1 | /////////////////////////////////////////////////////////////////////////////// |
---|
2 | version="$Id: gaussman.lib,v 1.37 2001-04-26 12:05:15 mschulze Exp $"; |
---|
3 | category="Singularities"; |
---|
4 | |
---|
5 | info=" |
---|
6 | LIBRARY: gaussman.lib Gauss-Manin Connection of a Singularity |
---|
7 | |
---|
8 | AUTHOR: Mathias Schulze, email: mschulze@mathematik.uni-kl.de |
---|
9 | |
---|
10 | OVERVIEW: A library to compute invariants related to the Gauss-Manin connection |
---|
11 | of a an isolated hypersurface singularity |
---|
12 | |
---|
13 | PROCEDURES: |
---|
14 | monodromy(f[,opt]); monodromy matrix, spectrum of monodromy of f |
---|
15 | vfiltration(f[,opt]); V-filtration on H''/H', singularity spectrum of f |
---|
16 | spectrum(f); singularity spectrum of f |
---|
17 | endfilt(poly f,list V); endomorphism filtration of filtration V |
---|
18 | spprint(list S); print spectrum S |
---|
19 | spadd(list S1,list S2); sum of spectra S1 and S2 |
---|
20 | spsub(list S1,list S2); difference of spectra S1 and S2 |
---|
21 | spmul(list S,int k); product of spectrum S and integer k |
---|
22 | spmul(list S,intvec k); linear combination of spectra S with coefficients k |
---|
23 | spissemicont(list S[,opt]); test spectrum S for semicontinuity |
---|
24 | spsemicont(list S0,list S[,opt]); relative semicontinuity of spectra S0 and S |
---|
25 | spmilnor(list S); milnor number of spectrum S |
---|
26 | spgeomgenus(list S); geometrical genus of spectrum S |
---|
27 | spgamma(list S); gamma invariant of spectrum S |
---|
28 | |
---|
29 | SEE ALSO: mondromy_lib, spectrum_lib |
---|
30 | |
---|
31 | KEYWORDS: singularities; Gauss-Manin connection; monodromy; spectrum; |
---|
32 | Brieskorn lattice; Hodge filtration; V-filtration |
---|
33 | "; |
---|
34 | |
---|
35 | LIB "linalg.lib"; |
---|
36 | |
---|
37 | /////////////////////////////////////////////////////////////////////////////// |
---|
38 | |
---|
39 | static proc maxintdif(ideal e) |
---|
40 | { |
---|
41 | dbprint(printlevel-voice+2,"//gaussman::maxintdif"); |
---|
42 | int i,j,id; |
---|
43 | int mid=0; |
---|
44 | for(i=ncols(e);i>=1;i--) |
---|
45 | { |
---|
46 | for(j=i-1;j>=1;j--) |
---|
47 | { |
---|
48 | id=int(e[i]-e[j]); |
---|
49 | if(id<0) |
---|
50 | { |
---|
51 | id=-id; |
---|
52 | } |
---|
53 | if(id>mid) |
---|
54 | { |
---|
55 | mid=id; |
---|
56 | } |
---|
57 | } |
---|
58 | } |
---|
59 | return(mid); |
---|
60 | } |
---|
61 | /////////////////////////////////////////////////////////////////////////////// |
---|
62 | |
---|
63 | static proc maxorddif(matrix H) |
---|
64 | { |
---|
65 | dbprint(printlevel-voice+2,"//gaussman::maxorddif"); |
---|
66 | int i,j,d; |
---|
67 | int d0,d1=-1,-1; |
---|
68 | for(i=nrows(H);i>=1;i--) |
---|
69 | { |
---|
70 | for(j=ncols(H);j>=1;j--) |
---|
71 | { |
---|
72 | d=ord(H[i,j]); |
---|
73 | if(d>=0) |
---|
74 | { |
---|
75 | if(d0<0||d<d0) |
---|
76 | { |
---|
77 | d0=d; |
---|
78 | } |
---|
79 | if(d1<0||d>d1) |
---|
80 | { |
---|
81 | d1=d; |
---|
82 | } |
---|
83 | } |
---|
84 | } |
---|
85 | } |
---|
86 | return(d1-d0); |
---|
87 | } |
---|
88 | /////////////////////////////////////////////////////////////////////////////// |
---|
89 | |
---|
90 | proc monodromy(poly f,list #) |
---|
91 | "USAGE: monodromy(f[,opt]); poly f, int opt |
---|
92 | ASSUME: basering has characteristic 0 and local ordering, |
---|
93 | f has isolated singularity at 0 |
---|
94 | RETURN: |
---|
95 | @format |
---|
96 | if opt==0: |
---|
97 | matrix M: exp(-2*pi*i*M) is a monodromy matrix of f |
---|
98 | if opt==1: |
---|
99 | ideal e: exp(-2*pi*i*e) are the eigenvalues of the monodromy of f |
---|
100 | default: opt=1 |
---|
101 | @end format |
---|
102 | SEE ALSO: mondromy_lib |
---|
103 | KEYWORDS: singularities; Gauss-Manin connection; monodromy |
---|
104 | EXAMPLE: example monodromy; shows an example |
---|
105 | " |
---|
106 | { |
---|
107 | if(charstr(basering)!="0") |
---|
108 | { |
---|
109 | ERROR("characteristic 0 expected"); |
---|
110 | } |
---|
111 | int n=nvars(basering)-1; |
---|
112 | for(int i=n+1;i>=1;i--) |
---|
113 | { |
---|
114 | if(var(i)>1) |
---|
115 | { |
---|
116 | ERROR("local ordering expected"); |
---|
117 | } |
---|
118 | } |
---|
119 | ideal J=jacob(f); |
---|
120 | ideal sJ=std(J); |
---|
121 | if(vdim(sJ)<=0) |
---|
122 | { |
---|
123 | if(vdim(sJ)==0) |
---|
124 | { |
---|
125 | ERROR("singularity at 0 expected"); |
---|
126 | } |
---|
127 | else |
---|
128 | { |
---|
129 | ERROR("isolated singularity at 0 expected"); |
---|
130 | } |
---|
131 | } |
---|
132 | int opt=1; |
---|
133 | if(size(#)>0) |
---|
134 | { |
---|
135 | if(typeof(#[1])=="int") |
---|
136 | { |
---|
137 | opt=#[1]; |
---|
138 | } |
---|
139 | } |
---|
140 | |
---|
141 | ideal m=kbase(sJ); |
---|
142 | int mu,modm=ncols(m),maxorddif(m); |
---|
143 | |
---|
144 | ideal w=f*m; |
---|
145 | matrix U=freemodule(mu); |
---|
146 | matrix A0[mu][mu],A,C,D; |
---|
147 | list l; |
---|
148 | module H,dH=freemodule(mu),freemodule(mu); |
