1 | /////////////////////////////////////////////////////////////////////////////// |
---|
2 | version="$Id: primdec.lib,v 1.105 2005-05-06 16:01:50 Singular Exp $"; |
---|
3 | category="Commutative Algebra"; |
---|
4 | info=" |
---|
5 | LIBRARY: primdec.lib Primary Decomposition and Radical of Ideals |
---|
6 | AUTHORS: Gerhard Pfister, pfister@mathematik.uni-kl.de (GTZ) |
---|
7 | @* Wolfram Decker, decker@math.uni-sb.de (SY) |
---|
8 | @* Hans Schoenemann, hannes@mathematik.uni-kl.de (SY) |
---|
9 | |
---|
10 | OVERVIEW: |
---|
11 | Algorithms for primary decomposition based on the ideas of |
---|
12 | Gianni, Trager and Zacharias (implementation by Gerhard Pfister), |
---|
13 | respectively based on the ideas of Shimoyama and Yokoyama (implementation |
---|
14 | by Wolfram Decker and Hans Schoenemann). |
---|
15 | @* The procedures are implemented to be used in characteristic 0. |
---|
16 | @* They also work in positive characteristic >> 0. |
---|
17 | @* In small characteristic and for algebraic extensions, primdecGTZ |
---|
18 | may not terminate. |
---|
19 | Algorithms for the computation of the radical based on the ideas of |
---|
20 | Krick, Logar and Kemper (implementation by Gerhard Pfister). |
---|
21 | |
---|
22 | PROCEDURES: |
---|
23 | Ann(M); annihilator of R^n/M, R=basering, M in R^n |
---|
24 | primdecGTZ(I); complete primary decomposition via Gianni,Trager,Zacharias |
---|
25 | primdecSY(I...); complete primary decomposition via Shimoyama-Yokoyama |
---|
26 | minAssGTZ(I); the minimal associated primes via Gianni,Trager,Zacharias |
---|
27 | minAssChar(I...); the minimal associated primes using characteristic sets |
---|
28 | testPrimary(L,k); tests the result of the primary decomposition |
---|
29 | radical(I); computes the radical of I via Krick/Logar and Kemper |
---|
30 | radicalEHV(I); computes the radical of I via Eisenbud,Huneke,Vasconcelos |
---|
31 | equiRadical(I); the radical of the equidimensional part of the ideal I |
---|
32 | prepareAss(I); list of radicals of the equidimensional components of I |
---|
33 | equidim(I); weak equidimensional decomposition of I |
---|
34 | equidimMax(I); equidimensional locus of I |
---|
35 | equidimMaxEHV(I); equidimensional locus of I via Eisenbud,Huneke,Vasconcelos |
---|
36 | zerodec(I); zerodimensional decomposition via Monico |
---|
37 | "; |
---|
38 | |
---|
39 | LIB "general.lib"; |
---|
40 | LIB "elim.lib"; |
---|
41 | LIB "poly.lib"; |
---|
42 | LIB "random.lib"; |
---|
43 | LIB "inout.lib"; |
---|
44 | LIB "matrix.lib"; |
---|
45 | LIB "triang.lib"; |
---|
46 | /////////////////////////////////////////////////////////////////////////////// |
---|
47 | // |
---|
48 | // Gianni/Trager/Zacharias |
---|
49 | // |
---|
50 | /////////////////////////////////////////////////////////////////////////////// |
---|
51 | |
---|
52 | static proc sat1 (ideal id, poly p) |
---|
53 | "USAGE: sat1(id,j); id ideal, j polynomial |
---|
54 | RETURN: saturation of id with respect to j (= union_(k=1...) of id:j^k) |
---|
55 | NOTE: result is a std basis in the basering |
---|
56 | " |
---|
57 | { |
---|
58 | int @k; |
---|
59 | ideal inew=std(id); |
---|
60 | ideal iold; |
---|
61 | intvec op=option(get); |
---|
62 | option(returnSB); |
---|
63 | while(specialIdealsEqual(iold,inew)==0 ) |
---|
64 | { |
---|
65 | iold=inew; |
---|
66 | inew=quotient(iold,p); |
---|
67 | @k++; |
---|
68 | } |
---|
69 | @k--; |
---|
70 | option(set,op); |
---|
71 | list L =inew,p^@k; |
---|
72 | return (L); |
---|
73 | } |
---|
74 | |
---|
75 | /////////////////////////////////////////////////////////////////////////////// |
---|
76 | |
---|
77 | static proc sat2 (ideal id, ideal h) |
---|
78 | "USAGE: sat2(id,j); id ideal, j polynomial |
---|
79 | RETURN: saturation of id with respect to j (= union_(k=1...) of id:j^k) |
---|
80 | NOTE: result is a std basis in the basering |
---|
81 | " |
---|
82 | { |
---|
83 | int @k,@i; |
---|
84 | def @P= basering; |
---|
85 | if(ordstr(basering)[1,2]!="dp") |
---|
86 | { |
---|
87 | execute("ring @Phelp=("+charstr(@P)+"),("+varstr(@P)+"),(C,dp);"); |
---|
88 | ideal inew=std(imap(@P,id)); |
---|
89 | ideal @h=imap(@P,h); |
---|
90 | } |
---|
91 | else |
---|
92 | { |
---|
93 | ideal @h=h; |
---|
94 | ideal inew=std(id); |
---|
95 | } |
---|
96 | ideal fac; |
