1 | ////////////////////////////////////////////////////////////////////////////// |
---|
2 | version="$Id: freegb.lib,v 1.8 2008-03-13 19:26:16 levandov Exp $"; |
---|
3 | category="Noncommutative"; |
---|
4 | info=" |
---|
5 | LIBRARY: ratgb.lib Twosided Noncommutative Groebner bases in Free Algebras |
---|
6 | AUTHOR: Viktor Levandovskyy, levandov@math.rwth-aachen.de |
---|
7 | |
---|
8 | PROCEDURES: |
---|
9 | freegbasis(list L, int n); compute two-sided Groebner basis of ideal, encoded via L, up to degree n |
---|
10 | lst2str(list L); convert a list (of modules) into polynomials in free algebra |
---|
11 | mod2str(module M); convert a module into a polynomial in free algebra |
---|
12 | " |
---|
13 | |
---|
14 | // this library computes two-sided GB of an ideal |
---|
15 | // in a free associative algebra |
---|
16 | |
---|
17 | // a monomial is encoded via a vector V |
---|
18 | // where V[1] = coefficient |
---|
19 | // V[1+i] = the corresponding symbol |
---|
20 | |
---|
21 | LIB "qhmoduli.lib"; // for Max |
---|
22 | |
---|
23 | proc vct2mono(vector v) |
---|
24 | { |
---|
25 | // produces a monomial in new vars |
---|
26 | // need a ring with new vars!! |
---|
27 | } |
---|
28 | |
---|
29 | proc lshift(module M, int s, string varing, def lpring) |
---|
30 | { |
---|
31 | // FINALLY IMPLEMENTED AS A PART OT THE CODE |
---|
32 | // shifts a poly from the ring @R to s positions |
---|
33 | // M lives in varing, the result in lpring |
---|
34 | // to be run from varing |
---|
35 | int i, j, k, sm, sv; |
---|
36 | vector v; |
---|
37 | // execute("setring "+lpring); |
---|
38 | setring lpring; |
---|
39 | poly @@p; |
---|
40 | ideal I; |
---|
41 | execute("setring "+varing); |
---|
42 | sm = ncols(M); |
---|
43 | for (i=1; i<=s; i++) |
---|
44 | { |
---|
45 | // modules, e.g. free polynomials |
---|
46 | for (j=1; j<=sm; j++) |
---|
47 | { |
---|
48 | //vectors, e.g. free monomials |
---|
49 | v = M[j]; |
---|
50 | sv = size(v); |
---|
51 | sp = "@@p = @@p + "; |
---|
52 | for (k=2; k<=sv; k++) |
---|
53 | { |
---|
54 | sp = sp + string(v[k])+"("+string(k-1+s)+")*"; |
---|
55 | } |
---|
56 | sp = sp + string(v[1])+";"; // coef; |
---|
57 | setring lpring; |
---|
58 | // execute("setring "+lpring); |
---|
59 | execute(sp); |
---|
60 | execute("setring "+varing); |
---|
61 | } |
---|
62 | setring lpring; |
---|
63 | // execute("setring "+lpring); |
---|
64 | I = I,@@p; |
---|
65 | @@p = 0; |
---|
66 | } |
---|
67 | setring lpring; |
---|
68 | //execute("setring "+lpring); |
---|
69 | export(I); |
---|
70 | // setring varing; |
---|
71 | execute("setring "+varing); |
---|
72 | } |
---|
73 | |
---|
74 | proc skip0(vector v) |
---|
75 | { |
---|
76 | // skips zeros in vector |
---|
77 | int sv = nrows(v); |
---|
78 | int sw = size(v); |
---|
79 | if (sv == sw) |
---|
80 | { |
---|
81 | return(v); |
---|
82 | } |
---|
83 | int i; |
---|
84 | int j=1; |
---|
85 | vector w; |
---|
86 | for (i=1; i<=sv; i++) |
---|
87 | { |
---|
88 | if (v[i] != 0) |
---|
89 | { |
---|
90 | w = w + v[i]*gen(j); |
---|
91 | j++; |
---|
92 | } |
---|
93 | } |
---|
94 | return(w); |
---|
95 | } |
---|
96 | |
---|
97 | proc lst2str(list L) |
---|
98 | "USAGE: lst2str(L); L a list of modules |
---|
99 | RETURN: list (of strings) |
---|
100 | PURPOSE: convert a list (of modules) into polynomials in free algebra |
---|
101 | EXAMPLE: example lst2str; shows examples |
---|
102 | " |
---|
103 | { |
---|
104 | // returns a list of strings |
---|
105 | // being sentences in words built from L |
---|
106 | int i; |
---|
107 | int s = size(L); |
---|
108 | list N; |
---|
109 | for(i=1; i<=s; i++) |
---|
110 | { |
---|
111 | if ((typeof(L[i]) == "module") || (typeof(L[i]) == "matrix") ) |
---|
112 | { |
---|
113 | N[i] = mod2str(L[i]); |
---|
114 | } |
---|
115 | else |
---|
116 | { |
---|
117 | "module or matrix expected in the list"; |
---|
118 | return(N); |
---|
119 | } |
---|
120 | } |
---|
121 | return(N); |
---|
122 | } |
---|
123 | example |
---|
124 | { |
---|
125 | "EXAMPLE:"; echo = 2; |
---|
126 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
127 | module M = [-1,x,y],[-7,y,y],[3,x,x]; |
---|
128 | module N = [1,x,y,x,y],[-2,y,x,y,x],[6,x,y,y,x,y]; |
---|
129 | list L; L[1] = M; L[2] = N; |
---|
130 | lst2str(L); |
---|
131 | } |
---|
132 | |
---|
133 | |
---|
134 | proc mod2str(module M) |
---|
135 | "USAGE: mod2str(M); M a module |
---|
136 | RETURN: string |
---|
137 | PURPOSE: convert a modules into a polynomial in free algebra |
---|
138 | EXAMPLE: example mod2str; shows examples |
---|
139 | " |
---|
140 | { |
---|
141 | // returns a string |
---|
142 | // a sentence in words built from M |
---|
143 | int i; |
---|
144 | int s = ncols(M); |
---|
145 | string t; |
---|
146 | string mp; |
---|
147 | for(i=1; i<=s; i++) |
---|
148 | { |
---|
149 | mp = vct2str(M[i]); |
---|
150 | if (mp[1] == "-") |
---|
151 | { |
---|
152 | t = t + mp; |
---|
153 | } |
---|
154 | else |
---|
155 | { |
---|
156 | t = t + "+" + mp; |
---|
157 | } |
---|
158 | } |
---|
159 | if (t[1]=="+") |
---|
160 | { |
---|
161 | t = t[2..size(t)]; // remove first "+" |
---|
162 | } |
---|
163 | return(t); |
---|
164 | } |
---|
165 | example |
---|
166 | { |
---|
167 | "EXAMPLE:"; echo = 2; |
---|
168 | ring r = 0,(x,y,z),(dp); |
---|
169 | module M = [1,x,y,x,y],[-2,y,x,y,x],[6,x,y,y,x,y]; |
