1 | ////////////////////////////////////////////////////////////////////////////// |
---|
2 | version="$Id$"; |
---|
3 | category="Noncommutative"; |
---|
4 | info=" |
---|
5 | LIBRARY: dmodapp.lib Applications of algebraic D-modules |
---|
6 | AUTHORS: Viktor Levandovskyy, levandov@math.rwth-aachen.de |
---|
7 | @* Daniel Andres, daniel.andres@math.rwth-aachen.de |
---|
8 | |
---|
9 | Support: DFG Graduiertenkolleg 1632 'Experimentelle und konstruktive Algebra' |
---|
10 | |
---|
11 | OVERVIEW: |
---|
12 | Let K be a field of characteristic 0, R = K[x1,...,xN] and |
---|
13 | D be the Weyl algebra in variables x1,...,xN,d1,...,dN. |
---|
14 | In this library there are the following procedures for algebraic D-modules: |
---|
15 | |
---|
16 | @* - given a cyclic representation D/I of a holonomic module and a polynomial |
---|
17 | F in R, it is proved that the localization of D/I with respect to the mult. |
---|
18 | closed set of all powers of F is a holonomic D-module. Thus we aim to compute |
---|
19 | its cyclic representaion D/L for an ideal L in D. The procedures for the |
---|
20 | localization are DLoc, SDLoc and DLoc0. |
---|
21 | |
---|
22 | @* - annihilator in D of a given polynomial F from R as well as |
---|
23 | of a given rational function G/F from Quot(R). These can be computed via |
---|
24 | procedures annPoly resp. annRat. |
---|
25 | |
---|
26 | @* - Groebner bases with respect to weights (according to (SST), given an |
---|
27 | arbitrary integer vector containing weights for variables, one computes the |
---|
28 | homogenization of a given ideal relative to this vector, then one computes a |
---|
29 | Groebner basis and returns the dehomogenization of the result), initial |
---|
30 | forms and initial ideals in Weyl algebras with respect to a given weight |
---|
31 | vector can be computed with GBWeight, inForm, initialMalgrange and |
---|
32 | initialIdealW. |
---|
33 | |
---|
34 | @* - restriction and integration of a holonomic module D/I. Suppose I |
---|
35 | annihilates a function F(x1,...,xn). Our aim is to compute an ideal J |
---|
36 | directly from I, which annihilates |
---|
37 | @* - F(0,...,0,xk,...,xn) in case of restriction or |
---|
38 | @* - the integral of F with respect to x1,...,xm in case of integration. |
---|
39 | The corresponding procedures are restrictionModule, restrictionIdeal, |
---|
40 | integralModule and integralIdeal. |
---|
41 | |
---|
42 | @* - characteristic varieties defined by ideals in Weyl algebras can be computed |
---|
43 | with charVariety and charInfo. |
---|
44 | |
---|
45 | @* - appelF1, appelF2 and appelF4 return ideals in parametric Weyl algebras, |
---|
46 | which annihilate corresponding Appel hypergeometric functions. |
---|
47 | |
---|
48 | |
---|
49 | References: |
---|
50 | @* (SST) Saito, Sturmfels, Takayama 'Groebner Deformations of Hypergeometric |
---|
51 | Differential Equations', Springer, 2000 |
---|
52 | @* (OTW) Oaku, Takayama, Walther 'A Localization Algorithm for D-modules', |
---|
53 | Journal of Symbolic Computation, 2000 |
---|
54 | @* (OT) Oaku, Takayama 'Algorithms for D-modules', |
---|
55 | Journal of Pure and Applied Algebra, 1998 |
---|
56 | |
---|
57 | |
---|
58 | PROCEDURES: |
---|
59 | |
---|
60 | annPoly(f); computes annihilator of a polynomial f in the corr. Weyl algebra |
---|
61 | annRat(f,g); computes annihilator of rational function f/g in corr. Weyl algebra |
---|
62 | DLoc(I,f); computes presentation of localization of D/I wrt symbolic power f^s |
---|
63 | SDLoc(I,f); computes generic presentation of the localization of D/I wrt f^s |
---|
64 | DLoc0(I,f); computes presentation of localization of D/I wrt f^s based on SDLoc |
---|
65 | |
---|
66 | GBWeight(I,u,v[,a,b]); computes Groebner basis of I wrt a weight vector |
---|
67 | initialMalgrange(f[,s,t,v]); computes Groebner basis of initial Malgrange ideal |
---|
68 | initialIdealW(I,u,v[,s,t]); computes initial ideal of wrt a given weight |
---|
69 | inForm(f,w); computes initial form of poly/ideal wrt a weight |
---|
70 | |
---|
71 | restrictionIdeal(I,w[,eng,m,G]); computes restriction ideal of I wrt w |
---|
72 | restrictionModule(I,w[,eng,m,G]); computes restriction module of I wrt w |
---|
73 | integralIdeal(I,w[,eng,m,G]); computes integral ideal of I wrt w |
---|
74 | integralModule(I,w[,eng,m,G]); computes integral module of I wrt w |
---|
75 | deRhamCohom(f[,eng,m]); computes basis of n-th de Rham cohom. group |
---|
76 | deRhamCohomIdeal(I[,w,eng,m,G]); computes basis of n-th de Rham cohom. group |
---|
77 | |
---|
78 | charVariety(I); computes characteristic variety of the ideal I |
---|
79 | charInfo(I); computes char. variety, singular locus and primary decomp. |
---|
80 | isFsat(I,F); checks whether the ideal I is F-saturated |
---|
81 | |
---|
82 | |
---|
83 | |
---|
84 | appelF1(); creates an ideal annihilating Appel F1 function |
---|
85 | appelF2(); creates an ideal annihilating Appel F2 function |
---|
86 | appelF4(); creates an ideal annihilating Appel F4 function |
---|
87 | |
---|
88 | fourier(I[,v]); applies Fourier automorphism to ideal |
---|
89 | inverseFourier(I[,v]); applies inverse Fourier automorphism to ideal |
---|
90 | |
---|
91 | bFactor(F); computes the roots of irreducible factors of an univariate poly |
---|
92 | intRoots(L); dismisses non-integer roots from list in bFactor format |
---|
93 | poly2list(f); decomposes the polynomial f into a list of terms and exponents |
---|
94 | fl2poly(L,s); reconstructs a monic univariate polynomial from its factorization |
---|
95 | |
---|
96 | insertGenerator(id,p[,k]); inserts an element into an ideal/module |
---|
97 | deleteGenerator(id,k); deletes the k-th element from an ideal/module |
---|
98 | |
---|
99 | engine(I,i); computes a Groebner basis with the algorithm specified by i |
---|
100 | isInt(n); checks whether number n is actually an int |
---|
101 | sortIntvec(v); sorts intvec |
---|
102 | |
---|
103 | KEYWORDS: D-module; annihilator of polynomial; annihilator of rational function; |
---|
104 | D-localization; localization of D-module; D-restriction; restriction of |
---|
105 | D-module; D-integration; integration of D-module; characteristic variety; |
---|
106 | Appel function; Appel hypergeometric function |
---|
107 | |
---|
108 | SEE ALSO: bfun_lib, dmod_lib, dmodvar_lib, gmssing_lib |
---|
109 | "; |
---|
110 | |
---|
111 | /* |
---|
112 | Changelog |
---|
113 | 21.09.10 by DA: |
---|
114 | - restructured library for better readability |
---|
115 | - new / improved procs: |
---|
116 | - toolbox: isInt, intRoots, sortIntvec |
---|
117 | - GB wrt weights: GBWeight, initialIdealW rewritten using GBWeight |
---|
118 | - restriction/integration: restrictionX, integralX where X in {Module, Ideal}, |
---|
119 | fourier, inverseFourier, deRhamCohom, deRhamCohomIdeal |
---|
120 | - characteristic variety: charVariety, charInfo |
---|
121 | - added keywords for features |
---|
122 | - reformated help strings and examples in existing procs |
---|
123 | - added SUPPORT in header |
---|
124 | - added reference (OT) |
---|
125 | |
---|
126 | 04.10.10 by DA: |
---|
127 | - incorporated suggestions by Oleksandr Motsak, among other: |
---|
128 | - bugfixes for fl2poly, sortIntvec, annPoly, GBWeight |
---|
129 | - enhanced functionality for deleteGenerator, inForm |
---|
130 | |
---|
131 | 11.10.10 by DA: |
---|
132 | - procedure bFactor now sorts the roots using new static procedure sortNumberIdeal |
---|
133 | |
---|
134 | 17.03.11 by DA: |
---|
135 | - bugfixes for inForm with polynomial input, typo in restrictionIdealEngine |
---|
136 | |
---|
137 | 06.06.12 by DA: |
---|
138 | - bugfix and documentation in deRhamCohomIdeal, output and |
---|
139 | documentation in deRhamCohom |
---|
140 | - changed charVariety: no homogenization is needed |
---|
141 | - changed inForm: code is much simpler using jet |
---|
142 | |
---|
143 | */ |
---|
144 | |
---|
145 | |
---|
146 | LIB "bfun.lib"; // for pIntersect etc |
---|
147 | LIB "dmod.lib"; // for SannfsBM etc |
---|
148 | LIB "general.lib"; // for sort |
---|
149 | LIB "gkdim.lib"; |
---|
150 | LIB "nctools.lib"; // for isWeyl etc |
---|
151 | LIB "poly.lib"; |
---|
152 | LIB "primdec.lib"; // for primdecGKZ |
---|
153 | LIB "qhmoduli.lib"; // for Max |
---|
154 | LIB "sing.lib"; // for slocus |
---|
155 | |
---|
156 | |
---|
157 | /////////////////////////////////////////////////////////////////////////////// |
---|
158 | // testing for consistency of the library: |
---|
159 | proc testdmodapp() |
---|
160 | { |
---|
161 | example annPoly; |
---|
162 | example annRat; |
---|
163 | example DLoc; |
---|
164 | example SDLoc; |
---|
165 | example DLoc0; |
---|
166 | example GBWeight; |
---|
167 | example initialMalgrange; |
---|
168 | example initialIdealW; |
---|
169 | example inForm; |
---|
170 | example restrictionIdeal; |
---|
171 | example restrictionModule; |
---|
172 | example integralIdeal; |
---|
173 | example integralModule; |
---|
174 | example deRhamCohom; |
---|
175 | example deRhamCohomIdeal; |
---|
176 | example charVariety; |
---|
177 | example charInfo; |
---|
178 | example isFsat; |
---|
179 | example appelF1; |
---|
180 | example appelF2; |
---|
181 | example appelF4; |
---|
182 | example fourier; |
---|
183 | example inverseFourier; |
---|
184 | example bFactor; |
---|
185 | example intRoots; |
---|
186 | example poly2list; |
---|
187 | example fl2poly; |
---|
188 | example insertGenerator; |
---|
189 | example deleteGenerator; |
---|
190 | example engine; |
---|
191 | example isInt; |
---|
192 | example sortIntvec; |
---|
193 | } |
---|
194 | |
---|
195 | |
---|
196 | // general assumptions //////////////////////////////////////////////////////// |
---|
197 | |
---|
198 | static proc dmodappAssumeViolation() |
---|
199 | { |
---|
200 | // char K <> 0 or qring |
---|
201 | if ( (size(ideal(basering)) >0) || (char(basering) >0) ) |
---|
202 | { |
---|
203 | ERROR("Basering is inappropriate: characteristic>0 or qring present"); |
---|
204 | } |
---|
205 | return(); |
---|
206 | } |
---|
207 | |
---|
208 | static proc dmodappMoreAssumeViolation() |
---|
209 | { |
---|
210 | // char K <> 0, qring |
---|
211 | dmodappAssumeViolation(); |
---|
212 | // no Weyl algebra |
---|
213 | if (isWeyl() == 0) |
---|
214 | { |
---|
215 | ERROR("Basering is not a Weyl algebra"); |
---|
216 | } |
---|
217 | // wrong sequence of vars |
---|
218 | int i,n; |
---|
219 | n = nvars(basering) div 2; |
---|
220 | for (i=1; i<=n; i++) |
---|
221 | { |
---|
222 | if (bracket(var(i+n),var(i))<>1) |
---|
223 | { |
---|
224 | ERROR(string(var(i+n))+" is not a differential operator for " +string(var(i))); |
---|
225 | } |
---|
226 | } |
---|
227 | return(); |
---|
228 | } |
---|
229 | |
---|
230 | static proc safeVarName (string s, string cv) |
---|
231 | // assumes 's' to be a valid variable name |
---|
232 | // returns valid var name string @@..@s |
---|
233 | { |
---|
234 | string S; |
---|
235 | if (cv == "v") { S = "," + "," + varstr(basering) + ","; } |
---|
236 | if (cv == "c") { S = "," + "," + charstr(basering) + ","; } |
---|
237 | if (cv == "cv") { S = "," + charstr(basering) + "," + varstr(basering) + ","; } |
---|
238 | s = "," + s + ","; |
---|
239 | while (find(S,s) <> 0) |
---|
240 | { |
---|
241 | s[1] = "@"; |
---|
242 | s = "," + s; |
---|
243 | } |
---|
244 | s = s[2..size(s)-1]; |
---|
245 | return(s) |
---|
246 | } |
---|
247 | |
---|
248 | static proc intLike (def i) |
---|
249 | { |
---|
250 | string str = typeof(i); |
---|
251 | if (str == "int" || str == "number" || str == "bigint") |
---|
252 | { |
---|
253 | return(1); |
---|
254 | } |
---|
255 | else |
---|
256 | { |
---|
257 | return(0); |
---|
258 | } |
---|
259 | } |
---|
260 | |
---|
261 | |
---|
262 | // toolbox //////////////////////////////////////////////////////////////////// |
---|
263 | |
---|
264 | proc engine(def I, int i) |
---|
265 | "USAGE: engine(I,i); I ideal/module/matrix, i an int |
---|
266 | RETURN: the same type as I |
---|
267 | PURPOSE: compute the Groebner basis of I with the algorithm, chosen via i |
---|
268 | NOTE: By default and if i=0, slimgb is used; otherwise std does the job. |
---|
269 | EXAMPLE: example engine; shows examples |
---|
270 | " |
---|
271 | { |
---|
272 | /* std - slimgb mix */ |
---|
273 | def J; |
---|
274 | // ideal J; |
---|
275 | if (i==0) |
---|
276 | { |
---|
277 | J = slimgb(I); |
---|
278 | } |
---|
279 | else |
---|
280 | { |
---|
281 | // without options -> strange! (ringlist?) |
---|
282 | intvec v = option(get); |
---|
283 | option(redSB); |
---|
284 | option(redTail); |
---|
285 | J = std(I); |
---|
286 | option(set, v); |
---|
287 | } |
---|
288 | return(J); |
---|
289 | } |
---|
290 | example |
---|
291 | { |
---|
292 | "EXAMPLE:"; echo = 2; |
---|
293 | ring r = 0,(x,y),Dp; |
---|
294 | ideal I = y*(x3-y2),x*(x3-y2); |
---|
295 | engine(I,0); // uses slimgb |
---|
296 | engine(I,1); // uses std |
---|
297 | } |
---|
298 | |
---|
299 | proc poly2list (poly f) |
---|
300 | "USAGE: poly2list(f); f a poly |
---|
301 | RETURN: list of exponents and corresponding terms of f |
---|
302 | PURPOSE: converts a poly to a list of pairs consisting of intvecs (1st entry) |
---|
303 | @* and polys (2nd entry), where the i-th pair contains the exponent of the |
---|
304 | @* i-th term of f and the i-th term (with coefficient) itself. |
---|
305 | EXAMPLE: example poly2list; shows examples |
---|
306 | " |
---|
307 | { |
---|
308 | list l; |
---|
309 | int i = 1; |
---|
310 | if (f == 0) // just for the zero polynomial |
---|
311 | { |
---|
312 | l[1] = list(leadexp(f), lead(f)); |
---|
313 | } |
---|
314 | else |
---|
315 | { |
---|
316 | l[size(f)] = list(); // memory pre-allocation |
---|
317 | while (f != 0) |
---|
318 | { |
---|
319 | l[i] = list(leadexp(f), lead(f)); |
---|
320 | f = f - lead(f); |
---|
321 | i++; |
---|
322 | } |
---|
323 | } |
---|
324 | return(l); |
---|
325 | } |
---|
326 | example |
---|
327 | { |
---|
328 | "EXAMPLE:"; echo = 2; |
---|
329 | ring r = 0,x,dp; |
---|
330 | poly F = x; |
---|
331 | poly2list(F); |
---|
332 | ring r2 = 0,(x,y),dp; |
---|
333 | poly F = x2y+5xy2; |
---|
334 | poly2list(F); |
---|
335 | poly2list(0); |
---|
336 | } |
---|
337 | |
---|
338 | proc fl2poly(list L, string s) |
---|
339 | "USAGE: fl2poly(L,s); L a list, s a string |
---|
340 | RETURN: poly |
---|
341 | PURPOSE: reconstruct a monic polynomial in one variable from its factorization |
---|
342 | ASSUME: s is a string with the name of some variable and |
---|
343 | @* L is supposed to consist of two entries: |
---|
344 | @* - L[1] of the type ideal with the roots of a polynomial |
---|
345 | @* - L[2] of the type intvec with the multiplicities of corr. roots |
---|
346 | EXAMPLE: example fl2poly; shows examples |
---|
347 | " |
---|
348 | { |
---|
349 | if (varNum(s)==0) |
---|
350 | { |
---|
351 | ERROR(s+ " is no variable in the basering"); |
---|
352 | } |
---|
353 | poly x = var(varNum(s)); |
---|
354 | poly P = 1; |
---|
355 | ideal RR = L[1]; |
---|
356 | int sl = ncols(RR); |
---|
357 | intvec IV = L[2]; |
---|
358 | if (sl <> nrows(IV)) |
---|
359 | { |
---|
360 | ERROR("number of roots doesn't match number of multiplicites"); |
---|
361 | } |
---|
362 | for(int i=1; i<=sl; i++) |
---|
363 | { |
---|
364 | if (IV[i] > 0) |
---|
365 | { |
---|
366 | P = P*((x-RR[i])^IV[i]); |
---|
367 | } |
---|
368 | else |
---|
369 | { |
---|
370 | printf("Ignored the root with incorrect multiplicity %s",string(IV[i])); |
---|
371 | } |
---|
372 | } |
---|
373 | return(P); |
---|