---|
97 | |
---|
98 | for(@i=1;@i<=ncols(@h);@i++) |
---|
99 | { |
---|
100 | if(deg(@h[@i])>0) |
---|
101 | { |
---|
102 | fac=fac+factorize(@h[@i],1); |
---|
103 | } |
---|
104 | } |
---|
105 | fac=simplify(fac,4); |
---|
106 | poly @f=1; |
---|
107 | if(deg(fac[1])>0) |
---|
108 | { |
---|
109 | ideal iold; |
---|
110 | for(@i=1;@i<=size(fac);@i++) |
---|
111 | { |
---|
112 | @f=@f*fac[@i]; |
---|
113 | } |
---|
114 | intvec op = option(get); |
---|
115 | option(returnSB); |
---|
116 | while(specialIdealsEqual(iold,inew)==0 ) |
---|
117 | { |
---|
118 | iold=inew; |
---|
119 | if(deg(iold[size(iold)])!=1) |
---|
120 | { |
---|
121 | inew=quotient(iold,@f); |
---|
122 | } |
---|
123 | else |
---|
124 | { |
---|
125 | inew=iold; |
---|
126 | } |
---|
127 | @k++; |
---|
128 | } |
---|
129 | option(set,op); |
---|
130 | @k--; |
---|
131 | } |
---|
132 | |
---|
133 | if(ordstr(@P)[1,2]!="dp") |
---|
134 | { |
---|
135 | setring @P; |
---|
136 | ideal inew=std(imap(@Phelp,inew)); |
---|
137 | poly @f=imap(@Phelp,@f); |
---|
138 | } |
---|
139 | list L =inew,@f^@k; |
---|
140 | return (L); |
---|
141 | } |
---|
142 | |
---|
143 | /////////////////////////////////////////////////////////////////////////////// |
---|
144 | |
---|
145 | |
---|
146 | proc minSat(ideal inew, ideal h) |
---|
147 | { |
---|
148 | int i,k; |
---|
149 | poly f=1; |
---|
150 | ideal iold,fac; |
---|
151 | list quotM,l; |
---|
152 | |
---|
153 | for(i=1;i<=ncols(h);i++) |
---|
154 | { |
---|
155 | if(deg(h[i])>0) |
---|
156 | { |
---|
157 | fac=fac+factorize(h[i],1); |
---|
158 | } |
---|
159 | } |
---|
160 | fac=simplify(fac,4); |
---|
161 | if(size(fac)==0) |
---|
162 | { |
---|
163 | l=inew,1; |
---|
164 | return(l); |
---|
165 | } |
---|
166 | fac=sort(fac)[1]; |
---|
167 | for(i=1;i<=size(fac);i++) |
---|
168 | { |
---|
169 | f=f*fac[i]; |
---|
170 | } |
---|
171 | quotM[1]=inew; |
---|
172 | quotM[2]=fac; |
---|
173 | quotM[3]=f; |
---|
174 | f=1; |
---|
175 | intvec op = option(get); |
---|
176 | option(returnSB); |
---|
177 | while(specialIdealsEqual(iold,quotM[1])==0) |
---|
178 | { |
---|
179 | if(k>0) |
---|
180 | { |
---|
181 | f=f*quotM[3]; |
---|
182 | } |
---|
183 | iold=quotM[1]; |
---|
184 | quotM=quotMin(quotM); |
---|
185 | k++; |
---|
186 | } |
---|
187 | option(set,op); |
---|
188 | l=quotM[1],f; |
---|
189 | return(l); |
---|
190 | } |
---|
191 | |
---|
192 | static proc quotMin(list tsil) |
---|
193 | { |
---|
194 | int i,j,k,action; |
---|
195 | ideal verg; |
---|
196 | list l; |
---|
197 | poly g; |
---|
198 | |
---|
199 | ideal laedi=tsil[1]; |
---|
200 | ideal fac=tsil[2]; |
---|
201 | poly f=tsil[3]; |
---|
202 | |
---|
203 | ideal star=quotient(laedi,f); |
---|
204 | |
---|
205 | if(specialIdealsEqual(star,laedi)) |
---|
206 | { |
---|
207 | l=star,fac,f; |
---|
208 | return(l); |
---|
209 | } |
---|
210 | |
---|
211 | action=1; |
---|
212 | |
---|
213 | while(action==1) |
---|
214 | { |
---|
215 | if(size(fac)==1) |
---|
216 | { |
---|
217 | action=0; |
---|
218 | break; |
---|
219 | } |
---|
220 | for(i=1;i<=size(fac);i++) |
---|
221 | { |
---|
222 | g=1; |
---|
223 | verg=laedi; |
---|
224 | |
---|
225 | for(j=1;j<=size(fac);j++) |
---|
226 | { |
---|
227 | if(i!=j) |
---|
228 | { |
---|
229 | g=g*fac[j]; |
---|
230 | } |
---|
231 | } |
---|
232 | |
---|
233 | verg=quotient(laedi,g); |
---|
234 | |
---|
235 | if(specialIdealsEqual(verg,star)==1) |
---|
236 | { |
---|
237 | f=g; |
---|
238 | fac[i]=0; |
---|
239 | fac=simplify(fac,2); |
---|
240 | break; |
---|
241 | } |
---|
242 | if(i==size(fac)) |
---|
243 | { |
---|
244 | action=0; |
---|
245 | } |
---|
246 | } |
---|
247 | } |
---|
248 | l=star,fac,f; |
---|
249 | return(l); |
---|
250 | } |
---|
251 | |
---|
252 | /////////////////////////////////////////////////////////////////////////////// |
---|
253 | |
---|
254 | static proc testFactor(list act,poly p) |
---|
255 | { |
---|
256 | poly keep=p; |
---|
257 | |
---|
258 | int i; |
---|
259 | poly q=act[1][1]^act[2][1]; |
---|
260 | for(i=2;i<=size(act[1]);i++) |
---|
261 | { |
---|
262 | q=q*act[1][i]^act[2][i]; |
---|
263 | } |
---|
264 | q=1/leadcoef(q)*q; |
---|
265 | p=1/leadcoef(p)*p; |
---|
266 | if(p-q!=0) |
---|
267 | { |
---|
268 | "ERROR IN FACTOR, please inform the authors"; |
---|
269 | } |
---|