---|
149 | module H0; |
---|
150 | int sdH=1; |
---|
151 | int k=-1; |
---|
152 | int j,K,N,mide; |
---|
153 | |
---|
154 | while(k<K||sdH>0) |
---|
155 | { |
---|
156 | k++; |
---|
157 | dbprint(printlevel-voice+2,"//gaussman::monodromy: k="+string(k)); |
---|
158 | |
---|
159 | dbprint(printlevel-voice+2,"//gaussman::monodromy: compute C"); |
---|
160 | C=coeffs(reduce(w,U,sJ),m); |
---|
161 | A0=A0+C*var(1)^k; |
---|
162 | |
---|
163 | if(sdH>0) |
---|
164 | { |
---|
165 | H0=H; |
---|
166 | dbprint(printlevel-voice+2,"//gaussman::monodromy: compute dH"); |
---|
167 | dH=jet(module(A0*dH+var(1)^2*diff(matrix(dH),var(1))),k); |
---|
168 | H=H*var(1)+dH; |
---|
169 | |
---|
170 | dbprint(printlevel-voice+2,"//gaussman::monodromy: test dH==0"); |
---|
171 | sdH=size(reduce(H,std(H0*var(1)))); |
---|
172 | if(sdH>0) |
---|
173 | { |
---|
174 | A0=A0-var(1); |
---|
175 | } |
---|
176 | else |
---|
177 | { |
---|
178 | dbprint(printlevel-voice+2, |
---|
179 | "//gaussman::monodromy: compute basis of saturation"); |
---|
180 | H=minbase(H0); |
---|
181 | int modH=maxorddif(H); |
---|
182 | K=modH+1; |
---|
183 | } |
---|
184 | } |
---|
185 | |
---|
186 | if(k==K&&sdH==0) |
---|
187 | { |
---|
188 | N=k-modH; |
---|
189 | dbprint(printlevel-voice+2, |
---|
190 | "//gaussman::monodromy: compute A on saturation"); |
---|
191 | l=division(H*var(1),A0*H+var(1)^2*diff(matrix(H),var(1))); |
---|
192 | A=jet(l[1],l[2],N-1); |
---|
193 | if(mide==0) |
---|
194 | { |
---|
195 | dbprint(printlevel-voice+2, |
---|
196 | "//gaussman::monodromy: compute eigenvalues e and"+ |
---|
197 | "multiplicities b of A"); |
---|
198 | l=system("eigenval",jet(A,0)); |
---|
199 | ideal e=l[1]; |
---|
200 | intvec b=l[2]; |
---|
201 | dbprint(printlevel-voice+2,"//gaussman::monodromy: e="+string(e)); |
---|
202 | dbprint(printlevel-voice+2,"//gaussman::monodromy: b="+string(b)); |
---|
203 | if(opt==1) |
---|
204 | { |
---|
205 | mide=maxintdif(e); |
---|
206 | K=K+mide; |
---|
207 | } |
---|
208 | } |
---|
209 | } |
---|
210 | |
---|
211 | if(k<K||sdH>0) |
---|
212 | { |
---|
213 | dbprint(printlevel-voice+2,"//gaussman::monodromy: divide by J"); |
---|
214 | l=division(J,ideal(matrix(w)-matrix(m)*C*U)); |
---|
215 | D=l[1]; |
---|
216 | |
---|
217 | dbprint(printlevel-voice+2,"//gaussman::monodromy: compute w/U"); |
---|
218 | for(j=mu;j>=1;j--) |
---|
219 | { |
---|
220 | if(l[2][j,j]!=0) |
---|
221 | { |
---|
222 | dbprint(printlevel-voice+2, |
---|
223 | "//gaussman::monodromy: compute U["+string(j)+"]"); |
---|
224 | U[j,j]=U[j,j]*l[2][j,j]; |
---|
225 | } |
---|
226 | dbprint(printlevel-voice+2, |
---|
227 | "//gaussman::monodromy: compute w["+string(j)+"]"); |
---|
228 | w[j]=0; |
---|
229 | for(i=n+1;i>=1;i--) |
---|
230 | { |
---|
231 | w[j]=w[j]+U[j,j]*diff(D[i,j],var(i))-diff(U[j,j],var(i))*D[i,j]; |
---|
232 | } |
---|
233 | } |
---|
234 | U=U*U; |
---|
235 | } |
---|
236 | } |
---|
237 | |
---|
238 | if(mide>0) |
---|
239 | { |
---|
240 | intvec ide; |
---|
241 | ide[mu]=0; |
---|
242 | module dU; |
---|
243 | matrix A0e; |
---|
244 | for(i=ncols(e);i>=1;i--) |
---|
245 | { |
---|
246 | for(j=i-1;j>=1;j--) |
---|
247 | { |
---|
248 | k=int(e[j]-e[i]); |
---|
249 | if(k>ide[i]) |
---|
250 | { |
---|
251 | ide[i]=k; |
---|
252 | } |
---|
253 | if(-k>ide[j]) |
---|
254 | { |
---|
255 | ide[j]=-k; |
---|
256 | } |
---|
257 | } |
---|
258 | } |
---|
259 | for(j,k=ncols(e),mu;j>=1;j--) |
---|
260 | { |
---|
261 | for(i=b[j];i>=1;i--) |
---|
262 | { |
---|
263 | ide[k]=ide[j]; |
---|
264 | k--; |
---|
265 | } |
---|
266 | } |
---|
267 | } |
---|
268 | while(mide>0) |
---|
269 | { |
---|
270 | dbprint(printlevel-voice+2,"//gaussman::monodromy: mide="+string(mide)); |
---|
271 | |
---|
272 | U=0; |
---|
273 | A0=jet(A,0); |
---|
274 | for(i=ncols(e);i>=1;i--) |
---|
275 | { |
---|
276 | A0e=freemodule(mu); |
---|
277 | for(j=n;j>=0;j--) // Potenzen von Matrizen? |
---|
278 | { |
---|
279 | A0e=A0e*(A0-e[i]); |
---|
280 | } |
---|
281 | dU=syz(A0e); |
---|
282 | U=dU+U; |
---|
283 | } |
---|
284 | A=division(U,A*U)[1]; |
---|
285 | |
---|
286 | for(i=mu;i>=1;i--) |
---|
287 | { |
---|
288 | for(j=mu;j>=1;j--) |
---|
289 | { |
---|
290 | if(ide[i]==0&&ide[j]>0) |
---|
291 | { |
---|
292 | A[i,j]=A[i,j]*var(1); |
---|
293 | } |
---|
294 | else |
---|
295 | { |
---|
296 | if(ide[i]>0&&ide[j]==0) |
---|
297 | { |
---|
298 | A[i,j]=A[i,j]/var(1); |
---|
299 | } |
---|
300 | } |
---|
301 | } |
---|
302 | } |
---|
303 | for(i=mu;i>=1;i--) |
---|
304 | { |
---|
305 | if(ide[i]>0) |
---|
306 | { |
---|
307 | A[i,i]=A[i,i]+1; |
---|
308 | e[i]=e[i]+1; |
---|
309 | ide[i]=ide[i]-1; |
---|
310 | } |
---|
311 | } |
---|
312 | mide--; |
---|
313 | } |
---|
314 | |
---|
315 | if(opt==1) |
---|
316 | { |
---|
317 | return(jet(A,0)); |
---|
318 | } |
---|
319 | else |
---|
320 | { |
---|
321 | return(e); |
---|
322 | } |
---|
323 | } |
---|
324 | example |
---|
325 | { "EXAMPLE:"; echo=2; |
---|
326 | ring R=0,(x,y),ds; |
---|
327 | poly f=x5+x2y2+y5; |
---|
