1 | ////////////////////////////////////////////////////////////////////////////// |
---|
2 | version="$Id: bfct.lib,v 1.12 2009-01-15 19:19:09 levandov Exp $"; |
---|
3 | category="Noncommutative"; |
---|
4 | info=" |
---|
5 | LIBRARY: bfct.lib Algorithms for b-functions and Bernstein-Sato polynomial |
---|
6 | AUTHORS: Daniel Andres, daniel.andres@math.rwth-aachen.de |
---|
7 | @* Viktor Levandovskyy, levandov@math.rwth-aachen.de |
---|
8 | |
---|
9 | THEORY: Given a polynomial ring R = K[x_1,...,x_n] and a polynomial F in R, |
---|
10 | @* one is interested in the global b-Function (also known as Bernstein-Sato |
---|
11 | @* polynomial) b(s) in K[s], defined to be the monic polynomial, satisfying a functional |
---|
12 | @* identity L * F^{s+1} = b(s) F^s, for some operator L in D[s]. Here, D stands for an |
---|
13 | @* n-th Weyl algebra K<x_1,...,x_n,d_1,...,d_n | d_j x_j = x_j d_j +1>. |
---|
14 | @* One is interested in the following data: |
---|
15 | @* - Bernstein-Sato polynomial b(s) in K[s], |
---|
16 | @* - the list of its roots, which are known to be rational |
---|
17 | @* - the multiplicities of the roots. |
---|
18 | @* References: Saito, Sturmfels, Takayama: Groebner Deformations of Hypergeometric |
---|
19 | @* Differential Equations (2000), Noro: An Efficient Modular Algorithm for Computing |
---|
20 | @* the Global b-function, (2002). |
---|
21 | |
---|
22 | |
---|
23 | MAIN PROCEDURES: |
---|
24 | |
---|
25 | bfct(f[,s,t,v]); computes the Bernstein-Sato polynomial of poly f |
---|
26 | bfctSyz(f[,r,s,t,u,v]); computes the Bernstein-Sato polynomial of poly f |
---|
27 | bfctAnn(f[,s]); computes the Bernstein-Sato polynomial of poly f |
---|
28 | bfctOneGB(f[,s,t]); computes the Bernstein-Sato polynomial of poly f |
---|
29 | bfctIdeal(I,w[,s,t]); computes the global b-function of ideal I w.r.t. the weight w |
---|
30 | pIntersect(f,I); intersection of the ideal I with the subalgebra K[f] for a poly f |
---|
31 | pIntersectSyz(f,I[,p,s,t]); intersection of the ideal I with the subalgebra K[f] for a poly f |
---|
32 | linReduce(f,I[,s]); reduces a poly by linear reductions only |
---|
33 | linReduceIdeal(I,[s,t]); reduces generators of ideal by linear reductions only |
---|
34 | linSyzSolve(I[,s]); computes a linear dependency of the elements of ideal I |
---|
35 | |
---|
36 | AUXILIARY PROCEDURES: |
---|
37 | |
---|
38 | allPositive(v); checks whether all entries of an intvec are positive |
---|
39 | scalarProd(v,w); computes the standard scalar product of two intvecs |
---|
40 | vec2poly(v[,i]); constructs an univariate poly with given coefficients |
---|
41 | |
---|
42 | SEE ALSO: dmod_lib, dmodapp_lib, gmssing_lib |
---|
43 | "; |
---|
44 | |
---|
45 | |
---|
46 | LIB "qhmoduli.lib"; // for Max |
---|
47 | LIB "dmod.lib"; // for SannfsBFCT etc |
---|
48 | LIB "dmodapp.lib"; // for initialIdealW etc |
---|
49 | |
---|
50 | |
---|
51 | /////////////////////////////////////////////////////////////////////////////// |
---|
52 | // testing for consistency of the library: |
---|
53 | proc testbfctlib () |
---|
54 | { |
---|
55 | // tests all procs for consistency |
---|
56 | "AUXILIARY PROCEDURES:"; |
---|
57 | example allPositive; |
---|
58 | example scalarProd; |
---|
59 | example vec2poly; |
---|
60 | "MAIN PROCEDURES:"; |
---|
61 | example bfct; |
---|
62 | example bfctSyz; |
---|
63 | example bfctAnn; |
---|
64 | example bfctOneGB; |
---|
65 | example bfctIdeal; |
---|
66 | example pIntersect; |
---|
67 | example pIntersectSyz; |
---|
68 | example linReduce; |
---|
69 | example linReduceIdeal; |
---|
70 | example linSyzSolve; |
---|
71 | } |
---|
72 | |
---|
73 | //--------------- auxiliary procedures --------------------------------------------------------- |
---|
74 | |
---|
75 | static proc gradedWeyl (intvec u,intvec v) |
---|
76 | "USAGE: gradedWeyl(u,v); u,v intvecs |
---|
77 | RETURN: a ring, the associated graded ring of the basering w.r.t. u and v |
---|
78 | PURPOSE: compute the associated graded ring of the basering w.r.t. u and v |
---|
79 | ASSUME: basering is a Weyl algebra |
---|
80 | EXAMPLE: example gradedWeyl; shows examples |
---|
81 | NOTE: u[i] is the weight of x(i), v[i] the weight of D(i). |
---|
82 | @* u+v has to be a non-negative intvec. |
---|
83 | " |
---|
84 | { |
---|
85 | int i; |
---|
86 | def save = basering; |
---|
87 | int n = nvars(save)/2; |
---|
88 | if (nrows(u)<>n || nrows(v)<>n) |
---|
89 | { |
---|
90 | ERROR("weight vectors have wrong dimension"); |
---|
91 | } |
---|
92 | intvec uv,gr; |
---|
93 | uv = u+v; |
---|
94 | for (i=1; i<=n; i++) |
---|
95 | { |
---|
96 | if (uv[i]>=0) |
---|
97 | { |
---|
98 | if (uv[i]==0) |
---|
99 | { |
---|
100 | gr[i] = 0; |
---|
101 | } |
---|
102 | else |
---|
103 | { |
---|
104 | gr[i] = 1; |
---|
105 | } |
---|
106 | } |
---|
107 | else |
---|
108 | { |
---|
109 | ERROR("the sum of the weight vectors has to be a non-negative intvec"); |
---|
110 | } |
---|
111 | } |
---|
112 | list l = ringlist(save); |
---|
113 | list l2 = l[2]; |
---|
114 | matrix l6 = l[6]; |
---|
115 | for (i=1; i<=n; i++) |
---|
116 | { |
---|
117 | if (gr[i] == 1) |
---|
118 | { |
---|
119 | l2[n+i] = "xi("+string(i)+")"; |
---|
120 | l6[i,n+i] = 0; |
---|
121 | } |
---|
122 | } |
---|
123 | l[2] = l2; |
---|
124 | l[6] = l6; |
---|
125 | def G = ring(l); |
---|
126 | return(G); |
---|
127 | } |
---|
128 | example |
---|
129 | { |
---|
130 | "EXAMPLE:"; echo = 2; |
---|
131 | ring @D = 0,(x,y,z,Dx,Dy,Dz),dp; |
---|
132 | def D = Weyl(); |
---|
133 | setring D; |
---|
134 | intvec u = -1,-1,1; intvec v = 2,1,1; |
---|
135 | def G = gradedWeyl(u,v); |
---|
136 | setring G; G; |
---|
137 | } |
---|
138 | |
---|
139 | |
---|
140 | proc allPositive (intvec v) |
---|
141 | "USAGE: allPositive(v); v an intvec |
---|
142 | RETURN: int, 1 if all components of v are positive, or 0 otherwise |
---|
143 | PURPOSE: check whether all components of an intvec are positive |
---|
144 | EXAMPLE: example allPositive; shows an example |
---|
145 | " |
---|
146 | { |
---|
147 | int i; |
---|
148 | for (i=1; i<=size(v); i++) |
---|
149 | { |
---|
150 | if (v[i]<=0) |
---|
151 | { |
---|
152 | return(0); |
---|
153 | break; |
---|
154 | } |
---|
155 | } |
---|
156 | return(1); |
---|
157 | } |
---|
158 | example |
---|
159 | { |
---|
160 | "EXAMPLE:"; echo = 2; |
---|
161 | intvec v = 1,2,3; |
---|
162 | allPositive(v); |
---|
163 | intvec w = 1,-2,3; |
---|
164 | allPositive(w); |
---|
165 | } |
---|
166 | |
---|
167 | static proc findFirst (list l, i) |
---|
168 | "USAGE: findFirst(l,i); l a list, i an argument of any type |
---|
169 | RETURN: int, the position of the first appearance of i in l, or 0 if i is not a member of l |
---|
170 | PURPOSE: check whether the second argument is a member of a list |
---|
171 | EXAMPLE: example findFirst; shows an example |
---|
172 | " |
---|
173 | { |
---|
174 | int j; |
---|
175 | for (j=1; j<=size(l); j++) |
---|
176 | { |
---|
177 | if (l[j]==i) |
---|
178 | { |
---|
179 | return(j); |
---|
180 | break; |
---|
181 | } |
---|
182 | } |
---|
183 | return(0); |