270 | } |
---|
271 | /////////////////////////////////////////////////////////////////////////////// |
---|
272 | |
---|
273 | static proc factor(poly p) |
---|
274 | "USAGE: factor(p) p poly |
---|
275 | RETURN: list=; |
---|
276 | NOTE: |
---|
277 | EXAMPLE: example factor; shows an example |
---|
278 | " |
---|
279 | { |
---|
280 | |
---|
281 | ideal @i; |
---|
282 | list @l; |
---|
283 | intvec @v,@w; |
---|
284 | int @j,@k,@n; |
---|
285 | |
---|
286 | if(deg(p)<=1) |
---|
287 | { |
---|
288 | @i=ideal(p); |
---|
289 | @v=1; |
---|
290 | @l[1]=@i; |
---|
291 | @l[2]=@v; |
---|
292 | return(@l); |
---|
293 | } |
---|
294 | if (size(p)==1) |
---|
295 | { |
---|
296 | @w=leadexp(p); |
---|
297 | for(@j=1;@j<=nvars(basering);@j++) |
---|
298 | { |
---|
299 | if(@w[@j]!=0) |
---|
300 | { |
---|
301 | @k++; |
---|
302 | @v[@k]=@w[@j]; |
---|
303 | @i=@i+ideal(var(@j)); |
---|
304 | } |
---|
305 | } |
---|
306 | @l[1]=@i; |
---|
307 | @l[2]=@v; |
---|
308 | return(@l); |
---|
309 | } |
---|
310 | // @l=factorize(p,2); |
---|
311 | @l=factorize(p); |
---|
312 | // if(npars(basering)>0) |
---|
313 | // { |
---|
314 | for(@j=1;@j<=size(@l[1]);@j++) |
---|
315 | { |
---|
316 | if(deg(@l[1][@j])==0) |
---|
317 | { |
---|
318 | @n++; |
---|
319 | } |
---|
320 | } |
---|
321 | if(@n>0) |
---|
322 | { |
---|
323 | if(@n==size(@l[1])) |
---|
324 | { |
---|
325 | @l[1]=ideal(1); |
---|
326 | @v=1; |
---|
327 | @l[2]=@v; |
---|
328 | } |
---|
329 | else |
---|
330 | { |
---|
331 | @k=0; |
---|
332 | int pleh; |
---|
333 | for(@j=1;@j<=size(@l[1]);@j++) |
---|
334 | { |
---|
335 | if(deg(@l[1][@j])!=0) |
---|
336 | { |
---|
337 | @k++; |
---|
338 | @i=@i+ideal(@l[1][@j]); |
---|
339 | if(size(@i)==pleh) |
---|
340 | { |
---|
341 | "//factorization error"; |
---|
342 | @l; |
---|
343 | @k--; |
---|
344 | @v[@k]=@v[@k]+@l[2][@j]; |
---|
345 | } |
---|
346 | else |
---|
347 | { |
---|
348 | pleh++; |
---|
349 | @v[@k]=@l[2][@j]; |
---|
350 | } |
---|
351 | } |
---|
352 | } |
---|
353 | @l[1]=@i; |
---|
354 | @l[2]=@v; |
---|
355 | } |
---|
356 | } |
---|
357 | // } |
---|
358 | return(@l); |
---|
359 | } |
---|
360 | example |
---|
361 | { "EXAMPLE:"; echo = 2; |
---|
362 | ring r = 0,(x,y,z),lp; |
---|
363 | poly p = (x+y)^2*(y-z)^3; |
---|
364 | list l = factor(p); |
---|
365 | l; |
---|
366 | ring r1 =(0,b,d,f,g),(a,c,e),lp; |
---|
367 | poly p =(1*d)*e^2+(1*d*f^2*g); |
---|
368 | list l = factor(p); |
---|
369 | l; |
---|
370 | ring r2 =(0,b,f,g),(a,c,e,d),lp; |
---|
371 | poly p =(1*d)*e^2+(1*d*f^2*g); |
---|
372 | list l = factor(p); |
---|
373 | l; |
---|
374 | } |
---|
375 | |
---|
376 | /////////////////////////////////////////////////////////////////////////////// |
---|
377 | |
---|
378 | proc idealsEqual( ideal k, ideal j) |
---|
379 | { |
---|
380 | return(stdIdealsEqual(std(k),std(j))); |
---|
381 | } |
---|
382 | |
---|
383 | static proc specialIdealsEqual( ideal k1, ideal k2) |
---|
384 | { |
---|
385 | int j; |
---|
386 | |
---|
387 | if(size(k1)==size(k2)) |
---|
388 | { |
---|
389 | for(j=1;j<=size(k1);j++) |
---|
390 | { |
---|
391 | if(leadexp(k1[j])!=leadexp(k2[j])) |
---|
392 | { |
---|
393 | return(0); |
---|
394 | } |
---|
395 | } |
---|
396 | return(1); |
---|
397 | } |
---|
398 | return(0); |
---|
399 | } |
---|
400 | |
---|
401 | static proc stdIdealsEqual( ideal k1, ideal k2) |
---|
402 | { |
---|
403 | int j; |
---|
404 | |
---|
405 | if(size(k1)==size(k2)) |
---|
406 | { |
---|
407 | for(j=1;j<=size(k1);j++) |
---|
408 | { |
---|
409 | if(leadexp(k1[j])!=leadexp(k2[j])) |
---|
410 | { |
---|
411 | return(0); |
---|
412 | } |
---|
413 | } |
---|
414 | attrib(k2,"isSB",1); |
---|
415 | if(size(reduce(k1,k2,1))==0) |
---|
416 | { |
---|
417 | return(1); |
---|
418 | } |
---|
419 | } |
---|
420 | return(0); |
---|
421 | } |
---|
422 | /////////////////////////////////////////////////////////////////////////////// |
---|
423 | |
---|
424 | proc primaryTest (ideal i, poly p) |
---|
425 | { |
---|
426 | int m=1; |
---|
427 | int n=nvars(basering); |
---|
428 | int e,f; |
---|
429 | poly t; |
---|
430 | ideal h; |
---|
431 | list act; |
---|
432 | |
---|
433 | ideal prm=p; |
---|
434 | attrib(prm,"isSB",1); |
---|
435 | |
---|
436 | while (n>1) |
---|
437 | { |
---|
438 | n=n-1; |
---|
439 | m=m+1; |
---|
440 | |
---|