328 | print(monodromy(f)); |
---|
329 | } |
---|
330 | /////////////////////////////////////////////////////////////////////////////// |
---|
331 | |
---|
332 | proc vfiltration(poly f,list #) |
---|
333 | "USAGE: vfiltration(f[,opt]); poly f, int opt |
---|
334 | ASSUME: basering has characteristic 0 and local ordering, |
---|
335 | f has isolated singularity at 0 |
---|
336 | RETURN: |
---|
337 | @format |
---|
338 | list V: V-filtration of f on H''/H' |
---|
339 | if opt==0 or opt==1: |
---|
340 | intvec V[1]: numerators of spectral numbers |
---|
341 | intvec V[2]: denominators of spectral numbers |
---|
342 | intvec V[3]: |
---|
343 | int V[3][i]: multiplicity of spectral number V[1][i]/V[2][i] |
---|
344 | if opt==1: |
---|
345 | list V[4]: |
---|
346 | module V[4][i]: vector space basis of V[1][i]/V[2][i]-th graded part |
---|
347 | in terms of V[5] |
---|
348 | ideal V[5]: monomial vector space basis |
---|
349 | default: opt=1 |
---|
350 | @end format |
---|
351 | NOTE: H' and H'' denote the Brieskorn lattices |
---|
352 | SEE ALSO: spectrum_lib |
---|
353 | KEYWORDS: singularities; Gauss-Manin connection; |
---|
354 | Brieskorn lattice; Hodge filtration; V-filtration; spectrum |
---|
355 | EXAMPLE: example vfiltration; shows an example |
---|
356 | " |
---|
357 | { |
---|
358 | if(charstr(basering)!="0") |
---|
359 | { |
---|
360 | ERROR("characteristic 0 expected"); |
---|
361 | } |
---|
362 | int n=nvars(basering)-1; |
---|
363 | for(int i=n+1;i>=1;i--) |
---|
364 | { |
---|
365 | if(var(i)>1) |
---|
366 | { |
---|
367 | ERROR("local ordering expected"); |
---|
368 | } |
---|
369 | } |
---|
370 | ideal J=jacob(f); |
---|
371 | ideal sJ=std(J); |
---|
372 | if(vdim(sJ)<=0) |
---|
373 | { |
---|
374 | if(vdim(sJ)==0) |
---|
375 | { |
---|
376 | ERROR("singularity at 0 expected"); |
---|
377 | } |
---|
378 | else |
---|
379 | { |
---|
380 | ERROR("isolated singularity at 0 expected"); |
---|
381 | } |
---|
382 | } |
---|
383 | int opt=1; |
---|
384 | if(size(#)>0) |
---|
385 | { |
---|
386 | if(typeof(#[1])=="int") |
---|
387 | { |
---|
388 | opt=#[1]; |
---|
389 | } |
---|
390 | } |
---|
391 | |
---|
392 | ideal m=kbase(sJ); |
---|
393 | int mu,modm=ncols(m),maxorddif(m); |
---|
394 | |
---|
395 | ideal w=f*m; |
---|
396 | matrix U=freemodule(mu); |
---|
397 | matrix A[mu][mu],C,D; |
---|
398 | list l; |
---|
399 | module H,dH=freemodule(mu),freemodule(mu); |
---|
400 | module H0; |
---|
401 | int sdH=1; |
---|
402 | int k=-1; |
---|
403 | int j,K; |
---|
404 | |
---|
405 | while(k<K||sdH>0) |
---|
406 | { |
---|
407 | k++; |
---|
408 | dbprint(printlevel-voice+2,"//gaussman::vfiltration: k="+string(k)); |
---|
409 | |
---|
410 | dbprint(printlevel-voice+2,"//gaussman::vfiltration: compute C"); |
---|
411 | C=coeffs(reduce(w,U,sJ),m); |
---|
412 | A=A+C*var(1)^k; |
---|
413 | |
---|
414 | if(sdH>0) |
---|
415 | { |
---|
416 | H0=H; |
---|
417 | dbprint(printlevel-voice+2,"//gaussman::vfiltration: compute dH"); |
---|
418 | dH=jet(module(A*dH+var(1)^2*diff(matrix(dH),var(1))),k); |
---|
419 | H=H*var(1)+dH; |
---|
420 | |
---|
421 | dbprint(printlevel-voice+2,"//gaussman::vfiltration: test dH==0"); |
---|
422 | sdH=size(reduce(H,std(H0*var(1)))); |
---|
423 | if(sdH>0) |
---|
424 | { |
---|
425 | A=A-var(1); |
---|
426 | } |
---|
427 | else |
---|
428 | { |
---|
429 | dbprint(printlevel-voice+2, |
---|
430 | "//gaussman::vfiltration: compute basis of saturation"); |
---|
431 | H=minbase(H0); |
---|
432 | int modH=maxorddif(H); |
---|
433 | K=modH+n+1; |
---|
434 | H0=freemodule(mu)*var(1)^k; |
---|
435 | } |
---|
436 | } |
---|
437 | |
---|
438 | if(k<K||sdH>0) |
---|
439 | { |
---|
440 | dbprint(printlevel-voice+2,"//gaussman::vfiltration: divide by J"); |
---|
441 | l=division(J,ideal(matrix(w)-matrix(m)*C*U)); |
---|
442 | D=l[1]; |
---|
443 | |
---|
444 | dbprint(printlevel-voice+2,"//gaussman::vfiltration: compute w/U"); |
---|
445 | for(j=mu;j>=1;j--) |
---|
446 | { |
---|
447 | if(l[2][j,j]!=0) |
---|
448 | { |
---|
449 | dbprint(printlevel-voice+2, |
---|
450 | "//gaussman::vfiltration: compute U["+string(j)+"]"); |
---|
451 | U[j,j]=U[j,j]*l[2][j,j]; |
---|
452 | } |
---|
453 | dbprint(printlevel-voice+2, |
---|
454 | "//gaussman::vfiltration: compute w["+string(j)+"]"); |
---|
455 | w[j]=0; |
---|
456 | for(i=n+1;i>=1;i--) |
---|
457 | { |
---|
458 | w[j]=w[j]+U[j,j]*diff(D[i,j],var(i))-diff(U[j,j],var(i))*D[i,j]; |
---|
459 | } |
---|
460 | } |
---|
461 | U=U*U; |
---|
462 | } |
---|
463 | } |
---|
464 | int N=k-modH; |
---|
465 | |
---|
466 | dbprint(printlevel-voice+2, |
---|
467 | "//gaussman::vfiltration: transform H0 to saturation"); |
---|
468 | l=division(H,H0); |
---|
469 | H0=jet(l[1],l[2],N-1); |
---|
470 | |
---|
471 | dbprint(printlevel-voice+2, |
---|
472 | "//gaussman::vfiltration: compute H0 as vector space V0"); |
---|
473 | dbprint(printlevel-voice+2, |
---|
474 | "//gaussman::vfiltration: compute H1 as vector space V1"); |
---|
475 | poly p; |
---|
476 | int i0,j0,i1,j1; |
---|