---|
184 | } |
---|
185 | // isin(list(1, 2, list(1)),2); // seems to be a bit buggy, |
---|
186 | // isin(list(1, 2, list(1)),3); // but works for the purposes |
---|
187 | // isin(list(1, 2, list(1)),list(1)); // of this library |
---|
188 | // isin(l,list(2)); |
---|
189 | example |
---|
190 | { |
---|
191 | "EXAMPLE:"; echo = 2; |
---|
192 | ring r = 0,x,dp; |
---|
193 | list l = 1,2,3; |
---|
194 | findFirst(l,4); |
---|
195 | findFirst(l,2); |
---|
196 | } |
---|
197 | |
---|
198 | proc scalarProd (intvec v, intvec w) |
---|
199 | "USAGE: scalarProd(v,w); v,w intvecs |
---|
200 | RETURN: int, the standard scalar product of v and w |
---|
201 | PURPOSE: computes the scalar product of two intvecs |
---|
202 | ASSUME: the arguments are of the same size |
---|
203 | EXAMPLE: example scalarProd; shows examples |
---|
204 | " |
---|
205 | { |
---|
206 | int i; int sp; |
---|
207 | if (size(v)!=size(w)) |
---|
208 | { |
---|
209 | ERROR("non-matching dimensions"); |
---|
210 | } |
---|
211 | else |
---|
212 | { |
---|
213 | for (i=1; i<=size(v);i++) |
---|
214 | { |
---|
215 | sp = sp + v[i]*w[i]; |
---|
216 | } |
---|
217 | } |
---|
218 | return(sp); |
---|
219 | } |
---|
220 | example |
---|
221 | { |
---|
222 | "EXAMPLE:"; echo = 2; |
---|
223 | intvec v = 1,2,3; |
---|
224 | intvec w = 4,5,6; |
---|
225 | scalarProd(v,w); |
---|
226 | } |
---|
227 | |
---|
228 | //-------------- main procedures ------------------------------------------------------- |
---|
229 | |
---|
230 | proc linReduceIdeal(ideal I, list #) |
---|
231 | "USAGE: linReduceIdeal(I [,s,t]); I an ideal, s,t optional ints |
---|
232 | RETURN: ideal/list, linear reductum (over field) of I by its elements |
---|
233 | PURPOSE: reduce a list of polys only by linear reductions (no monomial multiplications) |
---|
234 | NOTE: If s<>0, a list consisting of the reduced ideal and the coefficient |
---|
235 | @* vectors of the used reductions given as module is returned. |
---|
236 | @* Otherwise (and by default), only the reduced ideal is returned. |
---|
237 | @* If t=0 (and by default) all monomials are reduced (if possible), |
---|
238 | @* otherwise, only leading monomials are reduced. |
---|
239 | DISPLAY: If @code{printlevel}>=1, all debug messages will be printed. |
---|
240 | EXAMPLE: example linReduceIdeal; shows examples |
---|
241 | " |
---|
242 | { |
---|
243 | // #[1] = remembercoeffs |
---|
244 | // #[2] = redtail |
---|
245 | int ppl = printlevel - voice + 2; |
---|
246 | int remembercoeffs = 0; // default |
---|
247 | int redtail = 0; // default |
---|
248 | if (size(#)>0) |
---|
249 | { |
---|
250 | if (typeof(#[1])=="int" || typeof(#[1])=="number") |
---|
251 | { |
---|
252 | remembercoeffs = #[1]; |
---|
253 | } |
---|
254 | if (size(#)>1) |
---|
255 | { |
---|
256 | if (typeof(#[2])=="int" || typeof(#[2])=="number") |
---|
257 | { |
---|
258 | redtail = #[2]; |
---|
259 | } |
---|
260 | } |
---|
261 | } |
---|
262 | int sI = ncols(I); |
---|
263 | int sZeros = sI - size(I); |
---|
264 | dbprint(ppl,"ideal contains "+string(sZeros)+" zero(s)"); |
---|
265 | int i,j; |
---|
266 | ideal J,lmJ,ordJ; |
---|
267 | list lJ = sort(I); |
---|
268 | module M; // for the coefficients |
---|
269 | if (sZeros > 0) // I contains zeros |
---|
270 | { |
---|
271 | if (remembercoeffs <> 0) |
---|
272 | { |
---|
273 | j = 0; |
---|
274 | for (i=1; i<=sI; i++) |
---|
275 | { |
---|
276 | if (I[i] == 0) |
---|
277 | { |
---|
278 | j++; |
---|
279 | J[j] = 0; |
---|
280 | ordJ[j] = -1; |
---|
281 | M[j] = gen(i); |
---|
282 | } |
---|
283 | else |
---|
284 | { |
---|
285 | M[i+sZeros-j] = gen(lJ[2][i-j]+j); |
---|
286 | } |
---|
287 | } |
---|
288 | } |
---|
289 | else |
---|
290 | { |
---|
291 | for (i=1; i<=sZeros; i++) |
---|
292 | { |
---|
293 | J[i] = 0; |
---|
294 | ordJ[i] = -1; |
---|
295 | } |
---|
296 | } |
---|
297 | I = J,lJ[1]; |
---|
298 | } |
---|
299 | else |
---|
300 | { |
---|
301 | I = lJ[1]; |
---|
302 | if (remembercoeffs <> 0) |
---|
303 | { |
---|
304 | for (i=1; i<=size(lJ[1]); i++) { M[i+sZeros] = gen(lJ[2][i]); } |
---|
305 | } |
---|
306 | } |
---|
307 | dbprint(ppl,"initially sorted ideal:", I); |
---|
308 | if (remembercoeffs <> 0) { dbprint(ppl," used permutations:", M); } |
---|
309 | poly lm,c,redpoly,maxlmJ; |
---|
310 | J[sZeros+1] = I[sZeros+1]; |
---|
311 | lm = I[sZeros+1]; |
---|
312 | maxlmJ = leadmonom(lm); |
---|
313 | lmJ[sZeros+1] = maxlmJ; |
---|
314 | int ordlm,reduction,maxordJ; |
---|
315 | maxordJ = ord(lm); |
---|
316 | ordJ[sZeros+1] = maxordJ; |
---|
317 | vector v; |
---|
318 | for (i=sZeros+2; i<=sI; i++) |
---|
319 | { |
---|
320 | redpoly = I[i]; |
---|
321 | lm = leadmonom(redpoly); |
---|
322 | ordlm = ord(lm); |
---|
323 | reduction = 1; |
---|
324 | if (remembercoeffs <> 0) { v = M[i]; } |
---|
325 | while (reduction == 1) // while there was a reduction |
---|
326 | { |
---|
327 | reduction = 0; |
---|
328 | for (j=sZeros+1; j<i; j++) |
---|
329 | { |
---|
330 | if (lm == 0) { break; } |
---|
331 | if (ordlm > maxordJ) { break; } |
---|
332 | if (ordlm == ordJ[j]) |
---|
333 | { |
---|
334 | if (lm > maxlmJ) { break; } |
---|
335 | if (lm == lmJ[j]) |
---|
336 | { |
---|
337 | dbprint(ppl,"reducing " + string(redpoly)); |
---|
338 | dbprint(ppl," with " + string(J[j])); |
---|
339 | c = leadcoef(redpoly)/leadcoef(J[j]); |
---|
340 | redpoly = redpoly - c*J[j]; |
---|
341 | dbprint(ppl," to " + string(redpoly)); |
---|
342 | lm = leadmonom(redpoly); |
---|
343 | ordlm = ord(lm); |
---|
344 | if (remembercoeffs <> 0) { M[i] = M[i] - c * M[j]; } |
---|
345 | reduction = 1; |
---|
346 | } |
---|
347 | } |
---|
348 | } |
---|
349 | } |
---|
350 | for (j=sZeros+1; j<i; j++) |
---|
351 | { |
---|
352 | if (redpoly < J[j]) { break; } |
---|
353 | } |
---|
354 | J = insertGenerator(J,redpoly,j); |
---|
355 | lmJ = insertGenerator(lmJ,lm,j); |
---|
356 | ordJ = insertGenerator(ordJ,poly(ordlm),j); |
---|
357 | if (remembercoeffs <> 0) |
---|
358 | { |
---|
359 | v = M[i]; |
---|
360 | M = deleteGenerator(M,i); |
---|
361 | M = insertGenerator(M,v,j); |
---|
362 | } |
---|
363 | maxordJ = ord(J[i]); |
---|
364 | maxlmJ = lmJ[i]; |
---|
365 | } |
---|
366 | if (redtail <> 0) |
---|
367 | { |
---|
368 | dbprint(ppl,"finished reducing leading monomials:",J); |
---|
369 | if (remembercoeffs <> 0) { dbprint(ppl,"used reductions:",M); } |
---|
370 | int k; |
---|
371 | for (i=sZeros+1; i<=sI; i++) |
---|
372 | { |
---|
373 | lm = lmJ[i]; |
---|
374 | for (j=i+1; j<=sI; j++) |
---|
375 | { |
---|
376 | for (k=2; k<=size(J[j]); k++) // run over all terms in J[j] |
---|
377 | { |
---|
378 | if (ordJ[i] == ord(J[j][k])) |
---|
379 | { |
---|
380 | if (lm == normalize(J[j][k])) |
---|
381 | { |
---|
382 | c = leadcoef(J[j][k])/leadcoef(J[i]); |
---|
383 | dbprint(ppl,"reducing " + string(J[j])); |
---|
384 | dbprint(ppl," with " + string(J[i])); |
---|
385 | J[j] = J[j] - c*J[i]; |
---|
386 | dbprint(ppl," to " + string(J[j])); |
---|
387 | if (remembercoeffs <> 0) { M[j] = M[j] - c * M[i]; } |
---|
388 | } |
---|
389 | } |
---|
390 | } |
---|
391 | } |
---|