441 | //search for i[m] which has a power of var(n) as leading term |
---|
442 | if (n==1) |
---|
443 | { |
---|
444 | m=size(i); |
---|
445 | } |
---|
446 | else |
---|
447 | { |
---|
448 | while (lead(i[m])/var(n-1)==0) |
---|
449 | { |
---|
450 | m=m+1; |
---|
451 | } |
---|
452 | m=m-1; |
---|
453 | } |
---|
454 | //check whether i[m] =(c*var(n)+h)^e modulo prm for some |
---|
455 | //h in K[var(n+1),...,var(nvars(basering))], c in K |
---|
456 | //if not (0) is returned, else var(n)+h is added to prm |
---|
457 | |
---|
458 | e=deg(lead(i[m])); |
---|
459 | if(char(basering)!=0) |
---|
460 | { |
---|
461 | f=1; |
---|
462 | if(e mod char(basering)==0) |
---|
463 | { |
---|
464 | if ( voice >=15 ) |
---|
465 | { |
---|
466 | "// WARNING: The characteristic is perhaps too small to use"; |
---|
467 | "// the algorithm of Gianni/Trager/Zacharias."; |
---|
468 | "// This may result in an infinte loop"; |
---|
469 | "// loop in primaryTest, voice:",voice;""; |
---|
470 | } |
---|
471 | while(e mod char(basering)==0) |
---|
472 | { |
---|
473 | f=f*char(basering); |
---|
474 | e=e/char(basering); |
---|
475 | } |
---|
476 | |
---|
477 | } |
---|
478 | t=leadcoef(i[m])*e*var(n)^f+(i[m]-lead(i[m]))/var(n)^((e-1)*f); |
---|
479 | i[m]=poly(e)^e*leadcoef(i[m])^(e-1)*i[m]; |
---|
480 | if (reduce(i[m]-t^e,prm,1) !=0) |
---|
481 | { |
---|
482 | return(ideal(0)); |
---|
483 | } |
---|
484 | if(f>1) |
---|
485 | { |
---|
486 | act=factorize(t); |
---|
487 | if(size(act[1])>2) |
---|
488 | { |
---|
489 | return(ideal(0)); |
---|
490 | } |
---|
491 | if(deg(act[1][2])>1) |
---|
492 | { |
---|
493 | return(ideal(0)); |
---|
494 | } |
---|
495 | t=act[1][2]; |
---|
496 | } |
---|
497 | } |
---|
498 | else |
---|
499 | { |
---|
500 | t=leadcoef(i[m])*e*var(n)+(i[m]-lead(i[m]))/var(n)^(e-1); |
---|
501 | i[m]=poly(e)^e*leadcoef(i[m])^(e-1)*i[m]; |
---|
502 | if (reduce(i[m]-t^e,prm,1) !=0) |
---|
503 | { |
---|
504 | return(ideal(0)); |
---|
505 | } |
---|
506 | } |
---|
507 | |
---|
508 | h=interred(t); |
---|
509 | t=h[1]; |
---|
510 | |
---|
511 | prm = prm,t; |
---|
512 | attrib(prm,"isSB",1); |
---|
513 | } |
---|
514 | return(prm); |
---|
515 | } |
---|
516 | |
---|
517 | /////////////////////////////////////////////////////////////////////////////// |
---|
518 | proc gcdTest(ideal act) |
---|
519 | { |
---|
520 | int i,j; |
---|
521 | if(size(act)<=1) |
---|
522 | { |
---|
523 | return(0); |
---|
524 | } |
---|
525 | for (i=1;i<=size(act)-1;i++) |
---|
526 | { |
---|
527 | for(j=i+1;j<=size(act);j++) |
---|
528 | { |
---|
529 | if(deg(std(ideal(act[i],act[j]))[1])>0) |
---|
530 | { |
---|
531 | return(0); |
---|
532 | } |
---|
533 | } |
---|
534 | } |
---|
535 | return(1); |
---|
536 | } |
---|
537 | |
---|
538 | /////////////////////////////////////////////////////////////////////////////// |
---|
539 | static proc splitPrimary(list l,ideal ser,int @wr,list sact) |
---|
540 | { |
---|
541 | int i,j,k,s,r,w; |
---|
542 | list keepresult,act,keepprime; |
---|
543 | poly @f; |
---|
544 | int sl=size(l); |
---|
545 | for(i=1;i<=sl/2;i++) |
---|
546 | { |
---|
547 | if(sact[2][i]>1) |
---|
548 | { |
---|
549 | keepprime[i]=l[2*i-1]+ideal(sact[1][i]); |
---|
550 | } |
---|
551 | else |
---|
552 | { |
---|
553 | keepprime[i]=l[2*i-1]; |
---|
554 | } |
---|
555 | } |
---|
556 | i=0; |
---|
557 | while(i<size(l)/2) |
---|
558 | { |
---|
559 | i++; |
---|
560 | if((size(ser)>0)&&(size(reduce(ser,l[2*i-1],1))==0)) |
---|
561 | { |
---|
562 | l[2*i-1]=ideal(1); |
---|
563 | l[2*i]=ideal(1); |
---|
564 | continue; |
---|
565 | } |
---|
566 | |
---|
567 | if(size(l[2*i])==0) |
---|
568 | { |
---|
569 | if(homog(l[2*i-1])==1) |
---|
570 | { |
---|
571 | l[2*i]=maxideal(1); |
---|
572 | continue; |
---|
573 | } |
---|
574 | j=0; |
---|
575 | if(i<=sl/2) |
---|
576 | { |
---|
577 | j=1; |
---|
578 | } |
---|
579 | while(j<size(l[2*i-1])) |
---|
580 | { |
---|
581 | j++; |
---|
582 | act=factor(l[2*i-1][j]); |
---|
583 | r=size(act[1]); |
---|
584 | attrib(l[2*i-1],"isSB",1); |
---|
585 | if((r==1)&&(vdim(l[2*i-1])==deg(l[2*i-1][j]))) |
---|
586 | { |
---|
587 | l[2*i]=std(l[2*i-1],act[1][1]); |
---|
588 | break; |
---|
589 | } |
---|
590 | if((r==1)&&(act[2][1]>1)) |
---|
591 | { |
---|
592 | keepprime[i]=interred(keepprime[i]+ideal(act[1][1])); |
