1 | ////////////////////////////////////////////////////////////////////////////// |
---|
2 | version="$Id: dmod.lib,v 1.12 2007-01-23 15:03:30 Singular Exp $"; |
---|
3 | category="Noncommutative"; |
---|
4 | info=" |
---|
5 | LIBRARY: dmod.lib Algorithms for algebraic D-modules |
---|
6 | AUTHORS: Viktor Levandovskyy, levandov@risc.uni-linz.ac.at |
---|
7 | @* Jorge Martin Morales, jorge@unizar.es |
---|
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 ring R[1/F^s] for a natural number s. |
---|
11 | @* In fact, the ring R[1/F^s] has a structure of a D(R)-module, where D(R) |
---|
12 | @* is an n-th Weyl algebra K<x_1,...,x_n,d_1,...,d_n | d_j x_j = x_j d_j +1>. |
---|
13 | @* Constructively, one needs to find a left ideal I = I(F^s) in D(R), such |
---|
14 | @* that K[x_1,...,x_n,1/F^s] is isomorphic to D(R)/I as a D(R)-module. |
---|
15 | @* We often write just D for D(R) and D[s] for D tensor over K with K[s] |
---|
16 | @* One is interested in the following data: |
---|
17 | @* - Ann F^s = I = I(F^s) in D(R)[s], denoted by LD in the output |
---|
18 | @* - global Bernstein polynomial in K[s], denoted by bs, its minimal integer root s0 and |
---|
19 | @* the list of all roots of bs, which are rational, with their multiplicities is denoted by BS |
---|
20 | @* - Ann F^s0 = I(F^s0) in D(R), denoted by LD0 in the output (LD0 is a holonomic ideal in D(R)) |
---|
21 | @* - an operator in D(R)[s], denoted PS such that PS*F^(s+1) = bs*F^s holds in K[x_1,...,x_n,1/F^s]. |
---|
22 | |
---|
23 | @* We provide the following implementations: |
---|
24 | @* OT) the classical Ann F^s algorithm from Oaku and Takayama (J. Pure |
---|
25 | Applied Math., 1999), |
---|
26 | @* LOT) Levandovskyy's modification of the Oaku-Takayama algorithm (unpublished) |
---|
27 | @* BM) the Ann F^s algorithm by Briancon and Maisonobe (Remarques sur |
---|
28 | l'ideal de Bernstein associe a des polynomes, preprint, 2002) |
---|
29 | |
---|
30 | GUIDE: |
---|
31 | @* - Ann F^s = I = I(F^s) = LD in D(R)[s] can be computed by SannfsBM, SannfsOT, SannfsLOT |
---|
32 | @* - global Bernstein polynomial bs resp. BS in K[s] can be computed by bernsteinBM |
---|
33 | @* - Ann F^s0 = I(F^s0) = LD0 in D(R) can be computed by annfs0, annfsBM, annfsOT, annfsLOT |
---|
34 | @* - all the relevant data (LD, LD0, bs, PS) are computed by operatorBM |
---|
35 | |
---|
36 | PROCEDURES: |
---|
37 | |
---|
38 | annfs0(I,F [,eng]); compute Ann F^s0 in D and Bernstein poly from the known Ann F^s in D[s] |
---|
39 | annfsBM(F[,eng]); compute Ann F^s0 in D and Bernstein poly for a poly F (algorithm of Briancon-Maisonobe) |
---|
40 | annfsLOT(F[,eng]); compute Ann F^s0 in D and Bernstein poly for a poly F (Levandovskyy modification of the Oaku-Takayama algorithm) |
---|
41 | annfsOT(F[,eng]); compute Ann F^s0 in D and Bernstein poly for a poly F (algorithm of Oaku-Takayama) |
---|
42 | |
---|
43 | SannfsBM(F[,eng]); compute Ann F^s in D[s] for a poly F (algorithm of Briancon-Maisonobe) |
---|
44 | SannfsLOT(F[,eng]); compute Ann F^s in D[s] for a poly F (Levandovskyy modification of the Oaku-Takayama algorithm) |
---|
45 | SannfsOT(F[,eng]); compute Ann F^s in D[s] for a poly F (algorithm of Oaku-Takayama) |
---|
46 | |
---|
47 | bernsteinBM(F[,eng]); compute global Bernstein poly for a poly F (algorithm of Briancon-Maisonobe) |
---|
48 | operatorBM(F[,eng]); compute Ann F^s, Ann F^s0, BS and PS for a poly F (algorithm of Briancon-Maisonobe) |
---|
49 | annfsParamBM(F[,eng]); compute the generic Ann F^s and exceptional parametric constellations for a poly F with parametric coefficients (algorithm by Briancon and Maisonobe) |
---|
50 | |
---|
51 | annfsBMI(F[,eng]); compute Ann F^s and Bernstein ideal for a poly F=f1*..*fP (multivariate algorithm of Briancon-Maisonobe) |
---|
52 | |
---|
53 | |
---|
54 | AUXILIARY PROCEDURES: |
---|
55 | |
---|
56 | arrange(p); create a poly, describing a full hyperplane arrangement |
---|
57 | reiffen(p,q); create a poly, describing a Reiffen curve |
---|
58 | isHolonomic(M); check whether a module is holonomic |
---|
59 | convloc(L); replace global orderings with local in the ringlist L |
---|
60 | minIntRoot(P,fact); minimal integer root among the roots of a maximal ideal P |
---|
61 | "; |
---|
62 | |
---|
63 | LIB "nctools.lib"; |
---|
64 | LIB "elim.lib"; |
---|
65 | LIB "qhmoduli.lib"; // for Max |
---|
66 | LIB "gkdim.lib"; |
---|
67 | LIB "gmssing.lib"; |
---|
68 | LIB "control.lib"; // for genericity |
---|
69 | |
---|
70 | static proc engine(ideal I, int i) |
---|
71 | { |
---|
72 | /* std - slimgb mix */ |
---|
73 | ideal J; |
---|
74 | if (i==0) |
---|
75 | { |
---|
76 | J = slimgb(I); |
---|
77 | } |
---|
78 | else |
---|
79 | { |
---|
80 | // without options -> strange! (ringlist?) |
---|
81 | option(redSB); |
---|
82 | option(redTail); |
---|
83 | J = std(I); |
---|
84 | } |
---|
85 | return(J); |
---|
86 | } |
---|
87 | |
---|
88 | // alternative code for SannfsBM, rename from annfsBM to ALTannfsBM |
---|
89 | // is superfluos - will not be included in the official documentation |
---|
90 | proc ALTannfsBM (poly F, list #) |
---|
91 | "USAGE: annfsBM(f [,eng]); f a poly, eng an optional int |
---|
92 | RETURN: ring |
---|
93 | PURPOSE: compute the annihilator ideal of f^s in D[s], where D is the Weyl Algebra, according to the algorithm by Briancon and Maisonobe |
---|
94 | NOTE: activate this ring with the @code{setring} command. In this ring, |
---|
95 | @* - the ideal LD is the annihilator of f^s. |
---|
96 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
97 | @* otherwise, and by default @code{slimgb} is used. |
---|
98 | @* If printlevel=1, progress debug messages will be printed, |
---|
99 | @* if printlevel>=2, all the debug messages will be printed. |
---|
100 | EXAMPLE: example annfsBM; shows examples |
---|
101 | " |
---|
102 | { |
---|
103 | int eng = 0; |
---|
104 | if ( size(#)>0 ) |
---|
105 | { |
---|
106 | if ( typeof(#[1]) == "int" ) |
---|
107 | { |
---|
108 | eng = int(#[1]); |
---|
109 | } |
---|
110 | } |
---|
111 | // returns a list with a ring and an ideal LD in it |
---|
112 | int ppl = printlevel-voice+2; |
---|
113 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
114 | def save = basering; |
---|
115 | int N = nvars(basering); |
---|
116 | int Nnew = 2*N+2; |
---|
117 | int i,j; |
---|
118 | string s; |
---|
119 | list RL = ringlist(basering); |
---|
120 | list L, Lord; |
---|
121 | list tmp; |
---|
122 | intvec iv; |
---|
123 | L[1] = RL[1]; //char |
---|
124 | L[4] = RL[4]; //char, minpoly |
---|
125 | // check whether vars have admissible names |
---|
126 | list Name = RL[2]; |
---|
127 | list RName; |
---|
128 | RName[1] = "t"; |
---|
129 | RName[2] = "s"; |
---|
130 | for (i=1; i<=N; i++) |
---|
131 | { |
---|
132 | for(j=1; j<=size(RName); j++) |
---|
133 | { |
---|
134 | if (Name[i] == RName[j]) |
---|
135 | { |
---|
136 | ERROR("Variable names should not include t,s"); |
---|
137 | } |
---|
138 | } |
---|
139 | } |
---|
140 | // now, create the names for new vars |
---|
141 | list DName; |
---|
142 | for (i=1; i<=N; i++) |
---|
143 | { |
---|
144 | DName[i] = "D"+Name[i]; //concat |
---|
145 | } |
---|
146 | tmp[1] = "t"; |
---|
147 | tmp[2] = "s"; |
---|
148 | list NName = tmp + Name + DName; |
---|
149 | L[2] = NName; |
---|
150 | // Name, Dname will be used further |
---|
151 | kill NName; |
---|
152 | // block ord (lp(2),dp); |
---|
153 | tmp[1] = "lp"; // string |
---|
154 | iv = 1,1; |
---|
155 | tmp[2] = iv; //intvec |
---|
156 | Lord[1] = tmp; |
---|
157 | // continue with dp 1,1,1,1... |
---|
158 | tmp[1] = "dp"; // string |
---|
159 | s = "iv="; |
---|
160 | for (i=1; i<=Nnew; i++) |
---|
161 | { |
---|
162 | s = s+"1,"; |
---|
163 | } |
---|
164 | s[size(s)]= ";"; |
---|
165 | execute(s); |
---|
166 | kill s; |
---|
167 | tmp[2] = iv; |
---|
168 | Lord[2] = tmp; |
---|
169 | tmp[1] = "C"; |
---|
170 | iv = 0; |
---|
171 | tmp[2] = iv; |
---|
172 | Lord[3] = tmp; |
---|
173 | tmp = 0; |
---|
174 | L[3] = Lord; |
---|
175 | // we are done with the list |
---|
176 | def @R = ring(L); |
---|
177 | setring @R; |
---|
178 | matrix @D[Nnew][Nnew]; |
---|
179 | @D[1,2]=t; |
---|
180 | for(i=1; i<=N; i++) |
---|
181 | { |
---|
182 | @D[2+i,N+2+i]=1; |
---|
183 | } |
---|
184 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
185 | // L[6] = @D; |
---|
186 | ncalgebra(1,@D); |
---|
187 | dbprint(ppl,"// -1-1- the ring @R(t,s,_x,_Dx) is ready"); |
---|
188 | dbprint(ppl-1, @R); |
---|
189 | // create the ideal I |
---|
190 | poly F = imap(save,F); |
---|
191 | ideal I = t*F+s; |
---|
192 | poly p; |
---|
193 | for(i=1; i<=N; i++) |
---|
194 | { |
---|
195 | p = t; //t |
---|
196 | p = diff(F,var(2+i))*p; |
---|
197 | I = I, var(N+2+i) + p; |
---|
198 | } |
---|
199 | // -------- the ideal I is ready ---------- |
---|
200 | dbprint(ppl,"// -1-2- starting the elimination of t in @R"); |
---|
201 | dbprint(ppl-1, I); |
---|
202 | ideal J = engine(I,eng); |
---|
203 | ideal K = nselect(J,1); |
---|
204 | kill I,J; |
---|
205 | dbprint(ppl,"// -1-3- t is eliminated"); |
---|
206 | dbprint(ppl-1, K); //K is without t |
---|
207 | // create Dn[s], where Dn is the ordinary Weyl Algebra, and put the result into it, |
---|
208 | // thus creating the ring @R2 |
---|
209 | // keep: N, i,j,s, tmp, RL |
---|
210 | setring save; |
---|
211 | Nnew = 2*N+1; |
---|
212 | // list RL = ringlist(save); //is defined earlier |
---|
213 | kill Lord, tmp, iv; |
---|
214 | L = 0; |
---|
215 | list Lord, tmp; |
---|
216 | intvec iv; |
---|
217 | L[1] = RL[1]; |
---|
218 | L[4] = RL[4]; //char, minpoly |
---|
219 | // check whether vars have admissible names -> done earlier |
---|
220 | // list Name = RL[2] |
---|
221 | // DName is defined earlier |
---|
222 | tmp[1] = "s"; |
---|
223 | list NName = Name + DName + tmp; |
---|
224 | L[2] = NName; |
---|
225 | // dp ordering; |
---|
226 | string s = "iv="; |
---|
227 | for (i=1; i<=Nnew; i++) |
---|
228 | { |
---|
229 | s = s+"1,"; |
---|
230 | } |
---|
231 | s[size(s)] = ";"; |
---|
232 | execute(s); |
---|
233 | kill s; |
---|
234 | tmp = 0; |
---|
235 | tmp[1] = "dp"; //string |
---|
236 | tmp[2] = iv; //intvec |
---|
237 | Lord[1] = tmp; |
---|
238 | tmp[1] = "C"; |
---|
239 | iv = 0; |
---|
240 | tmp[2] = iv; |
---|
241 | Lord[2] = tmp; |
---|
242 | tmp = 0; |
---|
243 | L[3] = Lord; |
---|
244 | // we are done with the list |
---|
245 | // Add: Plural part |
---|
246 | def @R2 = ring(L); |
---|
247 | setring @R2; |
---|
248 | matrix @D[Nnew][Nnew]; |
---|
249 | for (i=1; i<=N; i++) |
---|
250 | { |
---|
251 | @D[i,N+i]=1; |
---|
252 | } |
---|
253 | ncalgebra(1,@D); |
---|
254 | dbprint(ppl,"// -2-1- the ring @R2(_x,_Dx,s) is ready"); |
---|
255 | dbprint(ppl-1, @R2); |
---|
256 | ideal K = imap(@R,K); |
---|
257 | option(redSB); |
---|
258 | //dbprint(ppl,"// -2-2- the final cosmetic std"); |
---|
259 | //K = engine(K,eng); //std does the job too |
---|
260 | // total cleanup |
---|
261 | kill @R; |
---|
262 | ideal LD = K; |
---|
263 | export LD; |
---|
264 | return(@R2); |
---|
265 | } |
---|
266 | example |
---|
267 | { |
---|
268 | "EXAMPLE:"; echo = 2; |
---|
269 | ring r = 0,(x,y,z,w),Dp; |
---|
270 | poly F = x^3+y^3+z^2*w; |
---|
271 | printlevel = 0; |
---|
272 | def A = ALTannfsBM(F); |
---|
273 | setring A; |
---|
274 | LD; |
---|
275 | } |
---|
276 | |
---|
277 | proc bernsteinBM(poly F, list #) |
---|
278 | "USAGE: bernsteinBM(f [,eng]); f a poly, eng an optional int |
---|
279 | RETURN: list of roots of the Bernstein polynomial b and its multiplicies |
---|
280 | PURPOSE: compute the global Bernstein-Sato polynomial for a hypersurface, defined by f, according to the algorithm by Briancon and Maisonobe |
---|
281 | NOTE: If eng <>0, @code{std} is used for Groebner basis computations, |
---|
282 | @* otherwise, and by default @code{slimgb} is used. |
---|
283 | @* If printlevel=1, progress debug messages will be printed, |
---|
284 | @* if printlevel>=2, all the debug messages will be printed. |
---|
285 | EXAMPLE: example bernsteinBM; shows examples |
---|
286 | " |
---|
287 | { |
---|
288 | int eng = 0; |
---|
289 | if ( size(#)>0 ) |
---|
290 | { |
---|
291 | if ( typeof(#[1]) == "int" ) |
---|
292 | { |
---|
293 | eng = int(#[1]); |
---|
294 | } |
---|
295 | } |
---|
296 | // returns a list with a ring and an ideal LD in it |
---|
297 | int ppl = printlevel-voice+2; |
---|
298 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
299 | def save = basering; |
---|
300 | int N = nvars(basering); |
---|
301 | int Nnew = 2*N+2; |
---|
302 | int i,j; |
---|
303 | string s; |
---|
304 | list RL = ringlist(basering); |
---|
305 | list L, Lord; |
---|
306 | list tmp; |
---|
307 | intvec iv; |
---|
308 | L[1] = RL[1]; //char |
---|
309 | L[4] = RL[4]; //char, minpoly |
---|
310 | // check whether vars have admissible names |
---|
311 | list Name = RL[2]; |
---|
312 | list RName; |
---|
313 | RName[1] = "t"; |
---|
314 | RName[2] = "s"; |
---|
315 | for (i=1; i<=N; i++) |
---|
316 | { |
---|
317 | for(j=1; j<=size(RName); j++) |
---|
318 | { |
---|
319 | if (Name[i] == RName[j]) |
---|
320 | { |
---|
321 | ERROR("Variable names should not include t,s"); |
---|
322 | } |
---|
323 | } |
---|
324 | } |
---|
325 | // now, create the names for new vars |
---|
326 | list DName; |
---|
327 | for (i=1; i<=N; i++) |
---|
328 | { |
---|
329 | DName[i] = "D"+Name[i]; //concat |
---|
330 | } |
---|
331 | tmp[1] = "t"; |
---|
332 | tmp[2] = "s"; |
---|
333 | list NName = tmp + Name + DName; |
---|
334 | L[2] = NName; |
---|
335 | // Name, Dname will be used further |
---|
336 | kill NName; |
---|
337 | // block ord (lp(2),dp); |
---|
338 | tmp[1] = "lp"; // string |
---|
339 | iv = 1,1; |
---|
340 | tmp[2] = iv; //intvec |
---|
341 | Lord[1] = tmp; |
---|
342 | // continue with dp 1,1,1,1... |
---|
343 | tmp[1] = "dp"; // string |
---|
344 | s = "iv="; |
---|
345 | for (i=1; i<=Nnew; i++) |
---|
346 | { |
---|
347 | s = s+"1,"; |
---|
348 | } |
---|
349 | s[size(s)]= ";"; |
---|
350 | execute(s); |
---|
351 | kill s; |
---|
352 | tmp[2] = iv; |
---|
353 | Lord[2] = tmp; |
---|
354 | tmp[1] = "C"; |
---|
355 | iv = 0; |
---|
356 | tmp[2] = iv; |
---|
357 | Lord[3] = tmp; |
---|
358 | tmp = 0; |
---|
359 | L[3] = Lord; |
---|
360 | // we are done with the list |
---|
361 | def @R = ring(L); |
---|
362 | setring @R; |
---|
363 | matrix @D[Nnew][Nnew]; |
---|
364 | @D[1,2]=t; |
---|
365 | for(i=1; i<=N; i++) |
---|
366 | { |
---|
367 | @D[2+i,N+2+i]=1; |
---|
368 | } |
---|
369 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
370 | // L[6] = @D; |
---|
371 | ncalgebra(1,@D); |
---|
372 | dbprint(ppl,"// -1-1- the ring @R(t,s,_x,_Dx) is ready"); |
---|
373 | dbprint(ppl-1, @R); |
---|
374 | // create the ideal I |
---|
375 | poly F = imap(save,F); |
---|
376 | ideal I = t*F+s; |
---|
377 | poly p; |
---|
378 | for(i=1; i<=N; i++) |
---|
379 | { |
---|
380 | p = t; //t |
---|
381 | p = diff(F,var(2+i))*p; |
---|
382 | I = I, var(N+2+i) + p; |
---|
383 | } |
---|
384 | // -------- the ideal I is ready ---------- |
---|
385 | dbprint(ppl,"// -1-2- starting the elimination of t in @R"); |
---|
386 | dbprint(ppl-1, I); |
---|
387 | ideal J = engine(I,eng); |
---|
388 | ideal K = nselect(J,1); |
---|
389 | kill I,J; |
---|
390 | dbprint(ppl,"// -1-3- t is eliminated"); |
---|
391 | dbprint(ppl-1, K); //K is without t |
---|
392 | // ----------- the ring @R2 ------------ |
---|
393 | // _x, _Dx,s; elim.ord for _x,_Dx. |
---|
394 | // keep: N, i,j,s, tmp, RL |
---|
395 | setring save; |
---|
396 | Nnew = 2*N+1; |
---|
397 | kill Lord, tmp, iv, RName; |
---|
398 | list Lord, tmp; |
---|
399 | intvec iv; |
---|
400 | L[1] = RL[1]; |
---|
401 | L[4] = RL[4]; //char, minpoly |
---|
402 | // check whether vars hava admissible names -> done earlier |
---|
403 | // now, create the names for new var |
---|
404 | tmp[1] = "s"; |
---|
405 | // DName is defined earlier |
---|
406 | list NName = Name + DName + tmp; |
---|
407 | L[2] = NName; |
---|
408 | tmp = 0; |
---|
409 | // block ord (dp(N),dp); |
---|
410 | string s = "iv="; |
---|
411 | for (i=1; i<=Nnew-1; i++) |
---|
412 | { |
---|
413 | s = s+"1,"; |
---|
414 | } |
---|
415 | s[size(s)]=";"; |
---|
416 | execute(s); |
---|
417 | tmp[1] = "dp"; //string |
---|
418 | tmp[2] = iv; //intvec |
---|
