bd7] | 1122 | |
---|
[8c4269a] | 1123 | while(b0<number(a[ncols(a)]-a[1])) |
---|
[0c4bd7] | 1124 | { |
---|
[1418c4] | 1125 | dbprint(printlevel-voice+2,"// find next possible index"); |
---|
[8c4269a] | 1126 | b1=number(a[ncols(a)]-a[1]); |
---|
| 1127 | for(j=ncols(a);j>=1;j--) |
---|
[0c4bd7] | 1128 | { |
---|
[8c4269a] | 1129 | for(i=ncols(a);i>=1;i--) |
---|
[0c4bd7] | 1130 | { |
---|
[8c4269a] | 1131 | b2=number(a[i]-a[j]); |
---|
| 1132 | if(b2>b0&&b2<b1) |
---|
[0c4bd7] | 1133 | { |
---|
[8c4269a] | 1134 | b1=b2; |
---|
[0c4bd7] | 1135 | } |
---|
| 1136 | else |
---|
| 1137 | { |
---|
[8c4269a] | 1138 | if(b2<=b0) |
---|
[0c4bd7] | 1139 | { |
---|
| 1140 | i=0; |
---|
| 1141 | } |
---|
| 1142 | } |
---|
| 1143 | } |
---|
| 1144 | } |
---|
[8c4269a] | 1145 | b0=b1; |
---|
[0c4bd7] | 1146 | |
---|
[a25a6a] | 1147 | l=ideal(); |
---|
[0c4bd7] | 1148 | for(k=ncols(m);k>=2;k--) |
---|
| 1149 | { |
---|
| 1150 | l=l+list(ideal()); |
---|
| 1151 | } |
---|
| 1152 | |
---|
[61549b] | 1153 | dbprint(printlevel-voice+2,"// collect conditions for EV["+string(b0)+"]"); |
---|
[8c4269a] | 1154 | j=ncols(a); |
---|
[a23a7fd] | 1155 | j0=mu; |
---|
| 1156 | while(j>=1) |
---|
[0c4bd7] | 1157 | { |
---|
| 1158 | i0=1; |
---|
| 1159 | i=1; |
---|
[8c4269a] | 1160 | while(i<ncols(a)&&a[i]<a[j]+b0) |
---|
[0c4bd7] | 1161 | { |
---|
| 1162 | i0=i0+d[i]; |
---|
| 1163 | i++; |
---|
| 1164 | } |
---|
[8c4269a] | 1165 | if(a[i]<a[j]+b0) |
---|
[0c4bd7] | 1166 | { |
---|
| 1167 | i0=i0+d[i]; |
---|
| 1168 | i++; |
---|
| 1169 | } |
---|
| 1170 | for(k=1;k<=ncols(m);k++) |
---|
| 1171 | { |
---|
| 1172 | M0=M[k]; |
---|
| 1173 | if(i0>1) |
---|
| 1174 | { |
---|
[a23a7fd] | 1175 | l[k]=l[k],M0[1..i0-1,j0-d[j]+1..j0]; |
---|
[0c4bd7] | 1176 | } |
---|
| 1177 | } |
---|
| 1178 | j0=j0-d[j]; |
---|
[a23a7fd] | 1179 | j--; |
---|
[0c4bd7] | 1180 | } |
---|
| 1181 | |
---|
[1418c4] | 1182 | dbprint(printlevel-voice+2,"// compose condition matrix"); |
---|
[0c4bd7] | 1183 | L=transpose(module(l[1])); |
---|
| 1184 | for(k=2;k<=ncols(m);k++) |
---|
| 1185 | { |
---|
| 1186 | L=L,transpose(module(l[k])); |
---|
| 1187 | } |
---|
| 1188 | |
---|
[1418c4] | 1189 | dbprint(printlevel-voice+2,"// compute kernel of condition matrix"); |
---|
[0c4bd7] | 1190 | v0=v0+list(syz(L)); |
---|
[8c4269a] | 1191 | a0=a0,b0; |
---|
[0c4bd7] | 1192 | } |
---|
| 1193 | |
---|
[1418c4] | 1194 | dbprint(printlevel-voice+2,"// compute graded parts"); |
---|
[0c4bd7] | 1195 | option(redSB); |
---|
| 1196 | for(i=1;i<size(v0);i++) |
---|
| 1197 | { |
---|
| 1198 | v0[i+1]=std(v0[i+1]); |
---|
| 1199 | v0[i]=std(reduce(v0[i],v0[i+1])); |
---|
| 1200 | } |