392 | } |
---|
393 | } |
---|
394 | if (remembercoeffs <> 0) { return(list(J,M)); } |
---|
395 | else { return(J); } |
---|
396 | } |
---|
397 | example |
---|
398 | { |
---|
399 | "EXAMPLE:"; echo = 2; |
---|
400 | ring r = 0,(x,y),dp; |
---|
401 | ideal I = 3,x+9,y4,y4+7x+2; |
---|
402 | linReduceIdeal(I); |
---|
403 | linReduceIdeal(I,0,1); |
---|
404 | list l = linReduceIdeal(I,1,1); l; |
---|
405 | module M = I; |
---|
406 | l[1] - ideal(M*l[2]); |
---|
407 | } |
---|
408 | |
---|
409 | proc linReduce(poly f, ideal I, list #) |
---|
410 | "USAGE: linReduce(f, I [,s,t]); f a poly, I an ideal, s,t optional ints |
---|
411 | RETURN: poly/list, linear reductum (over field) of f by elements from I |
---|
412 | PURPOSE: reduce a poly only by linear reductions (no monomial multiplications) |
---|
413 | NOTE: If s<>0, a list consisting of the reduced poly and the coefficient |
---|
414 | @* vector of the used reductions is returned, otherwise (and by default) |
---|
415 | @* only reduced poly is returned. |
---|
416 | @* If t=0 (and by default) all monomials are reduced (if possible), |
---|
417 | @* otherwise, only leading monomials are reduced. |
---|
418 | DISPLAY: If @code{printlevel}>=1, all debug messages will be printed. |
---|
419 | EXAMPLE: example linReduce; shows examples |
---|
420 | " |
---|
421 | { |
---|
422 | int ppl = printlevel - voice + 2; |
---|
423 | int remembercoeffs = 0; // default |
---|
424 | int redtail = 0; // default |
---|
425 | int prepareideal = 1; // default |
---|
426 | if (size(#)>0) |
---|
427 | { |
---|
428 | if (typeof(#[1])=="int" || typeof(#[1])=="number") |
---|
429 | { |
---|
430 | remembercoeffs = #[1]; |
---|
431 | } |
---|
432 | if (size(#)>1) |
---|
433 | { |
---|
434 | if (typeof(#[2])=="int" || typeof(#[2])=="number") |
---|
435 | { |
---|
436 | redtail = #[2]; |
---|
437 | } |
---|
438 | if (size(#)>2) |
---|
439 | { |
---|
440 | if (typeof(#[3])=="int" || typeof(#[3])=="number") |
---|
441 | { |
---|
442 | prepareideal = #[3]; |
---|
443 | } |
---|
444 | } |
---|
445 | } |
---|
446 | } |
---|
447 | int i,j,k; |
---|
448 | int sI = ncols(I); |
---|
449 | // pre-reduce I: |
---|
450 | module M; |
---|
451 | if (prepareideal <> 0) |
---|
452 | { |
---|
453 | if (remembercoeffs <> 0) |
---|
454 | { |
---|
455 | list sortedI = linReduceIdeal(I,1,redtail); |
---|
456 | I = sortedI[1]; |
---|
457 | M = sortedI[2]; |
---|
458 | dbprint(ppl,"prepared ideal:",I); |
---|
459 | dbprint(ppl," with operations:",M); |
---|
460 | } |
---|
461 | else |
---|
462 | { |
---|
463 | I = linReduceIdeal(I,0,redtail); |
---|
464 | } |
---|
465 | } |
---|
466 | else |
---|
467 | { |
---|
468 | if (remembercoeffs <> 0) |
---|
469 | { |
---|
470 | for (i=1; i<=sI; i++) |
---|
471 | { |
---|
472 | M[i] = gen(i); |
---|
473 | } |
---|
474 | } |
---|
475 | } |
---|
476 | ideal lmI,lcI,ordI; |
---|
477 | for (i=1; i<=sI; i++) |
---|
478 | { |
---|
479 | lmI[i] = leadmonom(I[i]); |
---|
480 | lcI[i] = leadcoef(I[i]); |
---|
481 | ordI[i] = ord(lmI[i]); |
---|
482 | } |
---|
483 | vector v; |
---|
484 | poly c; |
---|
485 | // === reduce leading monomials === |
---|
486 | poly lm = leadmonom(f); |
---|
487 | int ordf = ord(lm); |
---|
488 | for (i=sI; i>=1; i--) // I is sorted: smallest lm's on top |
---|
489 | { |
---|
490 | if (lm == 0) |
---|
491 | { |
---|
492 | break; |
---|
493 | } |
---|
494 | else |
---|
495 | { |
---|
496 | if (ordf == ordI[i]) |
---|
497 | { |
---|
498 | if (lm == lmI[i]) |
---|
499 | { |
---|
500 | c = leadcoef(f)/lcI[i]; |
---|
501 | f = f - c*I[i]; |
---|
502 | lm = leadmonom(f); |
---|
503 | ordf = ord(lm); |
---|
504 | if (remembercoeffs <> 0) |
---|
505 | { |
---|
506 | v = v - c * M[i]; |
---|
507 | } |
---|
508 | } |
---|
509 | } |
---|
510 | } |
---|
511 | } |
---|
512 | // === reduce tails as well === |
---|
513 | if (redtail <> 0) |
---|
514 | { |
---|
515 | dbprint(ppl,"poly after reduction of leading monomials:",f); |
---|
516 | for (i=1; i<=sI; i++) |
---|
517 | { |
---|
518 | dbprint(ppl,"testing ideal entry:",i); |
---|
519 | for (j=1; j<=size(f); j++) |
---|
520 | { |
---|
521 | if (ord(f[j]) == ordI[i]) |
---|
522 | { |
---|
523 | if (normalize(f[j]) == lmI[i]) |
---|
524 | { |
---|
525 | c = leadcoef(f[j])/lcI[i]; |
---|
526 | f = f - c*I[i]; |
---|
527 | dbprint(ppl,"reducing poly to ",f); |
---|
528 | if (remembercoeffs <> 0) |
---|
529 | { |
---|
530 | v = v - c * M[i]; |
---|
531 | } |
---|
532 | } |
---|
533 | } |
---|
534 | } |
---|
535 | } |
---|
536 | } |
---|
537 | if (remembercoeffs <> 0) |
---|
538 | { |
---|
539 | list l = f,v; |
---|
540 | return(l); |
---|
541 | } |
---|
542 | else |
---|
543 | { |
---|
544 | return(f); |
---|
545 | } |
---|
546 | } |
---|
547 | example |
---|
548 | { |
---|
549 | "EXAMPLE:"; echo = 2; |
---|
550 | ring r = 0,(x,y),dp; |
---|
551 | ideal I = 1,y,xy; |
---|
552 | poly f = 5xy+7y+3; |
---|
553 | poly g = 7x+5y+3; |
---|
554 | linReduce(g,I); |
---|
555 | linReduce(g,I,0,1); |
---|
556 | linReduce(f,I,1); |
---|
557 | f = x3 + y2 + x2 + y + x; |
---|
558 | I = x3 - y3, y3 - x2, x3 - y2, x2 - y, y2-x; |
---|
559 | list l = linReduce(f, I, 1); |
---|
560 | l; |
---|
561 | module M = I; |
---|
562 | f - (l[1] - (M*l[2])[1,1]); |
---|
563 | } |
---|
564 | |
---|
565 | proc linSyzSolve (ideal I, list #) |
---|
566 | "USAGE: linSyzSolve(I[,s]); I an ideal, s an optional int |
---|
567 | RETURN: vector, coefficient vector of a linear combination of 0 in the elements of I |
---|
568 | PURPOSE: compute a linear dependency between the elements of an ideal |
---|
569 | @* if such one exists |
---|
570 | NOTE: If s<>0, @code{std} is used for Groebner basis computations, |
---|
571 | @* otherwise, @code{slimgb} is used. |
---|
572 | @* By default, @code{slimgb} is used in char 0 and @code{std} in char >0. |
---|
573 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
574 | @* if printlevel>=2, all the debug messages will be printed. |
---|
575 | EXAMPLE: example linSyzSolve; shows examples |
---|
576 | " |
---|
577 | { |
---|
578 | int whichengine = 0; // default |
---|
579 | int enginespecified = 0; // default |
---|
580 | if (size(#)>0) |
---|
581 | { |
---|
582 | if (typeof(#[1])=="int" || typeof(#[1])=="number") |
---|
583 | { |
---|
584 | whichengine = int( #[1]); |
---|
585 | enginespecified = 1; |
---|
586 | } |
---|
587 | } |
---|
588 | int ppl = printlevel - voice +2; |
---|
589 | int sI = ncols(I); |
---|
590 | // check if we are done |
---|
591 | if (I[sI]==0) |
---|
592 | { |
---|
593 | vector v = gen(sI); |
---|
594 | } |
---|
595 | else |
---|
596 | { |
---|
597 | // ------- 1. introduce undefined coeffs ------------------ |
---|
598 | def save = basering; |
---|
599 | int p = char(save); |
---|
600 | if (enginespecified == 0) |
---|
601 | { |
---|
602 | if (p <> 0) |
---|
603 | { |
---|
604 | whichengine = 1; |
---|
605 | } |
---|
606 | } |
---|
607 | ring @A = p,(@a(1..sI)),lp; |
---|
608 | ring @aA = (p,@a(1..sI)), (@z),dp; |
---|
609 | // TODO: BUG! WHAT IF THE ORIGINAL RING ALREADY HAS SUCH VARIABLES/PARAMETERS!!!? |
---|