419 | Lord[1] = tmp; |
---|
420 | // continue with dp 1,1,1,1... |
---|
421 | tmp[1] = "dp"; //string |
---|
422 | s[size(s)] = ","; |
---|
423 | s = s+"1;"; |
---|
424 | execute(s); |
---|
425 | kill s; |
---|
426 | kill NName; |
---|
427 | tmp[2] = iv; |
---|
428 | Lord[2] = tmp; |
---|
429 | tmp[1] = "C"; |
---|
430 | iv = 0; |
---|
431 | tmp[2] = iv; |
---|
432 | Lord[3] = tmp; |
---|
433 | tmp = 0; |
---|
434 | L[3] = Lord; |
---|
435 | // we are done with the list. Now add a Plural part |
---|
436 | def @R2 = ring(L); |
---|
437 | setring @R2; |
---|
438 | matrix @D[Nnew][Nnew]; |
---|
439 | for (i=1; i<=N; i++) |
---|
440 | { |
---|
441 | @D[i,N+i]=1; |
---|
442 | } |
---|
443 | ncalgebra(1,@D); |
---|
444 | dbprint(ppl,"// -2-1- the ring @R2(_x,_Dx,s) is ready"); |
---|
445 | dbprint(ppl-1, @R2); |
---|
446 | ideal MM = maxideal(1); |
---|
447 | MM = 0,s,MM; |
---|
448 | map R01 = @R, MM; |
---|
449 | ideal K = R01(K); |
---|
450 | kill @R, R01; |
---|
451 | poly F = imap(save,F); |
---|
452 | K = K,F; |
---|
453 | dbprint(ppl,"// -2-2- starting the elimination of _x,_Dx in @R2"); |
---|
454 | dbprint(ppl-1, K); |
---|
455 | ideal M = engine(K,eng); |
---|
456 | ideal K2 = nselect(M,1,Nnew-1); |
---|
457 | kill K,M; |
---|
458 | dbprint(ppl,"// -2-3- _x,_Dx are eliminated in @R2"); |
---|
459 | dbprint(ppl-1, K2); |
---|
460 | // the ring @R3 and the search for minimal negative int s |
---|
461 | ring @R3 = 0,s,dp; |
---|
462 | dbprint(ppl,"// -3-1- the ring @R3(s) is ready"); |
---|
463 | ideal K3 = imap(@R2,K2); |
---|
464 | kill @R2; |
---|
465 | poly p = K3[1]; |
---|
466 | dbprint(ppl,"// -3-2- factorization"); |
---|
467 | list P = factorize(p); //with constants and multiplicities |
---|
468 | ideal bs; intvec m; //the Bernstein polynomial is monic, so we are not interested in constants |
---|
469 | for (i=2; i<= size(P[1]); i++) //we delete P[1][1] and P[2][1] |
---|
470 | { |
---|
471 | bs[i-1] = P[1][i]; |
---|
472 | m[i-1] = P[2][i]; |
---|
473 | } |
---|
474 | // "--------- b-function factorizes into ---------"; P; |
---|
475 | //int sP = minIntRoot(bs,1); |
---|
476 | //dbprint(ppl,"// -3-3- minimal interger root found"); |
---|
477 | //dbprint(ppl-1, sP); |
---|
478 | // convert factors to a list of their roots and multiplicities |
---|
479 | bs = normalize(bs); |
---|
480 | bs = -subst(bs,s,0); |
---|
481 | setring save; |
---|
482 | ideal bs = imap(@R3,bs); |
---|
483 | kill @R3; |
---|
484 | list BS = bs,m; |
---|
485 | return(BS); |
---|
486 | } |
---|
487 | example |
---|
488 | { |
---|
489 | "EXAMPLE:"; echo = 2; |
---|
490 | ring r = 0,(x,y,z,w),Dp; |
---|
491 | poly F = x^3+y^3+z^2*w; |
---|
492 | printlevel = 0; |
---|
493 | bernsteinBM(F); |
---|
494 | } |
---|
495 | |
---|
496 | // some changes |
---|
497 | proc annfsBM (poly F, list #) |
---|
498 | "USAGE: annfsBM(f [,eng]); f a poly, eng an optional int |
---|
499 | RETURN: ring |
---|
500 | PURPOSE: compute the D-module structure of basering[f^s], according |
---|
501 | to the algorithm by Briancon and Maisonobe |
---|
502 | NOTE: activate this ring with the @code{setring} command. In this ring, |
---|
503 | @* - the ideal LD is the needed D-mod structure, |
---|
504 | @* - the list BS is the list of roots and multiplicities of a Bernstein polynomial of f. |
---|
505 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
506 | @* otherwise, and by default @code{slimgb} is used. |
---|
507 | @* If printlevel=1, progress debug messages will be printed, |
---|
508 | @* if printlevel>=2, all the debug messages will be printed. |
---|
509 | EXAMPLE: example annfsBM; shows examples |
---|
510 | " |
---|
511 | { |
---|
512 | int eng = 0; |
---|
513 | if ( size(#)>0 ) |
---|
514 | { |
---|
515 | if ( typeof(#[1]) == "int" ) |
---|
516 | { |
---|
517 | eng = int(#[1]); |
---|
518 | } |
---|
519 | } |
---|
520 | // returns a list with a ring and an ideal LD in it |
---|
521 | int ppl = printlevel-voice+2; |
---|
522 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
523 | def save = basering; |
---|
524 | int N = nvars(basering); |
---|
525 | int Nnew = 2*N+2; |
---|
526 | int i,j; |
---|
527 | string s; |
---|
528 | list RL = ringlist(basering); |
---|
529 | list L, Lord; |
---|
530 | list tmp; |
---|
531 | intvec iv; |
---|
532 | L[1] = RL[1]; //char |
---|
533 | L[4] = RL[4]; //char, minpoly |
---|
534 | // check whether vars have admissible names |
---|
535 | list Name = RL[2]; |
---|
536 | list RName; |
---|
537 | RName[1] = "t"; |
---|
538 | RName[2] = "s"; |
---|
539 | for (i=1; i<=N; i++) |
---|
540 | { |
---|
541 | for(j=1; j<=size(RName); j++) |
---|
542 | { |
---|
543 | if (Name[i] == RName[j]) |
---|
544 | { |
---|
545 | ERROR("Variable names should not include t,s"); |
---|
546 | } |
---|
547 | } |
---|
548 | } |
---|
549 | // now, create the names for new vars |
---|
550 | list DName; |
---|
551 | for (i=1; i<=N; i++) |
---|
552 | { |
---|
553 | DName[i] = "D"+Name[i]; //concat |
---|
554 | } |
---|
555 | tmp[1] = "t"; |
---|
556 | tmp[2] = "s"; |
---|
557 | list NName = tmp + Name + DName; |
---|
558 | L[2] = NName; |
---|
559 | // Name, Dname will be used further |
---|
560 | kill NName; |
---|
561 | // block ord (lp(2),dp); |
---|
562 | tmp[1] = "lp"; // string |
---|
563 | iv = 1,1; |
---|
564 | tmp[2] = iv; //intvec |
---|
565 | Lord[1] = tmp; |
---|
566 | // continue with dp 1,1,1,1... |
---|
567 | tmp[1] = "dp"; // string |
---|
568 | s = "iv="; |
---|
569 | for (i=1; i<=Nnew; i++) |
---|
570 | { |
---|
571 | s = s+"1,"; |
---|
572 | } |
---|
573 | s[size(s)]= ";"; |
---|
574 | execute(s); |
---|
575 | kill s; |
---|
576 | tmp[2] = iv; |
---|
577 | Lord[2] = tmp; |
---|
578 | tmp[1] = "C"; |
---|
579 | iv = 0; |
---|
580 | tmp[2] = iv; |
---|
581 | Lord[3] = tmp; |
---|
582 | tmp = 0; |
---|
583 | L[3] = Lord; |
---|
584 | // we are done with the list |
---|
585 | def @R = ring(L); |
---|
586 | setring @R; |
---|
587 | matrix @D[Nnew][Nnew]; |
---|
588 | @D[1,2]=t; |
---|
589 | for(i=1; i<=N; i++) |
---|
590 | { |
---|
591 | @D[2+i,N+2+i]=1; |
---|
592 | } |
---|
593 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
594 | // L[6] = @D; |
---|
595 | ncalgebra(1,@D); |
---|
596 | dbprint(ppl,"// -1-1- the ring @R(t,s,_x,_Dx) is ready"); |
---|
597 | dbprint(ppl-1, @R); |
---|
598 | // create the ideal I |
---|
599 | poly F = imap(save,F); |
---|
600 | ideal I = t*F+s; |
---|
601 | poly p; |
---|
602 | for(i=1; i<=N; i++) |
---|
603 | { |
---|
604 | p = t; //t |
---|
605 | p = diff(F,var(2+i))*p; |
---|
606 | I = I, var(N+2+i) + p; |
---|
607 | } |
---|
608 | // -------- the ideal I is ready ---------- |
---|
609 | dbprint(ppl,"// -1-2- starting the elimination of t in @R"); |
---|
610 | dbprint(ppl-1, I); |
---|
611 | ideal J = engine(I,eng); |
---|
612 | ideal K = nselect(J,1); |
---|
613 | kill I,J; |
---|
614 | dbprint(ppl,"// -1-3- t is eliminated"); |
---|
615 | dbprint(ppl-1, K); //K is without t |
---|
616 | setring save; |
---|
617 | // ----------- the ring @R2 ------------ |
---|
618 | // _x, _Dx,s; elim.ord for _x,_Dx. |
---|
619 | // keep: N, i,j,s, tmp, RL |
---|
620 | Nnew = 2*N+1; |
---|
621 | kill Lord, tmp, iv, RName; |
---|
622 | list Lord, tmp; |
---|
623 | intvec iv; |
---|
624 | L[1] = RL[1]; |
---|
625 | L[4] = RL[4]; //char, minpoly |
---|
626 | // check whether vars hava admissible names -> done earlier |
---|
627 | // now, create the names for new var |
---|
628 | tmp[1] = "s"; |
---|
629 | // DName is defined earlier |
---|
630 | list NName = Name + DName + tmp; |
---|
631 | L[2] = NName; |
---|
632 | tmp = 0; |
---|
633 | // block ord (dp(N),dp); |
---|
634 | string s = "iv="; |
---|
635 | for (i=1; i<=Nnew-1; i++) |
---|
636 | { |
---|
637 | s = s+"1,"; |
---|
638 | } |
---|
639 | s[size(s)]=";"; |
---|
640 | execute(s); |
---|
641 | tmp[1] = "dp"; //string |
---|
642 | tmp[2] = iv; //intvec |
---|
643 | Lord[1] = tmp; |
---|
644 | // continue with dp 1,1,1,1... |
---|
645 | tmp[1] = "dp"; //string |
---|
646 | s[size(s)] = ","; |
---|
647 | s = s+"1;"; |
---|
648 | execute(s); |
---|
649 | kill s; |
---|
650 | kill NName; |
---|
651 | tmp[2] = iv; |
---|
652 | Lord[2] = tmp; |
---|
653 | tmp[1] = "C"; |
---|
654 | iv = 0; |
---|
655 | tmp[2] = iv; |
---|
656 | Lord[3] = tmp; |
---|
657 | tmp = 0; |
---|
658 | L[3] = Lord; |
---|
659 | // we are done with the list. Now add a Plural part |
---|
660 | def @R2 = ring(L); |
---|
661 | setring @R2; |
---|
662 | matrix @D[Nnew][Nnew]; |
---|
663 | for (i=1; i<=N; i++) |
---|
664 | { |
---|
665 | @D[i,N+i]=1; |
---|
666 | } |
---|
667 | ncalgebra(1,@D); |
---|
668 | dbprint(ppl,"// -2-1- the ring @R2(_x,_Dx,s) is ready"); |
---|
669 | dbprint(ppl-1, @R2); |
---|
670 | ideal MM = maxideal(1); |
---|
671 | MM = 0,s,MM; |
---|
672 | map R01 = @R, MM; |
---|
673 | ideal K = R01(K); |
---|
674 | poly F = imap(save,F); |
---|
675 | K = K,F; |
---|
676 | dbprint(ppl,"// -2-2- starting the elimination of _x,_Dx in @R2"); |
---|
677 | dbprint(ppl-1, K); |
---|
678 | ideal M = engine(K,eng); |
---|
679 | ideal K2 = nselect(M,1,Nnew-1); |
---|
680 | kill K,M; |
---|
681 | dbprint(ppl,"// -2-3- _x,_Dx are eliminated in @R2"); |
---|
682 | dbprint(ppl-1, K2); |
---|
683 | // the ring @R3 and the search for minimal negative int s |
---|
684 | ring @R3 = 0,s,dp; |
---|
685 | dbprint(ppl,"// -3-1- the ring @R3(s) is ready"); |
---|
686 | ideal K3 = imap(@R2,K2); |
---|
687 | poly p = K3[1]; |
---|
688 | dbprint(ppl,"// -3-2- factorization"); |
---|
689 | list P = factorize(p); //with constants and multiplicities |
---|
690 | ideal bs; intvec m; //the Bernstein polynomial is monic, so we are not interested in constants |
---|
691 | for (i=2; i<= size(P[1]); i++) //we delete P[1][1] and P[2][1] |
---|
692 | { |
---|
693 | bs[i-1] = P[1][i]; |
---|
694 | m[i-1] = P[2][i]; |
---|
695 | } |
---|
696 | // "--------- b-function factorizes into ---------"; P; |
---|
697 | int sP = minIntRoot(bs,1); |
---|
698 | dbprint(ppl,"// -3-3- minimal interger root found"); |
---|
699 | dbprint(ppl-1, sP); |
---|
700 | // convert factors to a list of their roots |
---|
701 | bs = normalize(bs); |
---|
702 | bs = -subst(bs,s,0); |
---|
703 | list BS = bs,m; |
---|
704 | //TODO: sort BS! |
---|
705 | // --------- substitute s found in the ideal --------- |
---|
706 | // --------- going back to @R and substitute --------- |
---|
707 | setring @R; |
---|
708 | ideal K2 = subst(K,s,sP); |
---|
709 | kill K; |
---|
710 | // create the ordinary Weyl algebra and put the result into it, |
---|
711 | // thus creating the ring @R4 |
---|
712 | // keep: N, i,j,s, tmp, RL |
---|
713 | setring save; |
---|
714 | Nnew = 2*N; |
---|
715 | // list RL = ringlist(save); //is defined earlier |
---|
716 | kill Lord, tmp, iv; |
---|
717 | L = 0; |
---|
718 | list Lord, tmp; |
---|
719 | intvec iv; |
---|
720 | L[1] = RL[1]; |
---|
721 | L[4] = RL[4]; //char, minpoly |
---|
722 | // check whether vars have admissible names -> done earlier |
---|
723 | // list Name = RL[2]M |
---|
724 | // DName is defined earlier |
---|
725 | list NName = Name + DName; |
---|
726 | L[2] = NName; |
---|
727 | // dp ordering; |
---|
728 | string s = "iv="; |
---|
729 | for (i=1; i<=Nnew; i++) |
---|
730 | { |
---|
731 | s = s+"1,"; |
---|
732 | } |
---|
733 | s[size(s)] = ";"; |
---|
734 | execute(s); |
---|
735 | kill s; |
---|
736 | tmp = 0; |
---|
737 | tmp[1] = "dp"; //string |
---|
738 | tmp[2] = iv; //intvec |
---|
739 | Lord[1] = tmp; |
---|
740 | tmp[1] = "C"; |
---|
741 | iv = 0; |
---|
742 | tmp[2] = iv; |
---|
743 | Lord[2] = tmp; |
---|
744 | tmp = 0; |
---|
745 | L[3] = Lord; |
---|
746 | // we are done with the list |
---|
747 | // Add: Plural part |
---|
748 | def @R4 = ring(L); |
---|
749 | setring @R4; |
---|
750 | matrix @D[Nnew][Nnew]; |
---|
751 | for (i=1; i<=N; i++) |
---|
752 | { |
---|
753 | @D[i,N+i]=1; |
---|
754 | } |
---|
755 | ncalgebra(1,@D); |
---|
756 | dbprint(ppl,"// -4-1- the ring @R4(_x,_Dx) is ready"); |
---|
757 | dbprint(ppl-1, @R4); |
---|
758 | ideal K4 = imap(@R,K2); |
---|
759 | option(redSB); |
---|
760 | dbprint(ppl,"// -4-2- the final cosmetic std"); |
---|
761 | K4 = engine(K4,eng); //std does the job too |
---|
762 | // total cleanup |
---|
763 | kill @R; |
---|
764 | kill @R2; |
---|
765 | list BS = imap(@R3,BS); |
---|
766 | export BS; |
---|
767 | kill @R3; |
---|
768 | ideal LD = K4; |
---|
769 | export LD; |
---|
770 | return(@R4); |
---|
771 | } |
---|
772 | example |
---|
773 | { |
---|
774 | "EXAMPLE:"; echo = 2; |
---|
775 | ring r = 0,(x,y,z),Dp; |
---|
776 | poly F = x^3+y^3+z^3; |
---|
777 | printlevel = 0; |
---|
778 | def A = annfsBM(F); |
---|
779 | setring A; |
---|
780 | LD; |
---|
781 | BS; |
---|
782 | } |
---|
783 | |
---|
784 | proc operatorBM(poly F, list #) |
---|
785 | "USAGE: operatorBM(f [,eng]); f a poly, eng an optional int |
---|
786 | RETURN: ring |
---|
787 | PURPOSE: compute the B-operator and other relevant data for Ann F^s, according to the algorithm by Briancon and Maisonobe |
---|
788 | NOTE: activate this ring with the @code{setring} command. In this ring D[s] |
---|
789 | @* - the polynomial F is the same as the input, |
---|
790 | @* - the ideal LD is the annihilator of f^s in Dn[s], |
---|
791 | @* - the ideal LD0 is the needed D-mod structure, where LD0 = LD|s=s0, |
---|
792 | @* - the polynomial bs is the global Bernstein polynomial of f in the variable s, |
---|
793 | @* - the list BS contains all the roots with multiplicities of the global Bernstein polynomial of f, |
---|
794 | @* - the polynomial PS is an operator in Dn[s] such that PS*f^(s+1) = bs*f^s. |
---|
795 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
796 | @* otherwise and by default @code{slimgb} is used. |
---|
797 | @* If printlevel=1, progress debug messages will be printed, |
---|
798 | @* if printlevel>=2, all the debug messages will be printed. |
---|
799 | EXAMPLE: example operatorBM; shows examples |
---|
800 | " |
---|
801 | { |
---|
802 | int eng = 0; |
---|
803 | if ( size(#)>0 ) |
---|
804 | { |
---|
805 | if ( typeof(#[1]) == "int" ) |
---|
806 | { |
---|
807 | eng = int(#[1]); |
---|
808 | } |
---|
809 | } |
---|
810 | // returns a list with a ring and an ideal LD in it |
---|
811 | int ppl = printlevel-voice+2; |
---|
812 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
813 | def save = basering; |
---|
814 | int N = nvars(basering); |
---|
815 | int Nnew = 2*N+2; |
---|
816 | int i,j; |
---|
817 | string s; |
---|
818 | list RL = ringlist(basering); |
---|
819 | list L, Lord; |
---|
820 | list tmp; |
---|
821 | intvec iv; |
---|
822 | L[1] = RL[1]; //char |
---|
823 | L[4] = RL[4]; //char, minpoly |
---|
824 | // check whether vars have admissible names |
---|
825 | list Name = RL[2]; |
---|
826 | list RName; |
---|
827 | RName[1] = "t"; |
---|
828 | RName[2] = "s"; |
---|
829 | for (i=1; i<=N; i++) |
---|
830 | { |
---|
831 | for(j=1; j<=size(RName); j++) |
---|
832 | { |
---|
833 | if (Name[i] == RName[j]) |
---|
834 | { |
---|
835 | ERROR("Variable names should not include t,s"); |
---|
836 | } |
---|
837 | } |
---|
838 | } |
---|
839 | // now, create the names for new vars |
---|
840 | list DName; |
---|
841 | for (i=1; i<=N; i++) |
---|
842 | { |
---|
843 | DName[i] = "D"+Name[i]; //concat |
---|
844 | } |
---|
845 | tmp[1] = "t"; |
---|
846 | tmp[2] = "s"; |
---|
847 | list NName = tmp + Name + DName; |
---|
848 | L[2] = NName; |
---|
849 | // Name, Dname will be used further |
---|
850 | kill NName; |
---|
851 | // block ord (lp(2),dp); |
---|
852 | tmp[1] = "lp"; // string |
---|
853 | iv = 1,1; |
---|
854 | tmp[2] = iv; //intvec |
---|
855 | Lord[1] = tmp; |
---|
856 | // continue with dp 1,1,1,1... |
---|
857 | tmp[1] = "dp"; // string |
---|
858 | s = "iv="; |
---|
859 | for (i=1; i<=Nnew; i++) |
---|
860 | { |
---|
861 | s = s+"1,"; |
---|
862 | } |
---|
863 | s[size(s)]= ";"; |
---|
864 | execute(s); |
---|
865 | kill s; |
---|
866 | tmp[2] = iv; |
---|
867 | Lord[2] = tmp; |
---|
868 | tmp[1] = "C"; |
---|
869 | iv = 0; |
---|
870 | tmp[2] = iv; |
---|
871 | Lord[3] = tmp; |
---|
872 | tmp = 0; |
---|
873 | L[3] = Lord; |
---|
874 | // we are done with the list |
---|
875 | def @R = ring(L); |
---|
876 | setring @R; |
---|
877 | matrix @D[Nnew][Nnew]; |
---|
878 | @D[1,2]=t; |
---|
879 | for(i=1; i<=N; i++) |
---|
880 | { |
---|