477 | matrix V0[mu*N][mu*N]; |
---|
478 | matrix V1[mu*N][mu*(N-1)]; |
---|
479 | for(i0=mu;i0>=1;i0--) |
---|
480 | { |
---|
481 | for(i1=mu;i1>=1;i1--) |
---|
482 | { |
---|
483 | p=H0[i1,i0]; |
---|
484 | while(p!=0) |
---|
485 | { |
---|
486 | j1=leadexp(p)[1]; |
---|
487 | for(j0=N-j1-1;j0>=0;j0--) |
---|
488 | { |
---|
489 | V0[i1+(j1+j0)*mu,i0+j0*mu]=V0[i1+(j1+j0)*mu,i0+j0*mu]+leadcoef(p); |
---|
490 | if(j1+j0+1<N) |
---|
491 | { |
---|
492 | V1[i1+(j1+j0+1)*mu,i0+j0*mu]= |
---|
493 | V1[i1+(j1+j0+1)*mu,i0+j0*mu]+leadcoef(p); |
---|
494 | } |
---|
495 | } |
---|
496 | p=p-lead(p); |
---|
497 | } |
---|
498 | } |
---|
499 | } |
---|
500 | |
---|
501 | dbprint(printlevel-voice+2, |
---|
502 | "//gaussman::vfiltration: compute A on saturation"); |
---|
503 | l=division(H*var(1),A*H+var(1)^2*diff(matrix(H),var(1))); |
---|
504 | A=jet(l[1],l[2],N-1); |
---|
505 | |
---|
506 | dbprint(printlevel-voice+2,"//gaussman::vfiltration: compute matrix M of A"); |
---|
507 | matrix M[mu*N][mu*N]; |
---|
508 | for(i0=mu;i0>=1;i0--) |
---|
509 | { |
---|
510 | for(i1=mu;i1>=1;i1--) |
---|
511 | { |
---|
512 | p=A[i1,i0]; |
---|
513 | while(p!=0) |
---|
514 | { |
---|
515 | j1=leadexp(p)[1]; |
---|
516 | for(j0=N-j1-1;j0>=0;j0--) |
---|
517 | { |
---|
518 | M[i1+(j0+j1)*mu,i0+j0*mu]=leadcoef(p); |
---|
519 | } |
---|
520 | p=p-lead(p); |
---|
521 | } |
---|
522 | } |
---|
523 | } |
---|
524 | for(i0=mu;i0>=1;i0--) |
---|
525 | { |
---|
526 | for(j0=N-1;j0>=0;j0--) |
---|
527 | { |
---|
528 | M[i0+j0*mu,i0+j0*mu]=M[i0+j0*mu,i0+j0*mu]+j0; |
---|
529 | } |
---|
530 | } |
---|
531 | |
---|
532 | dbprint(printlevel-voice+2, |
---|
533 | "//gaussman::vfiltration: compute eigenvalues eA of A"); |
---|
534 | ideal eA=system("eigenval",jet(A,0))[1]; |
---|
535 | dbprint(printlevel-voice+2,"//gaussman::vfiltration: eA="+string(eA)); |
---|
536 | |
---|
537 | dbprint(printlevel-voice+2, |
---|
538 | "//gaussman::vfiltration: compute eigenvalues eM of M"); |
---|
539 | ideal eM; |
---|
540 | k=0; |
---|
541 | intvec u; |
---|
542 | for(i=N;i>=1;i--) |
---|
543 | { |
---|
544 | u[i]=1; |
---|
545 | } |
---|
546 | i0=1; |
---|
547 | while(u[N]<=ncols(eA)) |
---|
548 | { |
---|
549 | for(i,i1=i0+1,i0;i<=N;i++) |
---|
550 | { |
---|
551 | if(eA[u[i]]+i<eA[u[i1]]+i1) |
---|
552 | { |
---|
553 | i1=i; |
---|
554 | } |
---|
555 | } |
---|
556 | k++; |
---|
557 | eM[k]=eA[u[i1]]+i1-1; |
---|
558 | u[i1]=u[i1]+1; |
---|
559 | if(u[i1]>ncols(eA)) |
---|
560 | { |
---|
561 | i0=i1+1; |
---|
562 | } |
---|
563 | } |
---|
564 | dbprint(printlevel-voice+2,"//gaussman::vfiltration: eM="+string(eM)); |
---|
565 | |
---|
566 | dbprint(printlevel-voice+2, |
---|
567 | "//gaussman::vfiltration: compute V-filtration on H0/H1"); |
---|
568 | ideal a; |
---|
569 | k=0; |
---|
570 | list V; |
---|
571 | matrix Me; |
---|
572 | V[ncols(eM)+1]=std(V1); |
---|
573 | intvec d; |
---|
574 | if(opt==0) |
---|
575 | { |
---|
576 | for(i=ncols(eM);number(eM[i])-1>number(n-1)/2;i--) |
---|
577 | { |
---|
578 | Me=freemodule(mu*N); |
---|
579 | for(i0=n;i0>=0;i0--) // Potenzen von Matrizen? |
---|
580 | { |
---|
581 | Me=Me*(M-eM[i]); |
---|
582 | } |
---|
583 | |
---|
584 | dbprint(printlevel-voice+2, |
---|
585 | "//gaussman::vfiltration: compute V["+string(i)+"]"); |
---|
586 | V1=module(V1)+syz(Me); |
---|
587 | V[i]=std(intersect(V1,V0)); |
---|
588 | |
---|
589 | if(size(V[i])>size(V[i+1])) |
---|
590 | { |
---|
591 | k++; |
---|
592 | a[k]=eM[i]-1; |
---|
593 | d[k]=size(V[i])-size(V[i+1]); |
---|
594 | } |
---|
595 | } |
---|
596 | |
---|
597 | dbprint(printlevel-voice+2, |
---|
598 | "//gaussman::vfiltration: symmetry index found"); |
---|
599 | j=k; |
---|
600 | |
---|
601 | if(number(eM[i])-1==number(n-1)/2) |
---|
602 | { |
---|
603 | Me=freemodule(mu*N); |
---|
604 | for(i0=n;i0>=0;i0--) // Potenzen von Matrizen? |
---|
605 | { |
---|
606 | Me=Me*(M-eM[i]); |
---|
607 | } |
---|
608 | |
---|
609 | dbprint(printlevel-voice+2, |
---|
610 | "//gaussman::vfiltration: compute V["+string(i)+"]"); |
---|
611 | V1=module(V1)+syz(Me); |
---|
612 | V[i]=std(intersect(V1,V0)); |
---|
613 | |
---|
614 | if(size(V[i])>size(V[i+1])) |
---|
615 | { |
---|
616 | k++; |
---|
617 | a[k]=eM[i]-1; |
---|
618 | d[k]=size(V[i])-size(V[i+1]); |
---|
619 | } |
---|
620 | } |
---|
621 | |
---|
622 | dbprint(printlevel-voice+2,"//gaussman::vfiltration: apply symmetry"); |
---|
623 | while(j>=1) |
---|
624 | { |
---|
625 | k++; |
---|
626 | a[k]=a[j]; |
---|
627 | a[j]=n-1-a[k]; |
---|
628 | d[k]=d[j]; |
---|
629 | j--; |
---|
630 | } |
---|
631 | |
---|
632 | return(list(a,d)); |
---|
633 | } |
---|
634 | else |
---|
635 | { |
---|
636 | list v; |
---|
637 | j=-1; |
---|
638 | for(i=ncols(eM);i>=1;i--) |
---|
639 | { |
---|
640 | Me=freemodule(mu*N); |
---|
641 | for(i0=n;i0>=0;i0--) // Potenzen von Matrizen? |
---|
642 | { |
---|
643 | Me=Me*(M-eM[i]); |
---|
644 | } |
---|
645 | |
---|
646 | dbprint(printlevel-voice+2, |
---|
647 | "//gaussman::vfiltration: compute V["+string(i)+"]"); |
---|