---|
[17f79e9] | 1201 | option(noredSB); |
---|
[0c4bd7] | 1202 | |
---|
[1418c4] | 1203 | dbprint(printlevel-voice+2,"// remove trivial graded parts"); |
---|
[0c4bd7] | 1204 | i=1; |
---|
| 1205 | while(size(v0[i])==0) |
---|
| 1206 | { |
---|
| 1207 | i++; |
---|
| 1208 | } |
---|
| 1209 | list v1=v0[i]; |
---|
[d5c289] | 1210 | intvec d1=size(v0[i]); |
---|
[8c4269a] | 1211 | ideal a1=a0[i]; |
---|
[0c4bd7] | 1212 | i++; |
---|
| 1213 | while(i<=size(v0)) |
---|
| 1214 | { |
---|
| 1215 | if(size(v0[i])>0) |
---|
| 1216 | { |
---|
| 1217 | v1=v1+list(v0[i]); |
---|
[d5c289] | 1218 | d1=d1,size(v0[i]); |
---|
[8c4269a] | 1219 | a1=a1,a0[i]; |
---|
[0c4bd7] | 1220 | } |
---|
| 1221 | i++; |
---|
| 1222 | } |
---|
[61549b] | 1223 | return(list(a1,d1,v1,m,g)); |
---|
[0c4bd7] | 1224 | } |
---|
| 1225 | example |
---|
| 1226 | { "EXAMPLE:"; echo=2; |
---|
| 1227 | ring R=0,(x,y),ds; |
---|
[86c1f0] | 1228 | poly t=x5+x2y2+y5; |
---|
[61549b] | 1229 | endvfilt(vfilt(t)); |
---|
[34a9eb1] | 1230 | } |
---|
| 1231 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1232 | |
---|
[2d3b6e6] | 1233 | proc sppnf(list sp) |
---|
| 1234 | "USAGE: sppnf(list(a,w[,m])); ideal a, intvec w, intvec m |
---|
| 1235 | ASSUME: ncols(e)==size(w)==size(m) |
---|
| 1236 | RETURN: order (a[i][,w[i]]) with multiplicity m[i] lexicographically |
---|
| 1237 | EXAMPLE: example sppnf; shows examples |
---|
[34a9eb1] | 1238 | " |
---|
| 1239 | { |
---|
[2d3b6e6] | 1240 | ideal a=sp[1]; |
---|
| 1241 | intvec w=sp[2]; |
---|
| 1242 | int n=ncols(a); |
---|
| 1243 | intvec m; |
---|
| 1244 | list V; |
---|
| 1245 | module v; |
---|
| 1246 | int i,j; |
---|
| 1247 | for(i=3;i<=size(sp);i++) |
---|
[34a9eb1] | 1248 | { |
---|
[2d3b6e6] | 1249 | if(typeof(sp[i])=="intvec") |
---|
| 1250 | { |
---|
| 1251 | m=sp[i]; |
---|
| 1252 | } |
---|
| 1253 | if(typeof(sp[i])=="module") |
---|
| 1254 | { |
---|
| 1255 | v=sp[i]; |
---|
| 1256 | for(j=n;j>=1;j--) |
---|
| 1257 | { |
---|
| 1258 | V[j]=module(v[j]); |
---|
| 1259 | } |
---|
| 1260 | } |
---|
| 1261 | if(typeof(sp[i])=="list") |
---|
| 1262 | { |
---|
| 1263 | V=sp[i]; |
---|
| 1264 | } |
---|
| 1265 | } |
---|
| 1266 | if(m==0) |
---|
| 1267 | { |
---|
| 1268 | for(i=n;i>=1;i--) |
---|
| 1269 | { |
---|
| 1270 | m[i]=1; |
---|
| 1271 | } |
---|
| 1272 | } |
---|
| 1273 | |
---|
| 1274 | int k; |
---|
| 1275 | ideal a0; |
---|
| 1276 | intvec w0,m0; |
---|
| 1277 | list V0; |
---|
| 1278 | number a1; |
---|
| 1279 | int w1,m1; |
---|
| 1280 | for(i=n;i>=1;i--) |
---|
| 1281 | { |
---|
| 1282 | if(m[i]!=0) |
---|
| 1283 | { |
---|
| 1284 | for(j=i-1;j>=1;j--) |
---|
| 1285 | { |
---|