881 | @D[2+i,N+2+i]=1; |
---|
882 | } |
---|
883 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
884 | // L[6] = @D; |
---|
885 | ncalgebra(1,@D); |
---|
886 | dbprint(ppl,"// -1-1- the ring @R(t,s,_x,_Dx) is ready"); |
---|
887 | dbprint(ppl-1, @R); |
---|
888 | // create the ideal I |
---|
889 | poly F = imap(save,F); |
---|
890 | ideal I = t*F+s; |
---|
891 | poly p; |
---|
892 | for(i=1; i<=N; i++) |
---|
893 | { |
---|
894 | p = t; //t |
---|
895 | p = diff(F,var(2+i))*p; |
---|
896 | I = I, var(N+2+i) + p; |
---|
897 | } |
---|
898 | // -------- the ideal I is ready ---------- |
---|
899 | dbprint(ppl,"// -1-2- starting the elimination of t in @R"); |
---|
900 | dbprint(ppl-1, I); |
---|
901 | ideal J = engine(I,eng); |
---|
902 | ideal K = nselect(J,1); |
---|
903 | kill I,J; |
---|
904 | dbprint(ppl,"// -1-3- t is eliminated"); |
---|
905 | dbprint(ppl-1, K); //K is without t |
---|
906 | setring save; |
---|
907 | // ----------- the ring @R2 ------------ |
---|
908 | // _x, _Dx,s; elim.ord for _x,_Dx. |
---|
909 | // keep: N, i,j,s, tmp, RL |
---|
910 | Nnew = 2*N+1; |
---|
911 | kill Lord, tmp, iv, RName; |
---|
912 | list Lord, tmp; |
---|
913 | intvec iv; |
---|
914 | L[1] = RL[1]; |
---|
915 | L[4] = RL[4]; //char, minpoly |
---|
916 | // check whether vars hava admissible names -> done earlier |
---|
917 | // now, create the names for new var |
---|
918 | tmp[1] = "s"; |
---|
919 | // DName is defined earlier |
---|
920 | list NName = Name + DName + tmp; |
---|
921 | L[2] = NName; |
---|
922 | tmp = 0; |
---|
923 | // block ord (dp(N),dp); |
---|
924 | string s = "iv="; |
---|
925 | for (i=1; i<=Nnew-1; i++) |
---|
926 | { |
---|
927 | s = s+"1,"; |
---|
928 | } |
---|
929 | s[size(s)]=";"; |
---|
930 | execute(s); |
---|
931 | tmp[1] = "dp"; //string |
---|
932 | tmp[2] = iv; //intvec |
---|
933 | Lord[1] = tmp; |
---|
934 | // continue with dp 1,1,1,1... |
---|
935 | tmp[1] = "dp"; //string |
---|
936 | s[size(s)] = ","; |
---|
937 | s = s+"1;"; |
---|
938 | execute(s); |
---|
939 | kill s; |
---|
940 | kill NName; |
---|
941 | tmp[2] = iv; |
---|
942 | Lord[2] = tmp; |
---|
943 | tmp[1] = "C"; |
---|
944 | iv = 0; |
---|
945 | tmp[2] = iv; |
---|
946 | Lord[3] = tmp; |
---|
947 | tmp = 0; |
---|
948 | L[3] = Lord; |
---|
949 | // we are done with the list. Now add a Plural part |
---|
950 | def @R2 = ring(L); |
---|
951 | setring @R2; |
---|
952 | matrix @D[Nnew][Nnew]; |
---|
953 | for (i=1; i<=N; i++) |
---|
954 | { |
---|
955 | @D[i,N+i]=1; |
---|
956 | } |
---|
957 | ncalgebra(1,@D); |
---|
958 | dbprint(ppl,"// -2-1- the ring @R2(_x,_Dx,s) is ready"); |
---|
959 | dbprint(ppl-1, @R2); |
---|
960 | ideal MM = maxideal(1); |
---|
961 | MM = 0,s,MM; |
---|
962 | map R01 = @R, MM; |
---|
963 | ideal K = R01(K); |
---|
964 | poly F = imap(save,F); |
---|
965 | K = K,F; |
---|
966 | dbprint(ppl,"// -2-2- starting the elimination of _x,_Dx in @R2"); |
---|
967 | dbprint(ppl-1, K); |
---|
968 | ideal M = engine(K,eng); |
---|
969 | ideal K2 = nselect(M,1,Nnew-1); |
---|
970 | kill K,M; |
---|
971 | dbprint(ppl,"// -2-3- _x,_Dx are eliminated in @R2"); |
---|
972 | dbprint(ppl-1, K2); |
---|
973 | // the ring @R3 and the search for minimal negative int s |
---|
974 | ring @R3 = 0,s,dp; |
---|
975 | dbprint(ppl,"// -3-1- the ring @R3(s) is ready"); |
---|
976 | ideal K3 = imap(@R2,K2); |
---|
977 | kill @R2; |
---|
978 | poly p = K3[1]; |
---|
979 | dbprint(ppl,"// -3-2- factorization"); |
---|
980 | list P = factorize(p); //with constants and multiplicities |
---|
981 | ideal bs; intvec m; //the Bernstein polynomial is monic, so we are not interested in constants |
---|
982 | for (i=2; i<= size(P[1]); i++) //we delete P[1][1] and P[2][1] |
---|
983 | { |
---|
984 | bs[i-1] = P[1][i]; |
---|
985 | m[i-1] = P[2][i]; |
---|
986 | } |
---|
987 | // "--------- b-function factorizes into ---------"; P; |
---|
988 | int sP = minIntRoot(bs,1); |
---|
989 | dbprint(ppl,"// -3-3- minimal interger root found"); |
---|
990 | dbprint(ppl-1, sP); |
---|
991 | // convert factors to a list of their roots with multiplicities |
---|
992 | bs = normalize(bs); |
---|
993 | bs = -subst(bs,s,0); |
---|
994 | list BS = bs,m; |
---|
995 | //TODO: sort BS! |
---|
996 | // --------- substitute s found in the ideal --------- |
---|
997 | // --------- going back to @R and substitute --------- |
---|
998 | setring @R; |
---|
999 | ideal K2 = subst(K,s,sP); |
---|
1000 | // create Dn[s], where Dn is the ordinary Weyl algebra, and put the result into it, |
---|
1001 | // thus creating the ring @R4 |
---|
1002 | // keep: N, i,j,s, tmp, RL |
---|
1003 | setring save; |
---|
1004 | Nnew = 2*N+1; |
---|
1005 | // list RL = ringlist(save); //is defined earlier |
---|
1006 | kill Lord, tmp, iv; |
---|
1007 | L = 0; |
---|
1008 | list Lord, tmp; |
---|
1009 | intvec iv; |
---|
1010 | L[1] = RL[1]; |
---|
1011 | L[4] = RL[4]; //char, minpoly |
---|
1012 | // check whether vars have admissible names -> done earlier |
---|
1013 | // list Name = RL[2] |
---|
1014 | // DName is defined earlier |
---|
1015 | tmp[1] = "s"; |
---|
1016 | list NName = Name + DName + tmp; |
---|
1017 | L[2] = NName; |
---|
1018 | // dp ordering; |
---|
1019 | string s = "iv="; |
---|
1020 | for (i=1; i<=Nnew; i++) |
---|
1021 | { |
---|
1022 | s = s+"1,"; |
---|
1023 | } |
---|
1024 | s[size(s)] = ";"; |
---|
1025 | execute(s); |
---|
1026 | kill s; |
---|
1027 | tmp = 0; |
---|
1028 | tmp[1] = "dp"; //string |
---|
1029 | tmp[2] = iv; //intvec |
---|
1030 | Lord[1] = tmp; |
---|
1031 | tmp[1] = "C"; |
---|
1032 | iv = 0; |
---|
1033 | tmp[2] = iv; |
---|
1034 | Lord[2] = tmp; |
---|
1035 | tmp = 0; |
---|
1036 | L[3] = Lord; |
---|
1037 | // we are done with the list |
---|
1038 | // Add: Plural part |
---|
1039 | def @R4 = ring(L); |
---|
1040 | setring @R4; |
---|
1041 | matrix @D[Nnew][Nnew]; |
---|
1042 | for (i=1; i<=N; i++) |
---|
1043 | { |
---|
1044 | @D[i,N+i]=1; |
---|
1045 | } |
---|
1046 | ncalgebra(1,@D); |
---|
1047 | dbprint(ppl,"// -4-1- the ring @R4(_x,_Dx,s) is ready"); |
---|
1048 | dbprint(ppl-1, @R4); |
---|
1049 | ideal LD0 = imap(@R,K2); |
---|
1050 | ideal LD = imap(@R,K); |
---|
1051 | kill @R; |
---|
1052 | poly bs = imap(@R3,p); |
---|
1053 | list BS = imap(@R3,BS); |
---|
1054 | kill @R3; |
---|
1055 | bs = normalize(bs); |
---|
1056 | poly F = imap(save,F); |
---|
1057 | dbprint(ppl,"// -4-2- starting the computation of PS via lift"); |
---|
1058 | //better liftstd, I didn't knot it works also for Plural, liftslimgb? |
---|
1059 | // liftstd may give extra coeffs in the resulting ideal |
---|
1060 | matrix T = lift(F+LD,bs); |
---|
1061 | poly PS = T[1,1]; |
---|
1062 | dbprint(ppl,"// -4-3- an operator PS found, PS*f^(s+1) = b(s)*f^s"); |
---|
1063 | dbprint(ppl-1,PS); |
---|
1064 | option(redSB); |
---|
1065 | dbprint(ppl,"// -4-4- the final cosmetic std"); |
---|
1066 | LD0 = engine(LD0,eng); //std does the job too |
---|
1067 | LD = engine(LD,eng); |
---|
1068 | export F,LD,LD0,bs,BS,PS; |
---|
1069 | return(@R4); |
---|
1070 | } |
---|
1071 | example |
---|
1072 | { |
---|
1073 | "EXAMPLE:"; echo = 2; |
---|
1074 | // ring r = 0,(x,y,z,w),Dp; |
---|
1075 | ring r = 0,(x,y,z),Dp; |
---|
1076 | // poly F = x^3+y^3+z^2*w; |
---|
1077 | poly F = x^3+y^3+z^3; |
---|
1078 | printlevel = 0; |
---|
1079 | def A = operatorBM(F); |
---|
1080 | setring A; |
---|
1081 | F; // the original polynomial itself |
---|
1082 | LD; // generic annihilator |
---|
1083 | LD0; // annihilator |
---|
1084 | bs; // normalized Bernstein poly |
---|
1085 | BS; // root and multiplicities of the Bernstein poly |
---|
1086 | PS; // the operator, s.t. PS*F^{s+1} = bs*F^s mod LD |
---|
1087 | reduce(PS*F-bs,LD); // check the property of PS |
---|
1088 | } |
---|
1089 | |
---|
1090 | proc annfsParamBM (poly F, list #) |
---|
1091 | "USAGE: annfsParamBM(f [,eng]); f a poly, eng an optional int |
---|
1092 | RETURN: ring |
---|
1093 | PURPOSE: compute the generic Ann F^s and exceptional parametric constellations of a polynomial with parametric coefficients, according to the algorithm by Briancon and Maisonobe |
---|
1094 | NOTE: activate this ring with the @code{setring} command. In this ring, |
---|
1095 | @* - the ideal LD is the D-module structure oa Ann F^s |
---|
1096 | @* - the ideal Param contains the list of the special parameters. |
---|
1097 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
1098 | @* otherwise, and by default @code{slimgb} is used. |
---|
1099 | @* If printlevel=1, progress debug messages will be printed, |
---|
1100 | @* if printlevel>=2, all the debug messages will be printed. |
---|
1101 | EXAMPLE: example annfsParamBM; shows examples |
---|
1102 | " |
---|
1103 | { |
---|
1104 | //PURPOSE: compute the list of all possible Bernstein-Sato polynomials for a polynomial with parametric coefficients, according to the algorithm by Briancon and Maisonobe |
---|
1105 | // @* - the list BS is the list of roots and multiplicities of a Bernstein polynomial of f. |
---|
1106 | // ***** not implented yet **** |
---|
1107 | int eng = 0; |
---|
1108 | if ( size(#)>0 ) |
---|
1109 | { |
---|
1110 | if ( typeof(#[1]) == "int" ) |
---|
1111 | { |
---|
1112 | eng = int(#[1]); |
---|
1113 | } |
---|
1114 | } |
---|
1115 | // returns a list with a ring and an ideal LD in it |
---|
1116 | int ppl = printlevel-voice+2; |
---|
1117 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
1118 | def save = basering; |
---|
1119 | int N = nvars(basering); |
---|
1120 | int Nnew = 2*N+2; |
---|
1121 | int i,j; |
---|
1122 | string s; |
---|
1123 | list RL = ringlist(basering); |
---|
1124 | list L, Lord; |
---|
1125 | list tmp; |
---|
1126 | intvec iv; |
---|
1127 | L[1] = RL[1]; //char |
---|
1128 | L[4] = RL[4]; //char, minpoly |
---|
1129 | // check whether vars have admissible names |
---|
1130 | list Name = RL[2]; |
---|
1131 | list RName; |
---|
1132 | RName[1] = "t"; |
---|
1133 | RName[2] = "s"; |
---|
1134 | for (i=1; i<=N; i++) |
---|
1135 | { |
---|
1136 | for(j=1; j<=size(RName); j++) |
---|
1137 | { |
---|
1138 | if (Name[i] == RName[j]) |
---|
1139 | { |
---|
1140 | ERROR("Variable names should not include t,s"); |
---|
1141 | } |
---|
1142 | } |
---|
1143 | } |
---|
1144 | // now, create the names for new vars |
---|
1145 | list DName; |
---|
1146 | for (i=1; i<=N; i++) |
---|
1147 | { |
---|
1148 | DName[i] = "D"+Name[i]; //concat |
---|
1149 | } |
---|
1150 | tmp[1] = "t"; |
---|
1151 | tmp[2] = "s"; |
---|
1152 | list NName = tmp + Name + DName; |
---|
1153 | L[2] = NName; |
---|
1154 | // Name, Dname will be used further |
---|
1155 | kill NName; |
---|
1156 | // block ord (lp(2),dp); |
---|
1157 | tmp[1] = "lp"; // string |
---|
1158 | iv = 1,1; |
---|
1159 | tmp[2] = iv; //intvec |
---|
1160 | Lord[1] = tmp; |
---|
1161 | // continue with dp 1,1,1,1... |
---|
1162 | tmp[1] = "dp"; // string |
---|
1163 | s = "iv="; |
---|
1164 | for (i=1; i<=Nnew; i++) |
---|
1165 | { |
---|
1166 | s = s+"1,"; |
---|
1167 | } |
---|
1168 | s[size(s)]= ";"; |
---|
1169 | execute(s); |
---|
1170 | kill s; |
---|
1171 | tmp[2] = iv; |
---|
1172 | Lord[2] = tmp; |
---|
1173 | tmp[1] = "C"; |
---|
1174 | iv = 0; |
---|
1175 | tmp[2] = iv; |
---|
1176 | Lord[3] = tmp; |
---|
1177 | tmp = 0; |
---|
1178 | L[3] = Lord; |
---|
1179 | // we are done with the list |
---|
1180 | def @R = ring(L); |
---|
1181 | setring @R; |
---|
1182 | matrix @D[Nnew][Nnew]; |
---|
1183 | @D[1,2]=t; |
---|
1184 | for(i=1; i<=N; i++) |
---|
1185 | { |
---|
1186 | @D[2+i,N+2+i]=1; |
---|
1187 | } |
---|
1188 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
1189 | // L[6] = @D; |
---|
1190 | ncalgebra(1,@D); |
---|
1191 | dbprint(ppl,"// -1-1- the ring @R(t,s,_x,_Dx) is ready"); |
---|
1192 | dbprint(ppl-1, @R); |
---|
1193 | // create the ideal I |
---|
1194 | poly F = imap(save,F); |
---|
1195 | ideal I = t*F+s; |
---|
1196 | poly p; |
---|
1197 | for(i=1; i<=N; i++) |
---|
1198 | { |
---|
1199 | p = t; //t |
---|
1200 | p = diff(F,var(2+i))*p; |
---|
1201 | I = I, var(N+2+i) + p; |
---|
1202 | } |
---|
1203 | // -------- the ideal I is ready ---------- |
---|
1204 | dbprint(ppl,"// -1-2- starting the elimination of t in @R"); |
---|
1205 | dbprint(ppl-1, I); |
---|
1206 | ideal J = engine(I,eng); |
---|
1207 | ideal K = nselect(J,1); |
---|
1208 | dbprint(ppl,"// -1-3- t is eliminated"); |
---|
1209 | dbprint(ppl-1, K); //K is without t |
---|
1210 | // ----- looking for special parameters ----- |
---|
1211 | dbprint(ppl,"// -2-1- starting the computation of the transformation matrix (via lift)"); |
---|
1212 | J = normalize(J); |
---|
1213 | matrix T = lift(I,J); //try also with liftstd |
---|
1214 | kill I,J; |
---|
1215 | dbprint(ppl,"// -2-2- the transformation matrix has been computed"); |
---|
1216 | dbprint(ppl-1, T); //T is the transformation matrix |
---|
1217 | dbprint(ppl,"// -2-3- genericity does the job"); |
---|
1218 | list lParam = genericity(T); |
---|
1219 | int ip = size(lParam); |
---|
1220 | int cip; |
---|
1221 | string sParam; |
---|
1222 | if (sParam[1]=="-") { sParam=""; } //genericity returns "-" |
---|
1223 | // if no parameters exist in a basering |
---|
1224 | for (cip=1; cip <= ip; cip++) |
---|
1225 | { |
---|
1226 | sParam = sParam + "," +lParam[cip]; |
---|
1227 | } |
---|
1228 | if (size(sParam) >=2) |
---|
1229 | { |
---|
1230 | sParam = sParam[2..size(sParam)]; // removes the 1st colon |
---|
1231 | } |
---|
1232 | export sParam; |
---|
1233 | kill T; |
---|
1234 | dbprint(ppl,"// -2-4- the special parameters has been computed"); |
---|
1235 | dbprint(ppl, sParam); |
---|
1236 | // create Dn[s], where Dn is the ordinary Weyl Algebra, and put the result into it, |
---|
1237 | // thus creating the ring @R2 |
---|
1238 | // keep: N, i,j,s, tmp, RL |
---|
1239 | setring save; |
---|
1240 | Nnew = 2*N+1; |
---|
1241 | // list RL = ringlist(save); //is defined earlier |
---|
1242 | kill Lord, tmp, iv; |
---|
1243 | L = 0; |
---|
1244 | list Lord, tmp; |
---|
1245 | intvec iv; |
---|
1246 | L[1] = RL[1]; |
---|
1247 | L[4] = RL[4]; //char, minpoly |
---|
1248 | // check whether vars have admissible names -> done earlier |
---|
1249 | // list Name = RL[2]M |
---|
1250 | // DName is defined earlier |
---|
1251 | tmp[1] = "s"; |
---|
1252 | list NName = Name + DName + tmp; |
---|
1253 | L[2] = NName; |
---|
1254 | // dp ordering; |
---|
1255 | string s = "iv="; |
---|
1256 | for (i=1; i<=Nnew; i++) |
---|
1257 | { |
---|
1258 | s = s+"1,"; |
---|
1259 | } |
---|
1260 | s[size(s)] = ";"; |
---|
1261 | execute(s); |
---|
1262 | kill s; |
---|
1263 | tmp = 0; |
---|
1264 | tmp[1] = "dp"; //string |
---|
1265 | tmp[2] = iv; //intvec |
---|
1266 | Lord[1] = tmp; |
---|
1267 | tmp[1] = "C"; |
---|
1268 | iv = 0; |
---|
1269 | tmp[2] = iv; |
---|
1270 | Lord[2] = tmp; |
---|
1271 | tmp = 0; |
---|
1272 | L[3] = Lord; |
---|
1273 | // we are done with the list |
---|
1274 | // Add: Plural part |
---|
1275 | def @R2 = ring(L); |
---|
1276 | setring @R2; |
---|
1277 | matrix @D[Nnew][Nnew]; |
---|
1278 | for (i=1; i<=N; i++) |
---|
1279 | { |
---|
1280 | @D[i,N+i]=1; |
---|
1281 | } |
---|
1282 | ncalgebra(1,@D); |
---|
1283 | dbprint(ppl,"// -3-1- the ring @R2(_x,_Dx,s) is ready"); |
---|
1284 | dbprint(ppl-1, @R2); |
---|
1285 | ideal K = imap(@R,K); |
---|
1286 | kill @R; |
---|
1287 | option(redSB); |
---|
1288 | dbprint(ppl,"// -3-2- the final cosmetic std"); |
---|
1289 | K = engine(K,eng); //std does the job too |
---|
1290 | ideal LD = K; |
---|
1291 | export LD; |
---|
1292 | if (sParam[1] == ",") |
---|
1293 | { |
---|
1294 | sParam = sParam[2..size(sParam)]; |
---|
1295 | } |
---|
1296 | // || ((sParam[1] == " ") && (sParam[2] == ","))) |
---|
1297 | execute("ideal Param ="+sParam+";"); |
---|
1298 | export Param; |
---|
1299 | kill sParam; |
---|
1300 | return(@R2); |
---|
1301 | } |
---|
1302 | example |
---|
1303 | { |
---|
1304 | "EXAMPLE:"; echo = 2; |
---|
1305 | ring r = (0,a,b),(x,y),Dp; |
---|
1306 | poly F = x^2 - (y-a)*(y-b); |
---|
1307 | printlevel = 0; |
---|
1308 | def A = annfsParamBM(F); setring A; |
---|
1309 | LD; |
---|
1310 | Param; |
---|
1311 | setring r; |
---|
1312 | poly G = x2-(y-a)^2; // try the exceptional value b=a of parameters |
---|
1313 | def B = annfsParamBM(G); setring B; |
---|
1314 | LD; |
---|
1315 | Param; |