648 | V1=module(V1)+syz(Me); |
---|
649 | V[i]=std(intersect(V1,V0)); |
---|
650 | |
---|
651 | if(size(V[i])>size(V[i+1])) |
---|
652 | { |
---|
653 | if(number(eM[i]-1)>=number(n-1)/2) |
---|
654 | { |
---|
655 | k++; |
---|
656 | a[k]=eM[i]-1; |
---|
657 | dbprint(printlevel-voice+2, |
---|
658 | "//gaussman::vfiltration: transform to V0"); |
---|
659 | v[k]=matrix(freemodule(ncols(V[i])),mu,mu*N)*division(V0,V[i])[1]; |
---|
660 | } |
---|
661 | else |
---|
662 | { |
---|
663 | if(j<0) |
---|
664 | { |
---|
665 | if(a[k]==number(n-1)/2) |
---|
666 | { |
---|
667 | j=k-1; |
---|
668 | } |
---|
669 | else |
---|
670 | { |
---|
671 | j=k; |
---|
672 | } |
---|
673 | } |
---|
674 | k++; |
---|
675 | a[k]=a[j]; |
---|
676 | a[j]=eM[i]-1; |
---|
677 | v[k]=v[j]; |
---|
678 | dbprint(printlevel-voice+2, |
---|
679 | "//gaussman::vfiltration: transform to V0"); |
---|
680 | v[j]=matrix(freemodule(ncols(V[i])),mu,mu*N)*division(V0,V[i])[1]; |
---|
681 | j--; |
---|
682 | } |
---|
683 | } |
---|
684 | } |
---|
685 | |
---|
686 | dbprint(printlevel-voice+2, |
---|
687 | "//gaussman::vfiltration: compute graded parts"); |
---|
688 | option(redSB); |
---|
689 | for(k=1;k<size(v);k++) |
---|
690 | { |
---|
691 | v[k]=std(reduce(v[k],std(v[k+1]))); |
---|
692 | d[k]=size(v[k]); |
---|
693 | } |
---|
694 | v[k]=std(v[k]); |
---|
695 | d[k]=size(v[k]); |
---|
696 | |
---|
697 | return(list(a,d,v,m)); |
---|
698 | } |
---|
699 | } |
---|
700 | example |
---|
701 | { "EXAMPLE:"; echo=2; |
---|
702 | ring R=0,(x,y),ds; |
---|
703 | poly f=x5+x2y2+y5; |
---|
704 | vfiltration(f); |
---|
705 | } |
---|
706 | /////////////////////////////////////////////////////////////////////////////// |
---|
707 | |
---|
708 | proc spectrum(poly f) |
---|
709 | "USAGE: spectrum(f); poly f |
---|
710 | ASSUME: basering has characteristic 0 and local ordering, |
---|
711 | f has isolated singularity at 0 |
---|
712 | RETURN: |
---|
713 | @format |
---|
714 | list S: singularity spectrum of f |
---|
715 | ideal S[1]: spectral numbers in increasing order |
---|
716 | intvec S[2]: |
---|
717 | int S[2][i]: multiplicity of spectral number S[1][i] |
---|
718 | @end format |
---|
719 | SEE ALSO: spectrum_lib |
---|
720 | KEYWORDS: singularities; Gauss-Manin connection; spectrum |
---|
721 | EXAMPLE: example spectrum; shows an example |
---|
722 | " |
---|
723 | { |
---|
724 | return(vfiltration(f,0)); |
---|
725 | } |
---|
726 | example |
---|
727 | { "EXAMPLE:"; echo=2; |
---|
728 | ring R=0,(x,y),ds; |
---|
729 | poly f=x5+x2y2+y5; |
---|
730 | spprint(spectrum(f)); |
---|
731 | } |
---|
732 | /////////////////////////////////////////////////////////////////////////////// |
---|
733 | |
---|
734 | proc endfilt(poly f,list V) |
---|
735 | "USAGE: endfilt(f,V); poly f, list V |
---|
736 | ASSUME: basering has characteristic 0 and local ordering, |
---|
737 | f has isolated singularity at 0 |
---|
738 | RETURN: |
---|
739 | @format |
---|
740 | list V1: endomorphim filtration of V on the Jacobian algebra of f |
---|
741 | ideal V1[1]: spectral numbers in increasing order |
---|
742 | intvec V1[2]: |
---|
743 | int V1[2][i]: multiplicity of spectral number V1[1][i] |
---|
744 | list V1[3]: |
---|
745 | module V1[3][i]: vector space basis of the V1[1][i]-th graded part |
---|
746 | in terms of V1[4] |
---|
747 | ideal V1[4]: monomial vector space basis |
---|
748 | @end format |
---|
749 | SEE ALSO: spectrum_lib |
---|
750 | KEYWORDS: singularities; Gauss-Manin connection; spectrum; |
---|
751 | Brieskorn lattice; Hodge filtration; V-filtration |
---|
752 | EXAMPLE: example endfilt; shows an example |
---|
753 | " |
---|
754 | { |
---|
755 | if(charstr(basering)!="0") |
---|
756 | { |
---|
757 | ERROR("characteristic 0 expected"); |
---|
758 | } |
---|
759 | int n=nvars(basering)-1; |
---|
760 | for(int i=n+1;i>=1;i--) |
---|
761 | { |
---|
762 | if(var(i)>1) |
---|
763 | { |
---|
764 | ERROR("local ordering expected"); |
---|
765 | } |
---|
766 | } |
---|
767 | ideal sJ=std(jacob(f)); |
---|
768 | if(vdim(sJ)<=0) |
---|
769 | { |
---|
770 | if(vdim(sJ)==0) |
---|
771 | { |
---|
772 | ERROR("singularity at 0 expected"); |
---|
773 | } |
---|
774 | else |
---|
775 | { |
---|
776 | ERROR("isolated singularity at 0 expected"); |
---|
777 | } |
---|
778 | } |
---|
779 | |
---|
780 | def a,d,v,m=V[1..4]; |
---|
781 | int mu=ncols(m); |
---|
782 | |
---|
783 | module V0=v[1]; |
---|
784 | for(i=2;i<=size(v);i++) |
---|
785 | { |
---|
786 | V0=V0,v[i]; |
---|
787 | } |
---|
788 | |
---|
789 | dbprint(printlevel-voice+2, |
---|
790 | "//gaussman::endfilt: compute multiplication in Jacobian algebra"); |
---|
791 | list M; |
---|
792 | matrix U=freemodule(ncols(m)); |
---|
793 | for(i=ncols(m);i>=1;i--) |
---|
794 | { |
---|
795 | M[i]=lift(V0,coeffs(reduce(m[i]*m,U,sJ),m)*V0); |
---|
796 | } |
---|
797 | |
---|
798 | int j,k,i0,j0,i1,j1; |
---|
799 | number b0=number(a[1]-a[ncols(a)]); |
---|
800 | number b1,b2; |
---|
801 | matrix M0; |
---|
802 | module L; |
---|