---|
593 | if(homog(keepprime[i])==1) |
---|
594 | { |
---|
595 | l[2*i]=maxideal(1); |
---|
596 | break; |
---|
597 | } |
---|
598 | } |
---|
599 | if(gcdTest(act[1])==1) |
---|
600 | { |
---|
601 | for(k=2;k<=r;k++) |
---|
602 | { |
---|
603 | keepprime[size(l)/2+k-1]=interred(keepprime[i]+ideal(act[1][k])); |
---|
604 | } |
---|
605 | keepprime[i]=interred(keepprime[i]+ideal(act[1][1])); |
---|
606 | for(k=1;k<=r;k++) |
---|
607 | { |
---|
608 | if(@wr==0) |
---|
609 | { |
---|
610 | keepresult[k]=std(l[2*i-1],act[1][k]^act[2][k]); |
---|
611 | } |
---|
612 | else |
---|
613 | { |
---|
614 | keepresult[k]=std(l[2*i-1],act[1][k]); |
---|
615 | } |
---|
616 | } |
---|
617 | l[2*i-1]=keepresult[1]; |
---|
618 | if(vdim(keepresult[1])==deg(act[1][1])) |
---|
619 | { |
---|
620 | l[2*i]=keepresult[1]; |
---|
621 | } |
---|
622 | if((homog(keepresult[1])==1)||(homog(keepprime[i])==1)) |
---|
623 | { |
---|
624 | l[2*i]=maxideal(1); |
---|
625 | } |
---|
626 | s=size(l)-2; |
---|
627 | for(k=2;k<=r;k++) |
---|
628 | { |
---|
629 | l[s+2*k-1]=keepresult[k]; |
---|
630 | keepprime[s/2+k]=interred(keepresult[k]+ideal(act[1][k])); |
---|
631 | if(vdim(keepresult[k])==deg(act[1][k])) |
---|
632 | { |
---|
633 | l[s+2*k]=keepresult[k]; |
---|
634 | } |
---|
635 | else |
---|
636 | { |
---|
637 | l[s+2*k]=ideal(0); |
---|
638 | } |
---|
639 | if((homog(keepresult[k])==1)||(homog(keepprime[s/2+k])==1)) |
---|
640 | { |
---|
641 | l[s+2*k]=maxideal(1); |
---|
642 | } |
---|
643 | } |
---|
644 | i--; |
---|
645 | break; |
---|
646 | } |
---|
647 | if(r>=2) |
---|
648 | { |
---|
649 | s=size(l); |
---|
650 | @f=act[1][1]; |
---|
651 | act=sat1(l[2*i-1],act[1][1]); |
---|
652 | if(deg(act[1][1])>0) |
---|
653 | { |
---|
654 | l[s+1]=std(l[2*i-1],act[2]); |
---|
655 | if(homog(l[s+1])==1) |
---|
656 | { |
---|
657 | l[s+2]=maxideal(1); |
---|
658 | } |
---|
659 | else |
---|
660 | { |
---|
661 | l[s+2]=ideal(0); |
---|
662 | } |
---|
663 | keepprime[s/2+1]=interred(keepprime[i]+ideal(@f)); |
---|
664 | if(homog(keepprime[s/2+1])==1) |
---|
665 | { |
---|
666 | l[s+2]=maxideal(1); |
---|
667 | } |
---|
668 | keepprime[i]=act[1]; |
---|
669 | l[2*i-1]=act[1]; |
---|
670 | attrib(l[2*i-1],"isSB",1); |
---|
671 | if(homog(l[2*i-1])==1) |
---|
672 | { |
---|
673 | l[2*i]=maxideal(1); |
---|
674 | } |
---|
675 | |
---|
676 | i--; |
---|
677 | break; |
---|
678 | } |
---|
679 | } |
---|
680 | } |
---|
681 | } |
---|
682 | } |
---|
683 | if(sl==size(l)) |
---|
684 | { |
---|
685 | return(l); |
---|
686 | } |
---|
687 | for(i=1;i<=size(l)/2;i++) |
---|
688 | { |
---|
689 | attrib(l[2*i-1],"isSB",1); |
---|
690 | |
---|
691 | if((size(ser)>0)&&(size(reduce(ser,l[2*i-1],1))==0)&&(deg(l[2*i-1][1])>0)) |
---|
692 | { |
---|
693 | "Achtung in split"; |
---|
694 | |
---|
695 | l[2*i-1]=ideal(1); |
---|
696 | l[2*i]=ideal(1); |
---|
697 | } |
---|
698 | if((size(l[2*i])==0)&&(specialIdealsEqual(keepprime[i],l[2*i-1])!=1)) |
---|
699 | { |
---|
700 | keepprime[i]=std(keepprime[i]); |
---|
701 | if(homog(keepprime[i])==1) |
---|
702 | { |
---|
703 | l[2*i]=maxideal(1); |
---|
704 | } |
---|
705 | else |
---|
706 | { |
---|
707 | act=zero_decomp(keepprime[i],ideal(0),@wr,1); |
---|
708 | if(size(act)==2) |
---|
709 | { |
---|
710 | l[2*i]=act[2]; |
---|
711 | } |
---|
712 | } |
---|
713 | } |
---|
714 | } |
---|
715 | return(l); |
---|
716 | } |
---|
717 | example |
---|
718 | { "EXAMPLE:"; echo=2; |
---|
719 | ring r = 32003,(x,y,z),lp; |
---|
720 | ideal i1=x*(x+1),yz,(z+1)*(z-1); |
---|
721 | ideal i2=xy,yz,(x-2)*(x+3); |
---|
722 | list l=i1,ideal(0),i2,ideal(0),i2,ideal(1); |
---|
723 | list l1=splitPrimary(l,ideal(0),0); |
---|
724 | l1; |
---|
725 | } |
---|
726 | /////////////////////////////////////////////////////////////////////////////// |
---|
727 | static proc splitCharp(list l) |
---|
728 | { |
---|
729 | if((char(basering)==0)||(npars(basering)>0)) |
---|
730 | { |
---|
731 | return(l); |
---|
732 | } |
---|
733 | def P=basering; |
---|
734 | int i,j,k,m,q,d,o; |
---|
735 | int n=nvars(basering); |
---|
736 | ideal s,t,u,sact; |
---|
737 | poly ni; |
---|
738 | string minp,gnir,va; |
---|
739 | list sa,keep,rp,keep1; |
---|
740 | for(i=1;i<=size(l)/2;i++) |
---|
741 | { |
---|
742 | if(size(l[2*i])==0) |
---|
743 | { |
---|