---|
1316 | } |
---|
1317 | |
---|
1318 | // *** the following example is nice, but too complicated for the documentation *** |
---|
1319 | // ring r = (0,a),(x,y,z),Dp; |
---|
1320 | // poly F = x^4+y^4+z^2+a*x*y*z; |
---|
1321 | // printlevel = 2; //0 |
---|
1322 | // def A = annfsParamBM(F); |
---|
1323 | // setring A; |
---|
1324 | // LD; |
---|
1325 | // Param; |
---|
1326 | |
---|
1327 | |
---|
1328 | proc annfsBMI(ideal F, list #) |
---|
1329 | "USAGE: annfsBMI(F [,eng]); F an ideal, eng an optional int |
---|
1330 | RETURN: ring |
---|
1331 | PURPOSE: compute the D-module structure of basering[f^s] where f = F[1]*..*F[P], |
---|
1332 | according to the algorithm by Briancon and Maisonobe. |
---|
1333 | NOTE: activate this ring with the @code{setring} command. In this ring, |
---|
1334 | @* - the ideal LD is the needed D-mod structure, |
---|
1335 | @* - the list BS is the Bernstein ideal of a polynomial f = F[1]*..*F[P]. |
---|
1336 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
1337 | @* otherwise, and by default @code{slimgb} is used. |
---|
1338 | @* If printlevel=1, progress debug messages will be printed, |
---|
1339 | @* if printlevel>=2, all the debug messages will be printed. |
---|
1340 | EXAMPLE: example annfsBMI; shows examples |
---|
1341 | " |
---|
1342 | { |
---|
1343 | int eng = 0; |
---|
1344 | if ( size(#)>0 ) |
---|
1345 | { |
---|
1346 | if ( typeof(#[1]) == "int" ) |
---|
1347 | { |
---|
1348 | eng = int(#[1]); |
---|
1349 | } |
---|
1350 | } |
---|
1351 | // returns a list with a ring and an ideal LD in it |
---|
1352 | int ppl = printlevel-voice+2; |
---|
1353 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
1354 | def save = basering; |
---|
1355 | int N = nvars(basering); |
---|
1356 | int P = size(F); //if F has some generators which are zero, int P = ncols(I); |
---|
1357 | int Nnew = 2*N+2*P; |
---|
1358 | int i,j; |
---|
1359 | string s; |
---|
1360 | list RL = ringlist(basering); |
---|
1361 | list L, Lord; |
---|
1362 | list tmp; |
---|
1363 | intvec iv; |
---|
1364 | L[1] = RL[1]; //char |
---|
1365 | L[4] = RL[4]; //char, minpoly |
---|
1366 | // check whether vars have admissible names |
---|
1367 | list Name = RL[2]; |
---|
1368 | list RName; |
---|
1369 | for (j=1; j<=P; j++) |
---|
1370 | { |
---|
1371 | RName[j] = "t("+string(j)+")"; |
---|
1372 | RName[j+P] = "s("+string(j)+")"; |
---|
1373 | } |
---|
1374 | for(i=1; i<=N; i++) |
---|
1375 | { |
---|
1376 | for(j=1; j<=size(RName); j++) |
---|
1377 | { |
---|
1378 | if (Name[i] == RName[j]) |
---|
1379 | { ERROR("Variable names should not include t(i),s(i)"); } |
---|
1380 | } |
---|
1381 | } |
---|
1382 | // now, create the names for new vars |
---|
1383 | list DName; |
---|
1384 | for(i=1; i<=N; i++) |
---|
1385 | { |
---|
1386 | DName[i] = "D"+Name[i]; //concat |
---|
1387 | } |
---|
1388 | list NName = RName + Name + DName; |
---|
1389 | L[2] = NName; |
---|
1390 | // Name, Dname will be used further |
---|
1391 | kill NName; |
---|
1392 | // block ord (lp(P),dp); |
---|
1393 | tmp[1] = "lp"; //string |
---|
1394 | s = "iv="; |
---|
1395 | for (i=1; i<=2*P; i++) |
---|
1396 | { |
---|
1397 | s = s+"1,"; |
---|
1398 | } |
---|
1399 | s[size(s)]= ";"; |
---|
1400 | execute(s); |
---|
1401 | tmp[2] = iv; //intvec |
---|
1402 | Lord[1] = tmp; |
---|
1403 | // continue with dp 1,1,1,1... |
---|
1404 | tmp[1] = "dp"; //string |
---|
1405 | s = "iv="; |
---|
1406 | for (i=1; i<=Nnew; i++) //actually i<=2*N |
---|
1407 | { |
---|
1408 | s = s+"1,"; |
---|
1409 | } |
---|
1410 | s[size(s)]= ";"; |
---|
1411 | execute(s); |
---|
1412 | kill s; |
---|
1413 | tmp[2] = iv; |
---|
1414 | Lord[2] = tmp; |
---|
1415 | tmp[1] = "C"; |
---|
1416 | iv = 0; |
---|
1417 | tmp[2] = iv; |
---|
1418 | Lord[3] = tmp; |
---|
1419 | tmp = 0; |
---|
1420 | L[3] = Lord; |
---|
1421 | // we are done with the list |
---|
1422 | def @R = ring(L); |
---|
1423 | setring @R; |
---|
1424 | matrix @D[Nnew][Nnew]; |
---|
1425 | for (i=1; i<=P; i++) |
---|
1426 | { |
---|
1427 | @D[i,i+P] = t(i); |
---|
1428 | } |
---|
1429 | for(i=1; i<=N; i++) |
---|
1430 | { |
---|
1431 | @D[2*P+i,2*P+N+i] = 1; |
---|
1432 | } |
---|
1433 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
1434 | // L[6] = @D; |
---|
1435 | ncalgebra(1,@D); |
---|
1436 | dbprint(ppl,"// -1-1- the ring @R(_t,_s,_x,_Dx) is ready"); |
---|
1437 | dbprint(ppl-1, @R); |
---|
1438 | // create the ideal I |
---|
1439 | ideal F = imap(save,F); |
---|
1440 | ideal I = t(1)*F[1]+s(1); |
---|
1441 | for (j=2; j<=P; j++) |
---|
1442 | { |
---|
1443 | I = I, t(j)*F[j]+s(j); |
---|
1444 | } |
---|
1445 | poly p,q; |
---|
1446 | for (i=1; i<=N; i++) |
---|
1447 | { |
---|
1448 | p=0; |
---|
1449 | for (j=1; j<=P; j++) |
---|
1450 | { |
---|
1451 | q = t(j); |
---|
1452 | q = diff(F[j],var(2*P+i))*q; |
---|
1453 | p = p + q; |
---|
1454 | } |
---|
1455 | I = I, var(2*P+N+i) + p; |
---|
1456 | } |
---|
1457 | // -------- the ideal I is ready ---------- |
---|
1458 | dbprint(ppl,"// -1-2- starting the elimination of "+string(t(1..P))+" in @R"); |
---|
1459 | dbprint(ppl-1, I); |
---|
1460 | ideal J = engine(I,eng); |
---|
1461 | ideal K = nselect(J,1,P); |
---|
1462 | kill I,J; |
---|
1463 | dbprint(ppl,"// -1-3- all t(i) are eliminated"); |
---|
1464 | dbprint(ppl-1, K); //K is without t(i) |
---|
1465 | // ----------- the ring @R2 ------------ |
---|
1466 | // _x, _Dx,s; elim.ord for _x,_Dx. |
---|
1467 | // keep: N, i,j,s, tmp, RL |
---|
1468 | setring save; |
---|
1469 | Nnew = 2*N+P; |
---|
1470 | kill Lord, tmp, iv, RName; |
---|
1471 | list Lord, tmp; |
---|
1472 | intvec iv; |
---|
1473 | L[1] = RL[1]; //char |
---|
1474 | L[4] = RL[4]; //char, minpoly |
---|
1475 | // check whether vars hava admissible names -> done earlier |
---|
1476 | // now, create the names for new var |
---|
1477 | for (j=1; j<=P; j++) |
---|
1478 | { |
---|
1479 | tmp[j] = "s("+string(j)+")"; |
---|
1480 | } |
---|
1481 | // DName is defined earlier |
---|
1482 | list NName = Name + DName + tmp; |
---|
1483 | L[2] = NName; |
---|
1484 | tmp = 0; |
---|
1485 | // block ord (dp(N),dp); |
---|
1486 | string s = "iv="; |
---|
1487 | for (i=1; i<=Nnew-P; i++) |
---|
1488 | { |
---|
1489 | s = s+"1,"; |
---|
1490 | } |
---|
1491 | s[size(s)]=";"; |
---|
1492 | execute(s); |
---|
1493 | tmp[1] = "dp"; //string |
---|
1494 | tmp[2] = iv; //intvec |
---|
1495 | Lord[1] = tmp; |
---|
1496 | // continue with dp 1,1,1,1... |
---|
1497 | tmp[1] = "dp"; //string |
---|
1498 | s[size(s)] = ","; |
---|
1499 | for (j=1; j<=P; j++) |
---|
1500 | { |
---|
1501 | s = s+"1,"; |
---|
1502 | } |
---|
1503 | s[size(s)]=";"; |
---|
1504 | execute(s); |
---|
1505 | kill s; |
---|
1506 | kill NName; |
---|
1507 | tmp[2] = iv; |
---|
1508 | Lord[2] = tmp; |
---|
1509 | tmp[1] = "C"; |
---|
1510 | iv = 0; |
---|
1511 | tmp[2] = iv; |
---|
1512 | Lord[3] = tmp; |
---|
1513 | tmp = 0; |
---|
1514 | L[3] = Lord; |
---|
1515 | // we are done with the list. Now add a Plural part |
---|
1516 | def @R2 = ring(L); |
---|
1517 | setring @R2; |
---|
1518 | matrix @D[Nnew][Nnew]; |
---|
1519 | for (i=1; i<=N; i++) |
---|
1520 | { |
---|
1521 | @D[i,N+i]=1; |
---|
1522 | } |
---|
1523 | ncalgebra(1,@D); |
---|
1524 | dbprint(ppl,"// -2-1- the ring @R2(_x,_Dx,_s) is ready"); |
---|
1525 | dbprint(ppl-1, @R2); |
---|
1526 | // ideal MM = maxideal(1); |
---|
1527 | // MM = 0,s,MM; |
---|
1528 | // map R01 = @R, MM; |
---|
1529 | // ideal K = R01(K); |
---|
1530 | ideal F = imap(save,F); // maybe ideal F = R01(I); ? |
---|
1531 | ideal K = imap(@R,K); // maybe ideal K = R01(I); ? |
---|
1532 | poly f=1; |
---|
1533 | for (j=1; j<=P; j++) |
---|
1534 | { |
---|
1535 | f = f*F[j]; |
---|
1536 | } |
---|
1537 | K = K,f; // to compute B (Bernstein-Sato ideal) |
---|
1538 | //j=2; // for example |
---|
1539 | //K = K,F[j]; // to compute Bj (see "On the computation of Bernstein-Sato ideals"; Castro, Ucha) |
---|
1540 | //K = K,F; // to compute Bsigma (see "On the computation of Bernstein-Sato ideals"; Castro, Ucha) |
---|
1541 | dbprint(ppl,"// -2-2- starting the elimination of _x,_Dx in @R2"); |
---|
1542 | dbprint(ppl-1, K); |
---|
1543 | ideal M = engine(K,eng); |
---|
1544 | ideal K2 = nselect(M,1,Nnew-P); |
---|
1545 | kill K,M; |
---|
1546 | dbprint(ppl,"// -2-3- _x,_Dx are eliminated in @R2"); |
---|
1547 | dbprint(ppl-1, K2); |
---|
1548 | // the ring @R3 and factorize |
---|
1549 | ring @R3 = 0,s(1..P),dp; |
---|
1550 | dbprint(ppl,"// -3-1- the ring @R3(_s) is ready"); |
---|
1551 | ideal K3 = imap(@R2,K2); |
---|
1552 | if (size(K3)==1) |
---|
1553 | { |
---|
1554 | poly p = K3[1]; |
---|
1555 | dbprint(ppl,"// -3-2- factorization"); |
---|
1556 | // Warning: now P is an integer |
---|
1557 | list Q = factorize(p); //with constants and multiplicities |
---|
1558 | ideal bs; intvec m; |
---|
1559 | for (i=2; i<=size(Q[1]); i++) //we delete Q[1][1] and Q[2][1] |
---|
1560 | { |
---|
1561 | bs[i-1] = Q[1][i]; |
---|
1562 | m[i-1] = Q[2][i]; |
---|
1563 | } |
---|
1564 | // "--------- Q-ideal factorizes into ---------"; list(bs,m); |
---|
1565 | list BS = bs,m; |
---|
1566 | } |
---|
1567 | else |
---|
1568 | { |
---|
1569 | // conjecture: the Bernstein ideal is principal |
---|
1570 | dbprint(ppl,"// -3-2- the Bernstein ideal is not principal"); |
---|
1571 | ideal BS = K3; |
---|
1572 | } |
---|
1573 | // create the ring @R4(_x,_Dx,_s) and put the result into it, |
---|
1574 | // _x, _Dx,s; ord "dp". |
---|
1575 | // keep: N, i,j,s, tmp, RL |
---|
1576 | setring save; |
---|
1577 | Nnew = 2*N+P; |
---|
1578 | // list RL = ringlist(save); //is defined earlier |
---|
1579 | kill Lord, tmp, iv; |
---|
1580 | L = 0; |
---|
1581 | list Lord, tmp; |
---|
1582 | intvec iv; |
---|
1583 | L[1] = RL[1]; //char |
---|
1584 | L[4] = RL[4]; //char, minpoly |
---|
1585 | // check whether vars hava admissible names -> done earlier |
---|
1586 | // now, create the names for new var |
---|
1587 | for (j=1; j<=P; j++) |
---|
1588 | { |
---|
1589 | tmp[j] = "s("+string(j)+")"; |
---|
1590 | } |
---|
1591 | // DName is defined earlier |
---|
1592 | list NName = Name + DName + tmp; |
---|
1593 | L[2] = NName; |
---|
1594 | tmp = 0; |
---|
1595 | // dp ordering; |
---|
1596 | string s = "iv="; |
---|
1597 | for (i=1; i<=Nnew; i++) |
---|
1598 | { |
---|
1599 | s = s+"1,"; |
---|
1600 | } |
---|
1601 | s[size(s)]=";"; |
---|
1602 | execute(s); |
---|
1603 | kill s; |
---|
1604 | kill NName; |
---|
1605 | tmp[1] = "dp"; //string |
---|
1606 | tmp[2] = iv; //intvec |
---|
1607 | Lord[1] = tmp; |
---|
1608 | tmp[1] = "C"; |
---|
1609 | iv = 0; |
---|
1610 | tmp[2] = iv; |
---|
1611 | Lord[2] = tmp; |
---|
1612 | tmp = 0; |
---|
1613 | L[3] = Lord; |
---|
1614 | // we are done with the list. Now add a Plural part |
---|
1615 | def @R4 = ring(L); |
---|
1616 | setring @R4; |
---|
1617 | matrix @D[Nnew][Nnew]; |
---|
1618 | for (i=1; i<=N; i++) |
---|
1619 | { |
---|
1620 | @D[i,N+i]=1; |
---|
1621 | } |
---|
1622 | ncalgebra(1,@D); |
---|
1623 | dbprint(ppl,"// -4-1- the ring @R4i(_x,_Dx,_s) is ready"); |
---|
1624 | dbprint(ppl-1, @R4); |
---|
1625 | ideal K4 = imap(@R,K); |
---|
1626 | option(redSB); |
---|
1627 | dbprint(ppl,"// -4-2- the final cosmetic std"); |
---|
1628 | K4 = engine(K4,eng); //std does the job too |
---|
1629 | // total cleanup |
---|
1630 | kill @R; |
---|
1631 | kill @R2; |
---|
1632 | def BS = imap(@R3,BS); |
---|
1633 | export BS; |
---|
1634 | kill @R3; |
---|
1635 | ideal LD = K4; |
---|
1636 | export LD; |
---|
1637 | return(@R4); |
---|
1638 | } |
---|
1639 | example |
---|
1640 | { |
---|
1641 | "EXAMPLE:"; echo = 2; |
---|
1642 | ring r = 0,(x,y),Dp; |
---|
1643 | ideal F = x,y,x+y; |
---|
1644 | printlevel = 0; |
---|
1645 | def A = annfsBMI(F); |
---|
1646 | setring A; |
---|
1647 | LD; |
---|
1648 | BS; |
---|
1649 | } |
---|
1650 | |
---|
1651 | proc annfsOT(poly F, list #) |
---|
1652 | "USAGE: annfsOT(f [,eng]); f a poly, eng an optional int |
---|
1653 | RETURN: ring |
---|
1654 | PURPOSE: compute the D-module structure of basering[f^s], according |
---|
1655 | to the algorithm by Oaku and Takayama |
---|
1656 | NOTE: activate this ring with the @code{setring} command. In this ring, |
---|
1657 | @* - the ideal LD is the needed D-mod structure, |
---|
1658 | @* - the list BS contains roots with multiplicities of a Bernstein polynomial of f. |
---|
1659 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
1660 | @* otherwise, and by default @code{slimgb} is used. |
---|
1661 | @* If printlevel=1, progress debug messages will be printed, |
---|
1662 | @* if printlevel>=2, all the debug messages will be printed. |
---|
1663 | EXAMPLE: example annfsOT; shows examples |
---|
1664 | " |
---|
1665 | { |
---|
1666 | int eng = 0; |
---|
1667 | if ( size(#)>0 ) |
---|
1668 | { |
---|
1669 | if ( typeof(#[1]) == "int" ) |
---|
1670 | { |
---|
1671 | eng = int(#[1]); |
---|
1672 | } |
---|
1673 | } |
---|
1674 | // returns a list with a ring and an ideal LD in it |
---|
1675 | int ppl = printlevel-voice+2; |
---|
1676 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
1677 | def save = basering; |
---|
1678 | int N = nvars(basering); |
---|
1679 | int Nnew = 2*(N+2); |
---|
1680 | int i,j; |
---|
1681 | string s; |
---|
1682 | list RL = ringlist(basering); |
---|
1683 | list L, Lord; |
---|
1684 | list tmp; |
---|
1685 | intvec iv; |
---|
1686 | L[1] = RL[1]; // char |
---|
1687 | L[4] = RL[4]; // char, minpoly |
---|
1688 | // check whether vars have admissible names |
---|
1689 | list Name = RL[2]; |
---|
1690 | list RName; |
---|
1691 | RName[1] = "u"; |
---|
1692 | RName[2] = "v"; |
---|
1693 | RName[3] = "t"; |
---|
1694 | RName[4] = "Dt"; |
---|
1695 | for(i=1;i<=N;i++) |
---|
1696 | { |
---|
1697 | for(j=1; j<=size(RName);j++) |
---|
1698 | { |
---|
1699 | if (Name[i] == RName[j]) |
---|
1700 | { |
---|
1701 | ERROR("Variable names should not include u,v,t,Dt"); |
---|
1702 | } |
---|
1703 | } |
---|
1704 | } |
---|
1705 | // now, create the names for new vars |
---|
1706 | tmp[1] = "u"; |
---|
1707 | tmp[2] = "v"; |
---|
1708 | list UName = tmp; |
---|
1709 | list DName; |
---|
1710 | for(i=1;i<=N;i++) |
---|
1711 | { |
---|
1712 | DName[i] = "D"+Name[i]; // concat |
---|
1713 | } |
---|
1714 | tmp = 0; |
---|
1715 | tmp[1] = "t"; |
---|
1716 | tmp[2] = "Dt"; |
---|
1717 | list NName = UName + tmp + Name + DName; |
---|
1718 | L[2] = NName; |
---|
1719 | tmp = 0; |
---|
1720 | // Name, Dname will be used further |
---|
1721 | kill UName; |
---|
1722 | kill NName; |
---|
1723 | // block ord (a(1,1),dp); |
---|
1724 | tmp[1] = "a"; // string |
---|
1725 | iv = 1,1; |
---|
1726 | tmp[2] = iv; //intvec |
---|
1727 | Lord[1] = tmp; |
---|
1728 | // continue with dp 1,1,1,1... |
---|
1729 | tmp[1] = "dp"; // string |
---|
1730 | s = "iv="; |
---|
1731 | for(i=1;i<=Nnew;i++) |
---|
1732 | { |
---|
1733 | s = s+"1,"; |
---|
1734 | } |
---|
1735 | s[size(s)]= ";"; |
---|
1736 | execute(s); |
---|
1737 | tmp[2] = iv; |
---|
1738 | Lord[2] = tmp; |
---|
1739 | tmp[1] = "C"; |
---|
1740 | iv = 0; |
---|
1741 | tmp[2] = iv; |
---|
1742 | Lord[3] = tmp; |
---|
1743 | tmp = 0; |
---|
1744 | L[3] = Lord; |
---|
1745 | // we are done with the list |
---|
1746 | def @R = ring(L); |
---|
1747 | setring @R; |
---|
1748 | matrix @D[Nnew][Nnew]; |
---|
1749 | @D[3,4]=1; |
---|
1750 | for(i=1; i<=N; i++) |
---|
1751 | { |
---|
1752 | @D[4+i,N+4+i]=1; |
---|
1753 | } |
---|
1754 | // @D[N+3,2*(N+2)]=1; old t,Dt stuff |
---|
1755 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
1756 | // L[6] = @D; |
---|
1757 | ncalgebra(1,@D); |
---|
1758 | dbprint(ppl,"// -1-1- the ring @R(u,v,t,Dt,_x,_Dx) is ready"); |
---|
1759 | dbprint(ppl-1, @R); |
---|
1760 | // create the ideal I |
---|
1761 | poly F = imap(save,F); |
---|
1762 | ideal I = u*F-t,u*v-1; |
---|
1763 | poly p; |
---|
1764 | for(i=1; i<=N; i++) |
---|
1765 | { |
---|
1766 | p = u*Dt; // u*Dt |
---|
1767 | p = diff(F,var(4+i))*p; |
---|
1768 | I = I, var(N+4+i) + p; |
---|
1769 | } |
---|
1770 | // -------- the ideal I is ready ---------- |
---|
1771 | dbprint(ppl,"// -1-2- starting the elimination of u,v in @R"); |
---|
1772 | dbprint(ppl-1, I); |
---|
1773 | ideal J = engine(I,eng); |
---|
1774 | ideal K = nselect(J,1,2); |
---|
1775 | dbprint(ppl,"// -1-3- u,v are eliminated"); |
---|