803 | list v0=freemodule(ncols(m)); |
---|
804 | ideal a0=b0; |
---|
805 | |
---|
806 | while(b0<number(a[ncols(a)]-a[1])) |
---|
807 | { |
---|
808 | dbprint(printlevel-voice+2, |
---|
809 | "//gaussman::endfilt: find next possible index"); |
---|
810 | b1=number(a[ncols(a)]-a[1]); |
---|
811 | for(j=ncols(a);j>=1;j--) |
---|
812 | { |
---|
813 | for(i=ncols(a);i>=1;i--) |
---|
814 | { |
---|
815 | b2=number(a[i]-a[j]); |
---|
816 | if(b2>b0&&b2<b1) |
---|
817 | { |
---|
818 | b1=b2; |
---|
819 | } |
---|
820 | else |
---|
821 | { |
---|
822 | if(b2<=b0) |
---|
823 | { |
---|
824 | i=0; |
---|
825 | } |
---|
826 | } |
---|
827 | } |
---|
828 | } |
---|
829 | b0=b1; |
---|
830 | |
---|
831 | list l=ideal(); |
---|
832 | for(k=ncols(m);k>=2;k--) |
---|
833 | { |
---|
834 | l=l+list(ideal()); |
---|
835 | } |
---|
836 | |
---|
837 | dbprint(printlevel-voice+2, |
---|
838 | "//gaussman::endfilt: collect conditions for V1["+string(b0)+"]"); |
---|
839 | j=ncols(a); |
---|
840 | j0=mu; |
---|
841 | while(j>=1) |
---|
842 | { |
---|
843 | i0=1; |
---|
844 | i=1; |
---|
845 | while(i<ncols(a)&&a[i]<a[j]+b0) |
---|
846 | { |
---|
847 | i0=i0+d[i]; |
---|
848 | i++; |
---|
849 | } |
---|
850 | if(a[i]<a[j]+b0) |
---|
851 | { |
---|
852 | i0=i0+d[i]; |
---|
853 | i++; |
---|
854 | } |
---|
855 | for(k=1;k<=ncols(m);k++) |
---|
856 | { |
---|
857 | M0=M[k]; |
---|
858 | if(i0>1) |
---|
859 | { |
---|
860 | l[k]=l[k],M0[1..i0-1,j0-d[j]+1..j0]; |
---|
861 | } |
---|
862 | } |
---|
863 | j0=j0-d[j]; |
---|
864 | j--; |
---|
865 | } |
---|
866 | |
---|
867 | dbprint(printlevel-voice+2, |
---|
868 | "//gaussman::endfilt: compose condition matrix"); |
---|
869 | L=transpose(module(l[1])); |
---|
870 | for(k=2;k<=ncols(m);k++) |
---|
871 | { |
---|
872 | L=L,transpose(module(l[k])); |
---|
873 | } |
---|
874 | |
---|
875 | dbprint(printlevel-voice+2, |
---|
876 | "//gaussman::endfilt: compute kernel of condition matrix"); |
---|
877 | v0=v0+list(syz(L)); |
---|
878 | a0=a0,b0; |
---|
879 | } |
---|
880 | |
---|
881 | dbprint(printlevel-voice+2,"//gaussman::endfilt: compute graded parts"); |
---|
882 | option(redSB); |
---|
883 | for(i=1;i<size(v0);i++) |
---|
884 | { |
---|
885 | v0[i+1]=std(v0[i+1]); |
---|
886 | v0[i]=std(reduce(v0[i],v0[i+1])); |
---|
887 | } |
---|
888 | |
---|
889 | dbprint(printlevel-voice+2, |
---|
890 | "//gaussman::endfilt: remove trivial graded parts"); |
---|
891 | i=1; |
---|
892 | while(size(v0[i])==0) |
---|
893 | { |
---|
894 | i++; |
---|
895 | } |
---|
896 | list v1=v0[i]; |
---|
897 | intvec d1=size(v0[i]); |
---|
898 | ideal a1=a0[i]; |
---|
899 | i++; |
---|
900 | while(i<=size(v0)) |
---|
901 | { |
---|
902 | if(size(v0[i])>0) |
---|
903 | { |
---|
904 | v1=v1+list(v0[i]); |
---|
905 | d1=d1,size(v0[i]); |
---|
906 | a1=a1,a0[i]; |
---|
907 | } |
---|
908 | i++; |
---|
909 | } |
---|
910 | return(list(a1,d1,v1,m)); |
---|
911 | } |
---|
912 | example |
---|
913 | { "EXAMPLE:"; echo=2; |
---|
914 | ring R=0,(x,y),ds; |
---|
915 | poly f=x5+x2y2+y5; |
---|
916 | endfilt(f,vfiltration(f)); |
---|
917 | } |
---|
918 | /////////////////////////////////////////////////////////////////////////////// |
---|
919 | |
---|
920 | proc spprint(list S) |
---|
921 | "USAGE: spprint(S); list S |
---|
922 | RETURN: string: spectrum S |
---|
923 | EXAMPLE: example spprint; shows an example |
---|
924 | " |
---|
925 | { |
---|
926 | string s; |
---|
927 | for(int i=1;i<size(S[2]);i++) |
---|
928 | { |
---|
929 | s=s+"("+string(S[1][i])+","+string(S[2][i])+"),"; |
---|
930 | } |
---|
931 | s=s+"("+string(S[1][i])+","+string(S[2][i])+")"; |
---|
932 | return(s); |
---|
933 | } |
---|
934 | example |
---|
935 | { "EXAMPLE:"; echo=2; |
---|
936 | ring R=0,(x,y),ds; |
---|
937 | list S=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
938 | spprint(S); |
---|
939 | } |
---|
940 | /////////////////////////////////////////////////////////////////////////////// |
---|
941 | |
---|
942 | proc spadd(list S1,list S2) |
---|
943 | "USAGE: spadd(S1,S2); list S1,S2 |
---|
944 | RETURN: list: sum of spectra S1 and S2 |
---|
945 | EXAMPLE: example spadd; shows an example |
---|
946 | " |
---|
947 | { |
---|
948 | ideal s; |
---|
949 | intvec m; |
---|
950 | int i,i1,i2=1,1,1; |
---|
951 | while(i1<=size(S1[2])||i2<=size(S2[2])) |
---|
952 | { |
---|
953 | if(i1<=size(S1[2])) |
---|
954 | { |
---|
955 | if(i2<=size(S2[2])) |
---|
956 | { |
---|
957 | if(number(S1[1][i1])<number(S2[1][i2])) |
---|
958 | { |
---|
959 | s[i]=S1[1][i1]; |
---|
960 | m[i]=S1[2][i1]; |
---|
961 | i++; |
---|
962 | i1++; |
---|
963 | } |
---|
964 | else |
---|
965 | { |
---|
966 | if(number(S1[1][i1])>number(S2[1][i2])) |
---|
967 | { |
---|
968 | s[i]=S2[1][i2]; |
---|
969 | m[i]=S2[2][i2]; |
---|
970 | i++; |
---|
971 | i2++; |
---|
972 | } |
---|
973 | else |
---|
974 | { |
---|
975 | if(S1[2][i1]+S2[2][i2]!=0) |
---|
976 | { |
---|
977 | s[i]=S1[1][i1]; |
---|
978 | m[i]=S1[2][i1]+S2[2][i2]; |
---|
979 | i++; |
---|
980 | } |
---|
981 | i1++; |
---|
982 | i2++; |
---|