| 1286 | if(m[j]!=0) |
---|
| 1287 | { |
---|
| 1288 | if(number(a[i])>number(a[j])|| |
---|
| 1289 | (number(a[i])==number(a[j])&&w[i]<w[j])) |
---|
| 1290 | { |
---|
| 1291 | a1=number(a[i]); |
---|
| 1292 | a[i]=a[j]; |
---|
| 1293 | a[j]=a1; |
---|
| 1294 | w1=w[i]; |
---|
| 1295 | w[i]=w[j]; |
---|
| 1296 | w[j]=w1; |
---|
| 1297 | m1=m[i]; |
---|
| 1298 | m[i]=m[j]; |
---|
| 1299 | m[j]=m1; |
---|
| 1300 | if(size(V)>0) |
---|
| 1301 | { |
---|
| 1302 | v=V[i]; |
---|
| 1303 | V[i]=V[j]; |
---|
| 1304 | V[j]=v; |
---|
| 1305 | } |
---|
| 1306 | } |
---|
| 1307 | if(number(a[i])==number(a[j])&&w[i]==w[j]) |
---|
| 1308 | { |
---|
| 1309 | m[i]=m[i]+m[j]; |
---|
| 1310 | m[j]=0; |
---|
| 1311 | if(size(V)>0) |
---|
| 1312 | { |
---|
| 1313 | V[i]=V[i]+V[j]; |
---|
| 1314 | } |
---|
| 1315 | } |
---|
| 1316 | } |
---|
| 1317 | } |
---|
| 1318 | k++; |
---|
| 1319 | a0[k]=a[i]; |
---|
| 1320 | w0[k]=w[i]; |
---|
| 1321 | m0[k]=m[i]; |
---|
| 1322 | if(size(V)>0) |
---|
| 1323 | { |
---|
| 1324 | V0[k]=V[i]; |
---|
| 1325 | } |
---|
| 1326 | } |
---|
| 1327 | } |
---|
| 1328 | |
---|
| 1329 | if(size(V0)>0) |
---|
| 1330 | { |
---|
| 1331 | n=size(V0); |
---|
| 1332 | module U=std(V0[n]); |
---|
| 1333 | for(i=n-1;i>=1;i--) |
---|
| 1334 | { |
---|
| 1335 | V0[i]=simplify(reduce(V0[i],U),1); |
---|
| 1336 | if(i>=2) |
---|
| 1337 | { |
---|
| 1338 | U=std(U+V0[i]); |
---|
| 1339 | } |
---|
| 1340 | } |
---|
| 1341 | } |
---|
| 1342 | |
---|
| 1343 | if(k>0) |
---|
| 1344 | { |
---|
| 1345 | sp=a0,w0,m0; |
---|
| 1346 | if(size(V0)>0) |
---|
| 1347 | { |
---|
| 1348 | sp[4]=V0; |
---|
| 1349 | } |
---|
[34a9eb1] | 1350 | } |
---|
[2d3b6e6] | 1351 | return(sp); |
---|
[34a9eb1] | 1352 | } |
---|
| 1353 | example |
---|
| 1354 | { "EXAMPLE:"; echo=2; |
---|
| 1355 | ring R=0,(x,y),ds; |
---|
[2d3b6e6] | 1356 | list sp=list(ideal(-1/2,-3/10,-3/10,-1/10,-1/10,0,1/10,1/10,3/10,3/10,1/2),intvec(2,1,1,1,1,1,1,1,1,1,0)); |
---|
| 1357 | sppprint(sppnf(sp)); |
---|
[34a9eb1] | 1358 | } |
---|
| 1359 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1360 | |
---|
[d70bc7] | 1361 | proc sppprint(list spp) |
---|
| 1362 | "USAGE: sppprint(spp); list spp |
---|
[275721f] | 1363 | RETURN: string s; spectral pairs spp |
---|
[e480544] | 1364 | EXAMPLE: example sppprint; shows examples |
---|
[8c4269a] | 1365 | " |
---|
| 1366 | { |
---|
| 1367 | string s; |
---|
[d70bc7] | 1368 | for(int i=1;i<size(spp[3]);i++) |
---|
[8c4269a] | 1369 | { |
---|
[d70bc7] | 1370 | s=s+"(("+string(spp[1][i])+","+string(spp[2][i])+"),"+string(spp[3][i])+"),"; |
---|
[8c4269a] | 1371 | } |
---|
[d70bc7] | 1372 | s=s+"(("+string(spp[1][i])+","+string(spp[2][i])+"),"+string(spp[3][i])+")"; |