---|
170 | mod2str(M); |
---|
171 | } |
---|
172 | |
---|
173 | proc vct2str(vector v) |
---|
174 | { |
---|
175 | int ppl = printlevel-voice+2; |
---|
176 | // for a word, encoded by v |
---|
177 | // produces a string for it |
---|
178 | v = skip0(v); |
---|
179 | number cf = leadcoef(v[1]); |
---|
180 | int s = size(v); |
---|
181 | string vs,vv,vp,err; |
---|
182 | int i,j,p,q; |
---|
183 | for (i=1; i<=s-1; i++) |
---|
184 | { |
---|
185 | p = IsVar(v[i+1]); |
---|
186 | if (p==0) |
---|
187 | { |
---|
188 | err = "Error: monomial expected at" + string(i+1); |
---|
189 | dbprint(ppl,err); |
---|
190 | return("_"); |
---|
191 | } |
---|
192 | if (p==1) |
---|
193 | { |
---|
194 | vs = vs + string(v[i+1]); |
---|
195 | } |
---|
196 | else //power |
---|
197 | { |
---|
198 | vv = string(v[i+1]); |
---|
199 | q = find(vv,"^"); |
---|
200 | if (q==0) |
---|
201 | { |
---|
202 | q = find(vv,string(p)); |
---|
203 | if (q==0) |
---|
204 | { |
---|
205 | err = "error in find for string "+vv; |
---|
206 | dbprint(ppl,err); |
---|
207 | return("_"); |
---|
208 | } |
---|
209 | } |
---|
210 | // q>0 |
---|
211 | vp = vv[1..q-1]; |
---|
212 | for(j=1;j<=p;j++) |
---|
213 | { |
---|
214 | vs = vs + vp; |
---|
215 | } |
---|
216 | } |
---|
217 | } |
---|
218 | string scf; |
---|
219 | if (cf == -1) |
---|
220 | { |
---|
221 | scf = "-"; |
---|
222 | } |
---|
223 | else |
---|
224 | { |
---|
225 | scf = string(cf); |
---|
226 | if (cf == 1) |
---|
227 | { |
---|
228 | scf = ""; |
---|
229 | } |
---|
230 | } |
---|
231 | vs = scf + vs; |
---|
232 | return(vs); |
---|
233 | } |
---|
234 | example |
---|
235 | { |
---|
236 | ring r = (0,a),(x,y3,z(1)),dp; |
---|
237 | vector v = [-7,x,y3^4,x2,z(1)^3]; |
---|
238 | vct2str(v); |
---|
239 | vector w = [-7a^5+6a,x,y3,y3,x,z(1),z(1)]; |
---|
240 | vct2str(w); |
---|
241 | } |
---|
242 | |
---|
243 | proc IsVar(poly p) |
---|
244 | { |
---|
245 | // checks whether p is a variable indeed |
---|
246 | // if it's a power of a variable, returns the power |
---|
247 | if (p==0) { return(0); } //"p=0"; |
---|
248 | poly q = leadmonom(p); |
---|
249 | if ( (p-lead(p)) !=0 ) { return(0); } // "p-lm(p)>0"; |
---|
250 | intvec v = leadexp(p); |
---|
251 | int s = size(v); |
---|
252 | int i=1; |
---|
253 | int cnt = 0; |
---|
254 | int pwr = 0; |
---|
255 | for (i=1; i<=s; i++) |
---|
256 | { |
---|
257 | if (v[i] != 0) |
---|
258 | { |
---|
259 | cnt++; |
---|
260 | pwr = v[i]; |
---|
261 | } |
---|
262 | } |
---|
263 | // "cnt:"; cnt; |
---|
264 | if (cnt==1) { return(pwr); } |
---|
265 | else { return(0); } |
---|
266 | } |
---|
267 | example |
---|
268 | { |
---|
269 | ring r = 0,(x,y),dp; |
---|
270 | poly f = xy+1; |
---|
271 | IsVar(f); |
---|
272 | poly g = xy; |
---|
273 | IsVar(g); |
---|
274 | poly h = y^3; |
---|
275 | IsVar(h); |
---|
276 | poly i = 1; |
---|
277 | IsVar(i); |
---|
278 | } |
---|
279 | |
---|
280 | // new conversion routines |
---|
281 | |
---|
282 | proc id2words(ideal I, int d) |
---|
283 | { |
---|
284 | // input: ideal I of polys in letter-place notation |
---|
285 | // in the ring with d real vars |
---|
286 | // output: the list of strings: associative words |
---|
287 | // extract names of vars |
---|
288 | int i,m,n; string s; string place = "(1)"; |
---|
289 | list lv; |
---|
290 | for(i=1; i<=d; i++) |
---|
291 | { |
---|
292 | s = string(var(i)); |
---|
293 | // get rid of place |
---|
294 | n = find(s, place); |
---|
295 | if (n>0) |
---|
296 | { |
---|
297 | s = s[1..n-1]; |
---|
298 | } |
---|
299 | lv[i] = s; |
---|
300 | } |
---|
301 | poly p,q; |
---|
302 | for (i=1; i<=ncols(I); i++) |
---|
303 | { |
---|
304 | if (I[i] != 0) |
---|
305 | { |
---|
306 | p = I[i]; |
---|
307 | while (p!=0) |
---|
308 | { |
---|
309 | q = leadmonom(p); |
---|
310 | |
---|
311 | } |
---|
312 | } |
---|
313 | } |
---|
314 | |
---|
315 | return(lv); |
---|
316 | } |
---|
317 | example |
---|
318 | { |
---|
319 | "EXAMPLE:"; echo = 2; |
---|
320 | ring r = 0,(x(1),y(1),z(1)),dp; |
---|
321 | ideal I = x(1)*y(2) -z(1)*x(2); |
---|
322 | id2words(I,3); |
---|
323 | } |
---|
324 | |
---|
325 | |
---|
326 | |
---|
327 | proc mono2word(poly p, int d) |
---|
328 | { |
---|
329 | |
---|
330 | } |
---|
331 | |
---|
332 | // given the element -7xy^2x, it is represented as [-7,x,y^2,x] or as [-7,x,y,y,x] |
---|
333 | // use the orig ord on (x,y,z) and expand it blockwise to (x(i),y(i),z(i)) |
---|
334 | |
---|
335 | // the correspondences: |
---|
336 | // monomial in K<x,y,z> <<--->> vector in R |
---|
337 | // polynomial in K<x,y,z> <<--->> list of vectors (matrix/module) in R |
---|
338 | // ideal in K<x,y,z> <<--->> list of matrices/modules in R |
---|
339 | |
---|
340 | |
---|
341 | // 1. form a new ring |
---|
342 | // 2. NOP |
---|
343 | // 3. compute GB -> with the kernel stuff |
---|
344 | // 4. skip shifted elts (check that no such exist?) |
---|
345 | // 5. go back to orig vars, produce strings/modules |
---|
346 | // 6. return the result |
---|
347 | |
---|
348 | proc freegbasis(list LM, int d) |
---|
349 | "USAGE: freegbasis(L, d); L a list of modules, d an integer |
---|
350 | RETURN: ring |
---|
351 | PURPOSE: compute the two-sided Groebner basis of an ideal, encoded by L in |
---|
352 | the free associative algebra, up to degree d |
---|