983 | } |
---|
984 | } |
---|
985 | } |
---|
986 | else |
---|
987 | { |
---|
988 | s[i]=S1[1][i1]; |
---|
989 | m[i]=S1[2][i1]; |
---|
990 | i++; |
---|
991 | i1++; |
---|
992 | } |
---|
993 | } |
---|
994 | else |
---|
995 | { |
---|
996 | s[i]=S2[1][i2]; |
---|
997 | m[i]=S2[2][i2]; |
---|
998 | i++; |
---|
999 | i2++; |
---|
1000 | } |
---|
1001 | } |
---|
1002 | return(list(s,m)); |
---|
1003 | } |
---|
1004 | example |
---|
1005 | { "EXAMPLE:"; echo=2; |
---|
1006 | ring R=0,(x,y),ds; |
---|
1007 | list S1=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
1008 | spprint(S1); |
---|
1009 | list S2=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1010 | spprint(S2); |
---|
1011 | spprint(spadd(S1,S2)); |
---|
1012 | } |
---|
1013 | /////////////////////////////////////////////////////////////////////////////// |
---|
1014 | |
---|
1015 | proc spsub(list S1,list S2) |
---|
1016 | "USAGE: spsub(S1,S2); list S1,S2 |
---|
1017 | RETURN: list: difference of spectra S1 and S2 |
---|
1018 | EXAMPLE: example spsub; shows an example |
---|
1019 | " |
---|
1020 | { |
---|
1021 | return(spadd(S1,spmul(S2,-1))); |
---|
1022 | } |
---|
1023 | example |
---|
1024 | { "EXAMPLE:"; echo=2; |
---|
1025 | ring R=0,(x,y),ds; |
---|
1026 | list S1=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
1027 | spprint(S1); |
---|
1028 | list S2=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1029 | spprint(S2); |
---|
1030 | spprint(spsub(S1,S2)); |
---|
1031 | } |
---|
1032 | /////////////////////////////////////////////////////////////////////////////// |
---|
1033 | |
---|
1034 | proc spmul(list #) |
---|
1035 | "USAGE: |
---|
1036 | @format |
---|
1037 | 1) spmul(S,k); list S, int k |
---|
1038 | 2) spmul(S,k); list S, intvec k |
---|
1039 | @end format |
---|
1040 | RETURN: |
---|
1041 | @format |
---|
1042 | 1) list: product of spectrum S and integer k |
---|
1043 | 2) list: linear combination of spectra S with coefficients k |
---|
1044 | @end format |
---|
1045 | EXAMPLE: example spmul; shows an example |
---|
1046 | " |
---|
1047 | { |
---|
1048 | if(size(#)==2) |
---|
1049 | { |
---|
1050 | if(typeof(#[1])=="list") |
---|
1051 | { |
---|
1052 | if(typeof(#[2])=="int") |
---|
1053 | { |
---|
1054 | return(list(#[1][1],#[1][2]*#[2])); |
---|
1055 | } |
---|
1056 | if(typeof(#[2])=="intvec") |
---|
1057 | { |
---|
1058 | list S0=list(ideal(),intvec(0)); |
---|
1059 | for(int i=size(#[2]);i>=1;i--) |
---|
1060 | { |
---|
1061 | S0=spadd(S0,spmul(#[1][i],#[2][i])); |
---|
1062 | } |
---|
1063 | return(S0); |
---|
1064 | } |
---|
1065 | } |
---|
1066 | } |
---|
1067 | return(list(ideal(),intvec(0))); |
---|
1068 | } |
---|
1069 | example |
---|
1070 | { "EXAMPLE:"; echo=2; |
---|
1071 | ring R=0,(x,y),ds; |
---|
1072 | list S=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
1073 | spprint(S); |
---|
1074 | spprint(spmul(S,2)); |
---|
1075 | list S1=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1076 | spprint(S1); |
---|
1077 | list S2=list(ideal(-1/3,0,1/3),intvec(1,2,1)); |
---|
1078 | spprint(S2); |
---|
1079 | spprint(spmul(list(S1,S2),intvec(1,2))); |
---|
1080 | } |
---|
1081 | /////////////////////////////////////////////////////////////////////////////// |
---|
1082 | |
---|
1083 | proc spissemicont(list S,list #) |
---|
1084 | "USAGE: spissemicont(S[,opt]); list S, int opt |
---|
1085 | RETURN: |
---|
1086 | @format |
---|
1087 | int k= |
---|
1088 | if opt==0: |
---|
1089 | 1, if sum of spectrum S over all intervals [a,a+1) is positive |
---|
1090 | 0, if sum of spectrum S over some interval [a,a+1) is negative |
---|
1091 | if opt==1: |
---|
1092 | 1, if sum of spectrum S over all intervals [a,a+1) and (a,a+1) is positive |
---|
1093 | 0, if sum of spectrum S over some interval [a,a+1) or (a,a+1) is negative |
---|
1094 | default: opt=0 |
---|
1095 | @end format |
---|
1096 | EXAMPLE: example spissemicont; shows an example |
---|
1097 | " |
---|
1098 | { |
---|
1099 | int opt=0; |
---|
1100 | if(size(#)>0) |
---|
1101 | { |
---|
1102 | if(typeof(#[1])=="int") |
---|
1103 | { |
---|
1104 | opt=1; |
---|
1105 | } |
---|
1106 | } |
---|
1107 | int i,j,k=1,1,0; |
---|
1108 | while(j<=size(S[2])) |
---|
1109 | { |
---|
1110 | while(j+1<=size(S[2])&&S[1][j]<S[1][i]+1) |
---|
1111 | { |
---|
1112 | k=k+S[2][j]; |
---|
1113 | j++; |
---|
1114 | } |
---|
1115 | if(j==size(S[2])&&S[1][j]<S[1][i]+1) |
---|
1116 | { |
---|
1117 | k=k+S[2][j]; |
---|
1118 | j++; |
---|
1119 | } |
---|
1120 | if(k<0) |
---|
1121 | { |
---|
1122 | return(0); |
---|
1123 | } |
---|
1124 | k=k-S[2][i]; |
---|
1125 | if(k<0&&opt==1) |
---|
1126 | { |
---|
1127 | return(0); |
---|
1128 | } |
---|
1129 | i++; |
---|
1130 | } |
---|
1131 | return(1); |
---|
1132 | } |
---|
1133 | example |
---|
1134 | { "EXAMPLE:"; echo=2; |
---|
1135 | ring R=0,(x,y),ds; |
---|
1136 | list S1=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
1137 | spprint(S1); |
---|
1138 | list S2=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1139 | spprint(S2); |