---|
[8c4269a] | 1373 | return(s); |
---|
| 1374 | } |
---|
| 1375 | example |
---|
| 1376 | { "EXAMPLE:"; echo=2; |
---|
| 1377 | ring R=0,(x,y),ds; |
---|
[d70bc7] | 1378 | list spp=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(2,1,1,1,1,1,0),intvec(1,2,2,1,2,2,1)); |
---|
| 1379 | sppprint(spp); |
---|
[8c4269a] | 1380 | } |
---|
| 1381 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1382 | |
---|
[d70bc7] | 1383 | proc spadd(list sp1,list sp2) |
---|
[275721f] | 1384 | "USAGE: spadd(sp1,sp2); list sp1, list sp2 |
---|
| 1385 | RETURN: list sp; sum of spectra sp1 and sp2 |
---|
[04b295] | 1386 | EXAMPLE: example spadd; shows examples |
---|
[8c4269a] | 1387 | " |
---|
| 1388 | { |
---|
| 1389 | ideal s; |
---|
| 1390 | intvec m; |
---|
| 1391 | int i,i1,i2=1,1,1; |
---|
[d70bc7] | 1392 | while(i1<=size(sp1[2])||i2<=size(sp2[2])) |
---|
[8c4269a] | 1393 | { |
---|
[d70bc7] | 1394 | if(i1<=size(sp1[2])) |
---|
[8c4269a] | 1395 | { |
---|
[d70bc7] | 1396 | if(i2<=size(sp2[2])) |
---|
[8c4269a] | 1397 | { |
---|
[d70bc7] | 1398 | if(number(sp1[1][i1])<number(sp2[1][i2])) |
---|
[8c4269a] | 1399 | { |
---|
[d70bc7] | 1400 | s[i]=sp1[1][i1]; |
---|
| 1401 | m[i]=sp1[2][i1]; |
---|
[8c4269a] | 1402 | i++; |
---|
| 1403 | i1++; |
---|
| 1404 | } |
---|
| 1405 | else |
---|
| 1406 | { |
---|
[d70bc7] | 1407 | if(number(sp1[1][i1])>number(sp2[1][i2])) |
---|
[8c4269a] | 1408 | { |
---|
[d70bc7] | 1409 | s[i]=sp2[1][i2]; |
---|
| 1410 | m[i]=sp2[2][i2]; |
---|
[8c4269a] | 1411 | i++; |
---|
| 1412 | i2++; |
---|
| 1413 | } |
---|
| 1414 | else |
---|
| 1415 | { |
---|
[d70bc7] | 1416 | if(sp1[2][i1]+sp2[2][i2]!=0) |
---|
[8c4269a] | 1417 | { |
---|
[d70bc7] | 1418 | s[i]=sp1[1][i1]; |
---|
| 1419 | m[i]=sp1[2][i1]+sp2[2][i2]; |
---|
[8c4269a] | 1420 | i++; |
---|
| 1421 | } |
---|
| 1422 | i1++; |
---|
| 1423 | i2++; |
---|
| 1424 | } |
---|
| 1425 | } |
---|
| 1426 | } |
---|
| 1427 | else |
---|
| 1428 | { |
---|
[d70bc7] | 1429 | s[i]=sp1[1][i1]; |
---|
| 1430 | m[i]=sp1[2][i1]; |
---|
[8c4269a] | 1431 | i++; |
---|
| 1432 | i1++; |
---|
| 1433 | } |
---|
| 1434 | } |
---|
| 1435 | else |
---|
| 1436 | { |
---|
[d70bc7] | 1437 | s[i]=sp2[1][i2]; |
---|
| 1438 | m[i]=sp2[2][i2]; |
---|
[8c4269a] | 1439 | i++; |
---|
| 1440 | i2++; |
---|
| 1441 | } |
---|
| 1442 | } |
---|
| 1443 | return(list(s,m)); |
---|
| 1444 | } |
---|
| 1445 | example |
---|
| 1446 | { "EXAMPLE:"; echo=2; |
---|
| 1447 | ring R=0,(x,y),ds; |
---|
[d70bc7] | 1448 | list sp1=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
| 1449 | spprint(sp1); |
---|