374 | } |
---|
375 | example |
---|
376 | { |
---|
377 | "EXAMPLE:"; echo = 2; |
---|
378 | ring r = 0,(x,y,z,s),Dp; |
---|
379 | ideal I = -1,-4/3,0,-5/3,-2; |
---|
380 | intvec mI = 2,1,2,1,1; |
---|
381 | list BS = I,mI; |
---|
382 | poly p = fl2poly(BS,"s"); |
---|
383 | p; |
---|
384 | factorize(p,2); |
---|
385 | } |
---|
386 | |
---|
387 | proc insertGenerator (list #) |
---|
388 | "USAGE: insertGenerator(id,p[,k]); |
---|
389 | @* id an ideal/module, p a poly/vector, k an optional int |
---|
390 | RETURN: of the same type as id |
---|
391 | PURPOSE: inserts p into id at k-th position and returns the enlarged object |
---|
392 | NOTE: If k is given, p is inserted at position k, otherwise (and by default), |
---|
393 | @* p is inserted at the beginning (k=1). |
---|
394 | SEE ALSO: deleteGenerator |
---|
395 | EXAMPLE: example insertGenerator; shows examples |
---|
396 | " |
---|
397 | { |
---|
398 | if (size(#) < 2) |
---|
399 | { |
---|
400 | ERROR("insertGenerator has to be called with at least 2 arguments (ideal/module,poly/vector)"); |
---|
401 | } |
---|
402 | string inp1 = typeof(#[1]); |
---|
403 | if (inp1 == "ideal" || inp1 == "module") |
---|
404 | { |
---|
405 | def id = #[1]; |
---|
406 | } |
---|
407 | else { ERROR("first argument has to be of type ideal or module"); } |
---|
408 | string inp2 = typeof(#[2]); |
---|
409 | if (inp2 == "poly" || inp2 == "vector") |
---|
410 | { |
---|
411 | def f = #[2]; |
---|
412 | } |
---|
413 | else { ERROR("second argument has to be of type poly/vector"); } |
---|
414 | if (inp1 == "ideal" && inp2 == "vector") |
---|
415 | { |
---|
416 | ERROR("second argument has to be a polynomial if first argument is an ideal"); |
---|
417 | } |
---|
418 | // don't check module/poly combination due to auto-conversion |
---|
419 | // if (inp1 == "module" && inp2 == "poly") |
---|
420 | // { |
---|
421 | // ERROR("second argument has to be a vector if first argument is a module"); |
---|
422 | // } |
---|
423 | int n = ncols(id); |
---|
424 | int k = 1; // default |
---|
425 | if (size(#)>=3) |
---|
426 | { |
---|
427 | if (intLike(#[3])) |
---|
428 | { |
---|
429 | k = int(#[3]); |
---|
430 | if (k<=0) |
---|
431 | { |
---|
432 | ERROR("third argument has to be positive"); |
---|
433 | } |
---|
434 | } |
---|
435 | else { ERROR("third argument has to be of type int"); } |
---|
436 | } |
---|
437 | execute(inp1 +" J;"); |
---|
438 | if (k == 1) { J = f,id; } |
---|
439 | else |
---|
440 | { |
---|
441 | if (k>n) |
---|
442 | { |
---|
443 | J = id; |
---|
444 | J[k] = f; |
---|
445 | } |
---|
446 | else // 1<k<=n |
---|
447 | { |
---|
448 | J[n+1] = id[n]; // preinit |
---|
449 | J[1..k-1] = id[1..k-1]; |
---|
450 | J[k] = f; |
---|
451 | J[k+1..n+1] = id[k..n]; |
---|
452 | } |
---|
453 | } |
---|
454 | return(J); |
---|
455 | } |
---|
456 | example |
---|
457 | { |
---|
458 | "EXAMPLE:"; echo = 2; |
---|
459 | ring r = 0,(x,y,z),dp; |
---|
460 | ideal I = x^2,z^4; |
---|
461 | insertGenerator(I,y^3); |
---|
462 | insertGenerator(I,y^3,2); |
---|
463 | module M = I*gen(3); |
---|
464 | insertGenerator(M,[x^3,y^2,z],2); |
---|
465 | insertGenerator(M,x+y+z,4); |
---|
466 | } |
---|
467 | |
---|
468 | proc deleteGenerator (def id, int k) |
---|
469 | "USAGE: deleteGenerator(id,k); id an ideal/module, k an int |
---|
470 | RETURN: of the same type as id |
---|
471 | PURPOSE: deletes the k-th generator from the first argument and returns |
---|
472 | @* the altered object |
---|
473 | SEE ALSO: insertGenerator |
---|
474 | EXAMPLE: example deleteGenerator; shows examples |
---|
475 | " |
---|
476 | { |
---|
477 | string inp1 = typeof(id); |
---|
478 | if (inp1 <> "ideal" && inp1 <> "module") |
---|
479 | { |
---|
480 | ERROR("first argument has to be of type ideal or module"); |
---|
481 | } |
---|
482 | execute(inp1 +" J;"); |
---|
483 | int n = ncols(id); |
---|
484 | if (n == 1 && k == 1) { return(J); } |
---|
485 | if (k<=0 || k>n) |
---|
486 | { |
---|
487 | ERROR("second argument has to be in the range 1,...,"+string(n)); |
---|
488 | } |
---|
489 | J[n-1] = 0; // preinit |
---|
490 | if (k == 1) { J = id[2..n]; } |
---|
491 | else |
---|
492 | { |
---|
493 | if (k == n) { J = id[1..n-1]; } |
---|
494 | else |
---|
495 | { |
---|
496 | J[1..k-1] = id[1..k-1]; |
---|
497 | J[k..n-1] = id[k+1..n]; |
---|
498 | } |
---|
499 | } |
---|
500 | return(J); |
---|
501 | } |
---|
502 | example |
---|
503 | { |
---|
504 | "EXAMPLE:"; echo = 2; |
---|
505 | ring r = 0,(x,y,z),dp; |
---|
506 | ideal I = x^2,y^3,z^4; |
---|
507 | deleteGenerator(I,2); |
---|
508 | module M = [x,y,z],[x2,y2,z2],[x3,y3,z3]; |
---|
509 | print(deleteGenerator(M,2)); |
---|
510 | M = M[1]; |
---|
511 | deleteGenerator(M,1); |
---|
512 | } |
---|
513 | |
---|
514 | static proc sortNumberIdeal (ideal I) |
---|
515 | // sorts ideal of constant polys (ie numbers), returns according permutation |
---|
516 | { |
---|
517 | int i; |
---|
518 | int nI = ncols(I); |
---|
519 | intvec dI; |
---|
520 | for (i=nI; i>0; i--) |
---|
521 | { |
---|
522 | dI[i] = int(denominator(leadcoef(I[i]))); |
---|
523 | } |
---|
524 | int lcmI = lcm(dI); |
---|
525 | for (i=nI; i>0; i--) |
---|
526 | { |
---|
527 | dI[i] = int(lcmI*leadcoef(I[i])); |
---|
528 | } |
---|
529 | intvec perm = sortIntvec(dI)[2]; |
---|
530 | return(perm); |
---|
531 | } |
---|
532 | example |
---|
533 | { |
---|
534 | "EXAMPLE:"; echo = 2; |
---|
535 | ring r = 0,s,dp; |
---|
536 | ideal I = -9/20,-11/20,-23/20,-19/20,-1,-13/10,-27/20,-13/20,-21/20,-17/20, |
---|
537 | -11/10,-9/10,-7/10; // roots of BS poly of reiffen(4,5) |
---|
538 | intvec v = sortNumberIdeal(I); v; |
---|
539 | I[v]; |
---|
540 | } |
---|
541 | |
---|
542 | proc bFactor (poly F) |
---|
543 | "USAGE: bFactor(f); f poly |
---|
544 | RETURN: list of ideal and intvec and possibly a string |
---|
545 | PURPOSE: tries to compute the roots of a univariate poly f |
---|
546 | NOTE: The output list consists of two or three entries: |
---|
547 | @* roots of f as an ideal, their multiplicities as intvec, and, |
---|
548 | @* if present, a third one being the product of all irreducible factors |
---|
549 | @* of degree greater than one, given as string. |
---|
550 | @* If f is the zero polynomial or if f has no roots in the ground field, |
---|
551 | @* this is encoded as root 0 with multiplicity 0. |
---|
552 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
553 | @* if printlevel>=2, all the debug messages will be printed. |
---|
554 | EXAMPLE: example bFactor; shows examples |
---|
555 | " |
---|
556 | { |
---|
557 | int ppl = printlevel - voice +2; |
---|
558 | def save = basering; |
---|
559 | ideal LI = variables(F); |
---|
560 | list L; |
---|
561 | int i = size(LI); |
---|
562 | if (i>1) { ERROR("poly has to be univariate")} |
---|
563 | if (i == 0) |
---|
564 | { |
---|
565 | dbprint(ppl,"// poly is constant"); |
---|
566 | L = list(ideal(0),intvec(0),string(F)); |
---|
567 | return(L); |
---|
568 | } |
---|
569 | poly v = LI[1]; |
---|
570 | L = ringlist(save); L = L[1..4]; |
---|
571 | L[2] = list(string(v)); |
---|
572 | L[3] = list(list("dp",intvec(1)),list("C",intvec(0))); |
---|
573 | def @S = ring(L); |
---|
574 | setring @S; |
---|
575 | poly ir = 1; poly F = imap(save,F); |
---|
576 | list L = factorize(F); |
---|
577 | ideal I = L[1]; |
---|
578 | intvec m = L[2]; |
---|
579 | ideal II; intvec mm; |
---|
580 | for (i=2; i<=ncols(I); i++) |
---|
581 | { |
---|
582 | if (deg(I[i]) > 1) |
---|
583 | { |
---|
584 | ir = ir * I[i]^m[i]; |
---|
585 | } |
---|
586 | else |
---|
587 | { |
---|
588 | II[size(II)+1] = I[i]; |
---|
589 | mm[size(II)] = m[i]; |
---|
590 | } |
---|
591 | } |
---|
592 | II = normalize(II); |
---|
593 | II = subst(II,var(1),0); |
---|
594 | II = -II; |
---|
595 | intvec perm = sortNumberIdeal(II); |
---|
596 | II = II[perm]; |
---|
597 | mm = mm[perm]; |
---|
598 | if (size(II)>0) |
---|
599 | { |
---|
600 | dbprint(ppl,"// found roots"); |
---|
601 | dbprint(ppl-1,string(II)); |
---|
602 | } |
---|
603 | else |
---|
604 | { |
---|
605 | dbprint(ppl,"// found no roots"); |
---|
606 | } |
---|
607 | L = list(II,mm); |
---|
608 | if (ir <> 1) |
---|
609 | { |
---|
610 | dbprint(ppl,"// found irreducible factors"); |
---|
611 | ir = cleardenom(ir); |
---|
612 | dbprint(ppl-1,string(ir)); |
---|
613 | L = L + list(string(ir)); |
---|
614 | } |
---|
615 | else |
---|
616 | { |
---|
617 | dbprint(ppl,"// no irreducible factors found"); |
---|
618 | } |
---|
619 | setring save; |
---|
620 | L = imap(@S,L); |
---|
621 | return(L); |
---|
622 | } |
---|
623 | example |
---|
624 | { |
---|
625 | "EXAMPLE:"; echo = 2; |
---|
626 | ring r = 0,(x,y),dp; |
---|
627 | bFactor((x^2-1)^2); |
---|
628 | bFactor((x^2+1)^2); |
---|
629 | bFactor((y^2+1/2)*(y+9)*(y-7)); |
---|
630 | bFactor(1); |
---|
631 | bFactor(0); |
---|
632 | } |
---|
633 | |
---|
634 | proc isInt (number n) |
---|
635 | "USAGE: isInt(n); n a number |
---|
636 | RETURN: int, 1 if n is an integer or 0 otherwise |
---|
637 | PURPOSE: check whether given object of type number is actually an int |
---|
638 | NOTE: Parameters are treated as integers. |
---|
639 | EXAMPLE: example isInt; shows an example |
---|
640 | " |
---|
641 | { |
---|
642 | number d = denominator(n); |
---|
643 | if (d<>1) |
---|
644 | { |
---|
645 | return(0); |
---|
646 | } |
---|
647 | else |
---|
648 | { |
---|
649 | return(1); |
---|
650 | } |
---|
651 | } |
---|
652 | example |
---|
653 | { |
---|
654 | "EXAMPLE:"; echo = 2; |
---|
655 | ring r = 0,x,dp; |
---|
656 | number n = 4/3; |
---|
657 | isInt(n); |
---|
658 | n = 11; |
---|
659 | isInt(n); |
---|
660 | } |
---|
661 | |
---|
662 | proc intRoots (list l) |
---|
663 | "USAGE: isInt(L); L a list |
---|
664 | RETURN: list |
---|
665 | PURPOSE: extracts integer roots from a list given in @code{bFactor} format |
---|
666 | ASSUME: The input list must be given in the format of @code{bFactor}. |
---|
667 | NOTE: Parameters are treated as integers. |
---|
668 | SEE ALSO: bFactor |
---|
669 | EXAMPLE: example intRoots; shows an example |
---|
670 | " |
---|
671 | { |
---|
672 | int wronginput; |
---|
673 | int sl = size(l); |
---|
674 | if (sl>0) |
---|
675 | { |
---|
676 | if (typeof(l[1])<>"ideal"){wronginput = 1;} |
---|
677 | if (sl>1) |
---|
678 | { |
---|
679 | if (typeof(l[2])<>"intvec"){wronginput = 1;} |
---|
680 | if (sl>2) |
---|
681 | { |
---|
682 | if (typeof(l[3])<>"string"){wronginput = 1;} |
---|
683 | if (sl>3){wronginput = 1;} |
---|
684 | } |
---|
685 | } |
---|
686 | } |
---|
687 | if (sl<2){wronginput = 1;} |
---|
688 | if (wronginput) |
---|
689 | { |
---|
690 | ERROR("Given list has wrong format."); |
---|
691 | } |
---|
692 | int i,j; |
---|
693 | ideal l1 = l[1]; |
---|
694 | int n = ncols(l1); |
---|
695 | j = 1; |
---|
696 | ideal I; |
---|
697 | intvec v; |
---|
698 | for (i=1; i<=n; i++) |
---|
699 | { |
---|
700 | if (size(l1[j])>1) // poly not number |
---|
701 | { |
---|
702 | ERROR("Ideal in list has wrong format."); |
---|
703 | } |
---|
704 | if (isInt(leadcoef(l1[i]))) |
---|
705 | { |
---|
706 | I[j] = l1[i]; |
---|
707 | v[j] = l[2][i]; |
---|
708 | j++; |
---|
709 | } |
---|
710 | } |
---|
711 | return(list(I,v)); |
---|
712 | } |
---|
713 | example |
---|
714 | { |
---|
715 | "EXAMPLE:"; echo = 2; |
---|
716 | ring r = 0,x,dp; |
---|
717 | list L = bFactor((x-4/3)*(x+3)^2*(x-5)^4); L; |
---|
718 | intRoots(L); |
---|
719 | } |
---|
720 | |
---|
721 | proc sortIntvec (intvec v) |
---|
722 | "USAGE: sortIntvec(v); v an intvec |
---|
723 | RETURN: list of two intvecs |
---|
724 | PURPOSE: sorts an intvec |
---|
725 | NOTE: In the output list L, the first entry consists of the entries of v |
---|
726 | @* satisfying L[1][i] >= L[1][i+1]. The second entry is a permutation |
---|
727 | @* such that v[L[2]] = L[1]. |
---|
728 | @* Unlike in the procedure @code{sort}, zeros are not dismissed. |
---|
729 | SEE ALSO: sort |
---|
730 | EXAMPLE: example sortIntvec; shows examples |
---|
731 | " |
---|
732 | { |
---|
733 | int i; |
---|
734 | intvec vpos,vzero,vneg,vv,sortv,permv; |
---|
735 | list l; |
---|
736 | for (i=1; i<=nrows(v); i++) |
---|
737 | { |
---|
738 | if (v[i]>0) |
---|
739 | { |
---|
740 | vpos = vpos,i; |
---|
741 | } |
---|
742 | else |
---|
743 | { |
---|
744 | if (v[i]==0) |
---|
745 | { |
---|
746 | vzero = vzero,i; |
---|
747 | } |
---|
748 | else // v[i]<0 |
---|
749 | { |
---|
750 | vneg = vneg,i; |
---|
751 | } |
---|
752 | } |
---|
753 | } |
---|
754 | if (size(vpos)>1) |
---|
755 | { |
---|
756 | vpos = vpos[2..size(vpos)]; |
---|
757 | vv = v[vpos]; |
---|
758 | l = sort(vv); |
---|
759 | vv = l[1]; |
---|
760 | vpos = vpos[l[2]]; |
---|
761 | sortv = vv[size(vv)..1]; |
---|
762 | permv = vpos[size(vv)..1]; |
---|
763 | } |
---|
764 | if (size(vzero)>1) |
---|
765 | { |
---|
766 | vzero = vzero[2..size(vzero)]; |
---|
767 | permv = permv,vzero; |
---|
768 | sortv = sortv,0:size(vzero); |
---|
769 | } |
---|
770 | if (size(vneg)>1) |
---|
771 | { |
---|
772 | vneg = vneg[2..size(vneg)]; |
---|
773 | vv = -v[vneg]; |
---|
774 | l = sort(vv); |
---|
775 | vv = -l[1]; |
---|
776 | vneg = vneg[l[2]]; |
---|
777 | sortv = sortv,vv; |
---|
778 | permv = permv,vneg; |
---|
779 | } |
---|
780 | if (permv[1]==0) |
---|
781 | { |
---|
782 | sortv = sortv[2..size(sortv)]; |
---|
783 | permv = permv[2..size(permv)]; |
---|
784 | } |
---|
785 | return(list(sortv,permv)); |
---|
786 | } |
---|
787 | example |
---|
788 | { |
---|
789 | "EXAMPLE:"; echo = 2; |
---|
790 | ring r = 0,x,dp; |
---|
791 | intvec v = -1,0,1,-2,0,2; |
---|
792 | list L = sortIntvec(v); L; |
---|
793 | v[L[2]]; |
---|
794 | v = -3,0; |
---|
795 | sortIntvec(v); |
---|
796 | v = 0,-3; |
---|
797 | sortIntvec(v); |
---|
798 | } |
---|
799 | |
---|
800 | |
---|
801 | // F-saturation /////////////////////////////////////////////////////////////// |
---|
802 | |
---|
803 | proc isFsat(ideal I, poly F) |
---|
804 | "USAGE: isFsat(I, F); I an ideal, F a poly |
---|
805 | RETURN: int, 1 if I is F-saturated and 0 otherwise |
---|
806 | PURPOSE: checks whether the ideal I is F-saturated |
---|
807 | NOTE: We check indeed that Ker(D--> F--> D/I) is 0, where D is the basering. |
---|
808 | EXAMPLE: example isFsat; shows examples |
---|
809 | " |
---|
810 | { |
---|
811 | /* checks whether I is F-saturated, that is Ke (D -F-> D/I) is 0 */ |
---|
812 | /* works in any algebra */ |
---|
813 | /* for simplicity : later check attrib */ |
---|
814 | /* returns 1 if I is F-sat */ |
---|
815 | if (attrib(I,"isSB")!=1) |
---|
816 | { |
---|
817 | I = groebner(I); |
---|
818 | } |
---|
819 | matrix @M = matrix(I); |
---|
820 | matrix @F[1][1] = F; |
---|
821 | def S = modulo(module(@F),module(@M)); |
---|
822 | S = NF(S,I); |
---|
823 | S = groebner(S); |
---|
824 | return( (gkdim(S) == -1) ); |
---|
825 | } |
---|
826 | example |
---|
827 | { |
---|
828 | "EXAMPLE:"; echo = 2; |
---|
829 | ring r = 0,(x,y),dp; |
---|
830 | poly G = x*(x-y)*y; |
---|
831 | def A = annfs(G); |
---|
832 | setring A; |
---|
833 | poly F = x3-y2; |
---|
834 | isFsat(LD,F); |
---|
835 | ideal J = LD*F; |
---|
836 | isFsat(J,F); |
---|
837 | } |
---|
838 | |
---|
839 | |
---|
840 | // annihilators /////////////////////////////////////////////////////////////// |
---|
841 | |
---|
842 | proc annRat(poly g, poly f) |
---|
843 | "USAGE: annRat(g,f); f, g polynomials |
---|
844 | RETURN: ring (a Weyl algebra) containing an ideal 'LD' |
---|
845 | PURPOSE: compute the annihilator of the rational function g/f in the |
---|
846 | @* corresponding Weyl algebra |
---|