353 | EXAMPLE: example freegbasis; shows examples |
---|
354 | " |
---|
355 | { |
---|
356 | // d = up to degree, will be shifted to d+1 |
---|
357 | if (d<1) {"bad d"; return(0);} |
---|
358 | |
---|
359 | int ppl = printlevel-voice+2; |
---|
360 | string err = ""; |
---|
361 | |
---|
362 | int i,j,s; |
---|
363 | def save = basering; |
---|
364 | // determine max no of places in the input |
---|
365 | int slm = size(LM); // numbers of polys in the ideal |
---|
366 | int sm; |
---|
367 | intvec iv; |
---|
368 | module M; |
---|
369 | for (i=1; i<=slm; i++) |
---|
370 | { |
---|
371 | // modules, e.g. free polynomials |
---|
372 | M = LM[i]; |
---|
373 | sm = ncols(M); |
---|
374 | for (j=1; j<=sm; j++) |
---|
375 | { |
---|
376 | //vectors, e.g. free monomials |
---|
377 | iv = iv, size(M[j])-1; // 1 place is reserved by the coeff |
---|
378 | } |
---|
379 | } |
---|
380 | int D = Max(iv); // max size of input words |
---|
381 | if (d<D) {"bad d"; return(LM);} |
---|
382 | D = D + d-1; |
---|
383 | // D = d; |
---|
384 | list LR = ringlist(save); |
---|
385 | list L, tmp; |
---|
386 | L[1] = LR[1]; // ground field |
---|
387 | L[4] = LR[4]; // quotient ideal |
---|
388 | tmp = LR[2]; // varnames |
---|
389 | s = size(LR[2]); |
---|
390 | for (i=1; i<=D; i++) |
---|
391 | { |
---|
392 | for (j=1; j<=s; j++) |
---|
393 | { |
---|
394 | tmp[i*s+j] = string(tmp[j])+"("+string(i+1)+")"; |
---|
395 | } |
---|
396 | } |
---|
397 | for (i=1; i<=s; i++) |
---|
398 | { |
---|
399 | tmp[i] = string(tmp[i])+"("+string(1)+")"; |
---|
400 | } |
---|
401 | L[2] = tmp; |
---|
402 | list OrigNames = LR[2]; |
---|
403 | // ordering: d blocks of the ord on r |
---|
404 | // try to get whether the ord on r is blockord itself |
---|
405 | s = size(LR[3]); |
---|
406 | if (s==2) |
---|
407 | { |
---|
408 | // not a blockord, 1 block + module ord |
---|
409 | tmp = LR[3][s]; // module ord |
---|
410 | for (i=1; i<=D; i++) |
---|
411 | { |
---|
412 | LR[3][s-1+i] = LR[3][1]; |
---|
413 | } |
---|
414 | LR[3][s+D] = tmp; |
---|
415 | } |
---|
416 | if (s>2) |
---|
417 | { |
---|
418 | // there are s-1 blocks |
---|
419 | int nb = s-1; |
---|
420 | tmp = LR[3][s]; // module ord |
---|
421 | for (i=1; i<=D; i++) |
---|
422 | { |
---|
423 | for (j=1; j<=nb; j++) |
---|
424 | { |
---|
425 | LR[3][i*nb+j] = LR[3][j]; |
---|
426 | } |
---|
427 | } |
---|
428 | // size(LR[3]); |
---|
429 | LR[3][nb*(D+1)+1] = tmp; |
---|
430 | } |
---|
431 | L[3] = LR[3]; |
---|
432 | def @R = ring(L); |
---|
433 | setring @R; |
---|
434 | ideal I; |
---|
435 | poly @p; |
---|
436 | s = size(OrigNames); |
---|
437 | // "s:";s; |
---|
438 | // convert LM to canonical vectors (no powers) |
---|
439 | setring save; |
---|
440 | kill M; // M was defined earlier |
---|
441 | module M; |
---|
442 | slm = size(LM); // numbers of polys in the ideal |
---|
443 | int sv,k,l; |
---|
444 | vector v; |
---|
445 | // poly p; |
---|
446 | string sp; |
---|
447 | setring @R; |
---|
448 | poly @@p=0; |
---|
449 | setring save; |
---|
450 | for (l=1; l<=slm; l++) |
---|
451 | { |
---|
452 | // modules, e.g. free polynomials |
---|
453 | M = LM[l]; |
---|
454 | sm = ncols(M); // in intvec iv the sizes are stored |
---|
455 | // modules, e.g. free polynomials |
---|
456 | for (j=1; j<=sm; j++) |
---|
457 | { |
---|
458 | //vectors, e.g. free monomials |
---|
459 | v = M[j]; |
---|
460 | sv = size(v); |
---|
461 | // "sv:";sv; |
---|
462 | sp = "@@p = @@p + "; |
---|
463 | for (k=2; k<=sv; k++) |
---|
464 | { |
---|
465 | sp = sp + string(v[k])+"("+string(k-1)+")*"; |
---|
466 | } |
---|
467 | sp = sp + string(v[1])+";"; // coef; |
---|
468 | setring @R; |
---|
469 | execute(sp); |
---|
470 | setring save; |
---|
471 | } |
---|
472 | setring @R; |
---|
473 | // "@@p:"; @@p; |
---|
474 | I = I,@@p; |
---|
475 | @@p = 0; |
---|
476 | setring save; |
---|
477 | } |
---|
478 | kill sp; |
---|
479 | // 3. compute GB |
---|
480 | setring @R; |
---|
481 | dbprint(ppl,"computing GB"); |
---|
482 | ideal J = system("freegb",I,d,nvars(save)); |
---|
483 | // ideal J = slimgb(I); |
---|
484 | dbprint(ppl,J); |
---|
485 | // 4. skip shifted elts |
---|
486 | ideal K = select1(J,1,s); // s = size(OrigNames) |
---|
487 | dbprint(ppl,K); |
---|
488 | dbprint(ppl, "done with GB"); |
---|
489 | // K contains vars x(1),...z(1) = images of originals |
---|
490 | // 5. go back to orig vars, produce strings/modules |
---|
491 | if (K[1] == 0) |
---|
492 | { |
---|
493 | "no reasonable output, GB gives 0"; |
---|
494 | return(0); |
---|
495 | } |
---|
496 | int sk = size(K); |
---|
497 | int sp, sx, a, b; |
---|
498 | intvec x; |
---|
499 | poly p,q; |
---|
500 | poly pn; |
---|
501 | // vars in 'save' |
---|
502 | setring save; |
---|
503 | module N; |
---|
504 | list LN; |
---|
505 | vector V; |
---|
506 | poly pn; |
---|
507 | // test and skip exponents >=2 |
---|
508 | setring @R; |
---|
509 | for(i=1; i<=sk; i++) |
---|
510 | { |
---|
511 | p = K[i]; |
---|
512 | while (p!=0) |
---|
513 | { |
---|
514 | q = lead(p); |
---|
515 | // "processing q:";q; |
---|
516 | x = leadexp(q); |
---|
517 | sx = size(x); |
---|
518 | for(k=1; k<=sx; k++) |
---|
519 | { |
---|
520 | if ( x[k] >= 2 ) |
---|
521 | { |
---|
522 | err = "skip: the value x[k] is " + string(x[k]); |
---|
523 | dbprint(ppl,err); |
---|
524 | // return(0); |