744 | if(deg(l[2*i-1][1])==vdim(l[2*i-1])) |
---|
745 | { |
---|
746 | l[2*i]=l[2*i-1]; |
---|
747 | } |
---|
748 | } |
---|
749 | } |
---|
750 | for(i=1;i<=size(l)/2;i++) |
---|
751 | { |
---|
752 | if(size(l[2*i])==0) |
---|
753 | { |
---|
754 | s=factorize(l[2*i-1][1],1); //vermeiden!!! |
---|
755 | t=l[2*i-1]; |
---|
756 | m=size(t); |
---|
757 | ni=s[1]; |
---|
758 | if(deg(ni)>1) |
---|
759 | { |
---|
760 | va=varstr(P); |
---|
761 | j=size(va); |
---|
762 | while(va[j]!=","){j--;} |
---|
763 | va=va[1..j-1]; |
---|
764 | gnir="ring RL=("+string(char(P))+","+string(var(n))+"),("+va+"),lp;"; |
---|
765 | execute(gnir); |
---|
766 | minpoly=leadcoef(imap(P,ni)); |
---|
767 | ideal act; |
---|
768 | ideal t=imap(P,t); |
---|
769 | |
---|
770 | for(k=2;k<=m;k++) |
---|
771 | { |
---|
772 | act=factorize(t[k],1); |
---|
773 | if(size(act)>1){break;} |
---|
774 | } |
---|
775 | setring P; |
---|
776 | sact=imap(RL,act); |
---|
777 | |
---|
778 | if(size(sact)>1) |
---|
779 | { |
---|
780 | sa=sat1(l[2*i-1],sact[1]); |
---|
781 | keep[size(keep)+1]=std(l[2*i-1],sa[2]); |
---|
782 | l[2*i-1]=std(sa[1]); |
---|
783 | l[2*i]=primaryTest(sa[1],sa[1][1]); |
---|
784 | } |
---|
785 | if((size(sact)==1)&&(m==2)) |
---|
786 | { |
---|
787 | l[2*i]=l[2*i-1]; |
---|
788 | attrib(l[2*i],"isSB",1); |
---|
789 | } |
---|
790 | if((size(sact)==1)&&(m>2)) |
---|
791 | { |
---|
792 | setring RL; |
---|
793 | option(redSB); |
---|
794 | t=std(t); |
---|
795 | |
---|
796 | list sp=zero_decomp(t,0,0); |
---|
797 | |
---|
798 | setring P; |
---|
799 | rp=imap(RL,sp); |
---|
800 | for(o=1;o<=size(rp);o++) |
---|
801 | { |
---|
802 | rp[o]=interred(simplify(rp[o],1)+ideal(ni)); |
---|
803 | } |
---|
804 | l[2*i-1]=rp[1]; |
---|
805 | l[2*i]=rp[2]; |
---|
806 | rp=delete(rp,1); |
---|
807 | rp=delete(rp,1); |
---|
808 | keep1=keep1+rp; |
---|
809 | option(noredSB); |
---|
810 | } |
---|
811 | kill RL; |
---|
812 | } |
---|
813 | } |
---|
814 | } |
---|
815 | if(size(keep)>0) |
---|
816 | { |
---|
817 | for(i=1;i<=size(keep);i++) |
---|
818 | { |
---|
819 | if(deg(keep[i][1])>0) |
---|
820 | { |
---|
821 | l[size(l)+1]=keep[i]; |
---|
822 | l[size(l)+1]=primaryTest(keep[i],keep[i][1]); |
---|
823 | } |
---|
824 | } |
---|
825 | } |
---|
826 | l=l+keep1; |
---|
827 | return(l); |
---|
828 | } |
---|
829 | |
---|
830 | /////////////////////////////////////////////////////////////////////////////// |
---|
831 | |
---|
832 | proc zero_decomp (ideal j,ideal ser,int @wr,list #) |
---|
833 | "USAGE: zero_decomp(j,ser,@wr); j,ser ideals, @wr=0 or 1 |
---|
834 | (@wr=0 for primary decomposition, @wr=1 for computaion of associated |
---|
835 | primes) |
---|
836 | RETURN: list = list of primary ideals and their radicals (at even positions |
---|
837 | in the list) if the input is zero-dimensional and a standardbases |
---|
838 | with respect to lex-ordering |
---|
839 | If ser!=(0) and ser is contained in j or if j is not zero-dimen- |
---|
840 | sional then ideal(1),ideal(1) is returned |
---|
841 | NOTE: Algorithm of Gianni/Trager/Zacharias |
---|
842 | EXAMPLE: example zero_decomp; shows an example |
---|
843 | " |
---|
844 | { |
---|
845 | def @P = basering; |
---|
846 | int uytrewq; |
---|
847 | int nva = nvars(basering); |
---|
848 | int @k,@s,@n,@k1,zz; |
---|
849 | list primary,lres0,lres1,act,@lh,@wh; |
---|
850 | map phi,psi,phi1,psi1; |
---|
851 | ideal jmap,jmap1,jmap2,helpprim,@qh,@qht,ser1; |
---|
852 | intvec @vh,@hilb; |
---|
853 | string @ri; |
---|
854 | poly @f; |
---|
855 | if (dim(j)>0) |
---|
856 | { |
---|
857 | primary[1]=ideal(1); |
---|
858 | primary[2]=ideal(1); |
---|
859 | return(primary); |
---|
860 | } |
---|
861 | j=interred(j); |
---|
862 | |
---|
863 | attrib(j,"isSB",1); |
---|
864 | |
---|
865 | if(vdim(j)==deg(j[1])) |
---|
866 | { |
---|
867 | act=factor(j[1]); |
---|
868 | for(@k=1;@k<=size(act[1]);@k++) |
---|
869 | { |
---|
870 | @qh=j; |
---|
871 | if(@wr==0) |
---|
872 | { |
---|
873 | @qh[1]=act[1][@k]^act[2][@k]; |
---|
874 | } |
---|
875 | else |
---|
876 | { |
---|
877 | @qh[1]=act[1][@k]; |
---|
878 | } |
---|
879 | primary[2*@k-1]=interred(@qh); |
---|
880 | @qh=j; |
---|
881 | @qh[1]=act[1][@k]; |
---|
882 | primary[2*@k]=interred(@qh); |
---|
883 | attrib( primary[2*@k-1],"isSB",1); |
---|
884 | |
---|