847 | ASSUME: basering is commutative and over a field of characteristic 0 |
---|
848 | NOTE: Activate the output ring with the @code{setring} command. |
---|
849 | @* In the output ring, the ideal 'LD' (in Groebner basis) is the |
---|
850 | @* annihilator of g/f. |
---|
851 | @* The algorithm uses the computation of Ann(f^{-1}) via D-modules, |
---|
852 | @* see (SST). |
---|
853 | DISPLAY: If printlevel =1, progress debug messages will be printed, |
---|
854 | @* if printlevel>=2, all the debug messages will be printed. |
---|
855 | SEE ALSO: annPoly |
---|
856 | EXAMPLE: example annRat; shows examples |
---|
857 | " |
---|
858 | { |
---|
859 | // assumption check |
---|
860 | dmodappAssumeViolation(); |
---|
861 | if (!isCommutative()) |
---|
862 | { |
---|
863 | ERROR("Basering must be commutative."); |
---|
864 | } |
---|
865 | // assumptions: f is not a constant |
---|
866 | if (f==0) { ERROR("the denominator f cannot be zero"); } |
---|
867 | if ((leadexp(f) == 0) && (size(f) < 2)) |
---|
868 | { |
---|
869 | // f = const, so use annPoly |
---|
870 | g = g/f; |
---|
871 | def @R = annPoly(g); |
---|
872 | return(@R); |
---|
873 | } |
---|
874 | // computes the annihilator of g/f |
---|
875 | def save = basering; |
---|
876 | int ppl = printlevel-voice+2; |
---|
877 | dbprint(ppl,"// -1-[annRat] computing the ann f^s"); |
---|
878 | def @R1 = SannfsBM(f); |
---|
879 | setring @R1; |
---|
880 | poly f = imap(save,f); |
---|
881 | int i,mir; |
---|
882 | int isr = 0; // checkRoot1(LD,f,1); // roots are negative, have to enter positive int |
---|
883 | if (!isr) |
---|
884 | { |
---|
885 | // -1 is not the root |
---|
886 | // find the m.i.r iteratively |
---|
887 | mir = 0; |
---|
888 | for(i=nvars(save)+1; i>=1; i--) |
---|
889 | { |
---|
890 | isr = checkRoot1(LD,f,i); |
---|
891 | if (isr) { mir =-i; break; } |
---|
892 | } |
---|
893 | if (mir ==0) |
---|
894 | { |
---|
895 | ERROR("No integer root found! Aborting computations, inform the authors!"); |
---|
896 | } |
---|
897 | // now mir == i is m.i.r. |
---|
898 | } |
---|
899 | else |
---|
900 | { |
---|
901 | // -1 is the m.i.r |
---|
902 | mir = -1; |
---|
903 | } |
---|
904 | dbprint(ppl,"// -2-[annRat] the minimal integer root is "); |
---|
905 | dbprint(ppl-1, mir); |
---|
906 | // use annfspecial |
---|
907 | dbprint(ppl,"// -3-[annRat] running annfspecial "); |
---|
908 | ideal AF = annfspecial(LD,f,mir,-1); // ann f^{-1} |
---|
909 | // LD = subst(LD,s,j); |
---|
910 | // LD = engine(LD,0); |
---|
911 | // modify the ring: throw s away |
---|
912 | // output ring comes from SannfsBM |
---|
913 | list U = ringlist(@R1); |
---|
914 | list tmp; // variables |
---|
915 | for(i=1; i<=size(U[2])-1; i++) |
---|
916 | { |
---|
917 | tmp[i] = U[2][i]; |
---|
918 | } |
---|
919 | U[2] = tmp; |
---|
920 | tmp = 0; |
---|
921 | tmp[1] = U[3][1]; // x,Dx block |
---|
922 | tmp[2] = U[3][3]; // module block |
---|
923 | U[3] = tmp; |
---|
924 | tmp = 0; |
---|
925 | tmp = U[1],U[2],U[3],U[4]; |
---|
926 | def @R2 = ring(tmp); |
---|
927 | setring @R2; |
---|
928 | // now supply with Weyl algebra relations |
---|
929 | int N = nvars(@R2) div 2; |
---|
930 | matrix @D[2*N][2*N]; |
---|
931 | for(i=1; i<=N; i++) |
---|
932 | { |
---|
933 | @D[i,N+i]=1; |
---|
934 | } |
---|
935 | def @R3 = nc_algebra(1,@D); |
---|
936 | setring @R3; |
---|
937 | dbprint(ppl,"// - -[annRat] ring without s is ready:"); |
---|
938 | dbprint(ppl-1,@R3); |
---|
939 | poly g = imap(save,g); |
---|
940 | matrix G[1][1] = g; |
---|
941 | matrix LL = matrix(imap(@R1,AF)); |
---|
942 | kill @R1; kill @R2; |
---|
943 | dbprint(ppl,"// -4-[annRat] running modulo"); |
---|
944 | ideal LD = modulo(G,LL); |
---|
945 | dbprint(ppl,"// -4-[annRat] running GB on the final result"); |
---|
946 | LD = engine(LD,0); |
---|
947 | export LD; |
---|
948 | return(@R3); |
---|
949 | } |
---|
950 | example |
---|
951 | { |
---|
952 | "EXAMPLE:"; echo = 2; |
---|
953 | ring r = 0,(x,y),dp; |
---|
954 | poly g = 2*x*y; poly f = x^2 - y^3; |
---|
955 | def B = annRat(g,f); |
---|
956 | setring B; |
---|
957 | LD; |
---|
958 | // Now, compare with the output of Macaulay2: |
---|
959 | ideal tst = 3*x*Dx + 2*y*Dy + 1, y^3*Dy^2 - x^2*Dy^2 + 6*y^2*Dy + 6*y, |
---|
960 | 9*y^2*Dx^2*Dy-4*y*Dy^3+27*y*Dx^2+2*Dy^2, 9*y^3*Dx^2-4*y^2*Dy^2+10*y*Dy -10; |
---|
961 | option(redSB); option(redTail); |
---|
962 | LD = groebner(LD); |
---|
963 | tst = groebner(tst); |
---|
964 | print(matrix(NF(LD,tst))); print(matrix(NF(tst,LD))); |
---|
965 | // So, these two answers are the same |
---|
966 | } |
---|
967 | |
---|
968 | proc annPoly(poly f) |
---|
969 | "USAGE: annPoly(f); f a poly |
---|
970 | RETURN: ring (a Weyl algebra) containing an ideal 'LD' |
---|
971 | PURPOSE: compute the complete annihilator ideal of f in the corresponding |
---|
972 | @* Weyl algebra |
---|
973 | ASSUME: basering is commutative and over a field of characteristic 0 |
---|
974 | NOTE: Activate the output ring with the @code{setring} command. |
---|
975 | @* In the output ring, the ideal 'LD' (in Groebner basis) is the |
---|
976 | @* annihilator. |
---|
977 | DISPLAY: If printlevel =1, progress debug messages will be printed, |
---|
978 | @* if printlevel>=2, all the debug messages will be printed. |
---|
979 | SEE ALSO: annRat |
---|
980 | EXAMPLE: example annPoly; shows examples |
---|
981 | " |
---|
982 | { |
---|
983 | // assumption check |
---|
984 | dmodappAssumeViolation(); |
---|
985 | if (!isCommutative()) |
---|
986 | { |
---|
987 | ERROR("Basering must be commutative."); |
---|
988 | } |
---|
989 | // computes a system of linear PDEs with polynomial coeffs for f |
---|
990 | def save = basering; |
---|
991 | list L = ringlist(save); |
---|
992 | list Name = L[2]; |
---|
993 | int N = nvars(save); |
---|
994 | int i; |
---|
995 | for (i=1; i<=N; i++) |
---|
996 | { |
---|
997 | Name[N+i] = safeVarName("D"+Name[i],"cv"); // concat |
---|
998 | } |
---|
999 | L[2] = Name; |
---|
1000 | def @R = ring(L); |
---|
1001 | setring @R; |
---|
1002 | def @@R = Weyl(); |
---|
1003 | setring @@R; |
---|
1004 | kill @R; |
---|
1005 | matrix M[1][N]; |
---|
1006 | for (i=1; i<=N; i++) |
---|
1007 | { |
---|
1008 | M[1,i] = var(N+i); |
---|
1009 | } |
---|
1010 | matrix F[1][1] = imap(save,f); |
---|
1011 | def I = modulo(module(F),module(M)); |
---|
1012 | ideal LD = I; |
---|
1013 | LD = groebner(LD); |
---|
1014 | export LD; |
---|
1015 | return(@@R); |
---|
1016 | } |
---|
1017 | example |
---|
1018 | { |
---|
1019 | "EXAMPLE:"; echo = 2; |
---|
1020 | ring r = 0,(x,y,z),dp; |
---|
1021 | poly f = x^2*z - y^3; |
---|
1022 | def A = annPoly(f); |
---|
1023 | setring A; // A is the 3rd Weyl algebra in 6 variables |
---|
1024 | LD; // the Groebner basis of annihilator |
---|
1025 | gkdim(LD); // must be 3 = 6/2, since A/LD is holonomic module |
---|
1026 | NF(Dy^4, LD); // must be 0 since Dy^4 clearly annihilates f |
---|
1027 | poly f = imap(r,f); |
---|
1028 | NF(LD*f,std(ideal(Dx,Dy,Dz))); // must be zero if LD indeed annihilates f |
---|
1029 | } |
---|
1030 | |
---|
1031 | |
---|
1032 | |
---|
1033 | // localizations ////////////////////////////////////////////////////////////// |
---|
1034 | |
---|
1035 | proc DLoc(ideal I, poly F) |
---|
1036 | "USAGE: DLoc(I, f); I an ideal, f a poly |
---|
1037 | RETURN: list of ideal and list |
---|
1038 | ASSUME: the basering is a Weyl algebra |
---|
1039 | PURPOSE: compute the presentation of the localization of D/I w.r.t. f^s |
---|
1040 | NOTE: In the output list L, |
---|
1041 | @* - L[1] is an ideal (given as Groebner basis), the presentation of the |
---|
1042 | @* localization, |
---|
1043 | @* - L[2] is a list containing roots with multiplicities of Bernstein |
---|
1044 | @* polynomial of (D/I)_f. |
---|
1045 | DISPLAY: If printlevel =1, progress debug messages will be printed, |
---|
1046 | @* if printlevel>=2, all the debug messages will be printed. |
---|
1047 | EXAMPLE: example DLoc; shows examples |
---|
1048 | " |
---|
1049 | { |
---|
1050 | /* runs SDLoc and DLoc0 */ |
---|
1051 | /* assume: run from Weyl algebra */ |
---|
1052 | dmodappAssumeViolation(); |
---|
1053 | if (!isWeyl()) |
---|
1054 | { |
---|
1055 | ERROR("Basering is not a Weyl algebra"); |
---|
1056 | } |
---|
1057 | int old_printlevel = printlevel; |
---|
1058 | printlevel=printlevel+1; |
---|
1059 | def @R = basering; |
---|
1060 | def @R2 = SDLoc(I,F); |
---|
1061 | setring @R2; |
---|
1062 | poly F = imap(@R,F); |
---|
1063 | def @R3 = DLoc0(LD,F); |
---|
1064 | setring @R3; |
---|
1065 | ideal bs = BS[1]; |
---|
1066 | intvec m = BS[2]; |
---|
1067 | setring @R; |
---|
1068 | ideal LD0 = imap(@R3,LD0); |
---|
1069 | ideal bs = imap(@R3,bs); |
---|
1070 | list BS; BS[1] = bs; BS[2] = m; |
---|
1071 | kill @R3; |
---|
1072 | printlevel = old_printlevel; |
---|
1073 | return(list(LD0,BS)); |
---|
1074 | } |
---|
1075 | example; |
---|
1076 | { |
---|
1077 | "EXAMPLE:"; echo = 2; |
---|
1078 | ring r = 0,(x,y,Dx,Dy),dp; |
---|
1079 | def R = Weyl(); setring R; // Weyl algebra in variables x,y,Dx,Dy |
---|
1080 | poly F = x2-y3; |
---|
1081 | ideal I = (y^3 - x^2)*Dx - 2*x, (y^3 - x^2)*Dy + 3*y^2; // I = Dx*F, Dy*F; |
---|
1082 | // I is not holonomic, since its dimension is not 4/2=2 |
---|
1083 | gkdim(I); |
---|
1084 | list L = DLoc(I, x2-y3); |
---|
1085 | L[1]; // localized module (R/I)_f is isomorphic to R/LD0 |
---|
1086 | L[2]; // description of b-function for localization |
---|
1087 | } |
---|
1088 | |
---|
1089 | proc DLoc0(ideal I, poly F) |
---|
1090 | "USAGE: DLoc0(I, f); I an ideal, f a poly |
---|
1091 | RETURN: ring (a Weyl algebra) containing an ideal 'LD0' and a list 'BS' |
---|
1092 | PURPOSE: compute the presentation of the localization of D/I w.r.t. f^s, |
---|
1093 | @* where D is a Weyl Algebra, based on the output of procedure SDLoc |
---|
1094 | ASSUME: the basering is similar to the output ring of SDLoc procedure |
---|
1095 | NOTE: activate the output ring with the @code{setring} command. In this ring, |
---|
1096 | @* - the ideal LD0 (given as Groebner basis) is the presentation of the |
---|
1097 | @* localization, |
---|
1098 | @* - the list BS contains roots and multiplicities of Bernstein |
---|
1099 | @* polynomial of (D/I)_f. |
---|
1100 | DISPLAY: If printlevel =1, progress debug messages will be printed, |
---|
1101 | @* if printlevel>=2, all the debug messages will be printed. |
---|
1102 | EXAMPLE: example DLoc0; shows examples |
---|
1103 | " |
---|
1104 | { |
---|
1105 | dmodappAssumeViolation(); |
---|
1106 | /* assume: to be run in the output ring of SDLoc */ |
---|
1107 | /* doing: add F, eliminate vars*Dvars, factorize BS */ |
---|
1108 | /* analogue to annfs0 */ |
---|
1109 | def @R2 = basering; |
---|
1110 | // we're in D_n[s], where the elim ord for s is set |
---|
1111 | ideal J = NF(I,std(F)); |
---|
1112 | // make leadcoeffs positive |
---|
1113 | int i; |
---|
1114 | for (i=1; i<= ncols(J); i++) |
---|
1115 | { |
---|
1116 | if (leadcoef(J[i]) <0 ) |
---|
1117 | { |
---|
1118 | J[i] = -J[i]; |
---|
1119 | } |
---|
1120 | } |
---|
1121 | J = J,F; |
---|
1122 | ideal M = groebner(J); |
---|
1123 | int Nnew = nvars(@R2); |
---|
1124 | ideal K2 = nselect(M,1..Nnew-1); |
---|
1125 | int ppl = printlevel-voice+2; |
---|
1126 | dbprint(ppl,"// -1-1- _x,_Dx are eliminated in basering"); |
---|
1127 | dbprint(ppl-1, K2); |
---|
1128 | // the ring @R3 and the search for minimal negative int s |
---|
1129 | ring @R3 = 0,s,dp; |
---|
1130 | dbprint(ppl,"// -2-1- the ring @R3 = K[s] is ready"); |
---|
1131 | ideal K3 = imap(@R2,K2); |
---|
1132 | poly p = K3[1]; |
---|
1133 | dbprint(ppl,"// -2-2- attempt the factorization"); |
---|
1134 | list PP = factorize(p); //with constants and multiplicities |
---|
1135 | ideal bs; intvec m; //the Bernstein polynomial is monic, so |
---|
1136 | // we are not interested in constants |
---|
1137 | for (i=2; i<= size(PP[1]); i++) //we delete P[1][1] and P[2][1] |
---|
1138 | { |
---|
1139 | bs[i-1] = PP[1][i]; |
---|
1140 | m[i-1] = PP[2][i]; |
---|
1141 | } |
---|
1142 | ideal bbs; int srat=0; int HasRatRoots = 0; |
---|
1143 | int sP; |
---|
1144 | for (i=1; i<= size(bs); i++) |
---|
1145 | { |
---|
1146 | if (deg(bs[i]) == 1) |
---|
1147 | { |
---|
1148 | bbs = bbs,bs[i]; |
---|
1149 | } |
---|
1150 | } |
---|
1151 | if (size(bbs)==0) |
---|
1152 | { |
---|
1153 | dbprint(ppl-1,"// -2-3- factorization: no rational roots"); |
---|
1154 | // HasRatRoots = 0; |
---|
1155 | HasRatRoots = 1; // s0 = -1 then |
---|
1156 | sP = -1; |
---|
1157 | // todo: return ideal with no subst and a b-function unfactorized |
---|
1158 | } |
---|
1159 | else |
---|
1160 | { |
---|
1161 | // exist rational roots |
---|
1162 | dbprint(ppl-1,"// -2-3- factorization: rational roots found"); |
---|
1163 | HasRatRoots = 1; |
---|
1164 | // dbprint(ppl-1,bbs); |
---|
1165 | bbs = bbs[2..ncols(bbs)]; |
---|
1166 | ideal P = bbs; |
---|
1167 | dbprint(ppl-1,P); |
---|
1168 | srat = size(bs) - size(bbs); |
---|
1169 | // define minIntRoot on linear factors or find out that it doesn't exist |
---|
1170 | intvec vP; |
---|
1171 | number nP; |
---|
1172 | P = normalize(P); // now leadcoef = 1 |
---|
1173 | P = ideal(matrix(lead(P))-matrix(P)); |
---|
1174 | sP = size(P); |
---|
1175 | int cnt = 0; |
---|
1176 | for (i=1; i<=sP; i++) |
---|
1177 | { |
---|
1178 | nP = leadcoef(P[i]); |
---|
1179 | if ( (nP - int(nP)) == 0 ) |
---|
1180 | { |
---|
1181 | cnt++; |
---|
1182 | vP[cnt] = int(nP); |
---|
1183 | } |
---|
1184 | } |
---|
1185 | // if ( size(vP)>=2 ) |
---|
1186 | // { |
---|
1187 | // vP = vP[2..size(vP)]; |
---|
1188 | // } |
---|
1189 | if ( size(vP)==0 ) |
---|
1190 | { |
---|
1191 | // no roots! |
---|
1192 | dbprint(ppl,"// -2-4- no integer root, setting s0 = -1"); |
---|
1193 | sP = -1; |
---|
1194 | // HasRatRoots = 0; // older stuff, here we do substitution |
---|
1195 | HasRatRoots = 1; |
---|
1196 | } |
---|
1197 | else |
---|
1198 | { |
---|
1199 | HasRatRoots = 1; |
---|
1200 | sP = -Max(-vP); |
---|
1201 | dbprint(ppl,"// -2-4- minimal integer root found"); |
---|
1202 | dbprint(ppl-1, sP); |
---|
1203 | // int sP = minIntRoot(bbs,1); |
---|
1204 | // P = normalize(P); |
---|
1205 | // bs = -subst(bs,s,0); |
---|
1206 | if (sP >=0) |
---|
1207 | { |
---|
1208 | dbprint(ppl,"// -2-5- nonnegative root, setting s0 = -1"); |
---|
1209 | sP = -1; |
---|
1210 | } |
---|
1211 | else |
---|
1212 | { |
---|
1213 | dbprint(ppl,"// -2-5- the root is negative"); |
---|
1214 | } |
---|
1215 | } |
---|
1216 | } |
---|
1217 | |
---|
1218 | if (HasRatRoots) |
---|
1219 | { |
---|
1220 | setring @R2; |
---|
1221 | K2 = subst(I,s,sP); |
---|
1222 | // IF min int root exists -> |
---|
1223 | // create the ordinary Weyl algebra and put the result into it, |
---|
1224 | // thus creating the ring @R5 |
---|
1225 | // ELSE : return the same ring with new objects |
---|
1226 | // keep: N, i,j,s, tmp, RL |
---|
1227 | Nnew = Nnew - 1; // former 2*N; |
---|
1228 | // list RL = ringlist(save); // is defined earlier |
---|
1229 | // kill Lord, tmp, iv; |
---|
1230 | list L = 0; |
---|
1231 | list Lord, tmp; |
---|
1232 | intvec iv; |
---|
1233 | list RL = ringlist(basering); |
---|
1234 | L[1] = RL[1]; |
---|
1235 | L[4] = RL[4]; //char, minpoly |
---|
1236 | // check whether vars have admissible names -> done earlier |
---|
1237 | // list Name = RL[2]M |
---|
1238 | // DName is defined earlier |
---|
1239 | list NName; // = RL[2]; // skip the last var 's' |
---|
1240 | for (i=1; i<=Nnew; i++) |
---|
1241 | { |
---|
1242 | NName[i] = RL[2][i]; |
---|