---|
525 | K[i] = 0; |
---|
526 | p = 0; |
---|
527 | q = 0; |
---|
528 | break; |
---|
529 | } |
---|
530 | } |
---|
531 | p = p - q; |
---|
532 | } |
---|
533 | } |
---|
534 | K = simplify(K,2); |
---|
535 | sk = size(K); |
---|
536 | for(i=1; i<=sk; i++) |
---|
537 | { |
---|
538 | // setring save; |
---|
539 | // V = 0; |
---|
540 | setring @R; |
---|
541 | p = K[i]; |
---|
542 | while (p!=0) |
---|
543 | { |
---|
544 | q = lead(p); |
---|
545 | err = "processing q:" + string(q); |
---|
546 | dbprint(ppl,err); |
---|
547 | x = leadexp(q); |
---|
548 | sx = size(x); |
---|
549 | pn = leadcoef(q); |
---|
550 | setring save; |
---|
551 | pn = imap(@R,pn); |
---|
552 | V = V + leadcoef(pn)*gen(1); |
---|
553 | for(k=1; k<=sx; k++) |
---|
554 | { |
---|
555 | if (x[k] ==1) |
---|
556 | { |
---|
557 | a = k / s; // block number=a+1, a!=0 |
---|
558 | b = k % s; // remainder |
---|
559 | // printf("a: %s, b: %s",a,b); |
---|
560 | if (b == 0) |
---|
561 | { |
---|
562 | // that is it's the last var in the block |
---|
563 | b = s; |
---|
564 | a = a-1; |
---|
565 | } |
---|
566 | V = V + var(b)*gen(a+2); |
---|
567 | } |
---|
568 | // else |
---|
569 | // { |
---|
570 | // printf("error: the value x[k] is %s", x[k]); |
---|
571 | // return(0); |
---|
572 | // } |
---|
573 | } |
---|
574 | err = "V: " + string(V); |
---|
575 | dbprint(ppl,err); |
---|
576 | // printf("V: %s", string(V)); |
---|
577 | N = N,V; |
---|
578 | V = 0; |
---|
579 | setring @R; |
---|
580 | p = p - q; |
---|
581 | pn = 0; |
---|
582 | } |
---|
583 | setring save; |
---|
584 | LN[i] = simplify(N,2); |
---|
585 | N = 0; |
---|
586 | } |
---|
587 | setring save; |
---|
588 | return(LN); |
---|
589 | } |
---|
590 | example |
---|
591 | { |
---|
592 | "EXAMPLE:"; echo = 2; |
---|
593 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
594 | module M = [-1,x,y],[-7,y,y],[3,x,x]; |
---|
595 | module N = [1,x,y,x],[-1,y,x,y]; |
---|
596 | list L; L[1] = M; L[2] = N; |
---|
597 | lst2str(L); |
---|
598 | def U = freegbasis(L,5); |
---|
599 | lst2str(U); |
---|
600 | } |
---|
601 | |
---|
602 | proc crs(list LM, int d) |
---|
603 | "USAGE: crs(L, d); L a list of modules, d an integer |
---|
604 | RETURN: ring |
---|
605 | PURPOSE: create a ring and shift the ideal |
---|
606 | EXAMPLE: example crs; shows examples |
---|
607 | " |
---|
608 | { |
---|
609 | // d = up to degree, will be shifted to d+1 |
---|
610 | if (d<1) {"bad d"; return(0);} |
---|
611 | |
---|
612 | int ppl = printlevel-voice+2; |
---|
613 | string err = ""; |
---|
614 | |
---|
615 | int i,j,s; |
---|
616 | def save = basering; |
---|
617 | // determine max no of places in the input |
---|
618 | int slm = size(LM); // numbers of polys in the ideal |
---|
619 | int sm; |
---|
620 | intvec iv; |
---|
621 | module M; |
---|
622 | for (i=1; i<=slm; i++) |
---|
623 | { |
---|
624 | // modules, e.g. free polynomials |
---|
625 | M = LM[i]; |
---|
626 | sm = ncols(M); |
---|
627 | for (j=1; j<=sm; j++) |
---|
628 | { |
---|
629 | //vectors, e.g. free monomials |
---|
630 | iv = iv, size(M[j])-1; // 1 place is reserved by the coeff |
---|
631 | } |
---|
632 | } |
---|
633 | int D = Max(iv); // max size of input words |
---|
634 | if (d<D) {"bad d"; return(LM);} |
---|
635 | D = D + d-1; |
---|
636 | // D = d; |
---|
637 | list LR = ringlist(save); |
---|
638 | list L, tmp; |
---|
639 | L[1] = LR[1]; // ground field |
---|
640 | L[4] = LR[4]; // quotient ideal |
---|
641 | tmp = LR[2]; // varnames |
---|
642 | s = size(LR[2]); |
---|
643 | for (i=1; i<=D; i++) |
---|
644 | { |
---|
645 | for (j=1; j<=s; j++) |
---|
646 | { |
---|
647 | tmp[i*s+j] = string(tmp[j])+"("+string(i)+")"; |
---|
648 | } |
---|
649 | } |
---|
650 | for (i=1; i<=s; i++) |
---|
651 | { |
---|
652 | tmp[i] = string(tmp[i])+"("+string(0)+")"; |
---|
653 | } |
---|
654 | L[2] = tmp; |
---|
655 | list OrigNames = LR[2]; |
---|
656 | // ordering: d blocks of the ord on r |
---|
657 | // try to get whether the ord on r is blockord itself |
---|
658 | s = size(LR[3]); |
---|
659 | if (s==2) |
---|
660 | { |
---|
661 | // not a blockord, 1 block + module ord |
---|
662 | tmp = LR[3][s]; // module ord |
---|
663 | for (i=1; i<=D; i++) |
---|
664 | { |
---|
665 | LR[3][s-1+i] = LR[3][1]; |
---|
666 | } |
---|
667 | LR[3][s+D] = tmp; |
---|
668 | } |
---|
669 | if (s>2) |
---|
670 | { |
---|
671 | // there are s-1 blocks |
---|
672 | int nb = s-1; |
---|
673 | tmp = LR[3][s]; // module ord |
---|
674 | for (i=1; i<=D; i++) |
---|
675 | { |
---|
676 | for (j=1; j<=nb; j++) |
---|
677 | { |
---|
678 | LR[3][i*nb+j] = LR[3][j]; |
---|
679 | } |
---|
680 | } |
---|
681 | // size(LR[3]); |
---|
682 | LR[3][nb*(D+1)+1] = tmp; |
---|
683 | } |
---|
684 | L[3] = LR[3]; |
---|
685 | def @R = ring(L); |
---|
686 | setring @R; |
---|
687 | ideal I; |
---|
688 | poly @p; |
---|
689 | s = size(OrigNames); |
---|
690 | // "s:";s; |
---|
691 | // convert LM to canonical vectors (no powers) |
---|
692 | setring save; |
---|
693 | kill M; // M was defined earlier |
---|
694 | module M; |
---|
695 | slm = size(LM); // numbers of polys in the ideal |
---|
696 | int sv,k,l; |
---|
697 | vector v; |
---|
698 | // poly p; |
---|
699 | string sp; |
---|
700 | setring @R; |
---|
701 | poly @@p=0; |
---|
702 | setring save; |
---|
703 | for (l=1; l<=slm; l++) |