610 | // TODO: YOU CAN OVERCOME THIS BY MEANS SIMILAR TO "chooseSafeVarName" FROM NEW "matrix.lib" |
---|
611 | def @B = save + @aA; |
---|
612 | setring @B; |
---|
613 | ideal I = imap(save,I); |
---|
614 | // ------- 2. form the linear system for the undef coeffs --- |
---|
615 | int i; poly W; ideal QQ; |
---|
616 | for (i=1; i<=sI; i++) |
---|
617 | { |
---|
618 | W = W + @a(i)*I[i]; |
---|
619 | } |
---|
620 | while (W!=0) |
---|
621 | { |
---|
622 | QQ = QQ,leadcoef(W); |
---|
623 | W = W - lead(W); |
---|
624 | } |
---|
625 | // QQ consists of polynomial expressions in @a(i) of type number |
---|
626 | setring @A; |
---|
627 | ideal QQ = imap(@B,QQ); |
---|
628 | // ------- 3. this QQ is a polynomial ideal, so "solve" the system ----- |
---|
629 | dbprint(ppl, "linSyzSolve: starting Groebner basis computation with engine:", whichengine); |
---|
630 | QQ = engine(QQ,whichengine); |
---|
631 | dbprint(ppl, "QQ after engine:", QQ); |
---|
632 | if (dim(QQ) == -1) |
---|
633 | { |
---|
634 | dbprint(ppl+1, "no solutions by linSyzSolve"); |
---|
635 | // output zeroes |
---|
636 | setring save; |
---|
637 | kill @A,@aA,@B; |
---|
638 | return(v); |
---|
639 | } |
---|
640 | // ------- 4. in order to get the numeric values ------- |
---|
641 | matrix AA = matrix(maxideal(1)); |
---|
642 | module MQQ = std(module(QQ)); |
---|
643 | AA = NF(AA,MQQ); // todo: we still receive NF warnings |
---|
644 | dbprint(ppl, "AA after NF:",AA); |
---|
645 | // "AA after NF:"; print(AA); |
---|
646 | for(i=1; i<=sI; i++) |
---|
647 | { |
---|
648 | AA = subst(AA,var(i),1); |
---|
649 | } |
---|
650 | dbprint(ppl, "AA after subst:",AA); |
---|
651 | // "AA after subst: "; print(AA); |
---|
652 | vector v = (module(transpose(AA)))[1]; |
---|
653 | setring save; |
---|
654 | vector v = imap(@A,v); |
---|
655 | kill @A,@aA,@B; |
---|
656 | } |
---|
657 | return(v); |
---|
658 | } |
---|
659 | example |
---|
660 | { |
---|
661 | "EXAMPLE:"; echo = 2; |
---|
662 | ring r = 0,x,dp; |
---|
663 | ideal I = x,2x; |
---|
664 | linSyzSolve(I); |
---|
665 | ideal J = x,x2; |
---|
666 | linSyzSolve(J); |
---|
667 | } |
---|
668 | |
---|
669 | proc pIntersect (poly s, ideal I) |
---|
670 | "USAGE: pIntersect(f, I); f a poly, I an ideal |
---|
671 | RETURN: vector, coefficient vector of the monic polynomial |
---|
672 | PURPOSE: compute the intersection of ideal I with the subalgebra K[f] |
---|
673 | ASSUME: I is given as Groebner basis. |
---|
674 | NOTE: If the intersection is zero, this proc might not terminate. |
---|
675 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
676 | @* if printlevel>=2, all the debug messages will be printed. |
---|
677 | EXAMPLE: example pIntersect; shows examples |
---|
678 | " |
---|
679 | { |
---|
680 | // assume I is given in Groebner basis |
---|
681 | if (attrib(I,"isSB") <> 1) |
---|
682 | { |
---|
683 | print("WARNING: The input has no SB attribute!"); |
---|
684 | print(" Treating it as if it were a Groebner basis and proceeding..."); |
---|
685 | attrib(I,"isSB",1); // set attribute for suppressing NF messages |
---|
686 | } |
---|
687 | int ppl = printlevel-voice+1; |
---|
688 | // ---case 1: I = basering--- |
---|
689 | if (size(I) == 1) |
---|
690 | { |
---|
691 | if (simplify(I,2)[1] == 1) |
---|
692 | { |
---|
693 | return(gen(2)); // = s |
---|
694 | } |
---|
695 | } |
---|
696 | def save = basering; |
---|
697 | int n = nvars(save); |
---|
698 | int i,j,k; |
---|
699 | // ---case 2: intersection is zero--- |
---|
700 | intvec degs = leadexp(s); |
---|
701 | intvec possdegbounds; |
---|
702 | list degI; |
---|
703 | i = 1; |
---|
704 | while (i <= ncols(I)) |
---|
705 | { |
---|
706 | if (i == ncols(I)+1) |
---|
707 | { |
---|
708 | break; |
---|
709 | } |
---|
710 | degI[i] = leadexp(I[i]); |
---|
711 | for (j=1; j<=n; j++) |
---|
712 | { |
---|
713 | if (degs[j] == 0) |
---|
714 | { |
---|
715 | if (degI[i][j] <> 0) |
---|
716 | { |
---|
717 | break; |
---|
718 | } |
---|
719 | } |
---|
720 | if (j == n) |
---|
721 | { |
---|
722 | k++; |
---|
723 | possdegbounds[k] = Max(degI[i]); |
---|
724 | } |
---|
725 | } |
---|
726 | i++; |
---|
727 | } |
---|
728 | int degbound = Min(possdegbounds); |
---|
729 | dbprint(ppl,"a lower bound for the degree of the insection is:"); |
---|
730 | dbprint(ppl,degbound); |
---|
731 | if (degbound == 0) // lm(s) does not appear in lm(I) |
---|
732 | { |
---|
733 | return(vector(0)); |
---|
734 | } |
---|
735 | // ---case 3: intersection is non-trivial--- |
---|
736 | ideal redNI = 1; |
---|
737 | vector v; |
---|
738 | list l,ll; |
---|
739 | l[1] = vector(0); |
---|
740 | poly toNF,tobracket,newNF,rednewNF,oldNF,secNF; |
---|
741 | i = 1; |
---|
742 | dbprint(ppl+1,"pIntersect starts..."); |
---|
743 | while (1) |
---|
744 | { |
---|
745 | dbprint(ppl,"testing degree: "+string(i)); |
---|
746 | if (i>1) |
---|
747 | { |
---|
748 | oldNF = newNF; |
---|
749 | tobracket = s^(i-1) - oldNF; |
---|
750 | if (tobracket==0) // todo bug in bracket? |
---|
751 | { |
---|
752 | toNF = 0; |
---|
753 | } |
---|
754 | else |
---|
755 | { |
---|
756 | toNF = bracket(tobracket,secNF); |
---|
757 | } |
---|
758 | newNF = NF(toNF+oldNF*secNF,I); // = NF(s^i,I) |
---|
759 | } |
---|
760 | else |
---|
761 | { |
---|
762 | newNF = NF(s,I); |
---|
763 | secNF = newNF; |
---|
764 | } |
---|
765 | ll = linReduce(newNF,redNI,1); |
---|
766 | rednewNF = ll[1]; |
---|
767 | l[i+1] = ll[2]; |
---|
768 | dbprint(ppl,"newNF is:", newNF); |
---|
769 | dbprint(ppl,"rednewNF is:", rednewNF); |
---|
770 | if (rednewNF != 0) // no linear dependency |
---|
771 | { |
---|
772 | redNI[i+1] = rednewNF; |
---|
773 | i++; |
---|
774 | } |
---|
775 | else // there is a linear dependency, hence we are done |
---|
776 | { |
---|
777 | dbprint(ppl+1,"the degree of the generator of the intersection is:", i); |
---|
778 | break; |
---|
779 | } |
---|
780 | } |
---|
781 | dbprint(ppl,"used linear reductions:", l); |
---|
782 | // we obtain the coefficients of the generator of the intersection by the used reductions: |
---|
783 | ring @R = 0,(a(1..i+1)),dp; |
---|
784 | setring @R; |
---|
785 | list l = imap(save,l); |
---|
786 | ideal C; |
---|
787 | for (j=1;j<=i+1;j++) |
---|
788 | { |
---|
789 | C[j] = 0; |
---|
790 | for (k=1;k<=j;k++) |
---|
791 | { |
---|
792 | C[j] = C[j]+l[j][k]*a(k); |
---|
793 | } |
---|
794 | } |
---|
795 | for (j=i;j>=1;j--) |
---|
796 | { |
---|
797 | C[i+1]= subst(C[i+1],a(j),a(j)+C[j]); |
---|
798 | } |
---|
799 | matrix m = coeffs(C[i+1],maxideal(1)); |
---|
800 | vector v = gen(i+1); |
---|
801 | for (j=1;j<=i+1;j++) |
---|
802 | { |
---|
803 | v = v + m[j,1]*gen(j); |
---|
804 | } |
---|
805 | setring save; |
---|
806 | v = imap(@R,v); |
---|
807 | kill @R; |
---|
808 | dbprint(ppl+1,"pIntersect finished"); |
---|
809 | return(v); |
---|
810 | } |
---|
811 | example |
---|
812 | { |
---|
813 | "EXAMPLE:"; echo = 2; |
---|
814 | ring r = 0,(x,y),dp; |
---|
815 | poly f = x^2+y^3+x*y^2; |
---|
816 | def D = initialMalgrange(f); |
---|
817 | setring D; |
---|
818 | inF; |
---|
819 | pIntersect(t*Dt,inF); |
---|
820 | } |
---|
821 | |
---|