1243 | } |
---|
1244 | L[2] = NName; |
---|
1245 | // dp ordering; |
---|
1246 | string s = "iv="; |
---|
1247 | for (i=1; i<=Nnew; i++) |
---|
1248 | { |
---|
1249 | s = s+"1,"; |
---|
1250 | } |
---|
1251 | s[size(s)] = ";"; |
---|
1252 | execute(s); |
---|
1253 | tmp = 0; |
---|
1254 | tmp[1] = "dp"; // string |
---|
1255 | tmp[2] = iv; // intvec |
---|
1256 | Lord[1] = tmp; |
---|
1257 | kill s; |
---|
1258 | tmp[1] = "C"; |
---|
1259 | iv = 0; |
---|
1260 | tmp[2] = iv; |
---|
1261 | Lord[2] = tmp; |
---|
1262 | tmp = 0; |
---|
1263 | L[3] = Lord; |
---|
1264 | // we are done with the list |
---|
1265 | // Add: Plural part |
---|
1266 | def @R4@ = ring(L); |
---|
1267 | setring @R4@; |
---|
1268 | int N = Nnew div 2; |
---|
1269 | matrix @D[Nnew][Nnew]; |
---|
1270 | for (i=1; i<=N; i++) |
---|
1271 | { |
---|
1272 | @D[i,N+i]=1; |
---|
1273 | } |
---|
1274 | def @R4 = nc_algebra(1,@D); |
---|
1275 | setring @R4; |
---|
1276 | kill @R4@; |
---|
1277 | dbprint(ppl,"// -3-1- the ring @R4 is ready"); |
---|
1278 | dbprint(ppl-1, @R4); |
---|
1279 | ideal K4 = imap(@R2,K2); |
---|
1280 | intvec vopt = option(get); |
---|
1281 | option(redSB); |
---|
1282 | dbprint(ppl,"// -3-2- the final cosmetic std"); |
---|
1283 | K4 = groebner(K4); // std does the job too |
---|
1284 | option(set,vopt); |
---|
1285 | // total cleanup |
---|
1286 | setring @R2; |
---|
1287 | ideal bs = imap(@R3,bs); |
---|
1288 | bs = -normalize(bs); // "-" for getting correct coeffs! |
---|
1289 | bs = subst(bs,s,0); |
---|
1290 | kill @R3; |
---|
1291 | setring @R4; |
---|
1292 | ideal bs = imap(@R2,bs); // only rationals are the entries |
---|
1293 | list BS; BS[1] = bs; BS[2] = m; |
---|
1294 | export BS; |
---|
1295 | // list LBS = imap(@R3,LBS); |
---|
1296 | // list BS; BS[1] = sbs; BS[2] = m; |
---|
1297 | // BS; |
---|
1298 | // export BS; |
---|
1299 | ideal LD0 = K4; |
---|
1300 | export LD0; |
---|
1301 | return(@R4); |
---|
1302 | } |
---|
1303 | else |
---|
1304 | { |
---|
1305 | /* SHOULD NEVER GET THERE */ |
---|
1306 | /* no rational/integer roots */ |
---|
1307 | /* return objects in the copy of current ring */ |
---|
1308 | setring @R2; |
---|
1309 | ideal LD0 = I; |
---|
1310 | poly BS = normalize(K2[1]); |
---|
1311 | export LD0; |
---|
1312 | export BS; |
---|
1313 | return(@R2); |
---|
1314 | } |
---|
1315 | } |
---|
1316 | example; |
---|
1317 | { |
---|
1318 | "EXAMPLE:"; echo = 2; |
---|
1319 | ring r = 0,(x,y,Dx,Dy),dp; |
---|
1320 | def R = Weyl(); setring R; // Weyl algebra in variables x,y,Dx,Dy |
---|
1321 | poly F = x2-y3; |
---|
1322 | ideal I = (y^3 - x^2)*Dx - 2*x, (y^3 - x^2)*Dy + 3*y^2; // I = Dx*F, Dy*F; |
---|
1323 | // moreover I is not holonomic, since its dimension is not 2 = 4/2 |
---|
1324 | gkdim(I); // 3 |
---|
1325 | def W = SDLoc(I,F); setring W; // creates ideal LD in W = R[s] |
---|
1326 | def U = DLoc0(LD, x2-y3); setring U; // compute in R |
---|
1327 | LD0; // Groebner basis of the presentation of localization |
---|
1328 | BS; // description of b-function for localization |
---|
1329 | } |
---|
1330 | |
---|
1331 | proc SDLoc(ideal I, poly F) |
---|
1332 | "USAGE: SDLoc(I, f); I an ideal, f a poly |
---|
1333 | RETURN: ring (basering extended by a new variable) containing an ideal 'LD' |
---|
1334 | PURPOSE: compute a generic presentation of the localization of D/I w.r.t. f^s |
---|
1335 | ASSUME: the basering D is a Weyl algebra over a field of characteristic 0 |
---|
1336 | NOTE: Activate this ring with the @code{setring} command. In this ring, |
---|
1337 | @* the ideal LD (given as Groebner basis) is the presentation of the |
---|
1338 | @* localization. |
---|
1339 | DISPLAY: If printlevel =1, progress debug messages will be printed, |
---|
1340 | @* if printlevel>=2, all the debug messages will be printed. |
---|
1341 | EXAMPLE: example SDLoc; shows examples |
---|
1342 | " |
---|
1343 | { |
---|
1344 | /* analogue to Sannfs */ |
---|
1345 | /* printlevel >=4 gives debug info */ |
---|
1346 | /* assume: we're in the Weyl algebra D in x1,x2,...,d1,d2,... */ |
---|
1347 | |
---|
1348 | dmodappAssumeViolation(); |
---|
1349 | if (!isWeyl()) |
---|
1350 | { |
---|
1351 | ERROR("Basering is not a Weyl algebra"); |
---|
1352 | } |
---|
1353 | def save = basering; |
---|
1354 | /* 1. create D <t, dt, s > as in LOT */ |
---|
1355 | /* ordering: eliminate t,dt */ |
---|
1356 | int ppl = printlevel-voice+2; |
---|
1357 | int N = nvars(save); N = N div 2; |
---|
1358 | int Nnew = 2*N + 3; // t,Dt,s |
---|
1359 | int i,j; |
---|
1360 | string s; |
---|
1361 | list RL = ringlist(save); |
---|
1362 | list L, Lord; |
---|
1363 | list tmp; |
---|
1364 | intvec iv; |
---|
1365 | L[1] = RL[1]; // char |
---|
1366 | L[4] = RL[4]; // char, minpoly |
---|
1367 | // check whether vars have admissible names |
---|
1368 | list Name = RL[2]; |
---|
1369 | list RName; |
---|
1370 | RName[1] = "@t"; |
---|
1371 | RName[2] = "@Dt"; |
---|
1372 | RName[3] = "@s"; |
---|
1373 | for(i=1;i<=N;i++) |
---|
1374 | { |
---|
1375 | for(j=1; j<=size(RName);j++) |
---|
1376 | { |
---|
1377 | if (Name[i] == RName[j]) |
---|
1378 | { |
---|
1379 | ERROR("Variable names should not include @t,@Dt,@s"); |
---|
1380 | } |
---|
1381 | } |
---|
1382 | } |
---|
1383 | // now, create the names for new vars |
---|
1384 | tmp = 0; |
---|
1385 | tmp[1] = "@t"; |
---|
1386 | tmp[2] = "@Dt"; |
---|
1387 | list SName ; SName[1] = "@s"; |
---|
1388 | list NName = tmp + Name + SName; |
---|
1389 | L[2] = NName; |
---|
1390 | tmp = 0; |
---|
1391 | kill NName; |
---|
1392 | // block ord (a(1,1),dp); |
---|
1393 | tmp[1] = "a"; // string |
---|
1394 | iv = 1,1; |
---|
1395 | tmp[2] = iv; //intvec |
---|
1396 | Lord[1] = tmp; |
---|
1397 | // continue with dp 1,1,1,1... |
---|
1398 | tmp[1] = "dp"; // string |
---|
1399 | s = "iv="; |
---|
1400 | for(i=1;i<=Nnew;i++) |
---|
1401 | { |
---|
1402 | s = s+"1,"; |
---|
1403 | } |
---|
1404 | s[size(s)]= ";"; |
---|
1405 | execute(s); |
---|
1406 | tmp[2] = iv; |
---|
1407 | Lord[2] = tmp; |
---|
1408 | tmp[1] = "C"; |
---|
1409 | iv = 0; |
---|
1410 | tmp[2] = iv; |
---|
1411 | Lord[3] = tmp; |
---|
1412 | tmp = 0; |
---|
1413 | L[3] = Lord; |
---|
1414 | // we are done with the list |
---|
1415 | def @R@ = ring(L); |
---|
1416 | setring @R@; |
---|
1417 | matrix @D[Nnew][Nnew]; |
---|
1418 | @D[1,2]=1; |
---|
1419 | for(i=1; i<=N; i++) |
---|
1420 | { |
---|
1421 | @D[2+i,N+2+i]=1; |
---|
1422 | } |
---|
1423 | // ADD [s,t]=-t, [s,Dt]=Dt |
---|
1424 | @D[1,Nnew] = -var(1); |
---|
1425 | @D[2,Nnew] = var(2); |
---|
1426 | def @R = nc_algebra(1,@D); |
---|
1427 | setring @R; |
---|
1428 | kill @R@; |
---|
1429 | dbprint(ppl,"// -1-1- the ring @R(@t,@Dt,_x,_Dx,@s) is ready"); |
---|
1430 | dbprint(ppl-1, @R); |
---|
1431 | poly F = imap(save,F); |
---|
1432 | ideal I = imap(save,I); |
---|
1433 | dbprint(ppl-1, "the ideal after map:"); |
---|
1434 | dbprint(ppl-1, I); |
---|
1435 | poly p = 0; |
---|
1436 | for(i=1; i<=N; i++) |
---|
1437 | { |
---|
1438 | p = diff(F,var(2+i))*@Dt + var(2+N+i); |
---|
1439 | dbprint(ppl-1, p); |
---|
1440 | I = subst(I,var(2+N+i),p); |
---|
1441 | dbprint(ppl-1, var(2+N+i)); |
---|
1442 | p = 0; |
---|
1443 | } |
---|
1444 | I = I, @t - F; |
---|
1445 | // t*Dt + s +1 reduced with t-f gives f*Dt + s |
---|
1446 | I = I, F*var(2) + var(Nnew); // @s |
---|
1447 | // -------- the ideal I is ready ---------- |
---|
1448 | dbprint(ppl,"// -1-2- starting the elimination of @t,@Dt in @R"); |
---|
1449 | dbprint(ppl-1, I); |
---|
1450 | // ideal J = engine(I,eng); |
---|
1451 | ideal J = groebner(I); |
---|
1452 | dbprint(ppl-1,"// -1-2-1- result of the elimination of @t,@Dt in @R"); |
---|
1453 | dbprint(ppl-1, J);; |
---|
1454 | ideal K = nselect(J,1..2); |
---|
1455 | dbprint(ppl,"// -1-3- @t,@Dt are eliminated"); |
---|
1456 | dbprint(ppl-1, K); // K is without t, Dt |
---|
1457 | K = groebner(K); // std does the job too |
---|
1458 | // now, we must change the ordering |
---|
1459 | // and create a ring without t, Dt |
---|
1460 | setring save; |
---|
1461 | // ----------- the ring @R3 ------------ |
---|
1462 | // _x, _Dx,s; elim.ord for _x,_Dx. |
---|
1463 | // keep: N, i,j,s, tmp, RL |
---|
1464 | Nnew = 2*N+1; |
---|
1465 | kill Lord, tmp, iv, RName; |
---|
1466 | list Lord, tmp; |
---|
1467 | intvec iv; |
---|
1468 | L[1] = RL[1]; |
---|
1469 | L[4] = RL[4]; // char, minpoly |
---|
1470 | // check whether vars hava admissible names -> done earlier |
---|
1471 | // now, create the names for new var |
---|
1472 | tmp[1] = "s"; |
---|
1473 | list NName = Name + tmp; |
---|
1474 | L[2] = NName; |
---|
1475 | tmp = 0; |
---|
1476 | // block ord (dp(N),dp); |
---|
1477 | // string s is already defined |
---|
1478 | s = "iv="; |
---|
1479 | for (i=1; i<=Nnew-1; i++) |
---|
1480 | { |
---|
1481 | s = s+"1,"; |
---|
1482 | } |
---|
1483 | s[size(s)]=";"; |
---|
1484 | execute(s); |
---|
1485 | tmp[1] = "dp"; // string |
---|
1486 | tmp[2] = iv; // intvec |
---|
1487 | Lord[1] = tmp; |
---|
1488 | // continue with dp 1,1,1,1... |
---|
1489 | tmp[1] = "dp"; // string |
---|
1490 | s[size(s)] = ","; |
---|
1491 | s = s+"1;"; |
---|
1492 | execute(s); |
---|
1493 | kill s; |
---|
1494 | kill NName; |
---|
1495 | tmp[2] = iv; |
---|
1496 | Lord[2] = tmp; |
---|
1497 | tmp[1] = "C"; iv = 0; tmp[2]=iv; |
---|
1498 | Lord[3] = tmp; tmp = 0; |
---|
1499 | L[3] = Lord; |
---|
1500 | // we are done with the list. Now add a Plural part |
---|
1501 | def @R2@ = ring(L); |
---|
1502 | setring @R2@; |
---|
1503 | matrix @D[Nnew][Nnew]; |
---|
1504 | for (i=1; i<=N; i++) |
---|
1505 | { |
---|
1506 | @D[i,N+i]=1; |
---|
1507 | } |
---|
1508 | def @R2 = nc_algebra(1,@D); |
---|
1509 | setring @R2; |
---|
1510 | kill @R2@; |
---|
1511 | dbprint(ppl,"// -2-1- the ring @R2(_x,_Dx,s) is ready"); |
---|
1512 | dbprint(ppl-1, @R2); |
---|
1513 | ideal MM = maxideal(1); |
---|
1514 | MM = 0,s,MM; |
---|
1515 | map R01 = @R, MM; |
---|
1516 | ideal K = R01(K); |
---|
1517 | // total cleanup |
---|
1518 | ideal LD = K; |
---|
1519 | // make leadcoeffs positive |
---|
1520 | for (i=1; i<= ncols(LD); i++) |
---|
1521 | { |
---|
1522 | if (leadcoef(LD[i]) <0 ) |
---|
1523 | { |
---|
1524 | LD[i] = -LD[i]; |
---|
1525 | } |
---|
1526 | } |
---|
1527 | export LD; |
---|
1528 | kill @R; |
---|
1529 | return(@R2); |
---|
1530 | } |
---|
1531 | example; |
---|
1532 | { |
---|
1533 | "EXAMPLE:"; echo = 2; |
---|
1534 | ring r = 0,(x,y,Dx,Dy),dp; |
---|
1535 | def R = Weyl(); // Weyl algebra on the variables x,y,Dx,Dy |
---|
1536 | setring R; |
---|
1537 | poly F = x2-y3; |
---|
1538 | ideal I = Dx*F, Dy*F; |
---|
1539 | // note, that I is not holonomic, since it's dimension is not 2 |
---|
1540 | gkdim(I); // 3, while dim R = 4 |
---|
1541 | def W = SDLoc(I,F); |
---|
1542 | setring W; // = R[s], where s is a new variable |
---|
1543 | LD; // Groebner basis of s-parametric presentation |
---|
1544 | } |
---|
1545 | |
---|
1546 | |
---|
1547 | // Groebner basis wrt weights and initial ideal business ////////////////////// |
---|
1548 | |
---|
1549 | proc GBWeight (ideal I, intvec u, intvec v, list #) |
---|
1550 | "USAGE: GBWeight(I,u,v [,s,t,w]); |
---|
1551 | @* I ideal, u,v intvecs, s,t optional ints, w an optional intvec |
---|
1552 | RETURN: ideal, Groebner basis of I w.r.t. the weights u and v |
---|
1553 | ASSUME: The basering is the n-th Weyl algebra over a field of characteristic 0 |
---|
1554 | @* and for all 1<=i<=n the identity var(i+n)*var(i)=var(i)*var(i+1)+1 |
---|
1555 | @* holds, i.e. the sequence of variables is given by |
---|
1556 | @* x(1),...,x(n),D(1),...,D(n), where D(i) is the differential operator |
---|
1557 | @* belonging to x(i). |
---|
1558 | PURPOSE: computes a Groebner basis with respect to given weights |
---|
1559 | NOTE: The weights u and v are understood as weight vectors for x(i) and D(i), |
---|
1560 | @* respectively. According to (SST), one computes the homogenization of a |
---|
1561 | @* given ideal relative to (u,v), then one computes a Groebner basis and |
---|
1562 | @* returns the dehomogenization of the result. |
---|
1563 | @* If s<>0, @code{std} is used for Groebner basis computations, |
---|
1564 | @* otherwise, and by default, @code{slimgb} is used. |
---|
1565 | @* If t<>0, a matrix ordering is used for Groebner basis computations, |
---|
1566 | @* otherwise, and by default, a block ordering is used. |
---|
1567 | @* If w is given and consists of exactly 2*n strictly positive entries, |
---|
1568 | @* w is used for constructing the weighted homogenized Weyl algebra, |
---|
1569 | @* see Noro (2002). Otherwise, and by default, the homogenization weight |
---|
1570 | @* (1,...,1) is used. |
---|
1571 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
1572 | @* if printlevel>=2, all the debug messages will be printed. |
---|
1573 | EXAMPLE: example GBWeight; shows examples |
---|
1574 | " |
---|
1575 | { |
---|
1576 | dmodappMoreAssumeViolation(); |
---|
1577 | int ppl = printlevel - voice +2; |
---|
1578 | def save = basering; |
---|
1579 | int n = nvars(save) div 2; |
---|
1580 | int whichengine = 0; // default |
---|
1581 | int methodord = 0; // default |
---|
1582 | intvec homogweights = 1:(2*n); // default |
---|
1583 | if (size(#)>0) |
---|
1584 | { |
---|
1585 | if (intLike(#[1])) |
---|
1586 | { |
---|
1587 | whichengine = int(#[1]); |
---|
1588 | } |
---|
1589 | if (size(#)>1) |
---|
1590 | { |
---|
1591 | if (intLike(#[2])) |
---|
1592 | { |
---|
1593 | methodord = int(#[2]); |
---|
1594 | } |
---|
1595 | if (size(#)>2) |
---|
1596 | { |
---|
1597 | if (typeof(#[3])=="intvec") |
---|
1598 | { |
---|
1599 | if (size(#[3])==2*n && allPositive(#[3])==1) |
---|
1600 | { |
---|
1601 | homogweights = #[3]; |
---|
1602 | } |
---|
1603 | else |
---|
1604 | { |
---|
1605 | print("// Homogenization weight vector must consist of positive entries and be"); |
---|
1606 | print("// of size " + string(n) + ". Using weight (1,...,1)."); |
---|
1607 | } |
---|
1608 | } |
---|
1609 | } |
---|
1610 | } |
---|
1611 | } |
---|
1612 | // 1. create homogenized Weyl algebra |
---|
1613 | // 1.1 create ordering |
---|
1614 | int i; |
---|
1615 | list RL = ringlist(save); |
---|
1616 | int N = 2*n+1; |
---|
1617 | intvec uv = u,v,0; |
---|
1618 | homogweights = homogweights,1; |
---|
1619 | list Lord = list(list("a",homogweights)); |
---|
1620 | list C0 = list("C",intvec(0)); |
---|
1621 | if (methodord == 0) // default: blockordering |
---|
1622 | { |
---|
1623 | Lord[5] = C0; |
---|
1624 | Lord[4] = list("lp",intvec(1)); |
---|
1625 | Lord[3] = list("dp",intvec(1:(N-1))); |
---|
1626 | Lord[2] = list("a",uv); |
---|
1627 | } |
---|
1628 | else // M() ordering |
---|
1629 | { |
---|
1630 | intmat @Ord[N][N]; |
---|
1631 | @Ord[1,1..N] = uv; @Ord[2,1..N] = 1:(N-1); |
---|
1632 | for (i=1; i<=N-2; i++) |
---|
1633 | { |
---|
1634 | @Ord[2+i,N - i] = -1; |
---|
1635 | } |
---|
1636 | dbprint(ppl-1,"// the ordering matrix:",@Ord); |
---|
1637 | Lord[2] = list("M",intvec(@Ord)); |
---|
1638 | Lord[3] = C0; |
---|
1639 | } |
---|
1640 | // 1.2 the homog var |
---|
1641 | list Lvar = RL[2]; Lvar[N] = safeVarName("h","cv"); |
---|
1642 | // 1.3 create commutative ring |
---|
1643 | list L@@Dh = RL; L@@Dh = L@@Dh[1..4]; |
---|
1644 | L@@Dh[2] = Lvar; L@@Dh[3] = Lord; |
---|
1645 | def @Dh = ring(L@@Dh); kill L@@Dh; |
---|
1646 | setring @Dh; |
---|
1647 | // 1.4 create non-commutative relations |
---|
1648 | matrix @relD[N][N]; |
---|
1649 | for (i=1; i<=n; i++) |
---|
1650 | { |
---|
1651 | @relD[i,n+i] = var(N)^(homogweights[i]+homogweights[n+i]); |