| 1450 | list sp2=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
| 1451 | spprint(sp2); |
---|
| 1452 | spprint(spadd(sp1,sp2)); |
---|
[8c4269a] | 1453 | } |
---|
| 1454 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1455 | |
---|
[d70bc7] | 1456 | proc spsub(list sp1,list sp2) |
---|
[275721f] | 1457 | "USAGE: spsub(sp1,sp2); list sp1, list sp2 |
---|
| 1458 | RETURN: list sp; difference of spectra sp1 and sp2 |
---|
[04b295] | 1459 | EXAMPLE: example spsub; shows examples |
---|
[8c4269a] | 1460 | " |
---|
| 1461 | { |
---|
[d70bc7] | 1462 | return(spadd(sp1,spmul(sp2,-1))); |
---|
[8c4269a] | 1463 | } |
---|
| 1464 | example |
---|
| 1465 | { "EXAMPLE:"; echo=2; |
---|
| 1466 | ring R=0,(x,y),ds; |
---|
[d70bc7] | 1467 | list sp1=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
| 1468 | spprint(sp1); |
---|
| 1469 | list sp2=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
| 1470 | spprint(sp2); |
---|
| 1471 | spprint(spsub(sp1,sp2)); |
---|
[8c4269a] | 1472 | } |
---|
| 1473 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1474 | |
---|
| 1475 | proc spmul(list #) |
---|
[275721f] | 1476 | "USAGE: spmul(sp0,k); list sp0, int[vec] k |
---|
| 1477 | RETURN: list sp; linear combination of spectra sp0 with coefficients k |
---|
[04b295] | 1478 | EXAMPLE: example spmul; shows examples |
---|
[8c4269a] | 1479 | " |
---|
| 1480 | { |
---|
| 1481 | if(size(#)==2) |
---|
| 1482 | { |
---|
| 1483 | if(typeof(#[1])=="list") |
---|
| 1484 | { |
---|
| 1485 | if(typeof(#[2])=="int") |
---|
| 1486 | { |
---|
| 1487 | return(list(#[1][1],#[1][2]*#[2])); |
---|
| 1488 | } |
---|
| 1489 | if(typeof(#[2])=="intvec") |
---|
| 1490 | { |
---|
[d70bc7] | 1491 | list sp0=list(ideal(),intvec(0)); |
---|
[8c4269a] | 1492 | for(int i=size(#[2]);i>=1;i--) |
---|
| 1493 | { |
---|
[d70bc7] | 1494 | sp0=spadd(sp0,spmul(#[1][i],#[2][i])); |
---|
[8c4269a] | 1495 | } |
---|
[d70bc7] | 1496 | return(sp0); |
---|
[8c4269a] | 1497 | } |
---|
| 1498 | } |
---|
| 1499 | } |
---|
| 1500 | return(list(ideal(),intvec(0))); |
---|
| 1501 | } |
---|
| 1502 | example |
---|
| 1503 | { "EXAMPLE:"; echo=2; |
---|
| 1504 | ring R=0,(x,y),ds; |
---|
[d70bc7] | 1505 | list sp=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
| 1506 | spprint(sp); |
---|
| 1507 | spprint(spmul(sp,2)); |
---|
| 1508 | list sp1=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
| 1509 | spprint(sp1); |
---|
| 1510 | list sp2=list(ideal(-1/3,0,1/3),intvec(1,2,1)); |
---|
| 1511 | spprint(sp2); |
---|
| 1512 | spprint(spmul(list(sp1,sp2),intvec(1,2))); |
---|
[8c4269a] | 1513 | } |
---|
| 1514 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1515 | |
---|