822 | proc pIntersectSyz (poly s, ideal II, list #) |
---|
823 | "USAGE: pIntersectSyz(f, I [,p,s,t]); f a poly, I an ideal, p, t optial ints, p a prime number |
---|
824 | RETURN: vector, coefficient vector of the monic polynomial |
---|
825 | PURPOSE: compute the intersection of an ideal I with the subalgebra K[f] |
---|
826 | ASSUME: I is given as Groebner basis. |
---|
827 | NOTE: If the intersection is zero, this procedure might not terminate. |
---|
828 | @* If p>0 is given, this proc computes the generator of the intersection in char p first |
---|
829 | @* and then only searches for a generator of the obtained degree in the basering. |
---|
830 | @* Otherwise, it searched for all degrees by computing syzygies. |
---|
831 | @* If s<>0, @code{std} is used for Groebner basis computations in char 0, |
---|
832 | @* otherwise, and by default, @code{slimgb} is used. |
---|
833 | @* If t<>0 and by default, @code{std} is used for Groebner basis computations in char >0, |
---|
834 | @* otherwise, @code{slimgb} is used. |
---|
835 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
836 | @* if printlevel>=2, all the debug messages will be printed. |
---|
837 | EXAMPLE: example pIntersectSyz; shows examples |
---|
838 | " |
---|
839 | { |
---|
840 | // assume I is given in Groebner basis |
---|
841 | ideal I = II; |
---|
842 | if (attrib(I,"isSB") <> 1) |
---|
843 | { |
---|
844 | print("WARNING: The input has no SB attribute!"); |
---|
845 | print(" Treating it as if it were a Groebner basis and proceeding..."); |
---|
846 | attrib(I,"isSB",1); // set attribute for suppressing NF messages |
---|
847 | } |
---|
848 | int ppl = printlevel-voice+2; |
---|
849 | int whichengine = 0; // default |
---|
850 | int modengine = 1; // default |
---|
851 | int solveincharp = 0; // default |
---|
852 | def save = basering; |
---|
853 | if (size(#)>0) |
---|
854 | { |
---|
855 | if (typeof(#[1])=="int" || typeof(#[1])=="number") |
---|
856 | { |
---|
857 | solveincharp = int(#[1]); |
---|
858 | } |
---|
859 | if (size(#)>1) |
---|
860 | { |
---|
861 | if (typeof(#[2])=="int" || typeof(#[2])=="number") |
---|
862 | { |
---|
863 | whichengine = int(#[2]); |
---|
864 | } |
---|
865 | if (size(#)>2) |
---|
866 | { |
---|
867 | if (typeof(#[3])=="int" || typeof(#[3])=="number") |
---|
868 | { |
---|
869 | modengine = int(#[3]); |
---|
870 | } |
---|
871 | } |
---|
872 | } |
---|
873 | } |
---|
874 | int i,j; |
---|
875 | vector v; |
---|
876 | poly tobracket,toNF,newNF,p; |
---|
877 | ideal NI = 1; |
---|
878 | newNF = NF(s,I); |
---|
879 | NI[2] = newNF; |
---|
880 | if (solveincharp<>0) |
---|
881 | { |
---|
882 | list l = ringlist(save); |
---|
883 | l[1] = solveincharp; |
---|
884 | matrix l5 = l[5]; |
---|
885 | matrix l6 = l[6]; |
---|
886 | def @Rp = ring(l); |
---|
887 | setring @Rp; |
---|
888 | list l = ringlist(@Rp); |
---|
889 | l[5] = fetch(save,l5); |
---|
890 | l[6] = fetch(save,l6); |
---|
891 | def Rp = ring(l); |
---|
892 | setring Rp; |
---|
893 | kill @Rp; |
---|
894 | dbprint(ppl+1,"solving in ring ", Rp); |
---|
895 | vector v; |
---|
896 | map phi = save,maxideal(1); |
---|
897 | poly s = phi(s); |
---|
898 | ideal NI = 1; |
---|
899 | setring save; |
---|
900 | } |
---|
901 | i = 1; |
---|
902 | dbprint(ppl+1,"pIntersectSyz starts..."); |
---|
903 | dbprint(ppl+1,"with ideal I=", I); |
---|
904 | while (1) |
---|
905 | { |
---|
906 | dbprint(ppl,"i:"+string(i)); |
---|
907 | if (i>1) |
---|
908 | { |
---|
909 | tobracket = s^(i-1)-NI[i]; |
---|
910 | if (tobracket!=0) |
---|
911 | { |
---|
912 | toNF = bracket(tobracket,NI[2]) + NI[i]*NI[2]; |
---|
913 | } |
---|
914 | else |
---|
915 | { |
---|
916 | toNF = NI[i]*NI[2]; |
---|
917 | } |
---|
918 | newNF = NF(toNF,I); |
---|
919 | NI[i+1] = newNF; |
---|
920 | } |
---|
921 | // look for a solution |
---|
922 | dbprint(ppl,"linSyzSolve starts with: "+string(matrix(NI))); |
---|
923 | if (solveincharp<>0) // modular method |
---|
924 | { |
---|
925 | setring Rp; |
---|
926 | NI[i+1] = phi(newNF); |
---|
927 | v = linSyzSolve(NI,modengine); |
---|
928 | if (v!=0) // there is a modular solution |
---|
929 | { |
---|
930 | dbprint(ppl,"got solution in char ",solveincharp," of degree " ,i); |
---|
931 | setring save; |
---|
932 | v = linSyzSolve(NI,whichengine); |
---|
933 | if (v==0) |
---|
934 | { |
---|
935 | break; |
---|
936 | } |
---|
937 | } |
---|
938 | else // no modular solution |
---|
939 | { |
---|
940 | setring save; |
---|
941 | v = 0; |
---|
942 | } |
---|
943 | } |
---|
944 | else // non-modular method |
---|
945 | { |
---|
946 | v = linSyzSolve(NI,whichengine); |
---|
947 | } |
---|
948 | matrix MM[1][nrows(v)] = matrix(v); |
---|
949 | dbprint(ppl,"linSyzSolve ready with: "+string(MM)); |
---|
950 | kill MM; |
---|
951 | // "linSyzSolve ready with"; print(v); |
---|
952 | if (v!=0) |
---|
953 | { |
---|
954 | // a solution: |
---|
955 | //check for the reality of the solution |
---|
956 | p = 0; |
---|
957 | for (j=1; j<=i+1; j++) |
---|
958 | { |
---|
959 | p = p + v[j]*NI[j]; |
---|
960 | } |
---|
961 | if (p!=0) |
---|
962 | { |
---|
963 | dbprint(ppl,"linSyzSolve: bad solution!"); |
---|
964 | } |
---|
965 | else |
---|
966 | { |
---|
967 | dbprint(ppl,"linSyzSolve: got solution!"); |
---|
968 | // "got solution!"; |
---|
969 | break; |
---|
970 | } |
---|
971 | } |
---|
972 | // no solution: |
---|
973 | i++; |
---|
974 | } |
---|
975 | dbprint(ppl+1,"pIntersectSyz finished"); |
---|
976 | return(v); |
---|
977 | } |
---|
978 | example |
---|
979 | { |
---|
980 | "EXAMPLE:"; echo = 2; |
---|
981 | ring r = 0,(x,y),dp; |
---|
982 | poly f = x^2+y^3+x*y^2; |
---|
983 | def D = initialMalgrange(f); |
---|
984 | setring D; |
---|
985 | inF; |
---|
986 | poly s = t*Dt; |
---|
987 | pIntersectSyz(s,inF); |
---|
988 | int p = prime(20000); |
---|
989 | pIntersectSyz(s,inF,p,0,0); |
---|
990 | } |
---|
991 | |
---|
992 | proc vec2poly (list #) |
---|
993 | "USAGE: vec2poly(v [,i]); v a vector or an intvec, i an optional int |
---|
994 | RETURN: poly, an univariate poly in i-th variable with coefficients given by v |
---|
995 | PURPOSE: constructs an univariate poly in K[var(i)] with given coefficients, |
---|
996 | @* such that the coefficient at var(i)^{j-1} is v[j]. |
---|
997 | NOTE: The optional argument i must be positive, by default i is 1. |
---|
998 | EXAMPLE: example vec2poly; shows examples |
---|
999 | " |
---|
1000 | { |
---|
1001 | def save = basering; |
---|
1002 | int i,ringvar; |
---|
1003 | ringvar = 1; // default |
---|
1004 | if (size(#) > 0) |
---|
1005 | { |
---|
1006 | if (typeof(#[1])=="vector" || typeof(#[1])=="intvec") |
---|
1007 | { |
---|
1008 | def v = #[1]; |
---|
1009 | } |
---|
1010 | else |
---|
1011 | { |
---|
1012 | ERROR("wrong input: expected vector/intvec expression"); |
---|
1013 | } |
---|
1014 | if (size(#) > 1) |
---|
1015 | { |
---|
1016 | if (typeof(#[2])=="int" || typeof(#[2])=="number") |
---|
1017 | { |
---|
1018 | ringvar = int(#[2]); |
---|
1019 | } |
---|
1020 | } |
---|
1021 | } |
---|
1022 | if (ringvar > nvars(save)) |
---|
1023 | { |
---|