---|
1652 | } |
---|
1653 | def Dh = nc_algebra(1,@relD); |
---|
1654 | setring Dh; kill @Dh; |
---|
1655 | dbprint(ppl-1,"// computing in ring",Dh); |
---|
1656 | // 2. Compute the initial ideal |
---|
1657 | ideal I = imap(save,I); |
---|
1658 | I = homog(I,var(N)); |
---|
1659 | // 2.1 the hard part: Groebner basis computation |
---|
1660 | dbprint(ppl, "// starting Groebner basis computation with engine: "+string(whichengine)); |
---|
1661 | I = engine(I, whichengine); |
---|
1662 | dbprint(ppl, "// finished Groebner basis computation"); |
---|
1663 | dbprint(ppl-1, "// ideal before dehomogenization is " +string(I)); |
---|
1664 | I = subst(I,var(N),1); // dehomogenization |
---|
1665 | setring save; |
---|
1666 | I = imap(Dh,I); kill Dh; |
---|
1667 | return(I); |
---|
1668 | } |
---|
1669 | example |
---|
1670 | { |
---|
1671 | "EXAMPLE:"; echo = 2; |
---|
1672 | ring r = 0,(x,y,Dx,Dy),dp; |
---|
1673 | def D2 = Weyl(); |
---|
1674 | setring D2; |
---|
1675 | ideal I = 3*x^2*Dy+2*y*Dx,2*x*Dx+3*y*Dy+6; |
---|
1676 | intvec u = -2,-3; |
---|
1677 | intvec v = -u; |
---|
1678 | GBWeight(I,u,v); |
---|
1679 | ideal J = std(I); |
---|
1680 | GBWeight(J,u,v); // same as above |
---|
1681 | u = 0,1; |
---|
1682 | GBWeight(I,u,v); |
---|
1683 | } |
---|
1684 | |
---|
1685 | proc inForm (def I, intvec w) |
---|
1686 | "USAGE: inForm(I,w); I ideal or poly, w intvec |
---|
1687 | RETURN: ideal, generated by initial forms of generators of I w.r.t. w, or |
---|
1688 | @* poly, initial form of input poly w.r.t. w |
---|
1689 | PURPOSE: computes the initial form of an ideal or a poly w.r.t. the weight w |
---|
1690 | NOTE: The size of the weight vector must be equal to the number of variables |
---|
1691 | @* of the basering. |
---|
1692 | EXAMPLE: example inForm; shows examples |
---|
1693 | " |
---|
1694 | { |
---|
1695 | string inp1 = typeof(I); |
---|
1696 | if ((inp1 <> "ideal") && (inp1 <> "poly")) |
---|
1697 | { |
---|
1698 | ERROR("first argument has to be an ideal or a poly"); |
---|
1699 | } |
---|
1700 | if (size(w) != nvars(basering)) |
---|
1701 | { |
---|
1702 | ERROR("weight vector has wrong dimension"); |
---|
1703 | } |
---|
1704 | ideal II = I; |
---|
1705 | int j; |
---|
1706 | poly g; |
---|
1707 | ideal J; |
---|
1708 | for (j=1; j<=ncols(II); j++) |
---|
1709 | { |
---|
1710 | g = II[j]; |
---|
1711 | J[j] = g - jet(g,deg(g,w)-1,w); |
---|
1712 | } |
---|
1713 | if (inp1 == "ideal") |
---|
1714 | { |
---|
1715 | return(J); |
---|
1716 | } |
---|
1717 | else |
---|
1718 | { |
---|
1719 | return(J[1]); |
---|
1720 | } |
---|
1721 | } |
---|
1722 | example |
---|
1723 | { |
---|
1724 | "EXAMPLE:"; echo = 2; |
---|
1725 | ring r = 0,(x,y,Dx,Dy),dp; |
---|
1726 | def D = Weyl(); setring D; |
---|
1727 | poly F = 3*x^2*Dy+2*y*Dx; |
---|
1728 | poly G = 2*x*Dx+3*y*Dy+6; |
---|
1729 | ideal I = F,G; |
---|
1730 | intvec w1 = -1,-1,1,1; |
---|
1731 | intvec w2 = -1,-2,1,2; |
---|
1732 | intvec w3 = -2,-3,2,3; |
---|
1733 | inForm(I,w1); |
---|
1734 | inForm(I,w2); |
---|
1735 | inForm(I,w3); |
---|
1736 | inForm(F,w1); |
---|
1737 | } |
---|
1738 | |
---|
1739 | proc initialIdealW(ideal I, intvec u, intvec v, list #) |
---|
1740 | "USAGE: initialIdealW(I,u,v [,s,t,w]); |
---|
1741 | @* I ideal, u,v intvecs, s,t optional ints, w an optional intvec |
---|
1742 | RETURN: ideal, GB of initial ideal of the input ideal wrt the weights u and v |
---|
1743 | ASSUME: The basering is the n-th Weyl algebra in characteristic 0 and for all |
---|
1744 | @* 1<=i<=n the identity var(i+n)*var(i)=var(i)*var(i+1)+1 holds, i.e. the |
---|
1745 | @* sequence of variables is given by x(1),...,x(n),D(1),...,D(n), |
---|
1746 | @* where D(i) is the differential operator belonging to x(i). |
---|
1747 | PURPOSE: computes the initial ideal with respect to given weights. |
---|
1748 | NOTE: u and v are understood as weight vectors for x(1..n) and D(1..n) |
---|
1749 | @* respectively. |
---|
1750 | @* If s<>0, @code{std} is used for Groebner basis computations, |
---|
1751 | @* otherwise, and by default, @code{slimgb} is used. |
---|
1752 | @* If t<>0, a matrix ordering is used for Groebner basis computations, |
---|
1753 | @* otherwise, and by default, a block ordering is used. |
---|
1754 | @* If w is given and consists of exactly 2*n strictly positive entries, |
---|
1755 | @* w is used as homogenization weight. |
---|
1756 | @* Otherwise, and by default, the homogenization weight (1,...,1) is used. |
---|
1757 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
1758 | @* if printlevel>=2, all the debug messages will be printed. |
---|
1759 | EXAMPLE: example initialIdealW; shows examples |
---|
1760 | " |
---|
1761 | { |
---|
1762 | // assumption check in GBWeight |
---|
1763 | int ppl = printlevel - voice + 2; |
---|
1764 | printlevel = printlevel + 1; |
---|
1765 | I = GBWeight(I,u,v,#); |
---|
1766 | printlevel = printlevel - 1; |
---|
1767 | intvec uv = u,v; |
---|
1768 | I = inForm(I,uv); |
---|
1769 | int eng; |
---|
1770 | if (size(#)>0) |
---|
1771 | { |
---|
1772 | if(typeof(#[1])=="int" || typeof(#[1])=="number") |
---|
1773 | { |
---|
1774 | eng = int(#[1]); |
---|
1775 | } |
---|
1776 | } |
---|
1777 | dbprint(ppl,"// starting cosmetic Groebner basis computation"); |
---|
1778 | I = engine(I,eng); |
---|
1779 | dbprint(ppl,"// finished cosmetic Groebner basis computation"); |
---|
1780 | return(I); |
---|
1781 | } |
---|
1782 | example |
---|
1783 | { |
---|
1784 | "EXAMPLE:"; echo = 2; |
---|
1785 | ring r = 0,(x,y,Dx,Dy),dp; |
---|
1786 | def D2 = Weyl(); |
---|
1787 | setring D2; |
---|
1788 | ideal I = 3*x^2*Dy+2*y*Dx,2*x*Dx+3*y*Dy+6; |
---|
1789 | intvec u = -2,-3; |
---|
1790 | intvec v = -u; |
---|
1791 | initialIdealW(I,u,v); |
---|
1792 | ideal J = std(I); |
---|
1793 | initialIdealW(J,u,v); // same as above |
---|
1794 | u = 0,1; |
---|
1795 | initialIdealW(I,u,v); |
---|
1796 | } |
---|
1797 | |
---|
1798 | proc initialMalgrange (poly f,list #) |
---|
1799 | "USAGE: initialMalgrange(f,[,a,b,v]); f poly, a,b optional ints, v opt. intvec |
---|
1800 | RETURN: ring, Weyl algebra induced by basering, extended by two new vars t,Dt |
---|
1801 | PURPOSE: computes the initial Malgrange ideal of a given polynomial w.r.t. the |
---|
1802 | @* weight vector (-1,0...,0,1,0,...,0) such that the weight of t is -1 |
---|
1803 | @* and the weight of Dt is 1. |
---|
1804 | ASSUME: The basering is commutative and over a field of characteristic 0. |
---|
1805 | NOTE: Activate the output ring with the @code{setring} command. |
---|
1806 | @* The returned ring contains the ideal 'inF', being the initial ideal |
---|
1807 | @* of the Malgrange ideal of f. |
---|
1808 | @* Varnames of the basering should not include t and Dt. |
---|
1809 | @* If a<>0, @code{std} is used for Groebner basis computations, |
---|
1810 | @* otherwise, and by default, @code{slimgb} is used. |
---|
1811 | @* If b<>0, a matrix ordering is used for Groebner basis computations, |
---|
1812 | @* otherwise, and by default, a block ordering is used. |
---|
1813 | @* If a positive weight vector v is given, the weight |
---|
1814 | @* (d,v[1],...,v[n],1,d+1-v[1],...,d+1-v[n]) is used for homogenization |
---|
1815 | @* computations, where d denotes the weighted degree of f. |
---|
1816 | @* Otherwise and by default, v is set to (1,...,1). See Noro (2002). |
---|
1817 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
1818 | @* if printlevel>=2, all the debug messages will be printed. |
---|
1819 | EXAMPLE: example initialMalgrange; shows examples |
---|
1820 | " |
---|
1821 | { |
---|
1822 | dmodappAssumeViolation(); |
---|
1823 | if (!isCommutative()) |
---|
1824 | { |
---|
1825 | ERROR("Basering must be commutative."); |
---|
1826 | } |
---|
1827 | int ppl = printlevel - voice + 2; |
---|
1828 | def save = basering; |
---|
1829 | int n = nvars(save); |
---|
1830 | int i; |
---|
1831 | int whichengine = 0; // default |
---|
1832 | int methodord = 0; // default |
---|
1833 | intvec u0 = 1:n; // default |
---|
1834 | if (size(#)>0) |
---|
1835 | { |
---|
1836 | if (intLike(#[1])) |
---|
1837 | { |
---|
1838 | whichengine = int(#[1]); |
---|
1839 | } |
---|
1840 | if (size(#)>1) |
---|
1841 | { |
---|
1842 | if (intLike(#[2])) |
---|
1843 | { |
---|
1844 | methodord = int(#[2]); |
---|
1845 | } |
---|
1846 | if (size(#)>2) |
---|
1847 | { |
---|
1848 | if ((typeof(#[3])=="intvec") && (size(#[3])==n) && (allPositive(#[3])==1)) |
---|
1849 | { |
---|
1850 | u0 = #[3]; |
---|
1851 | } |
---|
1852 | } |
---|
1853 | } |
---|
1854 | } |
---|
1855 | list RL = ringlist(save); |
---|
1856 | RL = RL[1..4]; // if basering is commutative nc_algebra |
---|
1857 | list C0 = list("C",intvec(0)); |
---|
1858 | // 1. create homogenization weights |
---|
1859 | // 1.1. get the weighted degree of f |
---|
1860 | list Lord = list(list("wp",u0),C0); |
---|
1861 | list L@r = RL; |
---|
1862 | L@r[3] = Lord; |
---|
1863 | def r = ring(L@r); kill L@r,Lord; |
---|
1864 | setring r; |
---|
1865 | poly f = imap(save,f); |
---|
1866 | int d = deg(f); |
---|
1867 | setring save; kill r; |
---|
1868 | // 1.2 the homogenization weights |
---|
1869 | intvec homogweights = d; |
---|
1870 | homogweights[n+2] = 1; |
---|
1871 | for (i=1; i<=n; i++) |
---|
1872 | { |
---|
1873 | homogweights[i+1] = u0[i]; |
---|
1874 | homogweights[n+2+i] = d+1-u0[i]; |
---|
1875 | } |
---|
1876 | // 2. create extended Weyl algebra |
---|
1877 | int N = 2*n+2; |
---|
1878 | // 2.1 create names for vars |
---|
1879 | string vart = safeVarName("t","cv"); |
---|
1880 | string varDt = safeVarName("D"+vart,"cv"); |
---|
1881 | while (varDt <> "D"+vart) |
---|
1882 | { |
---|
1883 | vart = safeVarName("@"+vart,"cv"); |
---|
1884 | varDt = safeVarName("D"+vart,"cv"); |
---|
1885 | } |
---|
1886 | list Lvar; |
---|
1887 | Lvar[1] = vart; Lvar[n+2] = varDt; |
---|
1888 | for (i=1; i<=n; i++) |
---|
1889 | { |
---|
1890 | Lvar[i+1] = string(var(i)); |
---|
1891 | Lvar[i+n+2] = safeVarName("D" + string(var(i)),"cv"); |
---|
1892 | } |
---|
1893 | // 2.2 create ordering |
---|
1894 | list Lord = list(list("dp",intvec(1:N)),C0); |
---|
1895 | // 2.3 create the (n+1)-th Weyl algebra |
---|
1896 | list L@D = RL; L@D[2] = Lvar; L@D[3] = Lord; |
---|
1897 | def @D = ring(L@D); kill L@D; |
---|
1898 | setring @D; |
---|
1899 | def D = Weyl(); |
---|
1900 | setring D; kill @D; |
---|
1901 | dbprint(ppl,"// the (n+1)-th Weyl algebra :" ,D); |
---|
1902 | // 3. compute the initial ideal |
---|
1903 | // 3.1 create the Malgrange ideal |
---|
1904 | poly f = imap(save,f); |
---|
1905 | ideal I = var(1)-f; |
---|
1906 | for (i=1; i<=n; i++) |
---|
1907 | { |
---|
1908 | I = I, var(n+2+i)+diff(f,var(i+1))*var(n+2); |
---|
1909 | } |
---|
1910 | // I = engine(I,whichengine); // todo efficient to compute GB wrt dp first? |
---|
1911 | // 3.2 computie the initial ideal |
---|
1912 | intvec w = 1,0:n; |
---|
1913 | printlevel = printlevel + 1; |
---|
1914 | I = initialIdealW(I,-w,w,whichengine,methodord,homogweights); |
---|
1915 | printlevel = printlevel - 1; |
---|
1916 | ideal inF = I; attrib(inF,"isSB",1); |
---|
1917 | export(inF); |
---|
1918 | setring save; |
---|
1919 | return(D); |
---|
1920 | } |
---|
1921 | example |
---|
1922 | { |
---|
1923 | "EXAMPLE:"; echo = 2; |
---|
1924 | ring r = 0,(x,y),dp; |
---|
1925 | poly f = x^2+y^3+x*y^2; |
---|
1926 | def D = initialMalgrange(f); |
---|
1927 | setring D; |
---|
1928 | inF; |
---|
1929 | setring r; |
---|
1930 | intvec v = 3,2; |
---|
1931 | def D2 = initialMalgrange(f,1,1,v); |
---|
1932 | setring D2; |
---|
1933 | inF; |
---|
1934 | } |
---|
1935 | |
---|
1936 | |
---|
1937 | // restriction and integration //////////////////////////////////////////////// |
---|
1938 | |
---|
1939 | static proc restrictionModuleEngine (ideal I, intvec w, list #) |
---|
1940 | // returns list L with 2 entries of type ideal |
---|
1941 | // L[1]=basis of free module, L[2]=generating system of submodule |
---|
1942 | // #=eng,m,G; eng=engine; m=min int root of bfctIdeal(I,w); G=GB of I wrt (-w,w) |
---|
1943 | { |
---|
1944 | dmodappMoreAssumeViolation(); |
---|
1945 | if (!isHolonomic(I)) |
---|
1946 | { |
---|
1947 | ERROR("Given ideal is not holonomic"); |
---|
1948 | } |
---|
1949 | int l0,l0set,Gset; |
---|
1950 | ideal G; |
---|
1951 | int whichengine = 0; // default |
---|
1952 | if (size(#)>0) |
---|
1953 | { |
---|
1954 | if (intLike(#[1])) |
---|
1955 | { |
---|
1956 | whichengine = int(#[1]); |
---|
1957 | } |
---|
1958 | if (size(#)>1) |
---|
1959 | { |
---|
1960 | if (intLike(#[2])) |
---|
1961 | { |
---|
1962 | l0 = int(#[2]); |
---|
1963 | l0set = 1; |
---|
1964 | } |
---|
1965 | if (size(#)>2) |
---|
1966 | { |
---|
1967 | if (typeof(#[3])=="ideal") |
---|
1968 | { |
---|
1969 | G = #[3]; |
---|
1970 | Gset = 1; |
---|
1971 | } |
---|
1972 | } |
---|
1973 | } |
---|
1974 | } |
---|
1975 | int ppl = printlevel; |
---|
1976 | int i,j,k; |
---|
1977 | int n = nvars(basering) div 2; |
---|
1978 | if (w == 0:size(w)) |
---|
1979 | { |
---|
1980 | ERROR("weight vector must not be zero"); |
---|
1981 | } |
---|
1982 | if (size(w)<>n) |
---|
1983 | { |
---|
1984 | ERROR("weight vector must have exactly " + string(n) + " entries"); |
---|
1985 | } |
---|
1986 | for (i=1; i<=n; i++) |
---|
1987 | { |
---|
1988 | if (w[i]<0) |
---|
1989 | { |
---|
1990 | ERROR("weight vector must not have negative entries"); |
---|
1991 | } |
---|
1992 | } |
---|
1993 | intvec ww = -w,w; |
---|
1994 | if (!Gset) |
---|
1995 | { |
---|
1996 | G = GBWeight(I,-w,w,whichengine); |
---|
1997 | dbprint(ppl,"// found GB wrt weight " +string(w)); |
---|
1998 | dbprint(ppl-1,"// " + string(G)); |
---|
1999 | } |
---|
2000 | if (!l0set) |
---|
2001 | { |
---|
2002 | ideal inG = inForm(G,ww); |
---|
2003 | inG = engine(inG,whichengine); |
---|
2004 | poly s = 0; |
---|
2005 | for (i=1; i<=n; i++) |
---|
2006 | { |
---|
2007 | s = s + w[i]*var(i)*var(i+n); |
---|
2008 | } |
---|
2009 | vector v = pIntersect(s,inG); |
---|
2010 | list L = bFactor(vec2poly(v)); |
---|
2011 | dbprint(ppl,"// found b-function of given ideal wrt weight " + string(w)); |
---|
2012 | dbprint(ppl-1,"// roots: "+string(L[1])); |
---|
2013 | dbprint(ppl-1,"// multiplicities: "+string(L[2])); |
---|
2014 | kill inG,v,s; |
---|
2015 | L = intRoots(L); // integral roots of b-function |
---|
2016 | if (L[2]==0:size(L[2])) // no integral roots |
---|
2017 | { |
---|
2018 | return(list(ideal(0),ideal(0))); |
---|
2019 | } |
---|
2020 | intvec v; |
---|
2021 | for (i=1; i<=ncols(L[1]); i++) |
---|
2022 | { |
---|
2023 | v[i] = int(L[1][i]); |
---|
2024 | } |
---|
2025 | l0 = Max(v); |
---|
2026 | dbprint(ppl,"// maximal integral root is " +string(l0)); |
---|
2027 | kill L,v; |
---|
2028 | } |
---|
2029 | if (l0 < 0) // maximal integral root is < 0 |
---|
2030 | { |
---|
2031 | return(list(ideal(0),ideal(0))); |
---|
2032 | } |
---|
2033 | intvec m; |
---|
2034 | for (i=ncols(G); i>0; i--) |
---|
2035 | { |
---|
2036 | m[i] = deg(G[i],ww); |
---|
2037 | } |
---|
2038 | dbprint(ppl,"// weighted degree of generators of GB is " +string(m)); |
---|
2039 | def save = basering; |
---|
2040 | list RL = ringlist(save); |
---|
2041 | RL = RL[1..4]; |
---|
2042 | list Lvar; |
---|
2043 | j = 1; |
---|
2044 | intvec neww; |
---|
2045 | for (i=1; i<=n; i++) |
---|
2046 | { |
---|
2047 | if (w[i]>0) |
---|
2048 | { |
---|
2049 | Lvar[j] = string(var(i+n)); |
---|
2050 | neww[j] = w[i]; |
---|
2051 | j++; |
---|
2052 | } |
---|
2053 | } |
---|
2054 | list Lord; |
---|
2055 | Lord[1] = list("dp",intvec(1:n)); |
---|