[d70bc7] | 1516 | proc spissemicont(list sp,list #) |
---|
[275721f] | 1517 | "USAGE: spissemicont(sp[,1]); list sp, int opt |
---|
[8c4269a] | 1518 | RETURN: |
---|
| 1519 | @format |
---|
| 1520 | int k= |
---|
[275721f] | 1521 | 1; if sum of sp is positive on all intervals [a,a+1) [and (a,a+1)] |
---|
| 1522 | 0; if sum of sp is negative on some interval [a,a+1) [or (a,a+1)] |
---|
[8c4269a] | 1523 | @end format |
---|
[04b295] | 1524 | EXAMPLE: example spissemicont; shows examples |
---|
[8c4269a] | 1525 | " |
---|
| 1526 | { |
---|
| 1527 | int opt=0; |
---|
| 1528 | if(size(#)>0) |
---|
| 1529 | { |
---|
| 1530 | if(typeof(#[1])=="int") |
---|
| 1531 | { |
---|
| 1532 | opt=1; |
---|
| 1533 | } |
---|
| 1534 | } |
---|
[96a9c70] | 1535 | int i,j,k; |
---|
| 1536 | i=1; |
---|
[74d9b7] | 1537 | while(i<=size(sp[2])-1) |
---|
[8c4269a] | 1538 | { |
---|
[96a9c70] | 1539 | j=i+1; |
---|
| 1540 | k=0; |
---|
| 1541 | while(j+1<=size(sp[2])&&number(sp[1][j])<=number(sp[1][i])+1) |
---|
[8c4269a] | 1542 | { |
---|
[96a9c70] | 1543 | if(opt==0||number(sp[1][j])<number(sp[1][i])+1) |
---|
[74d9b7] | 1544 | { |
---|
| 1545 | k=k+sp[2][j]; |
---|
| 1546 | } |
---|
[8c4269a] | 1547 | j++; |
---|
| 1548 | } |
---|
[96a9c70] | 1549 | if(j==size(sp[2])&&number(sp[1][j])<=number(sp[1][i])+1) |
---|
[8c4269a] | 1550 | { |
---|
[96a9c70] | 1551 | if(opt==0||number(sp[1][j])<number(sp[1][i])+1) |
---|
[74d9b7] | 1552 | { |
---|
| 1553 | k=k+sp[2][j]; |
---|
| 1554 | } |
---|
[8c4269a] | 1555 | } |
---|
[74d9b7] | 1556 | if(k<0) |
---|
[8c4269a] | 1557 | { |
---|
| 1558 | return(0); |
---|
| 1559 | } |
---|
[96a9c70] | 1560 | i++; |
---|
[8c4269a] | 1561 | } |
---|
| 1562 | return(1); |
---|
| 1563 | } |
---|
| 1564 | example |
---|
| 1565 | { "EXAMPLE:"; echo=2; |
---|
| 1566 | ring R=0,(x,y),ds; |
---|
[d70bc7] | 1567 | list sp1=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
| 1568 | spprint(sp1); |
---|
| 1569 | list sp2=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
| 1570 | spprint(sp2); |
---|
[74d9b7] | 1571 | spissemicont(spsub(sp1,spmul(sp2,3))); |
---|
| 1572 | spissemicont(spsub(sp1,spmul(sp2,4))); |
---|
[8c4269a] | 1573 | } |
---|
| 1574 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1575 | |
---|
[d70bc7] | 1576 | proc spsemicont(list sp0,list sp,list #) |
---|
[275721f] | 1577 | "USAGE: spsemicont(sp0,sp,k[,1]); list sp0, list sp |
---|
| 1578 | RETURN: |
---|
| 1579 | @format |
---|
| 1580 | list l; |
---|
[e9124e] | 1581 | intvec l[i]; if the spectra sp0 occur with multiplicities k |
---|
| 1582 | in a deformation of a [quasihomogeneous] singularity |
---|
[a8cc0a] | 1583 | with spectrum sp then k<=l[i] |