1024 | ERROR("var out of range"); |
---|
1025 | } |
---|
1026 | poly p; |
---|
1027 | for (i=1; i<=nrows(v); i++) |
---|
1028 | { |
---|
1029 | p = p + v[i]*(var(ringvar))^(i-1); |
---|
1030 | } |
---|
1031 | return(p); |
---|
1032 | } |
---|
1033 | example |
---|
1034 | { |
---|
1035 | "EXAMPLE:"; echo = 2; |
---|
1036 | ring r = 0,(x,y),dp; |
---|
1037 | vector v = gen(1) + 3*gen(3) + 22/9*gen(4); |
---|
1038 | intvec iv = 3,2,1; |
---|
1039 | vec2poly(v,2); |
---|
1040 | vec2poly(iv); |
---|
1041 | } |
---|
1042 | |
---|
1043 | static proc listofroots (list #) |
---|
1044 | { |
---|
1045 | def save = basering; |
---|
1046 | int n = nvars(save); |
---|
1047 | int i; |
---|
1048 | poly p; |
---|
1049 | if (typeof(#[1])=="vector") |
---|
1050 | { |
---|
1051 | vector b = #[1]; |
---|
1052 | for (i=1; i<=nrows(b); i++) |
---|
1053 | { |
---|
1054 | p = p + b[i]*(var(1))^(i-1); |
---|
1055 | } |
---|
1056 | } |
---|
1057 | else |
---|
1058 | { |
---|
1059 | p = #[1]; |
---|
1060 | } |
---|
1061 | int substitution = int(#[2]); |
---|
1062 | ring S = 0,s,dp; |
---|
1063 | ideal J; |
---|
1064 | for (i=1; i<=n; i++) |
---|
1065 | { |
---|
1066 | J[i] = s; |
---|
1067 | } |
---|
1068 | map @m = save,J; |
---|
1069 | poly p = @m(p); |
---|
1070 | if (substitution == 1) |
---|
1071 | { |
---|
1072 | p = subst(p,s,-s-1); |
---|
1073 | } |
---|
1074 | // the rest of this proc is nicked from bernsteinBM from dmod.lib |
---|
1075 | list P = factorize(p);//with constants and multiplicities |
---|
1076 | ideal bs; intvec m; //the Bernstein polynomial is monic, so we are not interested in constants |
---|
1077 | for (i=2; i<= size(P[1]); i++) //we delete P[1][1] and P[2][1] |
---|
1078 | { |
---|
1079 | bs[i-1] = P[1][i]; |
---|
1080 | m[i-1] = P[2][i]; |
---|
1081 | } |
---|
1082 | bs = normalize(bs); |
---|
1083 | bs = -subst(bs,s,0); |
---|
1084 | setring save; |
---|
1085 | ideal bs = imap(S,bs); |
---|
1086 | kill S; |
---|
1087 | list BS = bs,m; |
---|
1088 | return(BS); |
---|
1089 | } |
---|
1090 | |
---|
1091 | static proc bfctengine (poly f, int inorann, int whichengine, int methodord, int methodpIntersect, int pIntersectchar, int modengine, intvec u0) |
---|
1092 | { |
---|
1093 | int ppl = printlevel - voice +2; |
---|
1094 | int i; |
---|
1095 | def save = basering; |
---|
1096 | int n = nvars(save); |
---|
1097 | if (size(variables(f)) == 0) // f is constant |
---|
1098 | { |
---|
1099 | return(list(ideal(0),intvec(0))); |
---|
1100 | } |
---|
1101 | if (inorann == 0) // bfct using initial ideal |
---|
1102 | { |
---|
1103 | def D = initialMalgrange(f,whichengine,methodord,1,u0); |
---|
1104 | setring D; |
---|
1105 | ideal J = inF; |
---|
1106 | kill inF; |
---|
1107 | poly s = t*Dt; |
---|
1108 | } |
---|
1109 | else // bfct using Ann(f^s) |
---|
1110 | { |
---|
1111 | def D = SannfsBFCT(f,whichengine); |
---|
1112 | setring D; |
---|
1113 | ideal J = LD; |
---|
1114 | kill LD; |
---|
1115 | } |
---|
1116 | vector b; |
---|
1117 | // try it modular |
---|
1118 | if (methodpIntersect <> 0) // pIntersectSyz |
---|
1119 | { |
---|
1120 | if (pIntersectchar == 0) // pIntersectSyz::modular |
---|
1121 | { |
---|
1122 | int lb = 30000; |
---|
1123 | int ub = 536870909; |
---|
1124 | i = 1; |
---|
1125 | list usedprimes; |
---|
1126 | while (b == 0) |
---|
1127 | { |
---|
1128 | dbprint(ppl,"number of run in the loop: "+string(i)); |
---|
1129 | int q = prime(random(lb,ub)); |
---|
1130 | if (findFirst(usedprimes,q)==0) // if q was not already used |
---|
1131 | { |
---|
1132 | usedprimes = usedprimes,q; |
---|
1133 | dbprint(ppl,"used prime is: "+string(q)); |
---|
1134 | b = pIntersectSyz(s,J,q,whichengine,modengine); |
---|
1135 | } |
---|
1136 | i++; |
---|
1137 | } |
---|
1138 | } |
---|
1139 | else // pIntersectSyz::non-modular |
---|
1140 | { |
---|
1141 | b = pIntersectSyz(s,J,0,whichengine); |
---|
1142 | } |
---|
1143 | } |
---|
1144 | else // pIntersect: linReduce |
---|
1145 | { |
---|
1146 | b = pIntersect(s,J); |
---|
1147 | } |
---|
1148 | setring save; |
---|
1149 | vector b = imap(D,b); |
---|
1150 | if (inorann == 0) |
---|
1151 | { |
---|
1152 | list l = listofroots(b,1); |
---|
1153 | } |
---|
1154 | else |
---|
1155 | { |
---|
1156 | list l = listofroots(b,0); |
---|
1157 | } |
---|
1158 | return(l); |
---|
1159 | } |
---|
1160 | |
---|
1161 | proc bfct (poly f, list #) |
---|
1162 | "USAGE: bfct(f [,s,t,v]); f a poly, s,t optional ints, v an optional intvec |
---|
1163 | RETURN: list of ideal and intvec |
---|
1164 | PURPOSE: computes the roots of the Bernstein-Sato polynomial b(s) |
---|
1165 | @* for the hypersurface defined by f. |
---|
1166 | ASSUME: The basering is a commutative polynomial ring in char 0. |
---|
1167 | BACKGROUND: In this proc, the initial Malgrange ideal is computed according to the algorithm |
---|
1168 | @* by Masayuki Noro and then a system of linear equations is solved by linear reductions. |
---|
1169 | NOTE: In the output list, the ideal contains all the roots |
---|
1170 | @* and the intvec their multiplicities. |
---|
1171 | @* If s<>0, @code{std} is used for GB computations, |
---|
1172 | @* otherwise, and by default, @code{slimgb} is used. |
---|
1173 | @* If t<>0, a matrix ordering is used for Groebner basis computations, |
---|
1174 | @* otherwise, and by default, a block ordering is used. |
---|
1175 | @* If v is a positive weight vector, v is used for homogenization computations, |
---|
1176 | @* otherwise and by default, no weights are used. |
---|
1177 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
1178 | @* if printlevel>=2, all the debug messages will be printed. |
---|
1179 | EXAMPLE: example bfct; shows examples |
---|
1180 | " |
---|
1181 | { |
---|
1182 | int ppl = printlevel - voice +2; |
---|
1183 | int i; |
---|
1184 | int n = nvars(basering); |
---|
1185 | // in # we have two switches: |
---|
1186 | // one for the engine used for Groebner basis computations, |
---|
1187 | // one for M() ordering or its realization |
---|
1188 | // in # can also be the optional weight vector |
---|
1189 | int whichengine = 0; // default |
---|
1190 | int methodord = 0; // default |
---|
1191 | intvec u0 = 0; // default |
---|
1192 | if (size(#)>0) |
---|
1193 | { |
---|
1194 | if (typeof(#[1])=="int" || typeof(#[1])=="number") |
---|
1195 | { |
---|
1196 | whichengine = int(#[1]); |
---|
1197 | } |
---|
1198 | if (size(#)>1) |
---|
1199 | { |
---|
1200 | if (typeof(#[2])=="int" || typeof(#[2])=="number") |
---|
1201 | { |
---|
1202 | methodord = int(#[2]); |
---|
1203 | } |
---|
1204 | if (size(#)>2) |
---|
1205 | { |
---|
1206 | if (typeof(#[3])=="intvec" && size(#[3])==n && allPositive(#[3])==1) |
---|
1207 | { |
---|
1208 | u0 = #[3]; |
---|
1209 | } |
---|
1210 | } |
---|
1211 | } |
---|
1212 | } |
---|
1213 | list b = bfctengine(f,0,whichengine,methodord,0,0,0,u0); |
---|
1214 | return(b); |
---|
1215 | } |
---|
1216 | example |
---|
1217 | { |
---|
1218 | "EXAMPLE:"; echo = 2; |
---|
1219 | ring r = 0,(x,y),dp; |
---|
1220 | poly f = x^2+y^3+x*y^2; |
---|
1221 | bfct(f); |
---|
1222 | intvec v = 3,2; |