2056 | Lord[2] = list("C", intvec(0)); |
---|
2057 | RL[2] = Lvar; |
---|
2058 | RL[3] = Lord; |
---|
2059 | def r = ring(RL); |
---|
2060 | kill Lvar, Lord, RL; |
---|
2061 | setring r; |
---|
2062 | ideal B; |
---|
2063 | list Blist; |
---|
2064 | intvec mm = l0,-m+l0; |
---|
2065 | for (i=0; i<=Max(mm); i++) |
---|
2066 | { |
---|
2067 | B = weightKB(std(0),i,neww); |
---|
2068 | Blist[i+1] = B; |
---|
2069 | } |
---|
2070 | setring save; |
---|
2071 | list Blist = imap(r,Blist); |
---|
2072 | ideal ff = maxideal(1); |
---|
2073 | for (i=1; i<=n; i++) |
---|
2074 | { |
---|
2075 | if (w[i]<>0) |
---|
2076 | { |
---|
2077 | ff[i] = 0; |
---|
2078 | } |
---|
2079 | } |
---|
2080 | map f = save,ff; |
---|
2081 | ideal B,M; |
---|
2082 | poly p; |
---|
2083 | for (i=1; i<=size(G); i++) |
---|
2084 | { |
---|
2085 | for (j=1; j<=l0-m[i]+1; j++) |
---|
2086 | { |
---|
2087 | B = Blist[j]; |
---|
2088 | for (k=1; k<=ncols(B); k++) |
---|
2089 | { |
---|
2090 | p = B[k]*G[i]; |
---|
2091 | p = f(p); |
---|
2092 | M[size(M)+1] = p; |
---|
2093 | } |
---|
2094 | } |
---|
2095 | } |
---|
2096 | ideal Bl0 = Blist[1..(l0+1)]; |
---|
2097 | dbprint(ppl,"// found basis of free module"); |
---|
2098 | dbprint(ppl-1,"// " + string(Bl0)); |
---|
2099 | dbprint(ppl,"// found generators of submodule"); |
---|
2100 | dbprint(ppl-1,"// " + string(M)); |
---|
2101 | return(list(Bl0,M)); |
---|
2102 | } |
---|
2103 | |
---|
2104 | static proc restrictionModuleOutput (ideal B, ideal N, intvec w, int eng, string str) |
---|
2105 | // returns ring, which contains module "str" |
---|
2106 | { |
---|
2107 | int n = nvars(basering) div 2; |
---|
2108 | int i,j; |
---|
2109 | def save = basering; |
---|
2110 | // 1: create new ring |
---|
2111 | list RL = ringlist(save); |
---|
2112 | RL = RL[1..4]; |
---|
2113 | list V = RL[2]; |
---|
2114 | poly prodvar = 1; |
---|
2115 | int zeropresent; |
---|
2116 | j = 0; |
---|
2117 | for (i=1; i<=n; i++) |
---|
2118 | { |
---|
2119 | if (w[i]<>0) |
---|
2120 | { |
---|
2121 | V = delete(V,i-j); |
---|
2122 | V = delete(V,i-2*j-1+n); |
---|
2123 | j = j+1; |
---|
2124 | prodvar = prodvar*var(i)*var(i+n); |
---|
2125 | } |
---|
2126 | else |
---|
2127 | { |
---|
2128 | zeropresent = 1; |
---|
2129 | } |
---|
2130 | } |
---|
2131 | if (!zeropresent) // restrict/integrate all vars, return input ring |
---|
2132 | { |
---|
2133 | def newR = save; |
---|
2134 | } |
---|
2135 | else |
---|
2136 | { |
---|
2137 | RL[2] = V; |
---|
2138 | V = list(); |
---|
2139 | V[1] = list("C", intvec(0)); |
---|
2140 | V[2] = list("dp",intvec(1:(2*size(ideal(w))))); |
---|
2141 | RL[3] = V; |
---|
2142 | def @D = ring(RL); |
---|
2143 | setring @D; |
---|
2144 | def newR = Weyl(); |
---|
2145 | setring save; |
---|
2146 | kill @D; |
---|
2147 | } |
---|
2148 | // 2. get coker representation of module |
---|
2149 | module M = coeffs(N,B,prodvar); |
---|
2150 | if (zeropresent) |
---|
2151 | { |
---|
2152 | setring newR; |
---|
2153 | module M = imap(save,M); |
---|
2154 | } |
---|
2155 | M = engine(M,eng); |
---|
2156 | M = prune(M); |
---|
2157 | M = engine(M,eng); |
---|
2158 | execute("module " + str + " = M;"); |
---|
2159 | execute("export(" + str + ");"); |
---|
2160 | setring save; |
---|
2161 | return(newR); |
---|
2162 | } |
---|
2163 | |
---|
2164 | proc restrictionModule (ideal I, intvec w, list #) |
---|
2165 | "USAGE: restrictionModule(I,w,[,eng,m,G]); |
---|
2166 | @* I ideal, w intvec, eng and m optional ints, G optional ideal |
---|
2167 | RETURN: ring (a Weyl algebra) containing a module 'resMod' |
---|
2168 | ASSUME: The basering is the n-th Weyl algebra over a field of characteristic 0 |
---|
2169 | @* and for all 1<=i<=n the identity var(i+n)*var(i)=var(i)*var(i+1)+1 |
---|
2170 | @* holds, i.e. the sequence of variables is given by |
---|
2171 | @* x(1),...,x(n),D(1),...,D(n), where D(i) is the differential operator |
---|
2172 | @* belonging to x(i). |
---|
2173 | @* Further, assume that I is holonomic and that w is n-dimensional with |
---|
2174 | @* non-negative entries. |
---|
2175 | PURPOSE: computes the restriction module of a holonomic ideal to the subspace |
---|
2176 | @* defined by the variables corresponding to the non-zero entries of the |
---|
2177 | @* given intvec |
---|
2178 | NOTE: The output ring is the Weyl algebra defined by the zero entries of w. |
---|
2179 | @* It contains a module 'resMod' being the restriction module of I wrt w. |
---|
2180 | @* If there are no zero entries, the input ring is returned. |
---|
2181 | @* If eng<>0, @code{std} is used for Groebner basis computations, |
---|
2182 | @* otherwise, and by default, @code{slimgb} is used. |
---|
2183 | @* The minimal integer root of the b-function of I wrt the weight (-w,w) |
---|
2184 | @* can be specified via the optional argument m. |
---|
2185 | @* The optional argument G is used for specifying a Groebner Basis of I |
---|
2186 | @* wrt the weight (-w,w), that is, the initial form of G generates the |
---|
2187 | @* initial ideal of I wrt the weight (-w,w). |
---|
2188 | @* Further note, that the assumptions on m and G (if given) are not |
---|
2189 | @* checked. |
---|
2190 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
2191 | @* if printlevel>=2, all the debug messages will be printed. |
---|
2192 | EXAMPLE: example restrictionModule; shows examples |
---|
2193 | " |
---|
2194 | { |
---|
2195 | list L = restrictionModuleEngine(I,w,#); |
---|
2196 | int eng; |
---|
2197 | if (size(#)>0) |
---|
2198 | { |
---|
2199 | if (intLike(#[1])) |
---|
2200 | { |
---|
2201 | eng = int(#[1]); |
---|
2202 | } |
---|
2203 | } |
---|
2204 | def newR = restrictionModuleOutput(L[1],L[2],w,eng,"resMod"); |
---|
2205 | return(newR); |
---|
2206 | } |
---|
2207 | example |
---|
2208 | { |
---|
2209 | "EXAMPLE:"; echo = 2; |
---|
2210 | ring r = 0,(a,x,b,Da,Dx,Db),dp; |
---|
2211 | def D3 = Weyl(); |
---|
2212 | setring D3; |
---|
2213 | ideal I = a*Db-Dx+2*Da, x*Db-Da, x*Da+a*Da+b*Db+1, |
---|
2214 | x*Dx-2*x*Da-a*Da, b*Db^2+Dx*Da-Da^2+Db, |
---|
2215 | a*Dx*Da+2*x*Da^2+a*Da^2+b*Dx*Db+Dx+2*Da; |
---|
2216 | intvec w = 1,0,0; |
---|
2217 | def rm = restrictionModule(I,w); |
---|
2218 | setring rm; rm; |
---|
2219 | print(resMod); |
---|
2220 | } |
---|
2221 | |
---|
2222 | static proc restrictionIdealEngine (ideal I, intvec w, string cf, list #) |
---|
2223 | { |
---|
2224 | int eng; |
---|
2225 | if (size(#)>0) |
---|
2226 | { |
---|
2227 | if(intLike(#[1])) |
---|
2228 | { |
---|
2229 | eng = int(#[1]); |
---|
2230 | } |
---|
2231 | } |
---|
2232 | def save = basering; |
---|
2233 | if (cf == "restriction") |
---|
2234 | { |
---|
2235 | def newR = restrictionModule(I,w,#); |
---|
2236 | setring newR; |
---|
2237 | matrix M = resMod; |
---|
2238 | kill resMod; |
---|
2239 | } |
---|
2240 | if (cf == "integral") |
---|
2241 | { |
---|
2242 | def newR = integralModule(I,w,#); |
---|
2243 | setring newR; |
---|
2244 | matrix M = intMod; |
---|
2245 | kill intMod; |
---|
2246 | } |
---|
2247 | int i,r,c; |
---|
2248 | r = nrows(M); |
---|
2249 | c = ncols(M); |
---|
2250 | ideal J; |
---|
2251 | if (r == 1) // nothing to do |
---|
2252 | { |
---|
2253 | J = M; |
---|
2254 | } |
---|
2255 | else |
---|
2256 | { |
---|
2257 | matrix zm[r-1][1]; // zero matrix |
---|
2258 | matrix v[r-1][1]; |
---|
2259 | for (i=1; i<=c; i++) |
---|
2260 | { |
---|
2261 | if (M[1,i]<>0) |
---|
2262 | { |
---|
2263 | v = M[2..r,i]; |
---|
2264 | if (v == zm) |
---|
2265 | { |
---|
2266 | J[size(J)+1] = M[1,i]; |
---|
2267 | } |
---|
2268 | } |
---|
2269 | } |
---|
2270 | } |
---|
2271 | J = engine(J,eng); |
---|
2272 | if (cf == "restriction") |
---|
2273 | { |
---|
2274 | ideal resIdeal = J; |
---|
2275 | export(resIdeal); |
---|
2276 | } |
---|
2277 | if (cf == "integral") |
---|
2278 | { |
---|
2279 | ideal intIdeal = J; |
---|
2280 | export(intIdeal); |
---|
2281 | } |
---|
2282 | setring save; |
---|
2283 | return(newR); |
---|
2284 | } |
---|
2285 | |
---|
2286 | proc restrictionIdeal (ideal I, intvec w, list #) |
---|
2287 | "USAGE: restrictionIdeal(I,w,[,eng,m,G]); |
---|
2288 | @* I ideal, w intvec, eng and m optional ints, G optional ideal |
---|
2289 | RETURN: ring (a Weyl algebra) containing an ideal 'resIdeal' |
---|
2290 | ASSUME: The basering is the n-th Weyl algebra over a field of characteristic 0 |
---|
2291 | @* and for all 1<=i<=n the identity var(i+n)*var(i)=var(i)*var(i+1)+1 |
---|
2292 | @* holds, i.e. the sequence of variables is given by |
---|
2293 | @* x(1),...,x(n),D(1),...,D(n), where D(i) is the differential operator |
---|
2294 | @* belonging to x(i). |
---|
2295 | @* Further, assume that I is holonomic and that w is n-dimensional with |
---|
2296 | @* non-negative entries. |
---|
2297 | PURPOSE: computes the restriction ideal of a holonomic ideal to the subspace |
---|
2298 | @* defined by the variables corresponding to the non-zero entries of the |
---|
2299 | @* given intvec |
---|
2300 | NOTE: The output ring is the Weyl algebra defined by the zero entries of w. |
---|
2301 | @* It contains an ideal 'resIdeal' being the restriction ideal of I wrt w. |
---|
2302 | @* If there are no zero entries, the input ring is returned. |
---|
2303 | @* If eng<>0, @code{std} is used for Groebner basis computations, |
---|
2304 | @* otherwise, and by default, @code{slimgb} is used. |
---|
2305 | @* The minimal integer root of the b-function of I wrt the weight (-w,w) |
---|
2306 | @* can be specified via the optional argument m. |
---|
2307 | @* The optional argument G is used for specifying a Groebner basis of I |
---|
2308 | @* wrt the weight (-w,w), that is, the initial form of G generates the |
---|
2309 | @* initial ideal of I wrt the weight (-w,w). |
---|
2310 | @* Further note, that the assumptions on m and G (if given) are not |
---|
2311 | @* checked. |
---|
2312 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
2313 | @* if printlevel>=2, all the debug messages will be printed. |
---|
2314 | EXAMPLE: example restrictionIdeal; shows examples |
---|
2315 | " |
---|
2316 | { |
---|
2317 | def rm = restrictionIdealEngine(I,w,"restriction",#); |
---|
2318 | return(rm); |
---|
2319 | } |
---|
2320 | example |
---|
2321 | { |
---|
2322 | "EXAMPLE:"; echo = 2; |
---|
2323 | ring r = 0,(a,x,b,Da,Dx,Db),dp; |
---|
2324 | def D3 = Weyl(); |
---|
2325 | setring D3; |
---|
2326 | ideal I = a*Db-Dx+2*Da, |
---|
2327 | x*Db-Da, |
---|
2328 | x*Da+a*Da+b*Db+1, |
---|
2329 | x*Dx-2*x*Da-a*Da, |
---|
2330 | b*Db^2+Dx*Da-Da^2+Db, |
---|
2331 | a*Dx*Da+2*x*Da^2+a*Da^2+b*Dx*Db+Dx+2*Da; |
---|
2332 | intvec w = 1,0,0; |
---|
2333 | def D2 = restrictionIdeal(I,w); |
---|
2334 | setring D2; D2; |
---|
2335 | resIdeal; |
---|
2336 | } |
---|
2337 | |
---|
2338 | proc fourier (ideal I, list #) |
---|
2339 | "USAGE: fourier(I[,v]); I an ideal, v an optional intvec |
---|
2340 | RETURN: ideal |
---|
2341 | PURPOSE: computes the Fourier transform of an ideal in a Weyl algebra |
---|
2342 | ASSUME: The basering is the n-th Weyl algebra over a field of characteristic 0 |
---|
2343 | @* and for all 1<=i<=n the identity var(i+n)*var(i)=var(i)*var(i+1)+1 |
---|
2344 | @* holds, i.e. the sequence of variables is given by |
---|
2345 | @* x(1),...,x(n),D(1),...,D(n), where D(i) is the differential operator |
---|
2346 | @* belonging to x(i). |
---|
2347 | NOTE: The Fourier automorphism is defined by mapping x(i) to -D(i) and |
---|
2348 | @* D(i) to x(i). |
---|
2349 | @* If v is an intvec with entries ranging from 1 to n, the Fourier |
---|
2350 | @* transform of I restricted to the variables given by v is computed. |
---|
2351 | SEE ALSO: inverseFourier |
---|
2352 | EXAMPLE: example fourier; shows examples |
---|
2353 | " |
---|
2354 | { |
---|
2355 | dmodappMoreAssumeViolation(); |
---|
2356 | intvec v; |
---|
2357 | if (size(#)>0) |
---|
2358 | { |
---|
2359 | if(typeof(#[1])=="intvec") |
---|
2360 | { |
---|
2361 | v = #[1]; |
---|
2362 | } |
---|
2363 | } |
---|
2364 | int n = nvars(basering) div 2; |
---|
2365 | int i; |
---|
2366 | if(v <> 0:size(v)) |
---|
2367 | { |
---|
2368 | v = sortIntvec(v)[1]; |
---|
2369 | for (i=1; i<size(v); i++) |
---|
2370 | { |
---|
2371 | if (v[i] == v[i+1]) |
---|
2372 | { |
---|
2373 | ERROR("No double entries allowed in intvec"); |
---|
2374 | } |
---|
2375 | } |
---|
2376 | } |
---|
2377 | else |
---|
2378 | { |
---|
2379 | v = 1..n; |
---|
2380 | } |
---|
2381 | ideal m = maxideal(1); |
---|
2382 | for (i=1; i<=size(v); i++) |
---|
2383 | { |
---|
2384 | if (v[i]<0 || v[i]>n) |
---|
2385 | { |
---|
2386 | ERROR("Entries of intvec must range from 1 to "+string(n)); |
---|
2387 | } |
---|
2388 | m[v[i]] = -var(v[i]+n); |
---|
2389 | m[v[i]+n] = var(v[i]); |
---|
2390 | } |
---|
2391 | map F = basering,m; |
---|
2392 | ideal FI = F(I); |
---|
2393 | return(FI); |
---|
2394 | } |
---|
2395 | example |
---|
2396 | { |
---|
2397 | "EXAMPLE:"; echo = 2; |
---|
2398 | ring r = 0,(x,y,Dx,Dy),dp; |
---|
2399 | def D2 = Weyl(); |
---|
2400 | setring D2; |
---|
2401 | ideal I = x*Dx+2*y*Dy+2, x^2*Dx+y*Dx+2*x; |
---|
2402 | intvec v = 2; |
---|
2403 | fourier(I,v); |
---|
2404 | fourier(I); |
---|
2405 | } |
---|
2406 | |
---|
2407 | proc inverseFourier (ideal I, list #) |
---|
2408 | "USAGE: inverseFourier(I[,v]); I an ideal, v an optional intvec |
---|
2409 | RETURN: ideal |
---|
2410 | PURPOSE: computes the inverse Fourier transform of an ideal in a Weyl algebra |
---|
2411 | ASSUME: The basering is the n-th Weyl algebra over a field of characteristic 0 |
---|
2412 | @* and for all 1<=i<=n the identity var(i+n)*var(i)=var(i)*var(i+1)+1 |
---|
2413 | @* holds, i.e. the sequence of variables is given by |
---|
2414 | @* x(1),...,x(n),D(1),...,D(n), where D(i) is the differential operator |
---|
2415 | @* belonging to x(i). |
---|
2416 | NOTE: The Fourier automorphism is defined by mapping x(i) to -D(i) and |
---|
2417 | @* D(i) to x(i). |
---|
2418 | @* If v is an intvec with entries ranging from 1 to n, the inverse Fourier |
---|
2419 | @* transform of I restricted to the variables given by v is computed. |
---|
2420 | SEE ALSO: fourier |
---|
2421 | EXAMPLE: example inverseFourier; shows examples |
---|
2422 | " |
---|
2423 | { |
---|
2424 | dmodappMoreAssumeViolation(); |
---|
2425 | intvec v; |
---|
2426 | if (size(#)>0) |
---|
2427 | { |
---|
2428 | if(typeof(#[1])=="intvec") |
---|
2429 | { |
---|
2430 | v = #[1]; |
---|
2431 | } |
---|
2432 | } |
---|
2433 | int n = nvars(basering) div 2; |
---|
2434 | int i; |
---|
2435 | if(v <> 0:size(v)) |
---|
2436 | { |
---|
2437 | v = sortIntvec(v)[1]; |
---|
2438 | for (i=1; i<size(v); i++) |
---|
2439 | { |
---|
2440 | if (v[i] == v[i+1]) |
---|
2441 | { |
---|
2442 | ERROR("No double entries allowed in intvec"); |
---|
2443 | } |
---|
2444 | } |
---|
2445 | } |
---|
2446 | else |
---|
2447 | { |
---|
2448 | v = 1..n; |
---|
2449 | } |
---|
2450 | ideal m = maxideal(1); |
---|
2451 | for (i=1; i<=size(v); i++) |
---|
2452 | { |
---|
2453 | if (v[i]<0 || v[i]>n) |
---|
2454 | { |
---|
2455 | ERROR("Entries of intvec must range between 1 and "+string(n)); |
---|
2456 | } |
---|
2457 | m[v[i]] = var(v[i]+n); |
---|
2458 | m[v[i]+n] = -var(v[i]); |
---|
2459 | } |
---|
2460 | map F = basering,m; |
---|
2461 | ideal FI = F(I); |
---|
2462 | return(FI); |
---|
2463 | } |
---|
2464 | example |
---|
2465 | { |
---|
2466 | "EXAMPLE:"; echo = 2; |
---|
2467 | ring r = 0,(x,y,Dx,Dy),dp; |
---|
2468 | def D2 = Weyl(); |
---|
2469 | setring D2; |
---|
2470 | ideal I = x*Dx+2*y*Dy+2, x^2*Dx+y*Dx+2*x; |
---|
2471 | intvec v = 2; |
---|