---|
[275721f] | 1584 | @end format |
---|
[04b295] | 1585 | EXAMPLE: example spsemicont; shows examples |
---|
[8c4269a] | 1586 | " |
---|
| 1587 | { |
---|
| 1588 | list l,l0; |
---|
[38f6b33] | 1589 | int i,j,k; |
---|
[d70bc7] | 1590 | while(spissemicont(sp0,#)) |
---|
[8c4269a] | 1591 | { |
---|
[d70bc7] | 1592 | if(size(sp)>1) |
---|
[8c4269a] | 1593 | { |
---|
[d70bc7] | 1594 | l0=spsemicont(sp0,list(sp[1..size(sp)-1])); |
---|
[38f6b33] | 1595 | for(i=1;i<=size(l0);i++) |
---|
[8c4269a] | 1596 | { |
---|
[38f6b33] | 1597 | if(size(l)>0) |
---|
[e9124e] | 1598 | { |
---|
[38f6b33] | 1599 | j=1; |
---|
| 1600 | while(j<size(l)&&l[j]!=l0[i]) |
---|
[e9124e] | 1601 | { |
---|
[38f6b33] | 1602 | j++; |
---|
| 1603 | } |
---|
| 1604 | if(l[j]==l0[i]) |
---|
[e9124e] | 1605 | { |
---|
[d70bc7] | 1606 | l[j][size(sp)]=k; |
---|
[38f6b33] | 1607 | } |
---|
| 1608 | else |
---|
[e9124e] | 1609 | { |
---|
[d70bc7] | 1610 | l0[i][size(sp)]=k; |
---|
[38f6b33] | 1611 | l=l+list(l0[i]); |
---|
| 1612 | } |
---|
[e9124e] | 1613 | } |
---|
[38f6b33] | 1614 | else |
---|
[e9124e] | 1615 | { |
---|
[38f6b33] | 1616 | l=l0; |
---|
[e9124e] | 1617 | } |
---|
[8c4269a] | 1618 | } |
---|
| 1619 | } |
---|
[d70bc7] | 1620 | sp0=spsub(sp0,sp[size(sp)]); |
---|
[8c4269a] | 1621 | k++; |
---|
| 1622 | } |
---|
[d70bc7] | 1623 | if(size(sp)>1) |
---|
[8c4269a] | 1624 | { |
---|
| 1625 | return(l); |
---|
| 1626 | } |
---|
| 1627 | else |
---|
| 1628 | { |
---|
| 1629 | return(list(intvec(k-1))); |
---|
| 1630 | } |
---|
| 1631 | } |
---|
| 1632 | example |
---|
| 1633 | { "EXAMPLE:"; echo=2; |
---|
| 1634 | ring R=0,(x,y),ds; |
---|
[d70bc7] | 1635 | list sp0=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
| 1636 | spprint(sp0); |
---|
| 1637 | list sp1=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
| 1638 | spprint(sp1); |
---|
| 1639 | list sp2=list(ideal(-1/3,0,1/3),intvec(1,2,1)); |
---|
| 1640 | spprint(sp2); |
---|
| 1641 | list sp=sp1,sp2; |
---|
| 1642 | list l=spsemicont(sp0,sp); |
---|
[8c4269a] | 1643 | l; |
---|
[d70bc7] | 1644 | spissemicont(spsub(sp0,spmul(sp,l[1]))); |
---|
| 1645 | spissemicont(spsub(sp0,spmul(sp,l[1]-1))); |
---|
| 1646 | spissemicont(spsub(sp0,spmul(sp,l[1]+1))); |
---|
[8c4269a] | 1647 | } |
---|
| 1648 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1649 | |
---|
[d70bc7] | 1650 | proc spmilnor(list sp) |
---|
| 1651 | "USAGE: spmilnor(sp); list sp |
---|
[275721f] | 1652 | RETURN: int mu; Milnor number of spectrum sp |
---|
[04b295] | 1653 | EXAMPLE: example spmilnor; shows examples |
---|
[8960ec] | 1654 | " |
---|
| 1655 | { |
---|
[d70bc7] | 1656 | return(sum(sp[2])); |