885 | if((size(ser)>0)&&(size(reduce(ser,primary[2*@k-1],1))==0)) |
---|
886 | { |
---|
887 | primary[2*@k-1]=ideal(1); |
---|
888 | primary[2*@k]=ideal(1); |
---|
889 | } |
---|
890 | } |
---|
891 | return(primary); |
---|
892 | } |
---|
893 | |
---|
894 | if(homog(j)==1) |
---|
895 | { |
---|
896 | primary[1]=j; |
---|
897 | if((size(ser)>0)&&(size(reduce(ser,j,1))==0)) |
---|
898 | { |
---|
899 | primary[1]=ideal(1); |
---|
900 | primary[2]=ideal(1); |
---|
901 | return(primary); |
---|
902 | } |
---|
903 | if(dim(j)==-1) |
---|
904 | { |
---|
905 | primary[1]=ideal(1); |
---|
906 | primary[2]=ideal(1); |
---|
907 | } |
---|
908 | else |
---|
909 | { |
---|
910 | primary[2]=maxideal(1); |
---|
911 | } |
---|
912 | return(primary); |
---|
913 | } |
---|
914 | |
---|
915 | //the first element in the standardbase is factorized |
---|
916 | if(deg(j[1])>0) |
---|
917 | { |
---|
918 | act=factor(j[1]); |
---|
919 | testFactor(act,j[1]); |
---|
920 | } |
---|
921 | else |
---|
922 | { |
---|
923 | primary[1]=ideal(1); |
---|
924 | primary[2]=ideal(1); |
---|
925 | return(primary); |
---|
926 | } |
---|
927 | |
---|
928 | //with the factors new ideals (hopefully the primary decomposition) |
---|
929 | //are created |
---|
930 | if(size(act[1])>1) |
---|
931 | { |
---|
932 | if(size(#)>1) |
---|
933 | { |
---|
934 | primary[1]=ideal(1); |
---|
935 | primary[2]=ideal(1); |
---|
936 | primary[3]=ideal(1); |
---|
937 | primary[4]=ideal(1); |
---|
938 | return(primary); |
---|
939 | } |
---|
940 | for(@k=1;@k<=size(act[1]);@k++) |
---|
941 | { |
---|
942 | if(@wr==0) |
---|
943 | { |
---|
944 | primary[2*@k-1]=std(j,act[1][@k]^act[2][@k]); |
---|
945 | |
---|
946 | } |
---|
947 | else |
---|
948 | { |
---|
949 | primary[2*@k-1]=std(j,act[1][@k]); |
---|
950 | } |
---|
951 | if((act[2][@k]==1)&&(vdim(primary[2*@k-1])==deg(act[1][@k]))) |
---|
952 | { |
---|
953 | primary[2*@k] = primary[2*@k-1]; |
---|
954 | } |
---|
955 | else |
---|
956 | { |
---|
957 | |
---|
958 | primary[2*@k] = primaryTest(primary[2*@k-1],act[1][@k]); |
---|
959 | |
---|
960 | } |
---|
961 | } |
---|
962 | } |
---|
963 | else |
---|
964 | { |
---|
965 | primary[1]=j; |
---|
966 | if((size(#)>0)&&(act[2][1]>1)) |
---|
967 | { |
---|
968 | act[2]=1; |
---|
969 | primary[1]=std(primary[1],act[1][1]); |
---|
970 | } |
---|
971 | |
---|
972 | if((act[2][1]==1)&&(vdim(primary[1])==deg(act[1][1]))) |
---|
973 | { |
---|
974 | primary[2]=primary[1]; |
---|
975 | } |
---|
976 | else |
---|
977 | { |
---|
978 | primary[2]=primaryTest(primary[1],act[1][1]); |
---|
979 | } |
---|
980 | } |
---|
981 | |
---|
982 | if(size(#)==0) |
---|
983 | { |
---|
984 | primary=splitPrimary(primary,ser,@wr,act); |
---|
985 | } |
---|
986 | |
---|
987 | if((voice>=6)&&(char(basering)<=181)) |
---|
988 | { |
---|
989 | primary=splitCharp(primary); |
---|
990 | } |
---|
991 | |
---|
992 | if((@wr==2)&&(npars(basering)>0)&&(voice>=6)&&(char(basering)>0)) |
---|
993 | { |
---|
994 | //the prime decomposition of Yokoyama in characteristic p |
---|
995 | list ke,ek; |
---|
996 | @k=0; |
---|
997 | while(@k<size(primary)/2) |
---|
998 | { |
---|
999 | @k++; |
---|
1000 | if(size(primary[2*@k])==0) |
---|
1001 | { |
---|
1002 | ek=insepDecomp(primary[2*@k-1]); |
---|
1003 | primary=delete(primary,2*@k); |
---|
1004 | primary=delete(primary,2*@k-1); |
---|
1005 | @k--; |
---|
1006 | } |
---|
1007 | ke=ke+ek; |
---|
1008 | } |
---|
1009 | for(@k=1;@k<=size(ke);@k++) |
---|
1010 | { |
---|
1011 | primary[size(primary)+1]=ke[@k]; |
---|
1012 | primary[size(primary)+1]=ke[@k]; |
---|
1013 | } |
---|
1014 | } |
---|
1015 | |
---|
1016 | if(voice>=8){primary=extF(primary)}; |
---|
1017 | |
---|
1018 | //test whether all ideals in the decomposition are primary and |
---|
1019 | //in general position |
---|
1020 | //if not after a random coordinate transformation of the last |
---|
1021 | //variable the corresponding ideal is decomposed again. |
---|
1022 | if((npars(basering)>0)&&(voice>=8)) |
---|
1023 | { |
---|
1024 | poly randp; |
---|
1025 | for(zz=1;zz<nvars(basering);zz++) |
---|
1026 | { |
---|
1027 | randp=randp |
---|
1028 | +(random(0,5)*par(1)^2+random(0,5)*par(1)+random(0,5))*var(zz); |
---|
1029 | } |
---|
1030 | randp=randp+var(nvars(basering)); |
---|
1031 | } |
---|
1032 | @k=0; |
---|
1033 | while(@k<(size(primary)/2)) |