2472 | ideal FI = fourier(I); |
---|
2473 | inverseFourier(FI); |
---|
2474 | } |
---|
2475 | |
---|
2476 | proc integralModule (ideal I, intvec w, list #) |
---|
2477 | "USAGE: integralModule(I,w,[,eng,m,G]); |
---|
2478 | @* I ideal, w intvec, eng and m optional ints, G optional ideal |
---|
2479 | RETURN: ring (a Weyl algebra) containing a module 'intMod' |
---|
2480 | ASSUME: The basering is the n-th Weyl algebra over a field of characteristic 0 |
---|
2481 | @* and for all 1<=i<=n the identity var(i+n)*var(i)=var(i)*var(i+1)+1 |
---|
2482 | @* holds, i.e. the sequence of variables is given by |
---|
2483 | @* x(1),...,x(n),D(1),...,D(n), where D(i) is the differential operator |
---|
2484 | @* belonging to x(i). |
---|
2485 | @* Further, assume that I is holonomic and that w is n-dimensional with |
---|
2486 | @* non-negative entries. |
---|
2487 | PURPOSE: computes the integral module of a holonomic ideal w.r.t. the subspace |
---|
2488 | @* defined by the variables corresponding to the non-zero entries of the |
---|
2489 | @* given intvec |
---|
2490 | NOTE: The output ring is the Weyl algebra defined by the zero entries of w. |
---|
2491 | @* It contains a module 'intMod' being the integral module of I wrt w. |
---|
2492 | @* If there are no zero entries, the input ring is returned. |
---|
2493 | @* If eng<>0, @code{std} is used for Groebner basis computations, |
---|
2494 | @* otherwise, and by default, @code{slimgb} is used. |
---|
2495 | @* Let F(I) denote the Fourier transform of I w.r.t. w. |
---|
2496 | @* The minimal integer root of the b-function of F(I) w.r.t. the weight |
---|
2497 | @* (-w,w) can be specified via the optional argument m. |
---|
2498 | @* The optional argument G is used for specifying a Groebner Basis of F(I) |
---|
2499 | @* wrt the weight (-w,w), that is, the initial form of G generates the |
---|
2500 | @* initial ideal of F(I) w.r.t. the weight (-w,w). |
---|
2501 | @* Further note, that the assumptions on m and G (if given) are not |
---|
2502 | @* checked. |
---|
2503 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
2504 | @* if printlevel>=2, all the debug messages will be printed. |
---|
2505 | EXAMPLE: example integralModule; shows examples |
---|
2506 | " |
---|
2507 | { |
---|
2508 | int l0,l0set,Gset; |
---|
2509 | ideal G; |
---|
2510 | int whichengine = 0; // default |
---|
2511 | if (size(#)>0) |
---|
2512 | { |
---|
2513 | if (intLike(#[1])) |
---|
2514 | { |
---|
2515 | whichengine = int(#[1]); |
---|
2516 | } |
---|
2517 | if (size(#)>1) |
---|
2518 | { |
---|
2519 | if (intLike(#[2])) |
---|
2520 | { |
---|
2521 | l0 = int(#[2]); |
---|
2522 | l0set = 1; |
---|
2523 | } |
---|
2524 | if (size(#)>2) |
---|
2525 | { |
---|
2526 | if (typeof(#[3])=="ideal") |
---|
2527 | { |
---|
2528 | G = #[3]; |
---|
2529 | Gset = 1; |
---|
2530 | } |
---|
2531 | } |
---|
2532 | } |
---|
2533 | } |
---|
2534 | int ppl = printlevel; |
---|
2535 | int i; |
---|
2536 | int n = nvars(basering) div 2; |
---|
2537 | intvec v; |
---|
2538 | for (i=1; i<=n; i++) |
---|
2539 | { |
---|
2540 | if (w[i]>0) |
---|
2541 | { |
---|
2542 | if (v == 0:size(v)) |
---|
2543 | { |
---|
2544 | v[1] = i; |
---|
2545 | } |
---|
2546 | else |
---|
2547 | { |
---|
2548 | v[size(v)+1] = i; |
---|
2549 | } |
---|
2550 | } |
---|
2551 | } |
---|
2552 | ideal FI = fourier(I,v); |
---|
2553 | dbprint(ppl,"// computed Fourier transform of given ideal"); |
---|
2554 | dbprint(ppl-1,"// " + string(FI)); |
---|
2555 | list L; |
---|
2556 | if (l0set) |
---|
2557 | { |
---|
2558 | if (Gset) // l0 and G given |
---|
2559 | { |
---|
2560 | L = restrictionModuleEngine(FI,w,whichengine,l0,G); |
---|
2561 | } |
---|
2562 | else // l0 given, G not |
---|
2563 | { |
---|
2564 | L = restrictionModuleEngine(FI,w,whichengine,l0); |
---|
2565 | } |
---|
2566 | } |
---|
2567 | else // nothing given |
---|
2568 | { |
---|
2569 | L = restrictionModuleEngine(FI,w,whichengine); |
---|
2570 | } |
---|
2571 | ideal B,N; |
---|
2572 | B = inverseFourier(L[1],v); |
---|
2573 | N = inverseFourier(L[2],v); |
---|
2574 | def newR = restrictionModuleOutput(B,N,w,whichengine,"intMod"); |
---|
2575 | return(newR); |
---|
2576 | } |
---|
2577 | example |
---|
2578 | { |
---|
2579 | "EXAMPLE:"; echo = 2; |
---|
2580 | ring r = 0,(x,b,Dx,Db),dp; |
---|
2581 | def D2 = Weyl(); |
---|
2582 | setring D2; |
---|
2583 | ideal I = x*Dx+2*b*Db+2, x^2*Dx+b*Dx+2*x; |
---|
2584 | intvec w = 1,0; |
---|
2585 | def im = integralModule(I,w); |
---|
2586 | setring im; im; |
---|
2587 | print(intMod); |
---|
2588 | } |
---|
2589 | |
---|
2590 | proc integralIdeal (ideal I, intvec w, list #) |
---|
2591 | "USAGE: integralIdeal(I,w,[,eng,m,G]); |
---|
2592 | @* I ideal, w intvec, eng and m optional ints, G optional ideal |
---|
2593 | RETURN: ring (a Weyl algebra) containing an ideal 'intIdeal' |
---|
2594 | ASSUME: The basering is the n-th Weyl algebra over a field of characteristic 0 |
---|
2595 | @* and for all 1<=i<=n the identity var(i+n)*var(i)=var(i)*var(i+1)+1 |
---|
2596 | @* holds, i.e. the sequence of variables is given by |
---|
2597 | @* x(1),...,x(n),D(1),...,D(n), where D(i) is the differential operator |
---|
2598 | @* belonging to x(i). |
---|
2599 | @* Further, assume that I is holonomic and that w is n-dimensional with |
---|
2600 | @* non-negative entries. |
---|
2601 | PURPOSE: computes the integral ideal of a holonomic ideal w.r.t. the subspace |
---|
2602 | @* defined by the variables corresponding to the non-zero entries of the |
---|
2603 | @* given intvec. |
---|
2604 | NOTE: The output ring is the Weyl algebra defined by the zero entries of w. |
---|
2605 | @* It contains ideal 'intIdeal' being the integral ideal of I w.r.t. w. |
---|
2606 | @* If there are no zero entries, the input ring is returned. |
---|
2607 | @* If eng<>0, @code{std} is used for Groebner basis computations, |
---|
2608 | @* otherwise, and by default, @code{slimgb} is used. |
---|
2609 | @* The minimal integer root of the b-function of I wrt the weight (-w,w) |
---|
2610 | @* can be specified via the optional argument m. |
---|
2611 | @* The optional argument G is used for specifying a Groebner basis of I |
---|
2612 | @* wrt the weight (-w,w), that is, the initial form of G generates the |
---|
2613 | @* initial ideal of I wrt the weight (-w,w). |
---|
2614 | @* Further note, that the assumptions on m and G (if given) are not |
---|
2615 | @* checked. |
---|
2616 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
2617 | @* if printlevel>=2, all the debug messages will be printed. |
---|
2618 | EXAMPLE: example integralIdeal; shows examples |
---|
2619 | " |
---|
2620 | { |
---|
2621 | def im = restrictionIdealEngine(I,w,"integral",#); |
---|
2622 | return(im); |
---|
2623 | } |
---|
2624 | example |
---|
2625 | { |
---|
2626 | "EXAMPLE:"; echo = 2; |
---|
2627 | ring r = 0,(x,b,Dx,Db),dp; |
---|
2628 | def D2 = Weyl(); |
---|
2629 | setring D2; |
---|
2630 | ideal I = x*Dx+2*b*Db+2, x^2*Dx+b*Dx+2*x; |
---|
2631 | intvec w = 1,0; |
---|
2632 | def D1 = integralIdeal(I,w); |
---|
2633 | setring D1; D1; |
---|
2634 | intIdeal; |
---|
2635 | } |
---|
2636 | |
---|
2637 | proc deRhamCohomIdeal (ideal I, list #) |
---|
2638 | "USAGE: deRhamCohomIdeal (I[,w,eng,k,G]); |
---|
2639 | @* I ideal, w optional intvec, eng and k optional ints, G optional ideal |
---|
2640 | RETURN: ideal |
---|
2641 | ASSUME: The basering is the n-th Weyl algebra D over a field of characteristic |
---|
2642 | @* zero and for all 1<=i<=n the identity var(i+n)*var(i)=var(i)*var(i+1)+1 |
---|
2643 | @* holds, i.e. the sequence of variables is given by |
---|
2644 | @* x(1),...,x(n),D(1),...,D(n), where D(i) is the differential operator |
---|
2645 | @* belonging to x(i). |
---|
2646 | @* Further, assume that I is of special kind, namely let f in K[x] and |
---|
2647 | @* consider the module K[x,1/f]f^m, where m is smaller than or equal to |
---|
2648 | @* the minimal integer root of the Bernstein-Sato polynomial of f. |
---|
2649 | @* Since this module is known to be a holonomic D-module, it has a cyclic |
---|
2650 | @* presentation D/I. |
---|
2651 | PURPOSE: computes a basis of the n-th de Rham cohomology group of the complement |
---|
2652 | @* of the hypersurface defined by f |
---|
2653 | NOTE: The elements of the basis are of the form f^m*p, where p runs over the |
---|
2654 | @* entries of the returned ideal. |
---|
2655 | @* If I does not satisfy the assumptions described above, the result might |
---|
2656 | @* have no meaning. Note that I can be computed with @code{annfs}. |
---|
2657 | @* If w is an intvec with exactly n strictly positive entries, w is used |
---|
2658 | @* in the computation. Otherwise, and by default, w is set to (1,...,1). |
---|
2659 | @* If eng<>0, @code{std} is used for Groebner basis computations, |
---|
2660 | @* otherwise, and by default, @code{slimgb} is used. |
---|
2661 | @* Let F(I) denote the Fourier transform of I wrt w. |
---|
2662 | @* An integer smaller than or equal to the minimal integer root of the |
---|
2663 | @* b-function of F(I) wrt the weight (-w,w) can be specified via the |
---|
2664 | @* optional argument k. |
---|
2665 | @* The optional argument G is used for specifying a Groebner Basis of F(I) |
---|
2666 | @* wrt the weight (-w,w), that is, the initial form of G generates the |
---|
2667 | @* initial ideal of F(I) wrt the weight (-w,w). |
---|
2668 | @* Further note, that the assumptions on I, k and G (if given) are not |
---|
2669 | @* checked. |
---|
2670 | THEORY: (SST) pp. 232-235 |
---|
2671 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
2672 | @* if printlevel>=2, all the debug messages will be printed. |
---|
2673 | SEE ALSO: deRhamCohom |
---|
2674 | EXAMPLE: example deRhamCohomIdeal; shows examples |
---|
2675 | " |
---|
2676 | { |
---|
2677 | intvec w = 1:(nvars(basering) div 2); |
---|
2678 | int l0,l0set,Gset; |
---|
2679 | ideal G; |
---|
2680 | int whichengine = 0; // default |
---|
2681 | if (size(#)>0) |
---|
2682 | { |
---|
2683 | if (typeof(#[1])=="intvec") |
---|
2684 | { |
---|
2685 | if (allPositive(#[1])==1) |
---|
2686 | { |
---|
2687 | w = #[1]; |
---|
2688 | } |
---|
2689 | else |
---|
2690 | { |
---|
2691 | print("// Entries of intvec must be strictly positive"); |
---|
2692 | print("// Using weight " + string(w)); |
---|
2693 | } |
---|
2694 | if (size(#)>1) |
---|
2695 | { |
---|
2696 | if (intLike(#[2])) |
---|
2697 | { |
---|
2698 | whichengine = int(#[2]); |
---|
2699 | } |
---|
2700 | if (size(#)>2) |
---|
2701 | { |
---|
2702 | if (intLike(#[3])) |
---|
2703 | { |
---|
2704 | l0 = int(#[3]); |
---|
2705 | l0set = 1; |
---|
2706 | } |
---|
2707 | if (size(#)>3) |
---|
2708 | { |
---|
2709 | if (typeof(#[4])=="ideal") |
---|
2710 | { |
---|
2711 | G = #[4]; |
---|
2712 | Gset = 1; |
---|
2713 | } |
---|
2714 | } |
---|
2715 | } |
---|
2716 | } |
---|
2717 | } |
---|
2718 | } |
---|
2719 | int ppl = printlevel; |
---|
2720 | int i,j; |
---|
2721 | int n = nvars(basering) div 2; |
---|
2722 | intvec v; |
---|
2723 | for (i=1; i<=n; i++) |
---|
2724 | { |
---|
2725 | if (w[i]>0) |
---|
2726 | { |
---|
2727 | if (v == 0:size(v)) |
---|
2728 | { |
---|
2729 | v[1] = i; |
---|
2730 | } |
---|
2731 | else |
---|
2732 | { |
---|
2733 | v[size(v)+1] = i; |
---|
2734 | } |
---|
2735 | } |
---|
2736 | } |
---|
2737 | ideal FI = fourier(I,v); |
---|
2738 | dbprint(ppl,"// computed Fourier transform of given ideal"); |
---|
2739 | dbprint(ppl-1,"// " + string(FI)); |
---|
2740 | list L; |
---|
2741 | if (l0set) |
---|
2742 | { |
---|
2743 | if (Gset) // l0 and G given |
---|
2744 | { |
---|
2745 | L = restrictionModuleEngine(FI,w,whichengine,l0,G); |
---|
2746 | } |
---|
2747 | else // l0 given, G not |
---|
2748 | { |
---|
2749 | L = restrictionModuleEngine(FI,w,whichengine,l0); |
---|
2750 | } |
---|
2751 | } |
---|
2752 | else // nothing given |
---|
2753 | { |
---|
2754 | L = restrictionModuleEngine(FI,w,whichengine); |
---|
2755 | } |
---|
2756 | ideal B,N; |
---|
2757 | B = inverseFourier(L[1],v); |
---|
2758 | N = inverseFourier(L[2],v); |
---|
2759 | dbprint(ppl,"// computed integral module of given ideal"); |
---|
2760 | dbprint(ppl-1,"// " + string(B)); |
---|
2761 | dbprint(ppl-1,"// " + string(N)); |
---|
2762 | ideal DR; |
---|
2763 | poly p; |
---|
2764 | poly Dt = 1; |
---|
2765 | for (i=1; i<=n; i++) |
---|
2766 | { |
---|
2767 | Dt = Dt*var(i+n); |
---|
2768 | } |
---|
2769 | N = simplify(N,2+8); |
---|
2770 | printlevel = printlevel-1; |
---|
2771 | N = linReduceIdeal(N); |
---|
2772 | N = simplify(N,2+8); |
---|
2773 | for (i=1; i<=size(B); i++) |
---|
2774 | { |
---|
2775 | p = linReduce(B[i],N); |
---|
2776 | if (p<>0) |
---|
2777 | { |
---|
2778 | DR[size(DR)+1] = B[i]*Dt; |
---|
2779 | j=1; |
---|
2780 | while ((j<size(N)) && (p<N[j])) |
---|
2781 | { |
---|
2782 | j++; |
---|
2783 | } |
---|
2784 | N = insertGenerator(N,p,j+1); |
---|
2785 | } |
---|
2786 | } |
---|
2787 | printlevel = printlevel + 1; |
---|
2788 | return(DR); |
---|
2789 | } |
---|
2790 | example |
---|
2791 | { |
---|
2792 | "EXAMPLE:"; echo = 2; |
---|
2793 | ring r = 0,(x,y,z),dp; |
---|
2794 | poly F = x^3+y^3+z^3; |
---|
2795 | bfctAnn(F); // Bernstein-Sato poly of F has minimal integer root -2 |
---|
2796 | def W = annRat(1,F^2); // so we compute the annihilator of 1/F^2 |
---|
2797 | setring W; W; // Weyl algebra, contains LD = Ann(1/F^2) |
---|
2798 | LD; // K[x,y,z,1/F]F^(-2) is isomorphic to W/LD as W-module |
---|
2799 | deRhamCohomIdeal(LD); // we see that the K-dim is 2 |
---|
2800 | } |
---|
2801 | |
---|
2802 | proc deRhamCohom (poly f, list #) |
---|
2803 | "USAGE: deRhamCohom(f[,w,eng,m]); f poly, w optional intvec, |
---|
2804 | eng and m optional ints |
---|
2805 | RETURN: ring (a Weyl Algebra) containing a list 'DR' of ideal and int |
---|
2806 | ASSUME: Basering is commutative and over a field of characteristic 0. |
---|
2807 | PURPOSE: computes a basis of the n-th de Rham cohomology group of the complement |
---|
2808 | @* of the hypersurface defined by f, where n denotes the number of |
---|
2809 | @* variables of the basering |
---|
2810 | NOTE: The output ring is the n-th Weyl algebra. It contains a list 'DR' with |
---|
2811 | @* two entries (ideal J and int m) such that {f^m*J[i] : i=1..size(I)} is |
---|
2812 | @* a basis of the n-th de Rham cohomology group of the complement of the |
---|
2813 | @* hypersurface defined by f. |
---|
2814 | @* If w is an intvec with exactly n strictly positive entries, w is used |
---|
2815 | @* in the computation. Otherwise, and by default, w is set to (1,...,1). |
---|
2816 | @* If eng<>0, @code{std} is used for Groebner basis computations, |
---|
2817 | @* otherwise, and by default, @code{slimgb} is used. |
---|
2818 | @* If m is given, it is assumed to be less than or equal to the minimal |
---|
2819 | @* integer root of the Bernstein-Sato polynomial of f. This assumption is |
---|
2820 | @* not checked. If not specified, m is set to the minimal integer root of |
---|
2821 | @* the Bernstein-Sato polynomial of f. |
---|
2822 | THEORY: (SST) pp. 232-235 |
---|
2823 | DISPLAY: If printlevel=1, progress debug messages will be printed, |
---|
2824 | @* if printlevel>=2, all the debug messages will be printed. |
---|
2825 | SEE ALSO: deRhamCohomIdeal |
---|
2826 | EXAMPLE: example deRhamCohom; shows example |
---|
2827 | " |
---|
2828 | { |
---|
2829 | int ppl = printlevel - voice + 2; |
---|
2830 | def save = basering; |
---|
2831 | int n = nvars(save); |
---|
2832 | intvec w = 1:n; |
---|
2833 | int eng,l0,l0given; |
---|