---|
704 | { |
---|
705 | // modules, e.g. free polynomials |
---|
706 | M = LM[l]; |
---|
707 | sm = ncols(M); // in intvec iv the sizes are stored |
---|
708 | for (i=0; i<=d-iv[l]; i++) |
---|
709 | { |
---|
710 | // modules, e.g. free polynomials |
---|
711 | for (j=1; j<=sm; j++) |
---|
712 | { |
---|
713 | //vectors, e.g. free monomials |
---|
714 | v = M[j]; |
---|
715 | sv = size(v); |
---|
716 | // "sv:";sv; |
---|
717 | sp = "@@p = @@p + "; |
---|
718 | for (k=2; k<=sv; k++) |
---|
719 | { |
---|
720 | sp = sp + string(v[k])+"("+string(k-2+i)+")*"; |
---|
721 | } |
---|
722 | sp = sp + string(v[1])+";"; // coef; |
---|
723 | setring @R; |
---|
724 | execute(sp); |
---|
725 | setring save; |
---|
726 | } |
---|
727 | setring @R; |
---|
728 | // "@@p:"; @@p; |
---|
729 | I = I,@@p; |
---|
730 | @@p = 0; |
---|
731 | setring save; |
---|
732 | } |
---|
733 | } |
---|
734 | setring @R; |
---|
735 | export I; |
---|
736 | return(@R); |
---|
737 | } |
---|
738 | example |
---|
739 | { |
---|
740 | "EXAMPLE:"; echo = 2; |
---|
741 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
742 | module M = [-1,x,y],[-7,y,y],[3,x,x]; |
---|
743 | module N = [1,x,y,x],[-1,y,x,y]; |
---|
744 | list L; L[1] = M; L[2] = N; |
---|
745 | lst2str(L); |
---|
746 | def U = crs(L,5); |
---|
747 | setring U; U; |
---|
748 | I; |
---|
749 | } |
---|
750 | |
---|
751 | proc polylen(ideal I) |
---|
752 | { |
---|
753 | // returns the ideal of length of polys |
---|
754 | int i; |
---|
755 | intvec J; |
---|
756 | number s = 0; |
---|
757 | for(i=1;i<=ncols(I);i++) |
---|
758 | { |
---|
759 | J[i] = size(I[i]); |
---|
760 | s = s + J[i]; |
---|
761 | } |
---|
762 | printf("the sum of length %s",s); |
---|
763 | // print(s); |
---|
764 | return(J); |
---|
765 | } |
---|
766 | |
---|
767 | proc freegbRing(int d) |
---|
768 | "USAGE: freegbRing(d); d an integer |
---|
769 | RETURN: ring |
---|
770 | PURPOSE: creates a ring with d blocks of shifted original variables |
---|
771 | EXAMPLE: example freegbRing; shows examples |
---|
772 | " |
---|
773 | { |
---|
774 | // d = up to degree, will be shifted to d+1 |
---|
775 | if (d<1) {"bad d"; return(0);} |
---|
776 | |
---|
777 | int ppl = printlevel-voice+2; |
---|
778 | string err = ""; |
---|
779 | |
---|
780 | int i,j,s; |
---|
781 | def save = basering; |
---|
782 | int D = d-1; |
---|
783 | list LR = ringlist(save); |
---|
784 | list L, tmp; |
---|
785 | L[1] = LR[1]; // ground field |
---|
786 | L[4] = LR[4]; // quotient ideal |
---|
787 | tmp = LR[2]; // varnames |
---|
788 | s = size(LR[2]); |
---|
789 | for (i=1; i<=D; i++) |
---|
790 | { |
---|
791 | for (j=1; j<=s; j++) |
---|
792 | { |
---|
793 | tmp[i*s+j] = string(tmp[j])+"("+string(i+1)+")"; |
---|
794 | } |
---|
795 | } |
---|
796 | for (i=1; i<=s; i++) |
---|
797 | { |
---|
798 | tmp[i] = string(tmp[i])+"("+string(1)+")"; |
---|
799 | } |
---|
800 | L[2] = tmp; |
---|
801 | list OrigNames = LR[2]; |
---|
802 | // ordering: d blocks of the ord on r |
---|
803 | // try to get whether the ord on r is blockord itself |
---|
804 | // TODO: make L(2) ordering! exponent is maximally 2 |
---|
805 | s = size(LR[3]); |
---|
806 | if (s==2) |
---|
807 | { |
---|
808 | // not a blockord, 1 block + module ord |
---|
809 | tmp = LR[3][s]; // module ord |
---|
810 | for (i=1; i<=D; i++) |
---|
811 | { |
---|
812 | LR[3][s-1+i] = LR[3][1]; |
---|
813 | } |
---|
814 | LR[3][s+D] = tmp; |
---|
815 | } |
---|
816 | if (s>2) |
---|
817 | { |
---|
818 | // there are s-1 blocks |
---|
819 | int nb = s-1; |
---|
820 | tmp = LR[3][s]; // module ord |
---|
821 | for (i=1; i<=D; i++) |
---|
822 | { |
---|
823 | for (j=1; j<=nb; j++) |
---|
824 | { |
---|
825 | LR[3][i*nb+j] = LR[3][j]; |
---|
826 | } |
---|
827 | } |
---|
828 | // size(LR[3]); |
---|
829 | LR[3][nb*(D+1)+1] = tmp; |
---|
830 | } |
---|
831 | L[3] = LR[3]; |
---|
832 | def @R = ring(L); |
---|
833 | // setring @R; |
---|
834 | return (@R); |
---|
835 | } |
---|
836 | example |
---|
837 | { |
---|
838 | "EXAMPLE:"; echo = 2; |
---|
839 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
840 | def A = freegbRing(2); |
---|
841 | setring A; |
---|
842 | A; |
---|
843 | } |
---|
844 | |
---|
845 | |
---|
846 | proc ex_shift() |
---|
847 | { |
---|
848 | LIB "freegb.lib"; |
---|
849 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
850 | module M = [-1,x,y],[-7,y,y],[3,x,x]; |
---|
851 | module N = [1,x,y,x],[-1,y,x,y]; |
---|
852 | list L; L[1] = M; L[2] = N; |
---|
853 | lst2str(L); |
---|
854 | def U = crs(L,5); |
---|
855 | setring U; U; |
---|
856 | I; |
---|
857 | poly p = I[2]; // I[8]; |
---|
858 | p; |
---|
859 | system("stest",p,7,7,3); // error -> the world is ok |
---|
860 | poly q1 = system("stest",p,1,7,3); //ok |
---|
861 | poly q6 = system("stest",p,6,7,3); //ok |
---|
862 | system("btest",p,3); //ok |
---|
863 | system("btest",q1,3); //ok |
---|
864 | system("btest",q6,3); //ok |
---|
865 | } |
---|
866 | |
---|
867 | proc test_shrink() |
---|
868 | { |
---|
869 | LIB "freegb.lib"; |
---|
870 | ring r =0,(x,y,z),dp; |
---|
871 | int d = 5; |
---|
872 | def R = freegbRing(d); |
---|
873 | setring R; |
---|
874 | poly p1 = x(1)*y(2)*z(3); |
---|
875 | poly p2 = x(1)*y(4)*z(5); |
---|
876 | poly p3 = x(1)*y(1)*z(3); |
---|
877 | poly p4 = x(1)*y(2)*z(2); |
---|