---|
1223 | bfct(f,1,0,v); |
---|
1224 | } |
---|
1225 | |
---|
1226 | proc bfctSyz (poly f, list #) |
---|
1227 | "USAGE: bfctSyz(f [,r,s,t,u,v]); f a poly, r,s,t,u optional ints, v an optional intvec |
---|
1228 | RETURN: list of ideal and intvec |
---|
1229 | PURPOSE: computes the roots of the Bernstein-Sato polynomial b(s) |
---|
1230 | @* for the hypersurface defined by f |
---|
1231 | ASSUME: The basering is a commutative polynomial ring in char 0. |
---|
1232 | BACKGROUND: In this proc, the initial Malgrange ideal is computed according to the algorithm |
---|
1233 | @* by Masayuki Noro and then a system of linear equations is solved by computing syzygies. |
---|
1234 | NOTE: In the output list, the ideal contains all the roots |
---|
1235 | @* and the intvec their multiplicities. |
---|
1236 | @* If r<>0, @code{std} is used for GB computations in characteristic 0, |
---|
1237 | @* otherwise, and by default, @code{slimgb} is used. |
---|
1238 | @* If s<>0, a matrix ordering is used for GB computations, otherwise, |
---|
1239 | @* and by default, a block ordering is used. |
---|
1240 | @* If t<>0, the computation of the intersection is solely performed over |
---|
1241 | @* charasteristic 0, otherwise and by default, a modular method is used. |
---|
1242 | @* If u<>0 and by default, @code{std} is used for GB computations in |
---|
1243 | @* characteristic >0, otherwise, @code{slimgb} is used. |
---|
1244 | @* If v is a positive weight vector, v is used for homogenization |
---|
1245 | @* computations, otherwise and by default, no weights are used. |
---|
1246 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
1247 | @* if printlevel>=2, all the debug messages will be printed. |
---|
1248 | EXAMPLE: example bfctSyz; shows examples |
---|
1249 | " |
---|
1250 | { |
---|
1251 | int ppl = printlevel - voice +2; |
---|
1252 | int i; |
---|
1253 | // in # we have four switches: |
---|
1254 | // one for the engine used for Groebner basis computations in char 0, |
---|
1255 | // one for M() ordering or its realization |
---|
1256 | // one for a modular method when computing the intersection |
---|
1257 | // and one for the engine used for Groebner basis computations in char >0 |
---|
1258 | // in # can also be the optional weight vector |
---|
1259 | def save = basering; |
---|
1260 | int n = nvars(save); |
---|
1261 | int whichengine = 0; // default |
---|
1262 | int methodord = 0; // default |
---|
1263 | int pIntersectchar = 0; // default |
---|
1264 | int modengine = 1; // default |
---|
1265 | intvec u0 = 0; // default |
---|
1266 | if (size(#)>0) |
---|
1267 | { |
---|
1268 | if (typeof(#[1])=="int" || typeof(#[1])=="number") |
---|
1269 | { |
---|
1270 | whichengine = int(#[1]); |
---|
1271 | } |
---|
1272 | if (size(#)>1) |
---|
1273 | { |
---|
1274 | if (typeof(#[2])=="int" || typeof(#[2])=="number") |
---|
1275 | { |
---|
1276 | methodord = int(#[2]); |
---|
1277 | } |
---|
1278 | if (size(#)>2) |
---|
1279 | { |
---|
1280 | if (typeof(#[3])=="int" || typeof(#[3])=="number") |
---|
1281 | { |
---|
1282 | pIntersectchar = int(#[3]); |
---|
1283 | } |
---|
1284 | if (size(#)>3) |
---|
1285 | { |
---|
1286 | if (typeof(#[4])=="int" || typeof(#[4])=="number") |
---|
1287 | { |
---|
1288 | modengine = int(#[4]); |
---|
1289 | } |
---|
1290 | if (size(#)>4) |
---|
1291 | { |
---|
1292 | if (typeof(#[5])=="intvec" && size(#[5])==n && allPositive(#[5])==1) |
---|
1293 | { |
---|
1294 | u0 = #[5]; |
---|
1295 | } |
---|
1296 | } |
---|
1297 | } |
---|
1298 | } |
---|
1299 | } |
---|
1300 | } |
---|
1301 | list b = bfctengine(f,0,whichengine,methodord,1,pIntersectchar,modengine,u0); |
---|
1302 | return(b); |
---|
1303 | } |
---|
1304 | example |
---|
1305 | { |
---|
1306 | "EXAMPLE:"; echo = 2; |
---|
1307 | ring r = 0,(x,y),dp; |
---|
1308 | poly f = x^2+y^3+x*y^2; |
---|
1309 | bfctSyz(f); |
---|
1310 | intvec v = 3,2; |
---|
1311 | bfctSyz(f,0,1,1,0,v); |
---|
1312 | } |
---|
1313 | |
---|
1314 | proc bfctIdeal (ideal I, intvec w, list #) |
---|
1315 | "USAGE: bfctIdeal(I,w[,s,t]); I an ideal, w an intvec, s,t optional ints |
---|
1316 | RETURN: list of ideal and intvec |
---|
1317 | PURPOSE: computes the roots of the global b-function of I wrt the weight vector (-w,w). |
---|
1318 | ASSUME: The basering is the n-th Weyl algebra in char 0, where the sequence of |
---|
1319 | @* the variables is x(1),...,x(n),D(1),...,D(n). |
---|
1320 | BACKGROUND: In this proc, the initial ideal of I is computed according to the algorithm by |
---|
1321 | @* Masayuki Noro and then a system of linear equations is solved by linear reductions. |
---|
1322 | NOTE: In the output list, the ideal contains all the roots |
---|
1323 | @* and the intvec their multiplicities. |
---|
1324 | @* If s<>0, @code{std} is used for GB computations in characteristic 0, |
---|
1325 | @* otherwise, and by default, @code{slimgb} is used. |
---|
1326 | @* If t<>0, a matrix ordering is used for GB computations, otherwise, |
---|
1327 | @* and by default, a block ordering is used. |
---|
1328 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
1329 | @* if printlevel>=2, all the debug messages will be printed. |
---|
1330 | EXAMPLE: example bfctIdeal; shows examples |
---|
1331 | " |
---|
1332 | { |
---|
1333 | int ppl = printlevel - voice +2; |
---|
1334 | int i; |
---|
1335 | def save = basering; |
---|
1336 | int n = nvars(save)/2; |
---|
1337 | int whichengine = 0; // default |
---|
1338 | int methodord = 0; // default |
---|
1339 | if (size(#)>0) |
---|
1340 | { |
---|
1341 | if (typeof(#[1])=="int" || typeof(#[1])=="number") |
---|
1342 | { |
---|
1343 | whichengine = int(#[1]); |
---|
1344 | } |
---|
1345 | if (size(#)>1) |
---|
1346 | { |
---|
1347 | if (typeof(#[2])=="int" || typeof(#[2])=="number") |
---|
1348 | { |
---|
1349 | methodord = int(#[2]); |
---|
1350 | } |
---|
1351 | } |
---|
1352 | } |
---|
1353 | ideal J = initialIdealW(I,-w,w,whichengine,methodord); |
---|
1354 | poly s; |
---|
1355 | for (i=1; i<=n; i++) |
---|
1356 | { |
---|
1357 | s = s + w[i]*var(i)*var(n+i); |
---|
1358 | } |
---|
1359 | vector b = pIntersect(s,J); |
---|
1360 | list l = listofroots(b,0); |
---|
1361 | return(l); |
---|
1362 | } |
---|
1363 | example |
---|
1364 | { |
---|
1365 | "EXAMPLE:"; echo = 2; |
---|
1366 | ring @D = 0,(x,y,Dx,Dy),dp; |
---|
1367 | def D = Weyl(); |
---|
1368 | setring D; |
---|
1369 | ideal I = std(3*x^2*Dy+2*y*Dx,2*x*Dx+3*y*Dy+6); I; |
---|
1370 | intvec w1 = 1,1; |
---|
1371 | intvec w2 = 1,2; |
---|
1372 | intvec w3 = 2,3; |
---|
1373 | bfctIdeal(I,w1); |
---|
1374 | bfctIdeal(I,w2,1); |
---|
1375 | bfctIdeal(I,w3,0,1); |
---|
1376 | } |
---|
1377 | |
---|
1378 | proc bfctOneGB (poly f,list #) |
---|
1379 | "USAGE: bfctOneGB(f [,s,t]); f a poly, s,t optional ints |
---|
1380 | RETURN: list of ideal and intvec |
---|
1381 | PURPOSE: computes the roots of the Bernstein-Sato polynomial b(s) for the |
---|
1382 | @* hypersurface defined by f, using only one GB computation |
---|
1383 | ASSUME: The basering is a commutative polynomial ring in char 0. |
---|