1776 | dbprint(ppl-1, K); // K is without u,v |
---|
1777 | setring save; |
---|
1778 | // ------------ new ring @R2 ------------------ |
---|
1779 | // without u,v and with the elim.ord for t,Dt |
---|
1780 | // tensored with the K[s] |
---|
1781 | // keep: N, i,j,s, tmp, RL |
---|
1782 | Nnew = 2*N+2+1; |
---|
1783 | // list RL = ringlist(save); // is defined earlier |
---|
1784 | L = 0; // kill L; |
---|
1785 | kill Lord, tmp, iv, RName; |
---|
1786 | list Lord, tmp; |
---|
1787 | intvec iv; |
---|
1788 | L[1] = RL[1]; L[4] = RL[4]; // char, minpoly |
---|
1789 | // check whether vars have admissible names -> done earlier |
---|
1790 | // list Name = RL[2]; |
---|
1791 | list RName; |
---|
1792 | RName[1] = "t"; |
---|
1793 | RName[2] = "Dt"; |
---|
1794 | // now, create the names for new var (here, s only) |
---|
1795 | tmp[1] = "s"; |
---|
1796 | // DName is defined earlier |
---|
1797 | list NName = RName + Name + DName + tmp; |
---|
1798 | L[2] = NName; |
---|
1799 | tmp = 0; |
---|
1800 | // block ord (a(1,1),dp); |
---|
1801 | tmp[1] = "a"; iv = 1,1; tmp[2] = iv; //intvec |
---|
1802 | Lord[1] = tmp; |
---|
1803 | // continue with a(1,1,1,1)... |
---|
1804 | tmp[1] = "dp"; s = "iv="; |
---|
1805 | for(i=1; i<= Nnew; i++) |
---|
1806 | { |
---|
1807 | s = s+"1,"; |
---|
1808 | } |
---|
1809 | s[size(s)]= ";"; execute(s); |
---|
1810 | kill NName; |
---|
1811 | tmp[2] = iv; |
---|
1812 | Lord[2] = tmp; |
---|
1813 | // extra block for s |
---|
1814 | // tmp[1] = "dp"; iv = 1; |
---|
1815 | // s[size(s)]= ","; s = s + "1,1,1;"; execute(s); tmp[2] = iv; |
---|
1816 | // Lord[3] = tmp; |
---|
1817 | kill s; |
---|
1818 | tmp[1] = "C"; iv = 0; tmp[2] = iv; |
---|
1819 | Lord[3] = tmp; tmp = 0; |
---|
1820 | L[3] = Lord; |
---|
1821 | // we are done with the list. Now, add a Plural part |
---|
1822 | def @R2 = ring(L); |
---|
1823 | setring @R2; |
---|
1824 | matrix @D[Nnew][Nnew]; |
---|
1825 | @D[1,2] = 1; |
---|
1826 | for(i=1; i<=N; i++) |
---|
1827 | { |
---|
1828 | @D[2+i,2+N+i] = 1; |
---|
1829 | } |
---|
1830 | ncalgebra(1,@D); |
---|
1831 | dbprint(ppl,"// -2-1- the ring @R2(t,Dt,_x,_Dx,s) is ready"); |
---|
1832 | dbprint(ppl-1, @R2); |
---|
1833 | ideal MM = maxideal(1); |
---|
1834 | MM = 0,0,MM; |
---|
1835 | map R01 = @R, MM; |
---|
1836 | ideal K = R01(K); |
---|
1837 | // ideal K = imap(@R,K); // names of vars are important! |
---|
1838 | poly G = t*Dt+s+1; // s is a variable here |
---|
1839 | K = NF(K,std(G)),G; |
---|
1840 | // -------- the ideal K_(@R2) is ready ---------- |
---|
1841 | dbprint(ppl,"// -2-2- starting the elimination of t,Dt in @R2"); |
---|
1842 | dbprint(ppl-1, K); |
---|
1843 | ideal M = engine(K,eng); |
---|
1844 | ideal K2 = nselect(M,1,2); |
---|
1845 | dbprint(ppl,"// -2-3- t,Dt are eliminated"); |
---|
1846 | dbprint(ppl-1, K2); |
---|
1847 | // dbprint(ppl-1+1," -2-4- std of K2"); |
---|
1848 | // option(redSB); option(redTail); K2 = std(K2); |
---|
1849 | // K2; // without t,Dt, and with s |
---|
1850 | // -------- the ring @R3 ---------- |
---|
1851 | // _x, _Dx, s; elim.ord for _x,_Dx. |
---|
1852 | // keep: N, i,j,s, tmp, RL |
---|
1853 | setring save; |
---|
1854 | Nnew = 2*N+1; |
---|
1855 | // list RL = ringlist(save); // is defined earlier |
---|
1856 | // kill L; |
---|
1857 | kill Lord, tmp, iv, RName; |
---|
1858 | list Lord, tmp; |
---|
1859 | intvec iv; |
---|
1860 | L[1] = RL[1]; L[4] = RL[4]; // char, minpoly |
---|
1861 | // check whether vars have admissible names -> done earlier |
---|
1862 | // list Name = RL[2]; |
---|
1863 | // now, create the names for new var (here, s only) |
---|
1864 | tmp[1] = "s"; |
---|
1865 | // DName is defined earlier |
---|
1866 | list NName = Name + DName + tmp; |
---|
1867 | L[2] = NName; |
---|
1868 | tmp = 0; |
---|
1869 | // block ord (a(1,1...),dp); |
---|
1870 | string s = "iv="; |
---|
1871 | for(i=1; i<=Nnew-1; i++) |
---|
1872 | { |
---|
1873 | s = s+"1,"; |
---|
1874 | } |
---|
1875 | s[size(s)]= ";"; |
---|
1876 | execute(s); |
---|
1877 | tmp[1] = "a"; // string |
---|
1878 | tmp[2] = iv; //intvec |
---|
1879 | Lord[1] = tmp; |
---|
1880 | // continue with dp 1,1,1,1... |
---|
1881 | tmp[1] = "dp"; // string |
---|
1882 | s[size(s)]=","; s= s+"1;"; |
---|
1883 | execute(s); |
---|
1884 | kill s; |
---|
1885 | kill NName; |
---|
1886 | tmp[2] = iv; |
---|
1887 | Lord[2] = tmp; |
---|
1888 | tmp[1] = "C"; iv = 0; tmp[2] = iv; |
---|
1889 | Lord[3] = tmp; tmp = 0; |
---|
1890 | L[3] = Lord; |
---|
1891 | // we are done with the list. Now add a Plural part |
---|
1892 | def @R3 = ring(L); |
---|
1893 | setring @R3; |
---|
1894 | matrix @D[Nnew][Nnew]; |
---|
1895 | for(i=1; i<=N; i++) |
---|
1896 | { |
---|
1897 | @D[i,N+i]=1; |
---|
1898 | } |
---|
1899 | ncalgebra(1,@D); |
---|
1900 | dbprint(ppl,"// -3-1- the ring @R3(_x,_Dx,s) is ready"); |
---|
1901 | dbprint(ppl-1, @R3); |
---|
1902 | ideal MM = maxideal(1); |
---|
1903 | MM = 0,0,MM; |
---|
1904 | map R12 = @R2, MM; |
---|
1905 | ideal K = R12(K2); |
---|
1906 | poly F = imap(save,F); |
---|
1907 | K = K,F; |
---|
1908 | dbprint(ppl,"// -3-2- starting the elimination of _x,_Dx in @R3"); |
---|
1909 | dbprint(ppl-1, K); |
---|
1910 | ideal M = engine(K,eng); |
---|
1911 | ideal K3 = nselect(M,1,Nnew-1); |
---|
1912 | dbprint(ppl,"// -3-3- _x,_Dx are eliminated in @R3"); |
---|
1913 | dbprint(ppl-1, K3); |
---|
1914 | // the ring @R4 and the search for minimal negative int s |
---|
1915 | ring @R4 = 0,(s),dp; |
---|
1916 | dbprint(ppl,"// -4-1- the ring @R4 is ready"); |
---|
1917 | ideal K4 = imap(@R3,K3); |
---|
1918 | poly p = K4[1]; |
---|
1919 | dbprint(ppl,"// -4-2- factorization"); |
---|
1920 | //// ideal P = factorize(p,1); // without constants and multiplicities |
---|
1921 | list P = factorize(p); // with constants and multiplicities |
---|
1922 | ideal bs; intvec m; // the Bernstein polynomial is monic, so we are not interested in constants |
---|
1923 | for (i=2; i<=size(P[1]); i++) // we delete P[1][1] and P[2][1] |
---|
1924 | { |
---|
1925 | bs[i-1] = P[1][i]; |
---|
1926 | m[i-1] = P[2][i]; |
---|
1927 | } |
---|
1928 | // "------ b-function factorizes into ----------"; P; |
---|
1929 | //// int sP = minIntRoot(P, 1); |
---|
1930 | int sP = minIntRoot(bs,1); |
---|
1931 | dbprint(ppl,"// -4-3- minimal integer root found"); |
---|
1932 | dbprint(ppl-1, sP); |
---|
1933 | // convert factors to a list of their roots |
---|
1934 | // assume all factors are linear |
---|
1935 | //// ideal BS = normalize(P); |
---|
1936 | //// BS = subst(BS,s,0); |
---|
1937 | //// BS = -BS; |
---|
1938 | bs = normalize(bs); |
---|
1939 | bs = subst(bs,s,0); |
---|
1940 | bs = -bs; |
---|
1941 | list BS = bs,m; |
---|
1942 | // TODO: sort BS! |
---|
1943 | // ------ substitute s found in the ideal ------ |
---|
1944 | // ------- going back to @R2 and substitute -------- |
---|
1945 | setring @R2; |
---|
1946 | ideal K3 = subst(K2,s,sP); |
---|
1947 | // create the ordinary Weyl algebra and put the result into it, |
---|
1948 | // thus creating the ring @R5 |
---|
1949 | // keep: N, i,j,s, tmp, RL |
---|
1950 | setring save; |
---|
1951 | Nnew = 2*N; |
---|
1952 | // list RL = ringlist(save); // is defined earlier |
---|
1953 | kill Lord, tmp, iv; |
---|
1954 | L = 0; |
---|
1955 | list Lord, tmp; |
---|
1956 | intvec iv; |
---|
1957 | L[1] = RL[1]; L[4] = RL[4]; // char, minpoly |
---|
1958 | // check whether vars have admissible names -> done earlier |
---|
1959 | // list Name = RL[2]; |
---|
1960 | // DName is defined earlier |
---|
1961 | list NName = Name + DName; |
---|
1962 | L[2] = NName; |
---|
1963 | // dp ordering; |
---|
1964 | string s = "iv="; |
---|
1965 | for(i=1;i<=Nnew;i++) |
---|
1966 | { |
---|
1967 | s = s+"1,"; |
---|
1968 | } |
---|
1969 | s[size(s)]= ";"; |
---|
1970 | execute(s); |
---|
1971 | tmp = 0; |
---|
1972 | tmp[1] = "dp"; // string |
---|
1973 | tmp[2] = iv; //intvec |
---|
1974 | Lord[1] = tmp; |
---|
1975 | kill s; |
---|
1976 | tmp[1] = "C"; |
---|
1977 | iv = 0; |
---|
1978 | tmp[2] = iv; |
---|
1979 | Lord[2] = tmp; |
---|
1980 | tmp = 0; |
---|
1981 | L[3] = Lord; |
---|
1982 | // we are done with the list |
---|
1983 | // Add: Plural part |
---|
1984 | def @R5 = ring(L); |
---|
1985 | setring @R5; |
---|
1986 | matrix @D[Nnew][Nnew]; |
---|
1987 | for(i=1; i<=N; i++) |
---|
1988 | { |
---|
1989 | @D[i,N+i]=1; |
---|
1990 | } |
---|
1991 | ncalgebra(1,@D); |
---|
1992 | dbprint(ppl,"// -5-1- the ring @R5 is ready"); |
---|
1993 | dbprint(ppl-1, @R5); |
---|
1994 | ideal K5 = imap(@R2,K3); |
---|
1995 | option(redSB); |
---|
1996 | dbprint(ppl,"// -5-2- the final cosmetic std"); |
---|
1997 | K5 = engine(K5,eng); // std does the job too |
---|
1998 | // total cleanup |
---|
1999 | kill @R; |
---|
2000 | kill @R2; |
---|
2001 | kill @R3; |
---|
2002 | //// ideal BS = imap(@R4,BS); |
---|
2003 | list BS = imap(@R4,BS); |
---|
2004 | export BS; |
---|
2005 | ideal LD = K5; |
---|
2006 | kill @R4; |
---|
2007 | export LD; |
---|
2008 | return(@R5); |
---|
2009 | } |
---|
2010 | example |
---|
2011 | { |
---|
2012 | "EXAMPLE:"; echo = 2; |
---|
2013 | ring r = 0,(x,y,z),Dp; |
---|
2014 | poly F = x^2+y^3+z^5; |
---|
2015 | printlevel = 0; |
---|
2016 | def A = annfsOT(F); |
---|
2017 | setring A; |
---|
2018 | LD; |
---|
2019 | BS; |
---|
2020 | } |
---|
2021 | |
---|
2022 | |
---|
2023 | proc SannfsOT(poly F, list #) |
---|
2024 | "USAGE: SannfsOT(f [,eng]); f a poly, eng an optional int |
---|
2025 | RETURN: ring |
---|
2026 | PURPOSE: compute the D-module structure of basering[f^s], according to the 1st step of the algorithm by Oaku and Takayama in the ring D[s], where D is the Weyl algebra |
---|
2027 | NOTE: activate this ring with the @code{setring} command. |
---|
2028 | @* In the ring D[s], the ideal LD is the needed D-mod structure. |
---|
2029 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
2030 | @* otherwise, and by default @code{slimgb} is used. |
---|
2031 | @* If printlevel=1, progress debug messages will be printed, |
---|
2032 | @* if printlevel>=2, all the debug messages will be printed. |
---|
2033 | EXAMPLE: example SannfsOT; shows examples |
---|
2034 | " |
---|
2035 | { |
---|
2036 | int eng = 0; |
---|
2037 | if ( size(#)>0 ) |
---|
2038 | { |
---|
2039 | if ( typeof(#[1]) == "int" ) |
---|
2040 | { |
---|
2041 | eng = int(#[1]); |
---|
2042 | } |
---|
2043 | } |
---|
2044 | // returns a list with a ring and an ideal LD in it |
---|
2045 | int ppl = printlevel-voice+2; |
---|
2046 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
2047 | def save = basering; |
---|
2048 | int N = nvars(basering); |
---|
2049 | int Nnew = 2*(N+2); |
---|
2050 | int i,j; |
---|
2051 | string s; |
---|
2052 | list RL = ringlist(basering); |
---|
2053 | list L, Lord; |
---|
2054 | list tmp; |
---|
2055 | intvec iv; |
---|
2056 | L[1] = RL[1]; // char |
---|
2057 | L[4] = RL[4]; // char, minpoly |
---|
2058 | // check whether vars have admissible names |
---|
2059 | list Name = RL[2]; |
---|
2060 | list RName; |
---|
2061 | RName[1] = "u"; |
---|
2062 | RName[2] = "v"; |
---|
2063 | RName[3] = "t"; |
---|
2064 | RName[4] = "Dt"; |
---|
2065 | for(i=1;i<=N;i++) |
---|
2066 | { |
---|
2067 | for(j=1; j<=size(RName);j++) |
---|
2068 | { |
---|
2069 | if (Name[i] == RName[j]) |
---|
2070 | { |
---|
2071 | ERROR("Variable names should not include u,v,t,Dt"); |
---|
2072 | } |
---|
2073 | } |
---|
2074 | } |
---|
2075 | // now, create the names for new vars |
---|
2076 | tmp[1] = "u"; |
---|
2077 | tmp[2] = "v"; |
---|
2078 | list UName = tmp; |
---|
2079 | list DName; |
---|
2080 | for(i=1;i<=N;i++) |
---|
2081 | { |
---|
2082 | DName[i] = "D"+Name[i]; // concat |
---|
2083 | } |
---|
2084 | tmp = 0; |
---|
2085 | tmp[1] = "t"; |
---|
2086 | tmp[2] = "Dt"; |
---|
2087 | list NName = UName + tmp + Name + DName; |
---|
2088 | L[2] = NName; |
---|
2089 | tmp = 0; |
---|
2090 | // Name, Dname will be used further |
---|
2091 | kill UName; |
---|
2092 | kill NName; |
---|
2093 | // block ord (a(1,1),dp); |
---|
2094 | tmp[1] = "a"; // string |
---|
2095 | iv = 1,1; |
---|
2096 | tmp[2] = iv; //intvec |
---|
2097 | Lord[1] = tmp; |
---|
2098 | // continue with dp 1,1,1,1... |
---|
2099 | tmp[1] = "dp"; // string |
---|
2100 | s = "iv="; |
---|
2101 | for(i=1;i<=Nnew;i++) |
---|
2102 | { |
---|
2103 | s = s+"1,"; |
---|
2104 | } |
---|
2105 | s[size(s)]= ";"; |
---|
2106 | execute(s); |
---|
2107 | tmp[2] = iv; |
---|
2108 | Lord[2] = tmp; |
---|
2109 | tmp[1] = "C"; |
---|
2110 | iv = 0; |
---|
2111 | tmp[2] = iv; |
---|
2112 | Lord[3] = tmp; |
---|
2113 | tmp = 0; |
---|
2114 | L[3] = Lord; |
---|
2115 | // we are done with the list |
---|
2116 | def @R = ring(L); |
---|
2117 | setring @R; |
---|
2118 | matrix @D[Nnew][Nnew]; |
---|
2119 | @D[3,4]=1; |
---|
2120 | for(i=1; i<=N; i++) |
---|
2121 | { |
---|
2122 | @D[4+i,N+4+i]=1; |
---|
2123 | } |
---|
2124 | // @D[N+3,2*(N+2)]=1; old t,Dt stuff |
---|
2125 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
2126 | // L[6] = @D; |
---|
2127 | ncalgebra(1,@D); |
---|
2128 | dbprint(ppl,"// -1-1- the ring @R(u,v,t,Dt,_x,_Dx) is ready"); |
---|
2129 | dbprint(ppl-1, @R); |
---|
2130 | // create the ideal I |
---|
2131 | poly F = imap(save,F); |
---|
2132 | ideal I = u*F-t,u*v-1; |
---|
2133 | poly p; |
---|
2134 | for(i=1; i<=N; i++) |
---|
2135 | { |
---|
2136 | p = u*Dt; // u*Dt |
---|
2137 | p = diff(F,var(4+i))*p; |
---|
2138 | I = I, var(N+4+i) + p; |
---|
2139 | } |
---|
2140 | // -------- the ideal I is ready ---------- |
---|
2141 | dbprint(ppl,"// -1-2- starting the elimination of u,v in @R"); |
---|
2142 | dbprint(ppl-1, I); |
---|
2143 | ideal J = engine(I,eng); |
---|
2144 | ideal K = nselect(J,1,2); |
---|
2145 | dbprint(ppl,"// -1-3- u,v are eliminated"); |
---|
2146 | dbprint(ppl-1, K); // K is without u,v |
---|
2147 | |
---|
2148 | |
---|
2149 | setring save; |
---|
2150 | // ------------ new ring @R2 ------------------ |
---|
2151 | // without u,v and with the elim.ord for t,Dt |
---|
2152 | // tensored with the K[s] |
---|
2153 | // keep: N, i,j,s, tmp, RL |
---|
2154 | Nnew = 2*N+2+1; |
---|
2155 | // list RL = ringlist(save); // is defined earlier |
---|
2156 | L = 0; // kill L; |
---|
2157 | kill Lord, tmp, iv, RName; |
---|
2158 | list Lord, tmp; |
---|
2159 | intvec iv; |
---|
2160 | L[1] = RL[1]; L[4] = RL[4]; // char, minpoly |
---|
2161 | // check whether vars have admissible names -> done earlier |
---|
2162 | // list Name = RL[2]; |
---|
2163 | list RName; |
---|
2164 | RName[1] = "t"; |
---|
2165 | RName[2] = "Dt"; |
---|
2166 | // now, create the names for new var (here, s only) |
---|
2167 | tmp[1] = "s"; |
---|
2168 | // DName is defined earlier |
---|
2169 | list NName = RName + Name + DName + tmp; |
---|
2170 | L[2] = NName; |
---|
2171 | tmp = 0; |
---|
2172 | // block ord (a(1,1),dp); |
---|
2173 | tmp[1] = "a"; iv = 1,1; tmp[2] = iv; //intvec |
---|
2174 | Lord[1] = tmp; |
---|
2175 | // continue with a(1,1,1,1)... |
---|
2176 | tmp[1] = "dp"; s = "iv="; |
---|
2177 | for(i=1; i<= Nnew; i++) |
---|
2178 | { |
---|
2179 | s = s+"1,"; |
---|
2180 | } |
---|
2181 | s[size(s)]= ";"; execute(s); |
---|
2182 | kill NName; |
---|
2183 | tmp[2] = iv; |
---|
2184 | Lord[2] = tmp; |
---|
2185 | // extra block for s |
---|
2186 | // tmp[1] = "dp"; iv = 1; |
---|
2187 | // s[size(s)]= ","; s = s + "1,1,1;"; execute(s); tmp[2] = iv; |
---|
2188 | // Lord[3] = tmp; |
---|
2189 | kill s; |
---|
2190 | tmp[1] = "C"; iv = 0; tmp[2] = iv; |
---|
2191 | Lord[3] = tmp; tmp = 0; |
---|
2192 | L[3] = Lord; |
---|
2193 | // we are done with the list. Now, add a Plural part |
---|
2194 | def @R2 = ring(L); |
---|
2195 | setring @R2; |
---|
2196 | matrix @D[Nnew][Nnew]; |
---|
2197 | @D[1,2] = 1; |
---|
2198 | for(i=1; i<=N; i++) |
---|
2199 | { |
---|
2200 | @D[2+i,2+N+i] = 1; |
---|
2201 | } |
---|
2202 | ncalgebra(1,@D); |
---|
2203 | dbprint(ppl,"// -2-1- the ring @R2(t,Dt,_x,_Dx,s) is ready"); |
---|
2204 | dbprint(ppl-1, @R2); |
---|
2205 | ideal MM = maxideal(1); |
---|
2206 | MM = 0,0,MM; |
---|
2207 | map R01 = @R, MM; |
---|
2208 | ideal K = R01(K); |
---|
2209 | // ideal K = imap(@R,K); // names of vars are important! |
---|
2210 | poly G = t*Dt+s+1; // s is a variable here |
---|
2211 | K = NF(K,std(G)),G; |
---|
2212 | // -------- the ideal K_(@R2) is ready ---------- |