878 | poly p5 = x(3)*z(5); |
---|
879 | poly p6 = x(1)*y(1)*x(3)*z(5); |
---|
880 | poly p7 = x(1)*y(2)*x(3)*y(4)*z(5); |
---|
881 | poly p8 = p1+p2+p3+p4+p5 + p6 + p7; |
---|
882 | p1; system("shrinktest",p1,3); |
---|
883 | p2; system("shrinktest",p2,3); |
---|
884 | p3; system("shrinktest",p3,3); |
---|
885 | p4; system("shrinktest",p4,3); |
---|
886 | p5; system("shrinktest",p5,3); |
---|
887 | p6; system("shrinktest",p6,3); |
---|
888 | p7; system("shrinktest",p7,3); |
---|
889 | p8; system("shrinktest",p8,3); |
---|
890 | poly p9 = p1 + 2*p2 + 5*p5 + 7*p7; |
---|
891 | p9; system("shrinktest",p9,3); |
---|
892 | } |
---|
893 | |
---|
894 | proc ex2() |
---|
895 | { |
---|
896 | option(prot); |
---|
897 | LIB "freegb.lib"; |
---|
898 | ring r = 0,(x,y),dp; |
---|
899 | module M = [-1,x,y],[3,x,x]; // 3x^2 - xy |
---|
900 | def U = freegb(M,7); |
---|
901 | lst2str(U); |
---|
902 | } |
---|
903 | |
---|
904 | proc ex_nonhomog() |
---|
905 | { |
---|
906 | option(prot); |
---|
907 | LIB "freegb.lib"; |
---|
908 | ring r = 0,(x,y,h),dp; |
---|
909 | list L; |
---|
910 | module M; |
---|
911 | M = [-1,y,y],[1,x,x,x]; // x3-y2 |
---|
912 | L[1] = M; |
---|
913 | M = [1,x,h],[-1,h,x]; // xh-hx |
---|
914 | L[2] = M; |
---|
915 | M = [1,y,h],[-1,h,y]; // yh-hy |
---|
916 | L[3] = M; |
---|
917 | def U = freegb(L,4); |
---|
918 | lst2str(U); |
---|
919 | // strange elements in the basis |
---|
920 | } |
---|
921 | |
---|
922 | proc ex_nonhomog_comm() |
---|
923 | { |
---|
924 | option(prot); |
---|
925 | LIB "freegb.lib"; |
---|
926 | ring r = 0,(x,y),dp; |
---|
927 | module M = [-1,y,y],[1,x,x,x]; |
---|
928 | def U = freegb(M,5); |
---|
929 | lst2str(U); |
---|
930 | } |
---|
931 | |
---|
932 | proc ex_nonhomog_h() |
---|
933 | { |
---|
934 | option(prot); |
---|
935 | LIB "freegb.lib"; |
---|
936 | ring r = 0,(x,y,h),(a(1,1),dp); |
---|
937 | module M = [-1,y,y,h],[1,x,x,x]; // x3 - y2h |
---|
938 | def U = freegb(M,6); |
---|
939 | lst2str(U); |
---|
940 | } |
---|
941 | |
---|
942 | proc ex_nonhomog_h2() |
---|
943 | { |
---|
944 | option(prot); |
---|
945 | LIB "freegb.lib"; |
---|
946 | ring r = 0,(x,y,h),(dp); |
---|
947 | list L; |
---|
948 | module M; |
---|
949 | M = [-1,y,y,h],[1,x,x,x]; // x3 - y2h |
---|
950 | L[1] = M; |
---|
951 | M = [1,x,h],[-1,h,x]; // xh - hx |
---|
952 | L[2] = M; |
---|
953 | M = [1,y,h],[-1,h,y]; // yh - hy |
---|
954 | L[3] = M; |
---|
955 | def U = freegbasis(L,3); |
---|
956 | lst2str(U); |
---|
957 | // strange answer CHECK |
---|
958 | } |
---|
959 | |
---|
960 | |
---|
961 | proc ex_nonhomog_3() |
---|
962 | { |
---|
963 | option(prot); |
---|
964 | LIB "./freegb.lib"; |
---|
965 | ring r = 0,(x,y,z),(dp); |
---|
966 | list L; |
---|
967 | module M; |
---|
968 | M = [1,z,y],[-1,x]; // zy - x |
---|
969 | L[1] = M; |
---|
970 | M = [1,z,x],[-1,y]; // zx - y |
---|
971 | L[2] = M; |
---|
972 | M = [1,y,x],[-1,z]; // yx - z |
---|
973 | L[3] = M; |
---|
974 | lst2str(L); |
---|
975 | list U = freegb(L,4); |
---|
976 | lst2str(U); |
---|
977 | // strange answer CHECK |
---|
978 | } |
---|
979 | |
---|
980 | proc ex_densep_2() |
---|
981 | { |
---|
982 | option(prot); |
---|
983 | LIB "freegb.lib"; |
---|
984 | ring r = (0,a,b,c),(x,y),(Dp); // deglex |
---|
985 | module M = [1,x,x], [a,x,y], [b,y,x], [c,y,y]; |
---|
986 | lst2str(M); |
---|
987 | list U = freegb(M,5); |
---|
988 | lst2str(U); |
---|
989 | // a=b is important -> finite basis!!! |
---|
990 | module M = [1,x,x], [a,x,y], [a,y,x], [c,y,y]; |
---|
991 | lst2str(M); |
---|
992 | list U = freegb(M,5); |
---|
993 | lst2str(U); |
---|
994 | } |
---|
995 | |
---|
996 | |
---|
997 | // 1. form a new ring |
---|
998 | // 2. produce shifted generators |
---|
999 | // 3. compute GB |
---|
1000 | // 4. skip shifted elts |
---|
1001 | // 5. go back to orig vars, produce strings/modules |
---|
1002 | // 6. return the result |
---|
1003 | |
---|
1004 | proc freegbold(list LM, int d) |
---|
1005 | "USAGE: freegbold(L, d); L a list of modules, d an integer |
---|
1006 | RETURN: ring |
---|
1007 | PURPOSE: compute the two-sided Groebner basis of an ideal, encoded by L in |
---|
1008 | the free associative algebra, up to degree d |
---|
1009 | EXAMPLE: example freegbold; shows examples |
---|
1010 | " |
---|
1011 | { |
---|
1012 | // d = up to degree, will be shifted to d+1 |
---|
1013 | if (d<1) {"bad d"; return(0);} |
---|
1014 | |
---|
1015 | int ppl = printlevel-voice+2; |
---|
1016 | string err = ""; |
---|
1017 | |
---|
1018 | int i,j,s; |
---|
1019 | def save = basering; |
---|
1020 | // determine max no of places in the input |
---|
1021 | int slm = size(LM); // numbers of polys in the ideal |
---|
1022 | int sm; |
---|
1023 | intvec iv; |
---|
1024 | module M; |
---|
1025 | for (i=1; i<=slm; i++) |
---|
1026 | { |
---|
1027 | // modules, e.g. free polynomials |
---|
1028 | M = LM[i]; |
---|
1029 | sm = ncols(M); |
---|
1030 | for (j=1; j<=sm; j++) |
---|
1031 | { |
---|
1032 | //vectors, e.g. free monomials |
---|
1033 | iv = iv, size(M[j])-1; // 1 place is reserved by the coeff |
---|
1034 | } |
---|
1035 | } |
---|
1036 | int D = Max(iv); // max size of input words |
---|
1037 | if (d<D) {"bad d"; return(LM);} |
---|
1038 | D = D + d-1; |
---|
1039 | // D = d; |
---|
1040 | list LR = ringlist(save); |
---|
1041 | list L, tmp; |