1384 | BACKGROUND: In this proc, the initial Malgrange ideal is computed based on the |
---|
1385 | @* algorithm by Masayuki Noro and combined with an elimination ordering. |
---|
1386 | NOTE: In the output list, the ideal contains all the roots |
---|
1387 | @* and the intvec their multiplicities. |
---|
1388 | @* If s<>0, @code{std} is used for the GB computation, otherwise, |
---|
1389 | @* and by default, @code{slimgb} is used. |
---|
1390 | @* If t<>0, a matrix ordering is used for GB computations, |
---|
1391 | @* otherwise, and by default, a block ordering is used. |
---|
1392 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
1393 | @* if printlevel>=2, all the debug messages will be printed. |
---|
1394 | EXAMPLE: example bfctOneGB; shows examples |
---|
1395 | " |
---|
1396 | { |
---|
1397 | int ppl = printlevel - voice +2; |
---|
1398 | def save = basering; |
---|
1399 | int n = nvars(save); |
---|
1400 | int noofvars = 2*n+4; |
---|
1401 | int i; |
---|
1402 | int whichengine = 0; // default |
---|
1403 | int methodord = 0; // default |
---|
1404 | if (size(#)>0) |
---|
1405 | { |
---|
1406 | if (typeof(#[1])=="int" || typeof(#[1])=="number") |
---|
1407 | { |
---|
1408 | whichengine = int(#[1]); |
---|
1409 | } |
---|
1410 | if (size(#)>1) |
---|
1411 | { |
---|
1412 | if (typeof(#[2])=="int" || typeof(#[2])=="number") |
---|
1413 | { |
---|
1414 | methodord = int(#[2]); |
---|
1415 | } |
---|
1416 | } |
---|
1417 | } |
---|
1418 | intvec uv; |
---|
1419 | uv[n+3] = 1; |
---|
1420 | ring r = 0,(x(1..n)),dp; |
---|
1421 | poly f = fetch(save,f); |
---|
1422 | uv[1] = -1; uv[noofvars] = 0; |
---|
1423 | // for the ordering |
---|
1424 | intvec @a; @a = 1:noofvars; @a[2] = 2; |
---|
1425 | intvec @a2 = @a; @a2[2] = 0; @a2[2*n+4] = 0; |
---|
1426 | if (methodord == 0) // default: block ordering |
---|
1427 | { |
---|
1428 | ring Dh = 0,(t,s,x(n..1),Dt,D(n..1),h),(a(@a),a(@a2),a(uv),dp(noofvars-1),lp(1)); |
---|
1429 | } |
---|
1430 | else // M() ordering |
---|
1431 | { |
---|
1432 | intmat @Ord[noofvars][noofvars]; |
---|
1433 | @Ord[1,1..noofvars] = uv; |
---|
1434 | @Ord[2,1..noofvars] = 1:(noofvars-1); |
---|
1435 | for (i=1; i<=noofvars-2; i++) |
---|
1436 | { |
---|
1437 | @Ord[2+i,noofvars - i] = -1; |
---|
1438 | } |
---|
1439 | dbprint(ppl,"weights for ordering:",transpose(@a)); |
---|
1440 | dbprint(ppl,"the ordering matrix:",@Ord); |
---|
1441 | ring Dh = 0,(t,s,x(n..1),Dt,D(n..1),h),(a(@a),a(@a2),M(@Ord)); |
---|
1442 | } |
---|
1443 | dbprint(ppl,"the ring Dh:",Dh); |
---|
1444 | // non-commutative relations |
---|
1445 | matrix @relD[noofvars][noofvars]; |
---|
1446 | @relD[1,2] = t*h^2;// s*t = t*s+t*h^2 |
---|
1447 | @relD[2,n+3] = Dt*h^2;// Dt*s = s*Dt+h^2 |
---|
1448 | @relD[1,n+3] = h^2; |
---|
1449 | for (i=1; i<=n; i++) |
---|
1450 | { |
---|
1451 | @relD[i+2,n+3+i] = h^2; |
---|
1452 | } |
---|
1453 | dbprint(ppl,"nc relations:",@relD); |
---|
1454 | def DDh = nc_algebra(1,@relD); |
---|
1455 | setring DDh; |
---|
1456 | dbprint(ppl,"computing in ring",DDh); |
---|
1457 | ideal I; |
---|
1458 | poly f = imap(r,f); |
---|
1459 | kill r; |
---|
1460 | f = homog(f,h); |
---|
1461 | I = t - f, t*Dt - s; // defining the Malgrange ideal |
---|
1462 | for (i=1; i<=n; i++) |
---|
1463 | { |
---|
1464 | I = I, D(i)+diff(f,x(i))*Dt; |
---|
1465 | } |
---|
1466 | dbprint(ppl, "starting Groebner basis computation with engine:", whichengine); |
---|
1467 | I = engine(I, whichengine); |
---|
1468 | dbprint(ppl, "finished Groebner basis computation"); |
---|
1469 | dbprint(ppl, "I before dehomogenization is" ,I); |
---|
1470 | I = subst(I,h,1); // dehomogenization |
---|
1471 | dbprint(ppl, "I after dehomogenization is" ,I); |
---|
1472 | I = inForm(I,uv); // we are only interested in the initial form w.r.t. uv |
---|
1473 | dbprint(ppl, "the initial ideal:", string(matrix(I))); |
---|
1474 | intvec tonselect = 1; |
---|
1475 | for (i=3; i<=2*n+4; i++) { tonselect = tonselect,i; } |
---|
1476 | I = nselect(I,tonselect); |
---|
1477 | dbprint(ppl, "generators containing only s:", string(matrix(I))); |
---|
1478 | I = engine(I, whichengine); // is now a principal ideal; |
---|
1479 | setring save; |
---|
1480 | ideal J; J[2] = var(1); |
---|
1481 | map @m = DDh,J; |
---|
1482 | ideal I = @m(I); |
---|
1483 | poly p = I[1]; |
---|
1484 | list l = listofroots(p,1); |
---|
1485 | return(l); |
---|
1486 | } |
---|
1487 | example |
---|
1488 | { |
---|
1489 | "EXAMPLE:"; echo = 2; |
---|
1490 | ring r = 0,(x,y),dp; |
---|
1491 | poly f = x^2+y^3+x*y^2; |
---|
1492 | bfctOneGB(f); |
---|
1493 | bfctOneGB(f,1,1); |
---|
1494 | } |
---|
1495 | |
---|
1496 | proc bfctAnn (poly f, list #) |
---|
1497 | "USAGE: bfctAnn(f [,r]); f a poly, r an optional int |
---|
1498 | RETURN: list of ideal and intvec |
---|
1499 | PURPOSE: computes the roots of the Bernstein-Sato polynomial b(s) |
---|
1500 | @* for the hypersurface defined by f |
---|
1501 | ASSUME: The basering is a commutative polynomial ring in char 0. |
---|
1502 | BACKGROUND: In this proc, ann(f^s) is computed and then a system of linear |
---|
1503 | @* equations is solved by linear reductions. |
---|
1504 | NOTE: In the output list, the ideal contains all the roots |
---|
1505 | @* and the intvec their multiplicities. |
---|
1506 | @* If r<>0, @code{std} is used for GB computations, |
---|
1507 | @* otherwise, and by default, @code{slimgb} is used. |
---|
1508 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
1509 | @* if printlevel>=2, all the debug messages will be printed. |
---|
1510 | EXAMPLE: example bfctAnn; shows examples |
---|
1511 | " |
---|
1512 | { |
---|
1513 | def save = basering; |
---|
1514 | int ppl = printlevel - voice + 2; |
---|
1515 | int whichengine = 0; // default |
---|
1516 | if (size(#)>0) |
---|
1517 | { |
---|
1518 | if (typeof(#[1])=="int" || typeof(#[1])=="number") |
---|
1519 | { |
---|
1520 | whichengine = int(#[1]); |
---|
1521 | } |
---|
1522 | } |
---|
1523 | list b = bfctengine(f,1,whichengine,0,1,0,0,0); |
---|
1524 | return(b); |
---|
1525 | } |
---|
1526 | example |
---|
1527 | { |
---|
1528 | "EXAMPLE:"; echo = 2; |
---|
1529 | ring r = 0,(x,y),dp; |
---|
1530 | poly f = x^2+y^3+x*y^2; |
---|
1531 | bfctAnn(f); |
---|
1532 | } |
---|
1533 | |
---|
1534 | /* |
---|
1535 | //static proc hardexamples () |
---|
1536 | { |
---|
1537 | // some hard examples |
---|
1538 | ring r1 = 0,(x,y,z,w),dp; |
---|
1539 | // ab34 |
---|
1540 | poly ab34 = (z3+w4)*(3z2x+4w3y); |
---|
1541 | bfct(ab34); |
---|
1542 | // ha3 |
---|
1543 | poly ha3 = xyzw*(x+y)*(x+z)*(x+w)*(y+z+w); |
---|
1544 | bfct(ha3); |
---|
1545 | // ha4 |
---|
1546 | poly ha4 = xyzw*(x+y)*(x+z)*(x+w)*(y+z)*(y+w); |
---|
1547 | bfct(ha4); |
---|
1548 | // chal4: reiffen(4,5)*reiffen(5,4) |
---|
1549 | ring r2 = 0,(x,y),dp; |
---|
1550 | poly chal4 = (x4+xy4+y5)*(x5+x4y+y4); |
---|
1551 | bfct(chal4); |
---|
1552 | // (xy+z)*reiffen(4,5) |
---|
1553 | ring r3 = 0,(x,y,z),dp; |
---|
1554 | poly xyzreiffen45 = (xy+z)*(y4+yz4+z5); |
---|
1555 | bfct(xyzreiffen45); |
---|
1556 | } |
---|
1557 | */ |
---|