---|
1140 | spissemicont(spsub(S1,spmul(S2,5))); |
---|
1141 | spissemicont(spsub(S1,spmul(S2,5)),1); |
---|
1142 | spissemicont(spsub(S1,spmul(S2,6))); |
---|
1143 | } |
---|
1144 | /////////////////////////////////////////////////////////////////////////////// |
---|
1145 | |
---|
1146 | proc spsemicont(list S0,list S,list #) |
---|
1147 | "USAGE: spsemicont(S,k[,opt]); list S0, list S, int opt |
---|
1148 | RETURN: list of intvecs l: |
---|
1149 | spissemicont(sub(S0,spmul(S,k)),opt)==1 iff k<=l[i] for some i |
---|
1150 | NOTE: if the spectra S occur with multiplicities k in a deformation |
---|
1151 | of the [quasihomogeneous] spectrum S0 then |
---|
1152 | spissemicont(sub(S0,spmul(S,k))[,1])==1 |
---|
1153 | EXAMPLE: example spsemicont; shows an example |
---|
1154 | " |
---|
1155 | { |
---|
1156 | list l,l0; |
---|
1157 | int i,j,k; |
---|
1158 | while(spissemicont(S0,#)) |
---|
1159 | { |
---|
1160 | if(size(S)>1) |
---|
1161 | { |
---|
1162 | l0=spsemicont(S0,list(S[1..size(S)-1])); |
---|
1163 | for(i=1;i<=size(l0);i++) |
---|
1164 | { |
---|
1165 | if(size(l)>0) |
---|
1166 | { |
---|
1167 | j=1; |
---|
1168 | while(j<size(l)&&l[j]!=l0[i]) |
---|
1169 | { |
---|
1170 | j++; |
---|
1171 | } |
---|
1172 | if(l[j]==l0[i]) |
---|
1173 | { |
---|
1174 | l[j][size(S)]=k; |
---|
1175 | } |
---|
1176 | else |
---|
1177 | { |
---|
1178 | l0[i][size(S)]=k; |
---|
1179 | l=l+list(l0[i]); |
---|
1180 | } |
---|
1181 | } |
---|
1182 | else |
---|
1183 | { |
---|
1184 | l=l0; |
---|
1185 | } |
---|
1186 | } |
---|
1187 | } |
---|
1188 | S0=spsub(S0,S[size(S)]); |
---|
1189 | k++; |
---|
1190 | } |
---|
1191 | if(size(S)>1) |
---|
1192 | { |
---|
1193 | return(l); |
---|
1194 | } |
---|
1195 | else |
---|
1196 | { |
---|
1197 | return(list(intvec(k-1))); |
---|
1198 | } |
---|
1199 | } |
---|
1200 | example |
---|
1201 | { "EXAMPLE:"; echo=2; |
---|
1202 | ring R=0,(x,y),ds; |
---|
1203 | list S0=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
1204 | spprint(S0); |
---|
1205 | list S1=list(ideal(-1/6,1/6),intvec(1,1)); |
---|
1206 | spprint(S1); |
---|
1207 | list S2=list(ideal(-1/3,0,1/3),intvec(1,2,1)); |
---|
1208 | spprint(S2); |
---|
1209 | list S=S1,S2; |
---|
1210 | list l=spsemicont(S0,S); |
---|
1211 | l; |
---|
1212 | spissemicont(spsub(S0,spmul(S,l[1]))); |
---|
1213 | spissemicont(spsub(S0,spmul(S,l[1]-1))); |
---|
1214 | spissemicont(spsub(S0,spmul(S,l[1]+1))); |
---|
1215 | } |
---|
1216 | /////////////////////////////////////////////////////////////////////////////// |
---|
1217 | |
---|
1218 | proc spmilnor(list S) |
---|
1219 | "USAGE: spmilnor(S); list S |
---|
1220 | RETURN: int: Milnor number of spectrum S |
---|
1221 | EXAMPLE: example spmilnor; shows an example |
---|
1222 | " |
---|
1223 | { |
---|
1224 | return(sum(S[2])); |
---|
1225 | } |
---|
1226 | example |
---|
1227 | { "EXAMPLE:"; echo=2; |
---|
1228 | ring R=0,(x,y),ds; |
---|
1229 | list S=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
1230 | spprint(S); |
---|
1231 | spmilnor(S); |
---|
1232 | } |
---|
1233 | /////////////////////////////////////////////////////////////////////////////// |
---|
1234 | |
---|
1235 | proc spgeomgenus(list S) |
---|
1236 | "USAGE: spgeomgenus(S); list S |
---|
1237 | RETURN: int: geometrical genus of spectrum S |
---|
1238 | EXAMPLE: example spgeomgenus; shows an example |
---|
1239 | " |
---|
1240 | { |
---|
1241 | int g=0; |
---|
1242 | int i=1; |
---|
1243 | while(i+1<=size(S[2])&&number(S[1][i])<=number(0)) |
---|
1244 | { |
---|
1245 | g=g+S[2][i]; |
---|
1246 | i++; |
---|
1247 | } |
---|
1248 | if(i==size(S[2])&&number(S[1][i])<=number(0)) |
---|
1249 | { |
---|
1250 | g=g+S[2][i]; |
---|
1251 | } |
---|
1252 | return(g); |
---|
1253 | } |
---|
1254 | example |
---|
1255 | { "EXAMPLE:"; echo=2; |
---|
1256 | ring R=0,(x,y),ds; |
---|
1257 | list S=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
1258 | spprint(S); |
---|
1259 | spgeomgenus(S); |
---|
1260 | } |
---|
1261 | /////////////////////////////////////////////////////////////////////////////// |
---|
1262 | |
---|
1263 | proc spgamma(list S) |
---|
1264 | "USAGE: spgamma(S); list S |
---|
1265 | RETURN: number: gamma invariant of spectrum S |
---|
1266 | EXAMPLE: example spgamma; shows an example |
---|
1267 | " |
---|
1268 | { |
---|
1269 | int i,j; |
---|
1270 | number g=0; |
---|
1271 | for(i=1;i<=ncols(S[1]);i++) |
---|
1272 | { |
---|
1273 | for(j=1;j<=S[2][i];j++) |
---|
1274 | { |
---|
1275 | g=g+(number(S[1][i])-number(nvars(basering)-2)/2)^2; |
---|
1276 | } |
---|
1277 | } |
---|
1278 | g=-g/4+sum(S[2])*number(S[1][ncols(S[1])]-S[1][1])/48; |
---|
1279 | return(g); |
---|
1280 | } |
---|
1281 | example |
---|
1282 | { "EXAMPLE:"; echo=2; |
---|
1283 | ring R=0,(x,y),ds; |
---|
1284 | list S=list(ideal(-1/2,-3/10,-1/10,0,1/10,3/10,1/2),intvec(1,2,2,1,2,2,1)); |
---|
1285 | spprint(S); |
---|
1286 | spgamma(S); |
---|
1287 | } |
---|
1288 | /////////////////////////////////////////////////////////////////////////////// |
---|