2834 | if (size(#)>0) |
---|
2835 | { |
---|
2836 | if (typeof(#[1])=="intvec") |
---|
2837 | { |
---|
2838 | w = #[1]; |
---|
2839 | } |
---|
2840 | if (size(#)>1) |
---|
2841 | { |
---|
2842 | if(intLike(#[2])) |
---|
2843 | { |
---|
2844 | eng = int(#[2]); |
---|
2845 | } |
---|
2846 | if (size(#)>2) |
---|
2847 | { |
---|
2848 | if(intLike(#[3])) |
---|
2849 | { |
---|
2850 | l0 = int(#[3]); |
---|
2851 | l0given = 1; |
---|
2852 | } |
---|
2853 | } |
---|
2854 | } |
---|
2855 | } |
---|
2856 | if (!isCommutative()) |
---|
2857 | { |
---|
2858 | ERROR("Basering must be commutative."); |
---|
2859 | } |
---|
2860 | int i; |
---|
2861 | dbprint(ppl,"// Computing s-parametric annihilator Ann(f^s)..."); |
---|
2862 | def A = Sannfs(f); |
---|
2863 | setring A; |
---|
2864 | dbprint(ppl,"// ...done"); |
---|
2865 | dbprint(ppl-1,"// Got: " + string(LD)); |
---|
2866 | poly f = imap(save,f); |
---|
2867 | if (!l0given) |
---|
2868 | { |
---|
2869 | dbprint(ppl,"// Computing b-function of given poly..."); |
---|
2870 | ideal LDf = LD,f; |
---|
2871 | LDf = engine(LDf,eng); |
---|
2872 | vector v = pIntersect(var(2*n+1),LDf); // BS poly of f |
---|
2873 | list BS = bFactor(vec2poly(v)); |
---|
2874 | dbprint(ppl,"// ...done"); |
---|
2875 | dbprint(ppl-1,"// roots: " + string(BS[1])); |
---|
2876 | dbprint(ppl-1,"// multiplicities: " + string(BS[2])); |
---|
2877 | BS = intRoots(BS); |
---|
2878 | intvec iv; |
---|
2879 | for (i=1; i<=ncols(BS[1]); i++) |
---|
2880 | { |
---|
2881 | iv[i] = int(BS[1][i]); |
---|
2882 | } |
---|
2883 | l0 = Min(iv); |
---|
2884 | kill v,iv,BS,LDf; |
---|
2885 | } |
---|
2886 | dbprint(ppl,"// Computing Ann(f^" + string(l0) + ")..."); |
---|
2887 | LD = annfspecial(LD,f,l0,l0); // Ann(f^l0) |
---|
2888 | // create new ring without s |
---|
2889 | list RL = ringlist(A); |
---|
2890 | RL = RL[1..4]; |
---|
2891 | list Lt = RL[2]; |
---|
2892 | Lt = delete(Lt,2*n+1); |
---|
2893 | RL[2] = Lt; |
---|
2894 | Lt = RL[3]; |
---|
2895 | Lt = delete(Lt,2); |
---|
2896 | RL[3] = Lt; |
---|
2897 | def @B = ring(RL); |
---|
2898 | setring @B; |
---|
2899 | def B = Weyl(); |
---|
2900 | setring B; |
---|
2901 | kill @B; |
---|
2902 | ideal LD = imap(A,LD); |
---|
2903 | LD = engine(LD,eng); |
---|
2904 | dbprint(ppl,"// ...done"); |
---|
2905 | dbprint(ppl-1,"// Got: " + string(LD)); |
---|
2906 | kill A; |
---|
2907 | ideal DRJ = deRhamCohomIdeal(LD,w,eng); |
---|
2908 | list DR = DRJ,l0; |
---|
2909 | export(DR); |
---|
2910 | setring save; |
---|
2911 | return(B); |
---|
2912 | } |
---|
2913 | example |
---|
2914 | { |
---|
2915 | "EXAMPLE:"; echo = 2; |
---|
2916 | ring r = 0,(x,y,z),dp; |
---|
2917 | poly f = x^3+y^3+z^3; |
---|
2918 | def A = deRhamCohom(f); // we see that the K-dim is 2 |
---|
2919 | setring A; |
---|
2920 | DR; |
---|
2921 | } |
---|
2922 | |
---|
2923 | // Appel hypergeometric functions ///////////////////////////////////////////// |
---|
2924 | |
---|
2925 | proc appelF1() |
---|
2926 | "USAGE: appelF1(); |
---|
2927 | RETURN: ring (a parametric Weyl algebra) containing an ideal 'IAppel1' |
---|
2928 | PURPOSE: defines the ideal in a parametric Weyl algebra, |
---|
2929 | @* which annihilates Appel F1 hypergeometric function |
---|
2930 | NOTE: The output ring is a parametric Weyl algebra. It contains an ideal |
---|
2931 | @* 'IAappel1' annihilating Appel F1 hypergeometric function. |
---|
2932 | @* See (SST) p. 48. |
---|
2933 | EXAMPLE: example appelF1; shows example |
---|
2934 | " |
---|
2935 | { |
---|
2936 | // Appel F1, d = b', SST p.48 |
---|
2937 | ring @r = (0,a,b,c,d),(x,y,Dx,Dy),(a(0,0,1,1),a(0,0,1,0),dp); |
---|
2938 | def @S = Weyl(); |
---|
2939 | setring @S; |
---|
2940 | ideal IAppel1 = |
---|
2941 | (x*Dx)*(x*Dx+y*Dy+c-1) - x*(x*Dx+y*Dy+a)*(x*Dx+b), |
---|
2942 | (y*Dy)*(x*Dx+y*Dy+c-1) - y*(x*Dx+y*Dy+a)*(y*Dy+d), |
---|
2943 | (x-y)*Dx*Dy - d*Dx + b*Dy; |
---|
2944 | export IAppel1; |
---|
2945 | kill @r; |
---|
2946 | return(@S); |
---|
2947 | } |
---|
2948 | example |
---|
2949 | { |
---|
2950 | "EXAMPLE:"; echo = 2; |
---|
2951 | def A = appelF1(); |
---|
2952 | setring A; |
---|
2953 | IAppel1; |
---|
2954 | } |
---|
2955 | |
---|
2956 | proc appelF2() |
---|
2957 | "USAGE: appelF2(); |
---|
2958 | RETURN: ring (a parametric Weyl algebra) containing an ideal 'IAppel2' |
---|
2959 | PURPOSE: defines the ideal in a parametric Weyl algebra, |
---|
2960 | @* which annihilates Appel F2 hypergeometric function |
---|
2961 | NOTE: The output ring is a parametric Weyl algebra. It contains an ideal |
---|
2962 | @* 'IAappel2' annihilating Appel F2 hypergeometric function. |
---|
2963 | @* See (SST) p. 85. |
---|
2964 | EXAMPLE: example appelF2; shows example |
---|
2965 | " |
---|
2966 | { |
---|
2967 | // Appel F2, c = b', SST p.85 |
---|
2968 | ring @r = (0,a,b,c),(x,y,Dx,Dy),(a(0,0,1,1),a(0,0,1,0),dp); |
---|
2969 | def @S = Weyl(); |
---|
2970 | setring @S; |
---|
2971 | ideal IAppel2 = |
---|
2972 | (x*Dx)^2 - x*(x*Dx+y*Dy+a)*(x*Dx+b), |
---|
2973 | (y*Dy)^2 - y*(x*Dx+y*Dy+a)*(y*Dy+c); |
---|
2974 | export IAppel2; |
---|
2975 | kill @r; |
---|
2976 | return(@S); |
---|
2977 | } |
---|
2978 | example |
---|
2979 | { |
---|
2980 | "EXAMPLE:"; echo = 2; |
---|
2981 | def A = appelF2(); |
---|
2982 | setring A; |
---|
2983 | IAppel2; |
---|
2984 | } |
---|
2985 | |
---|
2986 | proc appelF4() |
---|
2987 | "USAGE: appelF4(); |
---|
2988 | RETURN: ring (a parametric Weyl algebra) containing an ideal 'IAppel4' |
---|
2989 | PURPOSE: defines the ideal in a parametric Weyl algebra, |
---|
2990 | @* which annihilates Appel F4 hypergeometric function |
---|
2991 | NOTE: The output ring is a parametric Weyl algebra. It contains an ideal |
---|
2992 | @* 'IAappel4' annihilating Appel F4 hypergeometric function. |
---|
2993 | @* See (SST) p. 39. |
---|
2994 | EXAMPLE: example appelF4; shows example |
---|
2995 | " |
---|
2996 | { |
---|
2997 | // Appel F4, d = c', SST, p. 39 |
---|
2998 | ring @r = (0,a,b,c,d),(x,y,Dx,Dy),(a(0,0,1,1),a(0,0,1,0),dp); |
---|
2999 | def @S = Weyl(); |
---|
3000 | setring @S; |
---|
3001 | ideal IAppel4 = |
---|
3002 | Dx*(x*Dx+c-1) - (x*Dx+y*Dy+a)*(x*Dx+y*Dy+b), |
---|
3003 | Dy*(y*Dy+d-1) - (x*Dx+y*Dy+a)*(x*Dx+y*Dy+b); |
---|
3004 | export IAppel4; |
---|
3005 | kill @r; |
---|
3006 | return(@S); |
---|
3007 | } |
---|
3008 | example |
---|
3009 | { |
---|
3010 | "EXAMPLE:"; echo = 2; |
---|
3011 | def A = appelF4(); |
---|
3012 | setring A; |
---|
3013 | IAppel4; |
---|
3014 | } |
---|
3015 | |
---|
3016 | |
---|
3017 | // characteric variety //////////////////////////////////////////////////////// |
---|
3018 | |
---|
3019 | proc charVariety(ideal I, list #) |
---|
3020 | "USAGE: charVariety(I [,eng]); I an ideal, eng an optional int |
---|
3021 | RETURN: ring (commutative) containing an ideal 'charVar' |
---|
3022 | PURPOSE: computes an ideal whose zero set is the characteristic variety of I in |
---|
3023 | @* the sense of D-module theory |
---|
3024 | ASSUME: The basering is the n-th Weyl algebra over a field of characteristic 0 |
---|
3025 | @* and for all 1<=i<=n the identity var(i+n)*var(i)=var(i)*var(i+1)+1 |
---|
3026 | @* holds, i.e. the sequence of variables is given by |
---|
3027 | @* x(1),...,x(n),D(1),...,D(n), where D(i) is the differential operator |
---|
3028 | @* belonging to x(i). |
---|
3029 | NOTE: The output ring is commutative. It contains an ideal 'charVar'. |
---|
3030 | @* If eng<>0, @code{std} is used for Groebner basis computations, |
---|
3031 | @* otherwise, and by default, @code{slimgb} is used. |
---|
3032 | DISPLAY: If @code{printlevel}=1, progress debug messages will be printed, |
---|
3033 | @* if @code{printlevel}>=2, all the debug messages will be printed. |
---|
3034 | SEE ALSO: charInfo |
---|
3035 | EXAMPLE: example charVariety; shows examples |
---|
3036 | " |
---|
3037 | { |
---|
3038 | // assumption check is done in GBWeight |
---|
3039 | int eng; |
---|
3040 | if (size(#)>0) |
---|
3041 | { |
---|
3042 | if (intLike(#[1])) |
---|
3043 | { |
---|
3044 | eng = int(#[1]); |
---|
3045 | } |
---|
3046 | } |
---|
3047 | int ppl = printlevel - voice + 2; |
---|
3048 | def save = basering; |
---|
3049 | int n = nvars(save) div 2; |
---|
3050 | intvec uv = (0:n),(1:n); |
---|
3051 | list RL = ringlist(save); |
---|
3052 | list L = RL[3]; |
---|
3053 | L = insert(L,list("a",uv)); |
---|
3054 | RL[3] = L; |
---|
3055 | // TODO printlevel |
---|
3056 | def Ra = ring(RL); |
---|
3057 | setring Ra; |
---|
3058 | dbprint(ppl,"// Starting Groebner basis computation..."); |
---|
3059 | ideal I = imap(save,I); |
---|
3060 | I = engine(I,eng); |
---|
3061 | dbprint(ppl,"// ... done."); |
---|
3062 | dbprint(ppl-1,"// Got: " + string(I)); |
---|
3063 | setring save; |
---|
3064 | RL = ringlist(save); |
---|
3065 | RL = RL[1..4]; |
---|
3066 | def newR = ring(RL); |
---|
3067 | setring newR; |
---|
3068 | ideal charVar = imap(Ra,I); |
---|
3069 | charVar = inForm(charVar,uv); |
---|
3070 | // charVar = groebner(charVar); |
---|
3071 | export(charVar); |
---|
3072 | setring save; |
---|
3073 | return(newR); |
---|
3074 | } |
---|
3075 | example |
---|
3076 | { |
---|
3077 | "EXAMPLE:"; echo = 2; |
---|
3078 | ring r = 0,(x,y),Dp; |
---|
3079 | poly F = x3-y2; |
---|
3080 | printlevel = 0; |
---|
3081 | def A = annfs(F); |
---|
3082 | setring A; // Weyl algebra |
---|
3083 | LD; // the annihilator of F |
---|
3084 | def CA = charVariety(LD); |
---|
3085 | setring CA; CA; // commutative ring |
---|
3086 | charVar; |
---|
3087 | dim(std(charVar)); // hence I is holonomic |
---|
3088 | } |
---|
3089 | |
---|
3090 | proc charInfo(ideal I) |
---|
3091 | "USAGE: charInfo(I); I an ideal |
---|
3092 | RETURN: ring (commut.) containing ideals 'charVar','singLoc' and list 'primDec' |
---|
3093 | PURPOSE: computes characteristic variety of I (in the sense of D-module theory), |
---|
3094 | @* its singular locus and primary decomposition |
---|
3095 | ASSUME: The basering is the n-th Weyl algebra over a field of characteristic 0 |
---|
3096 | @* and for all 1<=i<=n the identity var(i+n)*var(i)=var(i)*var(i+1)+1 |
---|
3097 | @* holds, i.e. the sequence of variables is given by |
---|
3098 | @* x(1),...,x(n),D(1),...,D(n), where D(i) is the differential operator |
---|
3099 | @* belonging to x(i). |
---|
3100 | NOTE: In the output ring, which is commutative: |
---|
3101 | @* - the ideal 'charVar' is the characteristic variety char(I), |
---|
3102 | @* - the ideal 'SingLoc' is the singular locus of char(I), |
---|
3103 | @* - the list 'primDec' is the primary decomposition of char(I). |
---|
3104 | DISPLAY: If @code{printlevel}=1, progress debug messages will be printed, |
---|
3105 | @* if @code{printlevel}>=2, all the debug messages will be printed. |
---|
3106 | EXAMPLE: example charInfo; shows examples |
---|
3107 | " |
---|
3108 | { |
---|
3109 | int ppl = printlevel - voice + 2; |
---|
3110 | def save = basering; |
---|
3111 | dbprint(ppl,"// computing characteristic variety..."); |
---|
3112 | def A = charVariety(I); |
---|
3113 | setring A; |
---|
3114 | dbprint(ppl,"// ...done"); |
---|
3115 | dbprint(ppl-1,"// Got: " + string(charVar)); |
---|
3116 | dbprint(ppl,"// computing singular locus..."); |
---|
3117 | ideal singLoc = slocus(charVar); |
---|
3118 | singLoc = groebner(singLoc); |
---|
3119 | dbprint(ppl,"// ...done"); |
---|
3120 | dbprint(ppl-1,"// Got: " + string(singLoc)); |
---|
3121 | dbprint(ppl,"// computing primary decomposition..."); |
---|
3122 | list primDec = primdecGTZ(charVar); |
---|
3123 | dbprint(ppl,"// ...done"); |
---|
3124 | //export(charVar,singLoc,primDec); |
---|
3125 | export(singLoc,primDec); |
---|
3126 | setring save; |
---|
3127 | return(A); |
---|
3128 | } |
---|
3129 | example |
---|
3130 | { |
---|
3131 | "EXAMPLE:"; echo = 2; |
---|
3132 | ring r = 0,(x,y),Dp; |
---|
3133 | poly F = x3-y2; |
---|
3134 | printlevel = 0; |
---|
3135 | def A = annfs(F); |
---|
3136 | setring A; // Weyl algebra |
---|
3137 | LD; // the annihilator of F |
---|
3138 | def CA = charInfo(LD); |
---|
3139 | setring CA; CA; // commutative ring |
---|
3140 | charVar; // characteristic variety |
---|
3141 | singLoc; // singular locus |
---|
3142 | primDec; // primary decomposition |
---|
3143 | } |
---|
3144 | |
---|
3145 | |
---|
3146 | // examples /////////////////////////////////////////////////////////////////// |
---|
3147 | |
---|
3148 | /* |
---|
3149 | static proc exCusp() |
---|
3150 | { |
---|
3151 | "EXAMPLE:"; echo = 2; |
---|
3152 | ring r = 0,(x,y,Dx,Dy),dp; |
---|
3153 | def R = Weyl(); setring R; |
---|
3154 | poly F = x2-y3; |
---|
3155 | ideal I = (y^3 - x^2)*Dx - 2*x, (y^3 - x^2)*Dy + 3*y^2; // I = Dx*F, Dy*F; |
---|
3156 | def W = SDLoc(I,F); |
---|
3157 | setring W; |
---|
3158 | LD; |
---|
3159 | def U = DLoc0(LD,x2-y3); |
---|
3160 | setring U; |
---|
3161 | LD0; |
---|
3162 | BS; |
---|
3163 | // the same with DLoc: |
---|
3164 | setring R; |
---|
3165 | DLoc(I,F); |
---|
3166 | } |
---|
3167 | |
---|
3168 | static proc exWalther1() |
---|
3169 | { |
---|
3170 | // p.18 Rem 3.10 |
---|
3171 | ring r = 0,(x,Dx),dp; |
---|
3172 | def R = nc_algebra(1,1); |
---|
3173 | setring R; |
---|
3174 | poly F = x; |
---|
3175 | ideal I = x*Dx+1; |
---|
3176 | def W = SDLoc(I,F); |
---|
3177 | setring W; |
---|
3178 | LD; |
---|
3179 | ideal J = LD, x; |
---|
3180 | eliminate(J,x*Dx); // must be [1]=s // agree! |
---|
3181 | // the same result with Dloc0: |
---|
3182 | def U = DLoc0(LD,x); |
---|
3183 | setring U; |
---|
3184 | LD0; |
---|
3185 | BS; |
---|
3186 | } |
---|
3187 | |
---|
3188 | static proc exWalther2() |
---|
3189 | { |
---|
3190 | // p.19 Rem 3.10 cont'd |
---|
3191 | ring r = 0,(x,Dx),dp; |
---|
3192 | def R = nc_algebra(1,1); |
---|
3193 | setring R; |
---|
3194 | poly F = x; |
---|
3195 | ideal I = (x*Dx)^2+1; |
---|
3196 | def W = SDLoc(I,F); |
---|
3197 | setring W; |
---|
3198 | LD; |
---|
3199 | ideal J = LD, x; |
---|
3200 | eliminate(J,x*Dx); // must be [1]=s^2+2*s+2 // agree! |
---|
3201 | // the same result with Dloc0: |
---|
3202 | def U = DLoc0(LD,x); |
---|
3203 | setring U; |
---|
3204 | LD0; |
---|
3205 | BS; |
---|
3206 | // almost the same with DLoc |
---|
3207 | setring R; |
---|
3208 | DLoc(I,F); |
---|
3209 | } |
---|
3210 | |
---|
3211 | static proc exWalther3() |
---|
3212 | { |
---|
3213 | // can check with annFs too :-) |
---|
3214 | // p.21 Ex 3.15 |
---|
3215 | LIB "nctools.lib"; |
---|
3216 | ring r = 0,(x,y,z,w,Dx,Dy,Dz,Dw),dp; |
---|
3217 | def R = Weyl(); |
---|
3218 | setring R; |
---|
3219 | poly F = x2+y2+z2+w2; |
---|
3220 | ideal I = Dx,Dy,Dz,Dw; |
---|
3221 | def W = SDLoc(I,F); |
---|
3222 | setring W; |
---|
3223 | LD; |
---|
3224 | ideal J = LD, x2+y2+z2+w2; |
---|
3225 | eliminate(J,x*y*z*w*Dx*Dy*Dz*Dw); // must be [1]=s^2+3*s+2 // agree |
---|
3226 | ring r2 = 0,(x,y,z,w),dp; |
---|
3227 | poly F = x2+y2+z2+w2; |
---|
3228 | def Z = annfs(F); |
---|
3229 | setring Z; |
---|
3230 | LD; |
---|
3231 | BS; |
---|
3232 | // the same result with Dloc0: |
---|
3233 | setring W; |
---|
3234 | def U = DLoc0(LD,x2+y2+z2+w2); |
---|
3235 | setring U; |
---|
3236 | LD0; BS; |
---|
3237 | // the same result with DLoc: |
---|
3238 | setring R; |
---|
3239 | DLoc(I,F); |
---|
3240 | } |
---|
3241 | |
---|
3242 | static proc ex_annRat() |
---|
3243 | { |
---|
3244 | // more complicated example for annRat |
---|
3245 | ring r = 0,(x,y,z),dp; |
---|
3246 | poly f = x3+y3+z3; // mir = -2 |
---|
3247 | poly g = x*y*z; |
---|
3248 | def A = annRat(g,f); |
---|
3249 | setring A; |
---|
3250 | } |
---|
3251 | */ |
---|