---|
1042 | L[1] = LR[1]; // ground field |
---|
1043 | L[4] = LR[4]; // quotient ideal |
---|
1044 | tmp = LR[2]; // varnames |
---|
1045 | s = size(LR[2]); |
---|
1046 | for (i=1; i<=D; i++) |
---|
1047 | { |
---|
1048 | for (j=1; j<=s; j++) |
---|
1049 | { |
---|
1050 | tmp[i*s+j] = string(tmp[j])+"("+string(i+1)+")"; |
---|
1051 | } |
---|
1052 | } |
---|
1053 | for (i=1; i<=s; i++) |
---|
1054 | { |
---|
1055 | tmp[i] = string(tmp[i])+"("+string(1)+")"; |
---|
1056 | } |
---|
1057 | L[2] = tmp; |
---|
1058 | list OrigNames = LR[2]; |
---|
1059 | // ordering: d blocks of the ord on r |
---|
1060 | // try to get whether the ord on r is blockord itself |
---|
1061 | // TODO: make L(2) ordering! exponent is maximally 2 |
---|
1062 | s = size(LR[3]); |
---|
1063 | if (s==2) |
---|
1064 | { |
---|
1065 | // not a blockord, 1 block + module ord |
---|
1066 | tmp = LR[3][s]; // module ord |
---|
1067 | for (i=1; i<=D; i++) |
---|
1068 | { |
---|
1069 | LR[3][s-1+i] = LR[3][1]; |
---|
1070 | } |
---|
1071 | LR[3][s+D] = tmp; |
---|
1072 | } |
---|
1073 | if (s>2) |
---|
1074 | { |
---|
1075 | // there are s-1 blocks |
---|
1076 | int nb = s-1; |
---|
1077 | tmp = LR[3][s]; // module ord |
---|
1078 | for (i=1; i<=D; i++) |
---|
1079 | { |
---|
1080 | for (j=1; j<=nb; j++) |
---|
1081 | { |
---|
1082 | LR[3][i*nb+j] = LR[3][j]; |
---|
1083 | } |
---|
1084 | } |
---|
1085 | // size(LR[3]); |
---|
1086 | LR[3][nb*(D+1)+1] = tmp; |
---|
1087 | } |
---|
1088 | L[3] = LR[3]; |
---|
1089 | def @R = ring(L); |
---|
1090 | setring @R; |
---|
1091 | ideal I; |
---|
1092 | poly @p; |
---|
1093 | s = size(OrigNames); |
---|
1094 | // "s:";s; |
---|
1095 | // convert LM to canonical vectors (no powers) |
---|
1096 | setring save; |
---|
1097 | kill M; // M was defined earlier |
---|
1098 | module M; |
---|
1099 | slm = size(LM); // numbers of polys in the ideal |
---|
1100 | int sv,k,l; |
---|
1101 | vector v; |
---|
1102 | // poly p; |
---|
1103 | string sp; |
---|
1104 | setring @R; |
---|
1105 | poly @@p=0; |
---|
1106 | setring save; |
---|
1107 | for (l=1; l<=slm; l++) |
---|
1108 | { |
---|
1109 | // modules, e.g. free polynomials |
---|
1110 | M = LM[l]; |
---|
1111 | sm = ncols(M); // in intvec iv the sizes are stored |
---|
1112 | for (i=0; i<=d-iv[l]; i++) |
---|
1113 | { |
---|
1114 | // modules, e.g. free polynomials |
---|
1115 | for (j=1; j<=sm; j++) |
---|
1116 | { |
---|
1117 | //vectors, e.g. free monomials |
---|
1118 | v = M[j]; |
---|
1119 | sv = size(v); |
---|
1120 | // "sv:";sv; |
---|
1121 | sp = "@@p = @@p + "; |
---|
1122 | for (k=2; k<=sv; k++) |
---|
1123 | { |
---|
1124 | sp = sp + string(v[k])+"("+string(k-1+i)+")*"; |
---|
1125 | } |
---|
1126 | sp = sp + string(v[1])+";"; // coef; |
---|
1127 | setring @R; |
---|
1128 | execute(sp); |
---|
1129 | setring save; |
---|
1130 | } |
---|
1131 | setring @R; |
---|
1132 | // "@@p:"; @@p; |
---|
1133 | I = I,@@p; |
---|
1134 | @@p = 0; |
---|
1135 | setring save; |
---|
1136 | } |
---|
1137 | } |
---|
1138 | kill sp; |
---|
1139 | // 3. compute GB |
---|
1140 | setring @R; |
---|
1141 | dbprint(ppl,"computing GB"); |
---|
1142 | // ideal J = groebner(I); |
---|
1143 | ideal J = slimgb(I); |
---|
1144 | dbprint(ppl,J); |
---|
1145 | // 4. skip shifted elts |
---|
1146 | ideal K = select1(J,1,s); // s = size(OrigNames) |
---|
1147 | dbprint(ppl,K); |
---|
1148 | dbprint(ppl, "done with GB"); |
---|
1149 | // K contains vars x(1),...z(1) = images of originals |
---|
1150 | // 5. go back to orig vars, produce strings/modules |
---|
1151 | if (K[1] == 0) |
---|
1152 | { |
---|
1153 | "no reasonable output, GB gives 0"; |
---|
1154 | return(0); |
---|
1155 | } |
---|
1156 | int sk = size(K); |
---|
1157 | int sp, sx, a, b; |
---|
1158 | intvec x; |
---|
1159 | poly p,q; |
---|
1160 | poly pn; |
---|
1161 | // vars in 'save' |
---|
1162 | setring save; |
---|
1163 | module N; |
---|
1164 | list LN; |
---|
1165 | vector V; |
---|
1166 | poly pn; |
---|
1167 | // test and skip exponents >=2 |
---|
1168 | setring @R; |
---|
1169 | for(i=1; i<=sk; i++) |
---|
1170 | { |
---|
1171 | p = K[i]; |
---|
1172 | while (p!=0) |
---|
1173 | { |
---|
1174 | q = lead(p); |
---|
1175 | // "processing q:";q; |
---|
1176 | x = leadexp(q); |
---|
1177 | sx = size(x); |
---|
1178 | for(k=1; k<=sx; k++) |
---|
1179 | { |
---|
1180 | if ( x[k] >= 2 ) |
---|
1181 | { |
---|
1182 | err = "skip: the value x[k] is " + string(x[k]); |
---|
1183 | dbprint(ppl,err); |
---|
1184 | // return(0); |
---|
1185 | K[i] = 0; |
---|
1186 | p = 0; |
---|
1187 | q = 0; |
---|
1188 | break; |
---|
1189 | } |
---|
1190 | } |
---|
1191 | p = p - q; |
---|
1192 | } |
---|
1193 | } |
---|
1194 | K = simplify(K,2); |
---|
1195 | sk = size(K); |
---|
1196 | for(i=1; i<=sk; i++) |
---|
1197 | { |
---|
1198 | // setring save; |
---|
1199 | // V = 0; |
---|
1200 | setring @R; |
---|
1201 | p = K[i]; |
---|
1202 | while (p!=0) |
---|
1203 | { |
---|
1204 | q = lead(p); |
---|
1205 | err = "processing q:" + string(q); |
---|
1206 | dbprint(ppl,err); |
---|
1207 | x = leadexp(q); |
---|
1208 | sx = size(x); |
---|
1209 | pn = leadcoef(q); |
---|
1210 | setring save; |
---|
1211 | pn = imap(@R,pn); |
---|
1212 | V = V + leadcoef(pn)*gen(1); |
---|
1213 | for(k=1; k<=sx; k++) |
---|
1214 | { |
---|
1215 | if (x[k] ==1) |