---|
2213 | dbprint(ppl,"// -2-2- starting the elimination of t,Dt in @R2"); |
---|
2214 | dbprint(ppl-1, K); |
---|
2215 | ideal M = engine(K,eng); |
---|
2216 | ideal K2 = nselect(M,1,2); |
---|
2217 | dbprint(ppl,"// -2-3- t,Dt are eliminated"); |
---|
2218 | dbprint(ppl-1, K2); |
---|
2219 | K2 = engine(K2,eng); |
---|
2220 | setring save; |
---|
2221 | // ----------- the ring @R3 ------------ |
---|
2222 | // _x, _Dx,s; elim.ord for _x,_Dx. |
---|
2223 | // keep: N, i,j,s, tmp, RL |
---|
2224 | Nnew = 2*N+1; |
---|
2225 | kill Lord, tmp, iv, RName; |
---|
2226 | list Lord, tmp; |
---|
2227 | intvec iv; |
---|
2228 | L[1] = RL[1]; |
---|
2229 | L[4] = RL[4]; // char, minpoly |
---|
2230 | // check whether vars hava admissible names -> done earlier |
---|
2231 | // now, create the names for new var |
---|
2232 | tmp[1] = "s"; |
---|
2233 | // DName is defined earlier |
---|
2234 | list NName = Name + DName + tmp; |
---|
2235 | L[2] = NName; |
---|
2236 | tmp = 0; |
---|
2237 | // block ord (dp(N),dp); |
---|
2238 | string s = "iv="; |
---|
2239 | for (i=1; i<=Nnew-1; i++) |
---|
2240 | { |
---|
2241 | s = s+"1,"; |
---|
2242 | } |
---|
2243 | s[size(s)]=";"; |
---|
2244 | execute(s); |
---|
2245 | tmp[1] = "dp"; // string |
---|
2246 | tmp[2] = iv; // intvec |
---|
2247 | Lord[1] = tmp; |
---|
2248 | // continue with dp 1,1,1,1... |
---|
2249 | tmp[1] = "dp"; // string |
---|
2250 | s[size(s)] = ","; |
---|
2251 | s = s+"1;"; |
---|
2252 | execute(s); |
---|
2253 | kill s; |
---|
2254 | kill NName; |
---|
2255 | tmp[2] = iv; |
---|
2256 | Lord[2] = tmp; |
---|
2257 | tmp[1] = "C"; iv = 0; tmp[2]=iv; |
---|
2258 | Lord[3] = tmp; tmp = 0; |
---|
2259 | L[3] = Lord; |
---|
2260 | // we are done with the list. Now add a Plural part |
---|
2261 | def @R3 = ring(L); |
---|
2262 | setring @R3; |
---|
2263 | matrix @D[Nnew][Nnew]; |
---|
2264 | for (i=1; i<=N; i++) |
---|
2265 | { |
---|
2266 | @D[i,N+i]=1; |
---|
2267 | } |
---|
2268 | ncalgebra(1,@D); |
---|
2269 | dbprint(ppl,"// -2-1- the ring @R3(_x,_Dx,s) is ready"); |
---|
2270 | dbprint(ppl-1, @R3); |
---|
2271 | ideal MM = maxideal(1); |
---|
2272 | MM = 0,s,MM; |
---|
2273 | map R01 = @R2, MM; |
---|
2274 | ideal K2 = R01(K2); |
---|
2275 | // total cleanup |
---|
2276 | ideal LD = K2; |
---|
2277 | // make leadcoeffs positive |
---|
2278 | for (i=1; i<= ncols(LD); i++) |
---|
2279 | { |
---|
2280 | if (leadcoef(LD[i]) <0 ) |
---|
2281 | { |
---|
2282 | LD[i] = -LD[i]; |
---|
2283 | } |
---|
2284 | } |
---|
2285 | export LD; |
---|
2286 | kill @R; |
---|
2287 | kill @R2; |
---|
2288 | return(@R3); |
---|
2289 | } |
---|
2290 | example |
---|
2291 | { |
---|
2292 | "EXAMPLE:"; echo = 2; |
---|
2293 | ring r = 0,(x,y,z),Dp; |
---|
2294 | poly F = x^3+y^3+z^3; |
---|
2295 | printlevel = 0; |
---|
2296 | def A = SannfsOT(F); |
---|
2297 | setring A; |
---|
2298 | LD; |
---|
2299 | } |
---|
2300 | |
---|
2301 | proc SannfsBM(poly F, list #) |
---|
2302 | "USAGE: SannfsBM(f [,eng]); f a poly, eng an optional int |
---|
2303 | RETURN: ring |
---|
2304 | PURPOSE: compute the D-module structure of basering[f^s], according to the 1st step of the algorithm by Briancon and Maisonobe in the ring D[s], where D is the Weyl algebra |
---|
2305 | NOTE: activate this ring with the @code{setring} command. |
---|
2306 | @* In the ring D[s], the ideal LD is the needed D-mod structure, |
---|
2307 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
2308 | @* otherwise, and by default @code{slimgb} is used. |
---|
2309 | @* If printlevel=1, progress debug messages will be printed, |
---|
2310 | @* if printlevel>=2, all the debug messages will be printed. |
---|
2311 | EXAMPLE: example SannfsBM; shows examples |
---|
2312 | " |
---|
2313 | { |
---|
2314 | int eng = 0; |
---|
2315 | if ( size(#)>0 ) |
---|
2316 | { |
---|
2317 | if ( typeof(#[1]) == "int" ) |
---|
2318 | { |
---|
2319 | eng = int(#[1]); |
---|
2320 | } |
---|
2321 | } |
---|
2322 | // returns a list with a ring and an ideal LD in it |
---|
2323 | int ppl = printlevel-voice+2; |
---|
2324 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
2325 | def save = basering; |
---|
2326 | int N = nvars(basering); |
---|
2327 | int Nnew = 2*N+2; |
---|
2328 | int i,j; |
---|
2329 | string s; |
---|
2330 | list RL = ringlist(basering); |
---|
2331 | list L, Lord; |
---|
2332 | list tmp; |
---|
2333 | intvec iv; |
---|
2334 | L[1] = RL[1]; // char |
---|
2335 | L[4] = RL[4]; // char, minpoly |
---|
2336 | // check whether vars have admissible names |
---|
2337 | list Name = RL[2]; |
---|
2338 | list RName; |
---|
2339 | RName[1] = "t"; |
---|
2340 | RName[2] = "s"; |
---|
2341 | for(i=1;i<=N;i++) |
---|
2342 | { |
---|
2343 | for(j=1; j<=size(RName);j++) |
---|
2344 | { |
---|
2345 | if (Name[i] == RName[j]) |
---|
2346 | { |
---|
2347 | ERROR("Variable names should not include t,s"); |
---|
2348 | } |
---|
2349 | } |
---|
2350 | } |
---|
2351 | // now, create the names for new vars |
---|
2352 | list DName; |
---|
2353 | for(i=1;i<=N;i++) |
---|
2354 | { |
---|
2355 | DName[i] = "D"+Name[i]; // concat |
---|
2356 | } |
---|
2357 | tmp[1] = "t"; |
---|
2358 | tmp[2] = "s"; |
---|
2359 | list NName = tmp + Name + DName; |
---|
2360 | L[2] = NName; |
---|
2361 | // Name, Dname will be used further |
---|
2362 | kill NName; |
---|
2363 | // block ord (lp(2),dp); |
---|
2364 | tmp[1] = "lp"; // string |
---|
2365 | iv = 1,1; |
---|
2366 | tmp[2] = iv; //intvec |
---|
2367 | Lord[1] = tmp; |
---|
2368 | // continue with dp 1,1,1,1... |
---|
2369 | tmp[1] = "dp"; // string |
---|
2370 | s = "iv="; |
---|
2371 | for(i=1;i<=Nnew;i++) |
---|
2372 | { |
---|
2373 | s = s+"1,"; |
---|
2374 | } |
---|
2375 | s[size(s)]= ";"; |
---|
2376 | execute(s); |
---|
2377 | kill s; |
---|
2378 | tmp[2] = iv; |
---|
2379 | Lord[2] = tmp; |
---|
2380 | tmp[1] = "C"; |
---|
2381 | iv = 0; |
---|
2382 | tmp[2] = iv; |
---|
2383 | Lord[3] = tmp; |
---|
2384 | tmp = 0; |
---|
2385 | L[3] = Lord; |
---|
2386 | // we are done with the list |
---|
2387 | def @R = ring(L); |
---|
2388 | setring @R; |
---|
2389 | matrix @D[Nnew][Nnew]; |
---|
2390 | @D[1,2]=t; |
---|
2391 | for(i=1; i<=N; i++) |
---|
2392 | { |
---|
2393 | @D[2+i,N+2+i]=1; |
---|
2394 | } |
---|
2395 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
2396 | // L[6] = @D; |
---|
2397 | ncalgebra(1,@D); |
---|
2398 | dbprint(ppl,"// -1-1- the ring @R(t,s,_x,_Dx) is ready"); |
---|
2399 | dbprint(ppl-1, @R); |
---|
2400 | // create the ideal I |
---|
2401 | poly F = imap(save,F); |
---|
2402 | ideal I = t*F+s; |
---|
2403 | poly p; |
---|
2404 | for(i=1; i<=N; i++) |
---|
2405 | { |
---|
2406 | p = t; // t |
---|
2407 | p = diff(F,var(2+i))*p; |
---|
2408 | I = I, var(N+2+i) + p; |
---|
2409 | } |
---|
2410 | // -------- the ideal I is ready ---------- |
---|
2411 | dbprint(ppl,"// -1-2- starting the elimination of t in @R"); |
---|
2412 | dbprint(ppl-1, I); |
---|
2413 | ideal J = engine(I,eng); |
---|
2414 | ideal K = nselect(J,1); |
---|
2415 | dbprint(ppl,"// -1-3- t is eliminated"); |
---|
2416 | dbprint(ppl-1, K); // K is without t |
---|
2417 | K = engine(K,eng); // std does the job too |
---|
2418 | // now, we must change the ordering |
---|
2419 | // and create a ring without t, Dt |
---|
2420 | setring save; |
---|
2421 | // ----------- the ring @R3 ------------ |
---|
2422 | // _x, _Dx,s; elim.ord for _x,_Dx. |
---|
2423 | // keep: N, i,j,s, tmp, RL |
---|
2424 | Nnew = 2*N+1; |
---|
2425 | kill Lord, tmp, iv, RName; |
---|
2426 | list Lord, tmp; |
---|
2427 | intvec iv; |
---|
2428 | L[1] = RL[1]; |
---|
2429 | L[4] = RL[4]; // char, minpoly |
---|
2430 | // check whether vars hava admissible names -> done earlier |
---|
2431 | // now, create the names for new var |
---|
2432 | tmp[1] = "s"; |
---|
2433 | // DName is defined earlier |
---|
2434 | list NName = Name + DName + tmp; |
---|
2435 | L[2] = NName; |
---|
2436 | tmp = 0; |
---|
2437 | // block ord (dp(N),dp); |
---|
2438 | string s = "iv="; |
---|
2439 | for (i=1; i<=Nnew-1; i++) |
---|
2440 | { |
---|
2441 | s = s+"1,"; |
---|
2442 | } |
---|
2443 | s[size(s)]=";"; |
---|
2444 | execute(s); |
---|
2445 | tmp[1] = "dp"; // string |
---|
2446 | tmp[2] = iv; // intvec |
---|
2447 | Lord[1] = tmp; |
---|
2448 | // continue with dp 1,1,1,1... |
---|
2449 | tmp[1] = "dp"; // string |
---|
2450 | s[size(s)] = ","; |
---|
2451 | s = s+"1;"; |
---|
2452 | execute(s); |
---|
2453 | kill s; |
---|
2454 | kill NName; |
---|
2455 | tmp[2] = iv; |
---|
2456 | Lord[2] = tmp; |
---|
2457 | tmp[1] = "C"; iv = 0; tmp[2]=iv; |
---|
2458 | Lord[3] = tmp; tmp = 0; |
---|
2459 | L[3] = Lord; |
---|
2460 | // we are done with the list. Now add a Plural part |
---|
2461 | def @R2 = ring(L); |
---|
2462 | setring @R2; |
---|
2463 | matrix @D[Nnew][Nnew]; |
---|
2464 | for (i=1; i<=N; i++) |
---|
2465 | { |
---|
2466 | @D[i,N+i]=1; |
---|
2467 | } |
---|
2468 | ncalgebra(1,@D); |
---|
2469 | dbprint(ppl,"// -2-1- the ring @R2(_x,_Dx,s) is ready"); |
---|
2470 | dbprint(ppl-1, @R2); |
---|
2471 | ideal MM = maxideal(1); |
---|
2472 | MM = 0,s,MM; |
---|
2473 | map R01 = @R, MM; |
---|
2474 | ideal K = R01(K); |
---|
2475 | // total cleanup |
---|
2476 | ideal LD = K; |
---|
2477 | // make leadcoeffs positive |
---|
2478 | for (i=1; i<= ncols(LD); i++) |
---|
2479 | { |
---|
2480 | if (leadcoef(LD[i]) <0 ) |
---|
2481 | { |
---|
2482 | LD[i] = -LD[i]; |
---|
2483 | } |
---|
2484 | } |
---|
2485 | export LD; |
---|
2486 | kill @R; |
---|
2487 | return(@R2); |
---|
2488 | } |
---|
2489 | example |
---|
2490 | { |
---|
2491 | "EXAMPLE:"; echo = 2; |
---|
2492 | ring r = 0,(x,y,z),Dp; |
---|
2493 | poly F = x^3+y^3+z^3; |
---|
2494 | printlevel = 0; |
---|
2495 | def A = SannfsBM(F); |
---|
2496 | setring A; |
---|
2497 | LD; |
---|
2498 | } |
---|
2499 | |
---|
2500 | proc SannfsLOT(poly F, list #) |
---|
2501 | "USAGE: SannfsLOT(f [,eng]); f a poly, eng an optional int |
---|
2502 | RETURN: ring |
---|
2503 | PURPOSE: compute the D-module structure of basering[f^s], according to the Levandovskyy's modification of the algorithm by Oaku and Takayama in the ring D[s], where D is the Weyl algebra |
---|
2504 | NOTE: activate this ring with the @code{setring} command. |
---|
2505 | @* In the ring D[s], the ideal LD is the needed D-mod structure. |
---|
2506 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
2507 | @* otherwise, and by default @code{slimgb} is used. |
---|
2508 | @* If printlevel=1, progress debug messages will be printed, |
---|
2509 | @* if printlevel>=2, all the debug messages will be printed. |
---|
2510 | EXAMPLE: example SannfsLOT; shows examples |
---|
2511 | " |
---|
2512 | { |
---|
2513 | int eng = 0; |
---|
2514 | if ( size(#)>0 ) |
---|
2515 | { |
---|
2516 | if ( typeof(#[1]) == "int" ) |
---|
2517 | { |
---|
2518 | eng = int(#[1]); |
---|
2519 | } |
---|
2520 | } |
---|
2521 | // returns a list with a ring and an ideal LD in it |
---|
2522 | int ppl = printlevel-voice+2; |
---|
2523 | // printf("plevel :%s, voice: %s",printlevel,voice); |
---|
2524 | def save = basering; |
---|
2525 | int N = nvars(basering); |
---|
2526 | // int Nnew = 2*(N+2); |
---|
2527 | int Nnew = 2*(N+1)+1; //removed u,v; added s |
---|
2528 | int i,j; |
---|
2529 | string s; |
---|
2530 | list RL = ringlist(basering); |
---|
2531 | list L, Lord; |
---|
2532 | list tmp; |
---|
2533 | intvec iv; |
---|
2534 | L[1] = RL[1]; // char |
---|
2535 | L[4] = RL[4]; // char, minpoly |
---|
2536 | // check whether vars have admissible names |
---|
2537 | list Name = RL[2]; |
---|
2538 | list RName; |
---|
2539 | // RName[1] = "u"; |
---|
2540 | // RName[2] = "v"; |
---|
2541 | RName[1] = "t"; |
---|
2542 | RName[2] = "Dt"; |
---|
2543 | for(i=1;i<=N;i++) |
---|
2544 | { |
---|
2545 | for(j=1; j<=size(RName);j++) |
---|
2546 | { |
---|
2547 | if (Name[i] == RName[j]) |
---|
2548 | { |
---|
2549 | ERROR("Variable names should not include t,Dt"); |
---|
2550 | } |
---|
2551 | } |
---|
2552 | } |
---|
2553 | // now, create the names for new vars |
---|
2554 | // tmp[1] = "u"; |
---|
2555 | // tmp[2] = "v"; |
---|
2556 | // list UName = tmp; |
---|
2557 | list DName; |
---|
2558 | for(i=1;i<=N;i++) |
---|
2559 | { |
---|
2560 | DName[i] = "D"+Name[i]; // concat |
---|
2561 | } |
---|
2562 | tmp = 0; |
---|
2563 | tmp[1] = "t"; |
---|
2564 | tmp[2] = "Dt"; |
---|
2565 | list SName ; SName[1] = "s"; |
---|
2566 | // list NName = UName + tmp + Name + DName; |
---|
2567 | list NName = tmp + Name + DName + SName; |
---|
2568 | L[2] = NName; |
---|
2569 | tmp = 0; |
---|
2570 | // Name, Dname will be used further |
---|
2571 | // kill UName; |
---|
2572 | kill NName; |
---|
2573 | // block ord (a(1,1),dp); |
---|
2574 | tmp[1] = "a"; // string |
---|
2575 | iv = 1,1; |
---|
2576 | tmp[2] = iv; //intvec |
---|
2577 | Lord[1] = tmp; |
---|
2578 | // continue with dp 1,1,1,1... |
---|
2579 | tmp[1] = "dp"; // string |
---|
2580 | s = "iv="; |
---|
2581 | for(i=1;i<=Nnew;i++) |
---|
2582 | { |
---|
2583 | s = s+"1,"; |
---|
2584 | } |
---|
2585 | s[size(s)]= ";"; |
---|
2586 | execute(s); |
---|
2587 | tmp[2] = iv; |
---|
2588 | Lord[2] = tmp; |
---|
2589 | tmp[1] = "C"; |
---|
2590 | iv = 0; |
---|
2591 | tmp[2] = iv; |
---|
2592 | Lord[3] = tmp; |
---|
2593 | tmp = 0; |
---|
2594 | L[3] = Lord; |
---|
2595 | // we are done with the list |
---|
2596 | def @R = ring(L); |
---|
2597 | setring @R; |
---|
2598 | matrix @D[Nnew][Nnew]; |
---|
2599 | @D[1,2]=1; |
---|
2600 | for(i=1; i<=N; i++) |
---|
2601 | { |
---|
2602 | @D[2+i,N+2+i]=1; |
---|
2603 | } |
---|
2604 | // ADD [s,t]=-t, [s,Dt]=Dt |
---|
2605 | @D[1,Nnew] = -var(1); |
---|
2606 | @D[2,Nnew] = var(2); |
---|
2607 | // @D[N+3,2*(N+2)]=1; old t,Dt stuff |
---|
2608 | // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
2609 | // L[6] = @D; |
---|
2610 | ncalgebra(1,@D); |
---|
2611 | dbprint(ppl,"// -1-1- the ring @R(t,Dt,_x,_Dx,s) is ready"); |
---|
2612 | dbprint(ppl-1, @R); |
---|
2613 | // create the ideal I |
---|
2614 | poly F = imap(save,F); |
---|
2615 | // ideal I = u*F-t,u*v-1; |
---|
2616 | ideal I = F-t; |
---|
2617 | poly p; |
---|
2618 | for(i=1; i<=N; i++) |
---|
2619 | { |
---|
2620 | // p = u*Dt; // u*Dt |
---|
2621 | p = Dt; |
---|
2622 | p = diff(F,var(2+i))*p; |
---|
2623 | I = I, var(N+2+i) + p; |
---|
2624 | } |
---|
2625 | // I = I, var(1)*var(2) + var(Nnew) +1; // reduce it with t-f!!! |
---|
2626 | // t*Dt + s +1 reduced with t-f gives f*Dt + s |
---|
2627 | I = I, F*var(2) + var(Nnew); |
---|
2628 | // -------- the ideal I is ready ---------- |
---|
2629 | dbprint(ppl,"// -1-2- starting the elimination of t,Dt in @R"); |
---|
2630 | dbprint(ppl-1, I); |
---|
2631 | ideal J = engine(I,eng); |
---|
2632 | ideal K = nselect(J,1,2); |
---|
2633 | dbprint(ppl,"// -1-3- t,Dt are eliminated"); |
---|
2634 | dbprint(ppl-1, K); // K is without t, Dt |
---|
2635 | K = engine(K,eng); // std does the job too |
---|
2636 | // now, we must change the ordering |
---|
2637 | // and create a ring without t, Dt |
---|
2638 | setring save; |
---|
2639 | // ----------- the ring @R3 ------------ |
---|
2640 | // _x, _Dx,s; elim.ord for _x,_Dx. |
---|
2641 | // keep: N, i,j,s, tmp, RL |
---|
2642 | Nnew = 2*N+1; |
---|
2643 | kill Lord, tmp, iv, RName; |
---|
2644 | list Lord, tmp; |
---|
2645 | intvec iv; |
---|
2646 | L[1] = RL[1]; |
---|
2647 | L[4] = RL[4]; // char, minpoly |
---|
2648 | // check whether vars hava admissible names -> done earlier |
---|
2649 | // now, create the names for new var |
---|
2650 | tmp[1] = "s"; |
---|
2651 | // DName is defined earlier |
---|
2652 | list NName = Name + DName + tmp; |
---|
2653 | L[2] = NName; |
---|
2654 | tmp = 0; |
---|
2655 | // block ord (dp(N),dp); |
---|
2656 | // string s is already defined |
---|
2657 | s = "iv="; |
---|
2658 | for (i=1; i<=Nnew-1; i++) |
---|
2659 | { |
---|
2660 | s = s+"1,"; |
---|
2661 | } |
---|
2662 | s[size(s)]=";"; |
---|
2663 | execute(s); |
---|
2664 | tmp[1] = "dp"; // string |
---|
2665 | tmp[2] = iv; // intvec |
---|
2666 | Lord[1] = tmp; |
---|
2667 | // continue with dp 1,1,1,1... |
---|
2668 | tmp[1] = "dp"; // string |
---|
2669 | s[size(s)] = ","; |
---|
2670 | s = s+"1;"; |
---|
2671 | execute(s); |
---|
2672 | kill s; |
---|
2673 | kill NName; |
---|
2674 | tmp[2] = iv; |
---|
2675 | Lord[2] = tmp; |