---|
[8960ec] | 1657 | } |
---|
| 1658 | example |
---|
| 1659 | { "EXAMPLE:"; echo=2; |
---|
| 1660 | ring R=0,(x,y),ds; |
---|
[d70bc7] | 1661 | list sp=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
| 1662 | spprint(sp); |
---|
| 1663 | spmilnor(sp); |
---|
[8960ec] | 1664 | } |
---|
| 1665 | /////////////////////////////////////////////////////////////////////////////// |
---|
| 1666 | |
---|
[d70bc7] | 1667 | proc spgeomgenus(list sp) |
---|
| 1668 | "USAGE: spgeomgenus(sp); list sp |
---|
[275721f] | 1669 | RETURN: int g; geometrical genus of spectrum sp |
---|
[04b295] | 1670 | EXAMPLE: example spgeomgenus; shows examples |
---|
[8c4269a] | 1671 | " |
---|
| 1672 | { |
---|
| 1673 | int g=0; |
---|
| 1674 | int i=1; |
---|
[d70bc7] | 1675 | while(i+1<=size(sp[2])&&number(sp[1][i])<=number(0)) |
---|
[8c4269a] | 1676 | { |
---|
[d70bc7] | 1677 | g=g+sp[2][i]; |
---|
[8c4269a] | 1678 | i++; |
---|
| 1679 | } |
---|
[d70bc7] | 1680 | if(i==size(sp[2])&&number(sp[1][i])<=number(0)) |
---|
[8c4269a] | 1681 | { |
---|
[d70bc7] | 1682 | g=g+sp[2][i]; |
---|
[8c4269a] | 1683 | } |
---|
| 1684 | return(g); |
---|
| 1685 | } |
---|
| 1686 | example |
---|
| 1687 | { "EXAMPLE:"; echo=2; |
---|
| 1688 | ring R=0,(x,y),ds; |
---|
[d70bc7] | 1689 | list sp=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
| 1690 | spprint(sp); |
---|
| 1691 | spgeomgenus(sp); |
---|
[cc3a04] | 1692 | } |
---|
| 1693 | /////////////////////////////////////////////////////////////////////////////// |
---|
[8960ec] | 1694 | |
---|
[d70bc7] | 1695 | proc spgamma(list sp) |
---|
| 1696 | "USAGE: spgamma(sp); list sp |
---|
[275721f] | 1697 | RETURN: number gamma; gamma invariant of spectrum sp |
---|
[04b295] | 1698 | EXAMPLE: example spgamma; shows examples |
---|
[8960ec] | 1699 | " |
---|
| 1700 | { |
---|
| 1701 | int i,j; |
---|
| 1702 | number g=0; |
---|
[d70bc7] | 1703 | for(i=1;i<=ncols(sp[1]);i++) |
---|
[8960ec] | 1704 | { |
---|
[d70bc7] | 1705 | for(j=1;j<=sp[2][i];j++) |
---|
[8960ec] | 1706 | { |
---|
[d70bc7] | 1707 | g=g+(number(sp[1][i])-number(nvars(basering)-2)/2)^2; |
---|
[8960ec] | 1708 | } |
---|
| 1709 | } |
---|
[d70bc7] | 1710 | g=-g/4+sum(sp[2])*number(sp[1][ncols(sp[1])]-sp[1][1])/48; |
---|
[8960ec] | 1711 | return(g); |
---|
| 1712 | } |
---|
| 1713 | example |
---|
| 1714 | { "EXAMPLE:"; echo=2; |
---|
| 1715 | ring R=0,(x,y),ds; |
---|
[d70bc7] | 1716 | list sp=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
| 1717 | spprint(sp); |
---|
| 1718 | spgamma(sp); |
---|
[8960ec] | 1719 | } |
---|
| 1720 | /////////////////////////////////////////////////////////////////////////////// |
---|