---|
1216 | { |
---|
1217 | a = k / s; // block number=a+1, a!=0 |
---|
1218 | b = k % s; // remainder |
---|
1219 | // printf("a: %s, b: %s",a,b); |
---|
1220 | if (b == 0) |
---|
1221 | { |
---|
1222 | // that is it's the last var in the block |
---|
1223 | b = s; |
---|
1224 | a = a-1; |
---|
1225 | } |
---|
1226 | V = V + var(b)*gen(a+2); |
---|
1227 | } |
---|
1228 | // else |
---|
1229 | // { |
---|
1230 | // printf("error: the value x[k] is %s", x[k]); |
---|
1231 | // return(0); |
---|
1232 | // } |
---|
1233 | } |
---|
1234 | err = "V: " + string(V); |
---|
1235 | dbprint(ppl,err); |
---|
1236 | // printf("V: %s", string(V)); |
---|
1237 | N = N,V; |
---|
1238 | V = 0; |
---|
1239 | setring @R; |
---|
1240 | p = p - q; |
---|
1241 | pn = 0; |
---|
1242 | } |
---|
1243 | setring save; |
---|
1244 | LN[i] = simplify(N,2); |
---|
1245 | N = 0; |
---|
1246 | } |
---|
1247 | setring save; |
---|
1248 | return(LN); |
---|
1249 | } |
---|
1250 | example |
---|
1251 | { |
---|
1252 | "EXAMPLE:"; echo = 2; |
---|
1253 | ring r = 0,(x,y,z),(dp(1),dp(2)); |
---|
1254 | module M = [-1,x,y],[-7,y,y],[3,x,x]; |
---|
1255 | module N = [1,x,y,x],[-1,y,x,y]; |
---|
1256 | list L; L[1] = M; L[2] = N; |
---|
1257 | lst2str(L); |
---|
1258 | def U = freegbold(L,5); |
---|
1259 | lst2str(U); |
---|
1260 | } |
---|
1261 | |
---|
1262 | proc sgb(ideal I, int d) |
---|
1263 | { |
---|
1264 | // new code |
---|
1265 | // map x_i to x_i(1) via map() |
---|
1266 | //LIB "freegb.lib"; |
---|
1267 | def save = basering; |
---|
1268 | //int d =7;// degree |
---|
1269 | int nv = nvars(save); |
---|
1270 | def R = freegbRing(d); |
---|
1271 | setring R; |
---|
1272 | int i; |
---|
1273 | ideal Imap; |
---|
1274 | for (i=1; i<=nv; i++) |
---|
1275 | { |
---|
1276 | Imap[i] = var(i); |
---|
1277 | } |
---|
1278 | //ideal I = x(1)*y(2), y(1)*x(2)+z(1)*z(2); |
---|
1279 | ideal I = x(1)*x(2),x(1)*y(2) + z(1)*x(2); |
---|
1280 | option(prot); |
---|
1281 | //option(teach); |
---|
1282 | ideal J = system("freegb",I,d,nv); |
---|
1283 | } |
---|
1284 | |
---|
1285 | |
---|
1286 | |
---|
1287 | static proc checkCeq() |
---|
1288 | { |
---|
1289 | ring r = 0,(x,y),Dp; |
---|
1290 | def A = freegbRing(4); |
---|
1291 | setring A; |
---|
1292 | A; |
---|
1293 | // I = x2-xy |
---|
1294 | ideal I = x(1)*x(2) - x(1)*y(2), x(2)*x(3) - x(2)*y(3), x(3)*x(4) - x(3)*y(4); |
---|
1295 | ideal C = x(2)-x(1),x(3)-x(2),x(4)-x(3),y(2)-y(1),y(3)-y(2),y(4)-y(3); |
---|
1296 | ideal K = I,C; |
---|
1297 | groebner(K); |
---|
1298 | } |
---|
1299 | |
---|
1300 | |
---|
1301 | proc exHom1() |
---|
1302 | { |
---|
1303 | // we start with |
---|
1304 | // z*y - x, z*x - y, y*x - z |
---|
1305 | LIB "freegb.lib"; |
---|
1306 | LIB "elim.lib"; |
---|
1307 | ring r = 0,(x,y,z,h),dp; |
---|
1308 | list L; |
---|
1309 | module M; |
---|
1310 | M = [1,z,y],[-1,x,h]; // zy - xh |
---|
1311 | L[1] = M; |
---|
1312 | M = [1,z,x],[-1,y,h]; // zx - yh |
---|
1313 | L[2] = M; |
---|
1314 | M = [1,y,x],[-1,z,h]; // yx - zh |
---|
1315 | L[3] = M; |
---|
1316 | lst2str(L); |
---|
1317 | def U = crs(L,4); |
---|
1318 | setring U; |
---|
1319 | I = I, |
---|
1320 | y(2)*h(3)+z(2)*x(3), y(3)*h(4)+z(3)*x(4), |
---|
1321 | y(2)*x(3)-z(2)*h(3), y(3)*x(4)-z(3)*h(4); |
---|
1322 | I = simplify(I,2); |
---|
1323 | ring r2 = 0,(x(0..4),y(0..4),z(0..4),h(0..4)),dp; |
---|
1324 | ideal J = imap(U,I); |
---|
1325 | // ideal K = homog(J,h); |
---|
1326 | option(redSB); |
---|
1327 | option(redTail); |
---|
1328 | ideal L = groebner(J); //(K); |
---|
1329 | ideal LL = sat(L,ideal(h))[1]; |
---|
1330 | ideal M = subst(LL,h,1); |
---|
1331 | M = simplify(M,2); |
---|
1332 | setring U; |
---|
1333 | ideal M = imap(r2,M); |
---|
1334 | lst2str(U); |
---|
1335 | } |
---|
1336 | |
---|
1337 | static proc test1() |
---|
1338 | { |
---|
1339 | LIB "freegb.lib"; |
---|
1340 | ring r = 0,(x,y),Dp; |
---|
1341 | int d = 10; // degree |
---|
1342 | def R = freegbRing(d); |
---|
1343 | setring R; |
---|
1344 | ideal I = x(1)*x(2) - y(1)*y(2); |
---|
1345 | option(prot); |
---|
1346 | option(teach); |
---|
1347 | ideal J = system("freegb",I,d,2); |
---|
1348 | J; |
---|
1349 | } |
---|
1350 | |
---|
1351 | static proc test2() |
---|
1352 | { |
---|
1353 | LIB "freegb.lib"; |
---|
1354 | ring r = 0,(x,y),Dp; |
---|
1355 | int d = 10; // degree |
---|
1356 | def R = freegbRing(d); |
---|
1357 | setring R; |
---|
1358 | ideal I = x(1)*x(2) - x(1)*y(2); |
---|
1359 | option(prot); |
---|
1360 | option(teach); |
---|
1361 | ideal J = system("freegb",I,d,2); |
---|
1362 | J; |
---|
1363 | } |
---|
1364 | |
---|
1365 | static proc test3() |
---|
1366 | { |
---|
1367 | LIB "freegb.lib"; |
---|
1368 | ring r = 0,(x,y,z),dp; |
---|
1369 | int d =5; // degree |
---|
1370 | def R = freegbRing(d); |
---|
1371 | setring R; |
---|
1372 | ideal I = x(1)*y(2), y(1)*x(2)+z(1)*z(2); |
---|
1373 | option(prot); |
---|
1374 | option(teach); |
---|
1375 | ideal J = system("freegb",I,d,3); |
---|
1376 | } |
---|
1377 | |
---|
1378 | proc schur2-3() |
---|
1379 | { |
---|
1380 | // nonhomog: |
---|
1381 | // h^4-10*h^2+9,f*e-e*f+h, h*2-e*h-2*e,h*f-f*h+2*f |
---|
1382 | // homogenized with t |
---|
1383 | // h^4-10*h^2*t^2+9*t^4,f*e-e*f+h*t, h*2-e*h-2*e*t,h*f-f*h+2*f*t, |
---|
1384 | // t*h - h*t, t*f - f*t, t*e - e*t |
---|
1385 | } |
---|