---|
2676 | tmp[1] = "C"; iv = 0; tmp[2]=iv; |
---|
2677 | Lord[3] = tmp; tmp = 0; |
---|
2678 | L[3] = Lord; |
---|
2679 | // we are done with the list. Now add a Plural part |
---|
2680 | def @R2 = ring(L); |
---|
2681 | setring @R2; |
---|
2682 | matrix @D[Nnew][Nnew]; |
---|
2683 | for (i=1; i<=N; i++) |
---|
2684 | { |
---|
2685 | @D[i,N+i]=1; |
---|
2686 | } |
---|
2687 | ncalgebra(1,@D); |
---|
2688 | dbprint(ppl,"// -2-1- the ring @R2(_x,_Dx,s) is ready"); |
---|
2689 | dbprint(ppl-1, @R2); |
---|
2690 | ideal MM = maxideal(1); |
---|
2691 | MM = 0,s,MM; |
---|
2692 | map R01 = @R, MM; |
---|
2693 | ideal K = R01(K); |
---|
2694 | // total cleanup |
---|
2695 | ideal LD = K; |
---|
2696 | // make leadcoeffs positive |
---|
2697 | for (i=1; i<= ncols(LD); i++) |
---|
2698 | { |
---|
2699 | if (leadcoef(LD[i]) <0 ) |
---|
2700 | { |
---|
2701 | LD[i] = -LD[i]; |
---|
2702 | } |
---|
2703 | } |
---|
2704 | export LD; |
---|
2705 | kill @R; |
---|
2706 | return(@R2); |
---|
2707 | } |
---|
2708 | example |
---|
2709 | { |
---|
2710 | "EXAMPLE:"; echo = 2; |
---|
2711 | ring r = 0,(x,y,z),Dp; |
---|
2712 | poly F = x^3+y^3+z^3; |
---|
2713 | printlevel = 0; |
---|
2714 | def A = SannfsLOT(F); |
---|
2715 | setring A; |
---|
2716 | LD; |
---|
2717 | } |
---|
2718 | |
---|
2719 | |
---|
2720 | proc annfsLOT(poly F, list #) |
---|
2721 | "USAGE: annfsLOT(F [,eng]); F a poly, eng an optional int |
---|
2722 | RETURN: ring |
---|
2723 | PURPOSE: compute the D-module structure of basering[f^s], according to the Levandovskyy's modification of the algorithm by Oaku and Takayama |
---|
2724 | NOTE: activate this ring with the @code{setring} command. In this ring, |
---|
2725 | @* - the ideal LD is the annihilator of f^s, |
---|
2726 | @* - the list BS contains the roots with multiplicities of a Bernstein polynomial of f. |
---|
2727 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
2728 | @* otherwise, and by default @code{slimgb} is used. |
---|
2729 | @* If printlevel=1, progress debug messages will be printed, |
---|
2730 | @* if printlevel>=2, all the debug messages will be printed. |
---|
2731 | EXAMPLE: example annfsLOT; shows examples |
---|
2732 | " |
---|
2733 | { |
---|
2734 | int eng = 0; |
---|
2735 | if ( size(#)>0 ) |
---|
2736 | { |
---|
2737 | if ( typeof(#[1]) == "int" ) |
---|
2738 | { |
---|
2739 | eng = int(#[1]); |
---|
2740 | } |
---|
2741 | } |
---|
2742 | printlevel=printlevel+1; |
---|
2743 | def save = basering; |
---|
2744 | def @A = SannfsLOT(F,eng); |
---|
2745 | setring @A; |
---|
2746 | poly F = imap(save,F); |
---|
2747 | def B = annfs0(LD,F,eng); |
---|
2748 | return(B); |
---|
2749 | } |
---|
2750 | example |
---|
2751 | { |
---|
2752 | "EXAMPLE:"; echo = 2; |
---|
2753 | ring r = 0,(x,y,z),Dp; |
---|
2754 | poly F = x^3+y^3+z^3; |
---|
2755 | printlevel = 0; |
---|
2756 | def A = annfsLOT(F); |
---|
2757 | setring A; |
---|
2758 | LD; |
---|
2759 | BS; |
---|
2760 | } |
---|
2761 | |
---|
2762 | proc annfs0(ideal I, poly F, list #) |
---|
2763 | "USAGE: annfs0(I, F [,eng]); I an ideal, F a poly, eng an optional int |
---|
2764 | RETURN: ring |
---|
2765 | PURPOSE: compute the annihilator ideal of f^s in the Weyl Algebra, based on the |
---|
2766 | output of procedures SannfsBM, SannfsOT or SannfsLOT |
---|
2767 | NOTE: activate this ring with the @code{setring} command. In this ring, |
---|
2768 | @* - the ideal LD is the annihilator of f^s, |
---|
2769 | @* - the list BS contains the roots with multiplicities of a Bernstein polynomial of f. |
---|
2770 | @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
2771 | @* otherwise, and by default @code{slimgb} is used. |
---|
2772 | @* If printlevel=1, progress debug messages will be printed, |
---|
2773 | @* if printlevel>=2, all the debug messages will be printed. |
---|
2774 | EXAMPLE: example annfs0; shows examples |
---|
2775 | " |
---|
2776 | { |
---|
2777 | int eng = 0; |
---|
2778 | if ( size(#)>0 ) |
---|
2779 | { |
---|
2780 | if ( typeof(#[1]) == "int" ) |
---|
2781 | { |
---|
2782 | eng = int(#[1]); |
---|
2783 | } |
---|
2784 | } |
---|
2785 | def @R2 = basering; |
---|
2786 | // we're in D_n[s], where the elim ord for s is set |
---|
2787 | ideal J = NF(I,std(F)); |
---|
2788 | // make leadcoeffs positive |
---|
2789 | int i; |
---|
2790 | for (i=1; i<= ncols(J); i++) |
---|
2791 | { |
---|
2792 | if (leadcoef(J[i]) <0 ) |
---|
2793 | { |
---|
2794 | J[i] = -J[i]; |
---|
2795 | } |
---|
2796 | } |
---|
2797 | J = J,F; |
---|
2798 | ideal M = engine(J,eng); |
---|
2799 | int Nnew = nvars(@R2); |
---|
2800 | ideal K2 = nselect(M,1,Nnew-1); |
---|
2801 | int ppl = printlevel-voice+2; |
---|
2802 | dbprint(ppl,"// -1-1- _x,_Dx are eliminated in basering"); |
---|
2803 | dbprint(ppl-1, K2); |
---|
2804 | // the ring @R3 and the search for minimal negative int s |
---|
2805 | ring @R3 = 0,s,dp; |
---|
2806 | dbprint(ppl,"// -2-1- the ring @R3 i.e. K[s] is ready"); |
---|
2807 | ideal K3 = imap(@R2,K2); |
---|
2808 | poly p = K3[1]; |
---|
2809 | dbprint(ppl,"// -2-2- factorization"); |
---|
2810 | // ideal P = factorize(p,1); //without constants and multiplicities |
---|
2811 | // "--------- b-function factorizes into ---------"; P; |
---|
2812 | // convert factors to the list of their roots with mults |
---|
2813 | // assume all factors are linear |
---|
2814 | // ideal BS = normalize(P); |
---|
2815 | // BS = subst(BS,s,0); |
---|
2816 | // BS = -BS; |
---|
2817 | list P = factorize(p); //with constants and multiplicities |
---|
2818 | int sP = minIntRoot(P[1],1); |
---|
2819 | ideal bs; intvec m; //the Bernstein polynomial is monic, so we are not interested in constants |
---|
2820 | for (i=2; i<= size(P[1]); i++) //we delete P[1][1] and P[2][1] |
---|
2821 | { |
---|
2822 | bs[i-1] = P[1][i]; |
---|
2823 | m[i-1] = P[2][i]; |
---|
2824 | } |
---|
2825 | bs = normalize(bs); |
---|
2826 | bs = -subst(bs,s,0); |
---|
2827 | dbprint(ppl,"// -2-3- minimal interger root found"); |
---|
2828 | dbprint(ppl-1, sP); |
---|
2829 | //TODO: sort BS! |
---|
2830 | // --------- substitute s found in the ideal --------- |
---|
2831 | // --------- going back to @R and substitute --------- |
---|
2832 | setring @R2; |
---|
2833 | K2 = subst(I,s,sP); |
---|
2834 | // create the ordinary Weyl algebra and put the result into it, |
---|
2835 | // thus creating the ring @R5 |
---|
2836 | // keep: N, i,j,s, tmp, RL |
---|
2837 | Nnew = Nnew - 1; // former 2*N; |
---|
2838 | // list RL = ringlist(save); // is defined earlier |
---|
2839 | // kill Lord, tmp, iv; |
---|
2840 | list L = 0; |
---|
2841 | list Lord, tmp; |
---|
2842 | intvec iv; |
---|
2843 | list RL = ringlist(basering); |
---|
2844 | L[1] = RL[1]; |
---|
2845 | L[4] = RL[4]; //char, minpoly |
---|
2846 | // check whether vars have admissible names -> done earlier |
---|
2847 | // list Name = RL[2]M |
---|
2848 | // DName is defined earlier |
---|
2849 | list NName; // = RL[2]; // skip the last var 's' |
---|
2850 | for (i=1; i<=Nnew; i++) |
---|
2851 | { |
---|
2852 | NName[i] = RL[2][i]; |
---|
2853 | } |
---|
2854 | L[2] = NName; |
---|
2855 | // dp ordering; |
---|
2856 | string s = "iv="; |
---|
2857 | for (i=1; i<=Nnew; i++) |
---|
2858 | { |
---|
2859 | s = s+"1,"; |
---|
2860 | } |
---|
2861 | s[size(s)] = ";"; |
---|
2862 | execute(s); |
---|
2863 | tmp = 0; |
---|
2864 | tmp[1] = "dp"; // string |
---|
2865 | tmp[2] = iv; // intvec |
---|
2866 | Lord[1] = tmp; |
---|
2867 | kill s; |
---|
2868 | tmp[1] = "C"; |
---|
2869 | iv = 0; |
---|
2870 | tmp[2] = iv; |
---|
2871 | Lord[2] = tmp; |
---|
2872 | tmp = 0; |
---|
2873 | L[3] = Lord; |
---|
2874 | // we are done with the list |
---|
2875 | // Add: Plural part |
---|
2876 | def @R4 = ring(L); |
---|
2877 | setring @R4; |
---|
2878 | int N = Nnew/2; |
---|
2879 | matrix @D[Nnew][Nnew]; |
---|
2880 | for (i=1; i<=N; i++) |
---|
2881 | { |
---|
2882 | @D[i,N+i]=1; |
---|
2883 | } |
---|
2884 | ncalgebra(1,@D); |
---|
2885 | dbprint(ppl,"// -3-1- the ring @R4 is ready"); |
---|
2886 | dbprint(ppl-1, @R4); |
---|
2887 | ideal K4 = imap(@R2,K2); |
---|
2888 | option(redSB); |
---|
2889 | dbprint(ppl,"// -3-2- the final cosmetic std"); |
---|
2890 | K4 = engine(K4,eng); // std does the job too |
---|
2891 | // total cleanup |
---|
2892 | ideal bs = imap(@R3,bs); |
---|
2893 | kill @R3; |
---|
2894 | list BS = bs,m; |
---|
2895 | export BS; |
---|
2896 | ideal LD = K4; |
---|
2897 | export LD; |
---|
2898 | return(@R4); |
---|
2899 | } |
---|
2900 | example |
---|
2901 | { "EXAMPLE:"; echo = 2; |
---|
2902 | ring r = 0,(x,y,z),Dp; |
---|
2903 | poly F = x^3+y^3+z^3; |
---|
2904 | printlevel = 0; |
---|
2905 | def A = SannfsBM(F); |
---|
2906 | // alternatively, one can use SannfsOT or SannfsLOT |
---|
2907 | setring A; |
---|
2908 | LD; |
---|
2909 | poly F = imap(r,F); |
---|
2910 | def B = annfs0(LD,F); |
---|
2911 | setring B; |
---|
2912 | LD; |
---|
2913 | BS; |
---|
2914 | } |
---|
2915 | |
---|
2916 | // proc annfsgms(poly F, list #) |
---|
2917 | // "USAGE: annfsgms(f [,eng]); f a poly, eng an optional int |
---|
2918 | // ASSUME: f has an isolated critical point at 0 |
---|
2919 | // RETURN: ring |
---|
2920 | // PURPOSE: compute the D-module structure of basering[f^s] |
---|
2921 | // NOTE: activate this ring with the @code{setring} command. In this ring, |
---|
2922 | // @* - the ideal LD is the needed D-mod structure, |
---|
2923 | // @* - the ideal BS is the list of roots of a Bernstein polynomial of f. |
---|
2924 | // @* If eng <>0, @code{std} is used for Groebner basis computations, |
---|
2925 | // @* otherwise (and by default) @code{slimgb} is used. |
---|
2926 | // @* If printlevel=1, progress debug messages will be printed, |
---|
2927 | // @* if printlevel>=2, all the debug messages will be printed. |
---|
2928 | // EXAMPLE: example annfsgms; shows examples |
---|
2929 | // " |
---|
2930 | // { |
---|
2931 | // LIB "gmssing.lib"; |
---|
2932 | // int eng = 0; |
---|
2933 | // if ( size(#)>0 ) |
---|
2934 | // { |
---|
2935 | // if ( typeof(#[1]) == "int" ) |
---|
2936 | // { |
---|
2937 | // eng = int(#[1]); |
---|
2938 | // } |
---|
2939 | // } |
---|
2940 | // int ppl = printlevel-voice+2; |
---|
2941 | // // returns a ring with the ideal LD in it |
---|
2942 | // def save = basering; |
---|
2943 | // // compute the Bernstein poly from gmssing.lib |
---|
2944 | // list RL = ringlist(basering); |
---|
2945 | // // in the descr. of the ordering, replace "p" by "s" |
---|
2946 | // list NL = convloc(RL); |
---|
2947 | // // create a ring with the ordering, converted to local |
---|
2948 | // def @LR = ring(NL); |
---|
2949 | // setring @LR; |
---|
2950 | // poly F = imap(save, F); |
---|
2951 | // ideal B = bernstein(F)[1]; |
---|
2952 | // // since B may not contain (s+1) [following gmssing.lib] |
---|
2953 | // // add it! |
---|
2954 | // B = B,-1; |
---|
2955 | // B = simplify(B,2+4); // erase zero and repeated entries |
---|
2956 | // // find the minimal integer value |
---|
2957 | // int S = minIntRoot(B,0); |
---|
2958 | // dbprint(ppl,"// -0- minimal integer root found"); |
---|
2959 | // dbprint(ppl-1,S); |
---|
2960 | // setring save; |
---|
2961 | // int N = nvars(basering); |
---|
2962 | // int Nnew = 2*(N+2); |
---|
2963 | // int i,j; |
---|
2964 | // string s; |
---|
2965 | // // list RL = ringlist(basering); |
---|
2966 | // list L, Lord; |
---|
2967 | // list tmp; |
---|
2968 | // intvec iv; |
---|
2969 | // L[1] = RL[1]; // char |
---|
2970 | // L[4] = RL[4]; // char, minpoly |
---|
2971 | // // check whether vars have admissible names |
---|
2972 | // list Name = RL[2]; |
---|
2973 | // list RName; |
---|
2974 | // RName[1] = "u"; |
---|
2975 | // RName[2] = "v"; |
---|
2976 | // RName[3] = "t"; |
---|
2977 | // RName[4] = "Dt"; |
---|
2978 | // for(i=1;i<=N;i++) |
---|
2979 | // { |
---|
2980 | // for(j=1; j<=size(RName);j++) |
---|
2981 | // { |
---|
2982 | // if (Name[i] == RName[j]) |
---|
2983 | // { |
---|
2984 | // ERROR("Variable names should not include u,v,t,Dt"); |
---|
2985 | // } |
---|
2986 | // } |
---|
2987 | // } |
---|
2988 | // // now, create the names for new vars |
---|
2989 | // // tmp[1] = "u"; tmp[2] = "v"; tmp[3] = "t"; tmp[4] = "Dt"; |
---|
2990 | // list UName = RName; |
---|
2991 | // list DName; |
---|
2992 | // for(i=1;i<=N;i++) |
---|
2993 | // { |
---|
2994 | // DName[i] = "D"+Name[i]; // concat |
---|
2995 | // } |
---|
2996 | // list NName = UName + Name + DName; |
---|
2997 | // L[2] = NName; |
---|
2998 | // tmp = 0; |
---|
2999 | // // Name, Dname will be used further |
---|
3000 | // kill UName; |
---|
3001 | // kill NName; |
---|
3002 | // // block ord (a(1,1),dp); |
---|
3003 | // tmp[1] = "a"; // string |
---|
3004 | // iv = 1,1; |
---|
3005 | // tmp[2] = iv; //intvec |
---|
3006 | // Lord[1] = tmp; |
---|
3007 | // // continue with dp 1,1,1,1... |
---|
3008 | // tmp[1] = "dp"; // string |
---|
3009 | // s = "iv="; |
---|
3010 | // for(i=1; i<=Nnew; i++) // need really all vars! |
---|
3011 | // { |
---|
3012 | // s = s+"1,"; |
---|
3013 | // } |
---|
3014 | // s[size(s)]= ";"; |
---|
3015 | // execute(s); |
---|
3016 | // tmp[2] = iv; |
---|
3017 | // Lord[2] = tmp; |
---|
3018 | // tmp[1] = "C"; |
---|
3019 | // iv = 0; |
---|
3020 | // tmp[2] = iv; |
---|
3021 | // Lord[3] = tmp; |
---|
3022 | // tmp = 0; |
---|
3023 | // L[3] = Lord; |
---|
3024 | // // we are done with the list |
---|
3025 | // def @R = ring(L); |
---|
3026 | // setring @R; |
---|
3027 | // matrix @D[Nnew][Nnew]; |
---|
3028 | // @D[3,4] = 1; // t,Dt |
---|
3029 | // for(i=1; i<=N; i++) |
---|
3030 | // { |
---|
3031 | // @D[4+i,4+N+i]=1; |
---|
3032 | // } |
---|
3033 | // // L[5] = matrix(UpOneMatrix(Nnew)); |
---|
3034 | // // L[6] = @D; |
---|
3035 | // ncalgebra(1,@D); |
---|
3036 | // dbprint(ppl,"// -1-1- the ring @R is ready"); |
---|
3037 | // dbprint(ppl-1,@R); |
---|
3038 | // // create the ideal |
---|
3039 | // poly F = imap(save,F); |
---|
3040 | // ideal I = u*F-t,u*v-1; |
---|
3041 | // poly p; |
---|
3042 | // for(i=1; i<=N; i++) |
---|
3043 | // { |
---|
3044 | // p = u*Dt; // u*Dt |
---|
3045 | // p = diff(F,var(4+i))*p; |
---|
3046 | // I = I, var(N+4+i) + p; // Dx, Dy |
---|
3047 | // } |
---|
3048 | // // add the relations between t,Dt and s |
---|
3049 | // // I = I, t*Dt+1+S; |
---|
3050 | // // -------- the ideal I is ready ---------- |
---|
3051 | // dbprint(ppl,"// -1-2- starting the elimination of u,v in @R"); |
---|
3052 | // ideal J = engine(I,eng); |
---|
3053 | // ideal K = nselect(J,1,2); |
---|
3054 | // dbprint(ppl,"// -1-3- u,v are eliminated in @R"); |
---|
3055 | // dbprint(ppl-1,K); // without u,v: not yet our answer |
---|
3056 | // //----- create a ring with elim.ord for t,Dt ------- |
---|
3057 | // setring save; |
---|
3058 | // // ------------ new ring @R2 ------------------ |
---|
3059 | // // without u,v and with the elim.ord for t,Dt |
---|
3060 | // // keep: N, i,j,s, tmp, RL |
---|
3061 | // Nnew = 2*N+2; |
---|
3062 | // // list RL = ringlist(save); // is defined earlier |
---|
3063 | // kill Lord,tmp,iv, RName; |
---|
3064 | // L = 0; |
---|
3065 | // list Lord, tmp; |
---|
3066 | // intvec iv; |
---|
3067 | // L[1] = RL[1]; // char |
---|
3068 | // L[4] = RL[4]; // char, minpoly |
---|
3069 | // // check whether vars have admissible names -> done earlier |
---|
3070 | // // list Name = RL[2]; |
---|
3071 | // list RName; |
---|
3072 | // RName[1] = "t"; |
---|
3073 | // RName[2] = "Dt"; |
---|
3074 | // // DName is defined earlier |
---|
3075 | // list NName = RName + Name + DName; |
---|
3076 | // L[2] = NName; |
---|
3077 | // tmp = 0; |
---|
3078 | // // block ord (a(1,1),dp); |
---|
3079 | // tmp[1] = "a"; // string |
---|
3080 | // iv = 1,1; |
---|
3081 | // tmp[2] = iv; //intvec |
---|
3082 | // Lord[1] = tmp; |
---|
3083 | // // continue with dp 1,1,1,1... |
---|
3084 | // tmp[1] = "dp"; // string |
---|
3085 | // s = "iv="; |
---|
3086 | // for(i=1;i<=Nnew;i++) |
---|
3087 | // { |
---|
3088 | // s = s+"1,"; |
---|
3089 | // } |
---|
3090 | // s[size(s)]= ";"; |
---|
3091 | // execute(s); |
---|
3092 | // kill s; |
---|
3093 | // kill NName; |
---|
3094 | // tmp[2] = iv; |
---|
3095 | // Lord[2] = tmp; |
---|
3096 | // tmp[1] = "C"; |
---|
3097 | // iv = 0; |
---|
3098 | // tmp[2] = iv; |
---|
3099 | // Lord[3] = tmp; |
---|
3100 | // tmp = 0; |
---|
3101 | // L[3] = Lord; |
---|
3102 | // // we are done with the list |
---|
3103 | // // Add: Plural part |
---|
3104 | // def @R2 = ring(L); |
---|
3105 | // setring @R2; |
---|
3106 | // matrix @D[Nnew][Nnew]; |
---|
3107 | // @D[1,2]=1; |
---|
3108 | // for(i=1; i<=N; i++) |
---|
3109 | // { |
---|
3110 | // @D[2+i,2+N+i]=1; |
---|
3111 | // } |
---|
3112 | // ncalgebra(1,@D); |
---|
3113 | // dbprint(ppl,"// -2-1- the ring @R2 is ready"); |
---|
3114 | // dbprint(ppl-1,@R2); |
---|
3115 | // ideal MM = maxideal(1); |
---|
3116 | // MM = 0,0,MM; |
---|
3117 | // map R01 = @R, MM; |
---|
3118 | // ideal K2 = R01(K); |
---|
3119 | // // add the relations between t,Dt and s |
---|
3120 | // // K2 = K2, t*Dt+1+S; |
---|
3121 | // poly G = t*Dt+S+1; |
---|
3122 | // K2 = NF(K2,std(G)),G; |
---|
3123 | // dbprint(ppl,"// -2-2- starting elimination for t,Dt in @R2"); |
---|
3124 | // ideal J = engine(K2,eng); |
---|
3125 | // ideal K = nselect(J,1,2); |
---|
3126 | // dbprint(ppl,"// -2-3- t,Dt are eliminated"); |
---|
3127 | // dbprint(ppl-1,K); |
---|
3128 | // // "------- produce a final result --------"; |
---|
3129 | // // ----- create the ordinary Weyl algebra and put |
---|
3130 | // // ----- the result into it |
---|
3131 | // // ------ create the ring @R5 |
---|
3132 | // // keep: N, i,j,s, tmp, RL |
---|
3133 | // setring save; |
---|
3134 | // Nnew = 2*N; |
---|
3135 | // // list RL = ringlist(save); // is defined earlier |
---|
3136 | // kill Lord, tmp, iv; |
---|
3137 | // // kill L; |
---|
3138 | // L = 0; |
---|
3139 | // list Lord, tmp; |
---|
3140 | // intvec iv; |
---|
3141 | // L[1] = RL[1]; // char |
---|
3142 | // L[4] = RL[4]; // char, minpoly |
---|
3143 | // // check whether vars have admissible names -> done earlier |
---|
3144 | // // list Name = RL[2]; |
---|
3145 | // // DName is defined earlier |
---|
3146 | // list NName = Name + DName; |
---|
3147 | // L[2] = NName; |
---|
3148 | // // dp ordering; |
---|
3149 | // string s = "iv="; |
---|
3150 | // for(i=1;i<=2*N;i++) |
---|
3151 | // { |
---|
3152 | // s = s+"1,"; |
---|
3153 | // } |
---|
3154 | // s[size(s)]= ";"; |
---|
3155 | // execute(s); |
---|
3156 | // tmp = 0; |
---|
3157 | // tmp[1] = "dp"; // string |
---|
3158 | // tmp[2] = iv; //intvec |
---|
3159 | // Lord[1] = tmp; |
---|
3160 | // kill s; |
---|
3161 | // tmp[1] = "C"; |
---|
3162 | // iv = 0; |
---|
3163 | // tmp[2] = iv; |
---|
3164 | // Lord[2] = tmp; |
---|
3165 | // tmp = 0; |
---|
3166 | // L[3] = Lord; |
---|
3167 | // // we are done with the list |
---|
3168 | // // Add: Plural part |
---|
3169 | // def @R5 = ring(L); |
---|
3170 | // setring @R5; |
---|
3171 | // matrix @D[Nnew][Nnew]; |
---|
3172 | // for(i=1; i<=N; i++) |
---|
3173 | // { |
---|
3174 | // @D[i,N+i]=1; |
---|
3175 | // } |
---|
3176 | // ncalgebra(1,@D); |
---|
3177 | // dbprint(ppl,"// -3-1- the ring @R5 is ready"); |
---|
3178 | // dbprint(ppl-1,@R5); |
---|
3179 | // ideal K5 = imap(@R2,K); |
---|
3180 | // option(redSB); |
---|
3181 | // dbprint(ppl,"// -3-2- the final cosmetic std"); |
---|
3182 | // K5 = engine(K5,eng); // std does the job too |
---|
3183 | // // total cleanup |
---|
3184 | // kill @R; |
---|
3185 | // kill @R2; |
---|
3186 | // ideal LD = K5; |
---|
3187 | // ideal BS = imap(@LR,B); |
---|
3188 | // kill @LR; |
---|
3189 | // export BS; |
---|
3190 | // export LD; |
---|
3191 | // return(@R5); |
---|
3192 | // } |
---|
3193 | // example |
---|
3194 | // { |
---|
3195 | // "EXAMPLE:"; echo = 2; |
---|
3196 | // ring r = 0,(x,y,z),Dp; |
---|
3197 | // poly F = x^2+y^3+z^5; |
---|
3198 | // def A = annfsgms(F); |
---|
3199 | // setring A; |
---|
3200 | // LD; |
---|
3201 | // print(matrix(BS)); |
---|
3202 | // } |
---|
3203 | |
---|
3204 | |
---|
3205 | proc convloc(list @NL) |
---|
3206 | "USAGE: convloc(L); L a list |
---|
3207 | RETURN: list |
---|
3208 | PURPOSE: convert a ringlist L into another ringlist, |
---|
3209 | where all the 'p' orderings are replaced with the 's' orderings. |
---|
3210 | ASSUME: L is a result of a ringlist command |
---|
3211 | EXAMPLE: example minIntRoot; shows examples |
---|
3212 | " |
---|
3213 | { |
---|
3214 | list NL = @NL; |
---|
3215 | // given a ringlist, returns a new ringlist, |
---|
3216 | // where all the p-orderings are replaced with s-ord's |
---|
3217 | int i,j,k,found; |
---|
3218 | int nblocks = size(NL[3]); |
---|
3219 | for(i=1; i<=nblocks; i++) |
---|
3220 | { |
---|
3221 | for(j=1; j<=size(NL[3][i]); j++) |
---|
3222 | { |
---|
3223 | if (typeof(NL[3][i][j]) == "string" ) |
---|
3224 | { |
---|
3225 | found = 0; |
---|
3226 | for (k=1; k<=size(NL[3][i][j]); k++) |
---|
3227 | { |
---|
3228 | if (NL[3][i][j][k]=="p") |
---|
3229 | { |
---|
3230 | NL[3][i][j][k]="s"; |
---|
3231 | found = 1; |
---|
3232 | // printf("replaced at %s,%s,%s",i,j,k); |
---|
3233 | } |
---|
3234 | } |
---|
3235 | } |
---|
3236 | } |
---|
3237 | } |
---|
3238 | return(NL); |
---|
3239 | } |
---|
3240 | example |
---|
3241 | { |
---|
3242 | "EXAMPLE:"; echo = 2; |
---|
3243 | ring r = 0,(x,y,z),(Dp(2),dp(1)); |
---|
3244 | list L = ringlist(r); |
---|
3245 | list N = convloc(L); |
---|
3246 | def rs = ring(N); |
---|
3247 | setring rs; |
---|
3248 | rs; |
---|
3249 | } |
---|
3250 | |
---|
3251 | proc minIntRoot(ideal P, int fact) |
---|
3252 | "USAGE: minIntRoot(P, fact); P an ideal, fact an int |
---|
3253 | RETURN: int |
---|
3254 | PURPOSE: minimal integer root of a maximal ideal P |
---|
3255 | NOTE: if fact==1, P is the result of some 'factorize' call, |
---|
3256 | @* else P is treated as the result of bernstein::gmssing.lib |
---|
3257 | @* in both cases without constants and multiplicities |
---|
3258 | EXAMPLE: example minIntRoot; shows examples |
---|
3259 | " |
---|
3260 | { |
---|
3261 | // ideal P = factorize(p,1); |
---|
3262 | // or ideal P = bernstein(F)[1]; |
---|
3263 | intvec vP; |
---|
3264 | number nP; |
---|
3265 | int i; |
---|
3266 | if ( fact ) |
---|
3267 | { |
---|
3268 | // the result comes from "factorize" |
---|
3269 | P = normalize(P); // now leadcoef = 1 |
---|
3270 | P = lead(P)-P; |
---|
3271 | // nP = leadcoef(P[i]-lead(P[i])); // for 1 var only, extract the coeff |
---|
3272 | } |
---|
3273 | else |
---|
3274 | { |
---|
3275 | // bernstein deletes -1 usually |
---|
3276 | P = P, -1; |
---|
3277 | } |
---|
3278 | // for both situations: |
---|
3279 | // now we have an ideal of fractions of type "number" |
---|
3280 | int sP = size(P); |
---|
3281 | for(i=1; i<=sP; i++) |
---|
3282 | { |
---|
3283 | nP = leadcoef(P[i]); |
---|
3284 | if ( (nP - int(nP)) == 0 ) |
---|
3285 | { |
---|
3286 | vP = vP,int(nP); |
---|
3287 | } |
---|
3288 | } |
---|
3289 | if ( size(vP)>=2 ) |
---|
3290 | { |
---|
3291 | vP = vP[2..size(vP)]; |
---|
3292 | } |
---|
3293 | sP = -Max(-vP); |
---|
3294 | if (sP == 0) |
---|
3295 | { |
---|
3296 | "Warning: zero root!"; |
---|
3297 | } |
---|
3298 | return(sP); |
---|
3299 | } |
---|
3300 | example |
---|
3301 | { |
---|
3302 | "EXAMPLE:"; echo = 2; |
---|
3303 | ring r = 0,(x,y),ds; |
---|
3304 | poly f1 = x*y*(x+y); |
---|
3305 | ideal I1 = bernstein(f1)[1]; // a local Bernstein poly |
---|
3306 | minIntRoot(I1,0); |
---|
3307 | poly f2 = x2-y3; |
---|
3308 | ideal I2 = bernstein(f2)[1]; |
---|
3309 | minIntRoot(I2,0); |
---|
3310 | // now we illustrate the behaviour of factorize |
---|
3311 | // together with a global ordering |
---|
3312 | ring r2 = 0,x,dp; |
---|
3313 | poly f3 = 9*(x+2/3)*(x+1)*(x+4/3); //global b-poly of f1=x*y*(x+y) |
---|
3314 | ideal I3 = factorize(f3,1); |
---|
3315 | minIntRoot(I3,1); |
---|
3316 | // and a more interesting situation |
---|
3317 | ring s = 0,(x,y,z),ds; |
---|
3318 | poly f = x3 + y3 + z3; |
---|
3319 | ideal I = bernstein(f)[1]; |
---|
3320 | minIntRoot(I,0); |
---|
3321 | } |
---|
3322 | |
---|
3323 | proc isHolonomic(def M) |
---|
3324 | "USAGE: isHolonomic(M); M an ideal/module/matrix |
---|
3325 | RETURN: int, 1 if M is holonomic and 0 otherwise |
---|
3326 | PURPOSE: check the modules for the property of holonomy |
---|
3327 | NOTE: M is holonomic, if 2*dim(M) = dim(R), where R is a |
---|
3328 | ground ring; dim stays for Gelfand-Kirillov dimension |
---|
3329 | EXAMPLE: example isHolonomic; shows examples |
---|
3330 | " |
---|
3331 | { |
---|
3332 | if ( (typeof(M) != "ideal") && (typeof(M) != "module") && (typeof(M) != "matrix") ) |
---|
3333 | { |
---|
3334 | // print(typeof(M)); |
---|
3335 | ERROR("an argument of type ideal/module/matrix expected"); |
---|
3336 | } |
---|
3337 | if (attrib(M,"isSB")!=1) |
---|
3338 | { |
---|
3339 | M = std(M); |
---|
3340 | } |
---|
3341 | int dimR = gkdim(std(0)); |
---|
3342 | int dimM = gkdim(M); |
---|
3343 | return( (dimR==2*dimM) ); |
---|
3344 | } |
---|
3345 | example |
---|
3346 | { |
---|
3347 | "EXAMPLE:"; echo = 2; |
---|
3348 | ring R = 0,(x,y),dp; |
---|
3349 | poly F = x*y*(x+y); |
---|
3350 | def A = annfsBM(F,0); |
---|
3351 | setring A; |
---|
3352 | LD; |
---|
3353 | isHolonomic(LD); |
---|
3354 | ideal I = std(LD[1]); |
---|
3355 | I; |
---|
3356 | isHolonomic(I); |
---|
3357 | } |
---|
3358 | |
---|
3359 | proc reiffen(int p, int q) |
---|
3360 | "USAGE: reiffen(p, q); int p, int q |
---|
3361 | RETURN: ring |
---|
3362 | PURPOSE: set up the polynomial, describing a Reiffen curve |
---|
3363 | NOTE: activate this ring with the @code{setring} command and find the |
---|
3364 | curve as a polynomial RC |
---|
3365 | @* a Reiffen curve is defined as F = x^p + y^q + xy^{q-1}, q >= p+1 >= 5 |
---|
3366 | ASSUME: q >= p+1 >= 5. Otherwise an error message is returned |
---|
3367 | EXAMPLE: example reiffen; shows examples |
---|
3368 | " |
---|
3369 | { |
---|
3370 | // a Reiffen curve is defined as |
---|
3371 | // F = x^p + y^q +x*y^{q-1}, q \geq p+1 \geq 5 |
---|
3372 | |
---|
3373 | if ( (p<4) || (q<5) || (q-p<1) ) |
---|
3374 | { |
---|
3375 | ERROR("Some of conditions p>=4, q>=5 or q>=p+1 is not satisfied!"); |
---|
3376 | } |
---|
3377 | ring @r = 0,(x,y),dp; |
---|
3378 | poly RC = y^q +x^p + x*y^(q-1); |
---|
3379 | export RC; |
---|
3380 | return(@r); |
---|
3381 | } |
---|
3382 | example |
---|
3383 | { |
---|
3384 | "EXAMPLE:"; echo = 2; |
---|
3385 | def r = reiffen(4,5); |
---|
3386 | setring r; |
---|
3387 | RC; |
---|
3388 | } |
---|
3389 | |
---|
3390 | proc arrange(int p) |
---|
3391 | "USAGE: arrange(p); int p |
---|
3392 | RETURN: ring |
---|
3393 | PURPOSE: set up the polynomial, describing a hyperplane arrangement |
---|
3394 | NOTE: must be executed in a ring |
---|
3395 | ASSUME: basering is present |
---|
3396 | EXAMPLE: example arrange; shows examples |
---|
3397 | " |
---|
3398 | { |
---|
3399 | int UseBasering = 0 ; |
---|
3400 | if (p<2) |
---|
3401 | { |
---|
3402 | ERROR("p>=2 is required!"); |
---|
3403 | } |
---|
3404 | if ( nameof(basering)!="basering" ) |
---|
3405 | { |
---|
3406 | if (p > nvars(basering)) |
---|
3407 | { |
---|
3408 | ERROR("too big p"); |
---|
3409 | } |
---|
3410 | else |
---|
3411 | { |
---|
3412 | def @r = basering; |
---|
3413 | UseBasering = 1; |
---|
3414 | } |
---|
3415 | } |
---|
3416 | else |
---|
3417 | { |
---|
3418 | // no basering found |
---|
3419 | ERROR("no basering found!"); |
---|
3420 | // ring @r = 0,(x(1..p)),dp; |
---|
3421 | } |
---|
3422 | int i,j,sI; |
---|
3423 | poly q = 1; |
---|
3424 | list ar; |
---|
3425 | ideal tmp; |
---|
3426 | tmp = ideal(var(1)); |
---|
3427 | ar[1] = tmp; |
---|
3428 | for (i = 2; i<=p; i++) |
---|
3429 | { |
---|
3430 | // add i-nary sums to the product |
---|
3431 | ar = indAR(ar,i); |
---|
3432 | } |
---|
3433 | for (i = 1; i<=p; i++) |
---|
3434 | { |
---|
3435 | tmp = ar[i]; sI = size(tmp); |
---|
3436 | for (j = 1; j<=sI; j++) |
---|
3437 | { |
---|
3438 | q = q*tmp[j]; |
---|
3439 | } |
---|
3440 | } |
---|
3441 | if (UseBasering) |
---|
3442 | { |
---|
3443 | return(q); |
---|
3444 | } |
---|
3445 | // poly AR = q; export AR; |
---|
3446 | // return(@r); |
---|
3447 | } |
---|
3448 | example |
---|
3449 | { |
---|
3450 | "EXAMPLE:"; echo = 2; |
---|
3451 | ring X = 0,(x,y,z,t),dp; |
---|
3452 | poly q = arrange(3); |
---|
3453 | factorize(q,1); |
---|
3454 | } |
---|
3455 | |
---|
3456 | static proc indAR(list L, int n) |
---|
3457 | "USAGE: indAR(L,n); list L, int n |
---|
3458 | RETURN: list |
---|
3459 | PURPOSE: computes arrangement inductively, using L and varn(n) as the |
---|
3460 | next variable |
---|
3461 | ASSUME: L has a structure of an arrangement |
---|
3462 | EXAMPLE: example indAR; shows examples |
---|
3463 | " |
---|
3464 | { |
---|
3465 | if ( (n<2) || (n>nvars(basering)) ) |
---|
3466 | { |
---|
3467 | ERROR("incorrect n"); |
---|
3468 | } |
---|
3469 | int sl = size(L); |
---|
3470 | list K; |
---|
3471 | ideal tmp; |
---|
3472 | poly @t = L[sl][1] + var(n); //1 elt |
---|
3473 | K[sl+1] = ideal(@t); |
---|
3474 | tmp = L[1]+var(n); |
---|
3475 | K[1] = tmp; tmp = 0; |
---|
3476 | int i,j,sI; |
---|
3477 | ideal I; |
---|
3478 | for(i=sl; i>=2; i--) |
---|
3479 | { |
---|
3480 | I = L[i-1]; sI = size(I); |
---|
3481 | for(j=1; j<=sI; j++) |
---|
3482 | { |
---|
3483 | I[j] = I[j] + var(n); |
---|
3484 | } |
---|
3485 | tmp = L[i],I; |
---|
3486 | K[i] = tmp; |
---|
3487 | I = 0; tmp = 0; |
---|
3488 | } |
---|
3489 | kill I; kill tmp; |
---|
3490 | return(K); |
---|
3491 | } |
---|
3492 | example |
---|
3493 | { |
---|
3494 | "EXAMPLE:"; echo = 2; |
---|
3495 | ring r = 0,(x,y,z,t,v),dp; |
---|
3496 | list L; |
---|
3497 | L[1] = ideal(x); |
---|
3498 | list K = indAR(L,2); |
---|
3499 | K; |
---|
3500 | list M = indAR(K,3); |
---|
3501 | M; |
---|
3502 | M = indAR(M,4); |
---|
3503 | M; |
---|
3504 | } |
---|
3505 | |
---|
3506 | static proc exCheckGenericity() |
---|
3507 | { |
---|
3508 | LIB "control.lib"; |
---|
3509 | ring r = (0,a,b,c),x,dp; |
---|
3510 | poly p = (x-a)*(x-b)*(x-c); |
---|
3511 | def A = annfsBM(p); |
---|
3512 | setring A; |
---|
3513 | ideal J = slimgb(LD); |
---|
3514 | matrix T = lift(LD,J); |
---|
3515 | T = normalize(T); |
---|
3516 | genericity(T); |
---|
3517 | // Ann =x^3*Dx+3*x^2*t*Dt+(-a-b-c)*x^2*Dx+(-2*a-2*b-2*c)*x*t*Dt+3*x^2+(a*b+a*c+b*c)*x*Dx+(a*b+a*c+b*c)*t*Dt+(-2*a-2*b-2*c)*x+(-a*b*c)*Dx+(a*b+a*c+b*c) |
---|
3518 | // genericity: g = a2-ab-ac+b2-bc+c2 =0 |
---|
3519 | // g = (a -(b+c)/2)^2 + (3/4)*(b-c)^2; |
---|
3520 | // g ==0 <=> a=b=c |
---|
3521 | // indeed, Ann = (x-a)^2*(x*Dx+3*t*Dt+(-a)*Dx+3) |
---|
3522 | // -------------------------------------------- |
---|
3523 | // BUT a direct computation shows |
---|
3524 | // when a=b=c, |
---|
3525 | // Ann = x*Dx+3*t*Dt+(-a)*Dx+3 |
---|
3526 | } |
---|
3527 | |
---|
3528 | static proc exOT_17() |
---|
3529 | { |
---|
3530 | // Oaku-Takayama, p.208 |
---|
3531 | ring R = 0,(x,y),dp; |
---|
3532 | poly F = x^3-y^2; // x^2+x*y+y^2; |
---|
3533 | option(prot); |
---|
3534 | option(mem); |
---|
3535 | // option(redSB); |
---|
3536 | def A = annfsOT(F,0); |
---|
3537 | setring A; |
---|
3538 | LD; |
---|
3539 | gkdim(LD); // a holonomic check |
---|
3540 | // poly F = x^3-y^2; // = x^7 - y^5; // x^3-y^4; // x^5 - y^4; |
---|
3541 | } |
---|
3542 | |
---|
3543 | static proc exOT_16() |
---|
3544 | { |
---|
3545 | // Oaku-Takayama, p.208 |
---|
3546 | ring R = 0,(x),dp; |
---|
3547 | poly F = x*(1-x); |
---|
3548 | option(prot); |
---|
3549 | option(mem); |
---|
3550 | // option(redSB); |
---|
3551 | def A = annfsOT(F,0); |
---|
3552 | setring A; |
---|
3553 | LD; |
---|
3554 | gkdim(LD); // a holonomic check |
---|
3555 | } |
---|
3556 | |
---|
3557 | static proc ex_bcheck() |
---|
3558 | { |
---|
3559 | ring R = 0,(x,y),dp; |
---|
3560 | poly F = x*y*(x+y); |
---|
3561 | option(prot); |
---|
3562 | option(mem); |
---|
3563 | int eng = 0; |
---|
3564 | // option(redSB); |
---|
3565 | def A = annfsOT(F,eng); |
---|
3566 | setring A; |
---|
3567 | LD; |
---|
3568 | } |
---|
3569 | |
---|
3570 | static proc ex_bcheck2() |
---|
3571 | { |
---|
3572 | ring R = 0,(x,y),dp; |
---|
3573 | poly F = x*y*(x+y); |
---|
3574 | int eng = 0; |
---|
3575 | def A = annfsOT(F,eng); |
---|
3576 | setring A; |
---|
3577 | LD; |
---|
3578 | } |
---|
3579 | |
---|
3580 | static proc ex_BMI() |
---|
3581 | { |
---|
3582 | ring r = 0,(x,y),Dp; |
---|
3583 | poly F1 = (x2-y3)*(x3-y2); |
---|
3584 | poly F2 = (x2-y3)*(xy4+y5+x4); |
---|
3585 | } |
---|