---|
1034 | { |
---|
1035 | @k++; |
---|
1036 | if (size(primary[2*@k])==0) |
---|
1037 | { |
---|
1038 | for(zz=1;zz<size(primary[2*@k-1])-1;zz++) |
---|
1039 | { |
---|
1040 | attrib(primary[2*@k-1],"isSB",1); |
---|
1041 | if(vdim(primary[2*@k-1])==deg(primary[2*@k-1][zz])) |
---|
1042 | { |
---|
1043 | primary[2*@k]=primary[2*@k-1]; |
---|
1044 | } |
---|
1045 | } |
---|
1046 | } |
---|
1047 | } |
---|
1048 | |
---|
1049 | @k=0; |
---|
1050 | ideal keep; |
---|
1051 | while(@k<(size(primary)/2)) |
---|
1052 | { |
---|
1053 | @k++; |
---|
1054 | if (size(primary[2*@k])==0) |
---|
1055 | { |
---|
1056 | |
---|
1057 | jmap=randomLast(100); |
---|
1058 | jmap1=maxideal(1); |
---|
1059 | jmap2=maxideal(1); |
---|
1060 | @qht=primary[2*@k-1]; |
---|
1061 | if((npars(basering)>0)&&(voice>=10)) |
---|
1062 | { |
---|
1063 | jmap[size(jmap)]=randp; |
---|
1064 | } |
---|
1065 | |
---|
1066 | for(@n=2;@n<=size(primary[2*@k-1]);@n++) |
---|
1067 | { |
---|
1068 | if(deg(lead(primary[2*@k-1][@n]))==1) |
---|
1069 | { |
---|
1070 | for(zz=1;zz<=nva;zz++) |
---|
1071 | { |
---|
1072 | if(lead(primary[2*@k-1][@n])/var(zz)!=0) |
---|
1073 | { |
---|
1074 | jmap1[zz]=-1/leadcoef(primary[2*@k-1][@n])*primary[2*@k-1][@n] |
---|
1075 | +2/leadcoef(primary[2*@k-1][@n])*lead(primary[2*@k-1][@n]); |
---|
1076 | jmap2[zz]=primary[2*@k-1][@n]; |
---|
1077 | @qht[@n]=var(zz); |
---|
1078 | |
---|
1079 | } |
---|
1080 | } |
---|
1081 | jmap[nva]=subst(jmap[nva],lead(primary[2*@k-1][@n]),0); |
---|
1082 | } |
---|
1083 | } |
---|
1084 | if(size(subst(jmap[nva],var(1),0)-var(nva))!=0) |
---|
1085 | { |
---|
1086 | // jmap[nva]=subst(jmap[nva],var(1),0); |
---|
1087 | //hier geaendert +untersuchen!!!!!!!!!!!!!! |
---|
1088 | } |
---|
1089 | phi1=@P,jmap1; |
---|
1090 | phi=@P,jmap; |
---|
1091 | for(@n=1;@n<=nva;@n++) |
---|
1092 | { |
---|
1093 | jmap[@n]=-(jmap[@n]-2*var(@n)); |
---|
1094 | } |
---|
1095 | psi=@P,jmap; |
---|
1096 | psi1=@P,jmap2; |
---|
1097 | @qh=phi(@qht); |
---|
1098 | |
---|
1099 | //=================== the new part ============================ |
---|
1100 | |
---|
1101 | @qh=groebner(@qh); |
---|
1102 | |
---|
1103 | //============================================================= |
---|
1104 | // if(npars(@P)>0) |
---|
1105 | // { |
---|
1106 | // @ri= "ring @Phelp =" |
---|
1107 | // +string(char(@P))+", |
---|
1108 | // ("+varstr(@P)+","+parstr(@P)+",@t),(C,dp);"; |
---|
1109 | // } |
---|
1110 | // else |
---|
1111 | // { |
---|
1112 | // @ri= "ring @Phelp =" |
---|
1113 | // +string(char(@P))+",("+varstr(@P)+",@t),(C,dp);"; |
---|
1114 | // } |
---|
1115 | // execute(@ri); |
---|
1116 | // ideal @qh=homog(imap(@P,@qht),@t); |
---|
1117 | // |
---|
1118 | // ideal @qh1=std(@qh); |
---|
1119 | // @hilb=hilb(@qh1,1); |
---|
1120 | // @ri= "ring @Phelp1 =" |
---|
1121 | // +string(char(@P))+",("+varstr(@Phelp)+"),(C,lp);"; |
---|
1122 | // execute(@ri); |
---|
1123 | // ideal @qh=homog(imap(@P,@qh),@t); |
---|
1124 | // kill @Phelp; |
---|
1125 | // @qh=std(@qh,@hilb); |
---|
1126 | // @qh=subst(@qh,@t,1); |
---|
1127 | // setring @P; |
---|
1128 | // @qh=imap(@Phelp1,@qh); |
---|
1129 | // kill @Phelp1; |
---|
1130 | // @qh=clearSB(@qh); |
---|
1131 | // attrib(@qh,"isSB",1); |
---|
1132 | //============================================================= |
---|
1133 | |
---|
1134 | ser1=phi1(ser); |
---|
1135 | |
---|
1136 | @lh=zero_decomp (@qh,phi(ser1),@wr); |
---|
1137 | |
---|
1138 | |
---|
1139 | kill lres0; |
---|
1140 | list lres0; |
---|
1141 | if(size(@lh)==2) |
---|
1142 | { |
---|
1143 | helpprim=@lh[2]; |
---|
1144 | lres0[1]=primary[2*@k-1]; |
---|
1145 | ser1=psi(helpprim); |
---|
1146 | lres0[2]=psi1(ser1); |
---|
1147 | if(size(reduce(lres0[2],lres0[1],1))==0) |
---|
1148 | { |
---|
1149 | primary[2*@k]=primary[2*@k-1]; |
---|
1150 | continue; |
---|
1151 | } |
---|
1152 | } |
---|
1153 | else |
---|
1154 | { |
---|
1155 | |
---|
1156 | lres1=psi(@lh); |
---|
1157 | lres0=psi1(lres1); |
---|
1158 | } |
---|
1159 | |
---|
1160 | //=================== the new part ============================ |
---|
1161 | |
---|
1162 | primary=delete(primary,2*@k-1); |
---|
1163 | primary=delete(primary,2*@k-1); |
---|
1164 | @k--; |
---|
1165 | if(size(lres0)==2) |
---|
1166 | { |
---|
1167 | lres0[2]=groebner(lres0[2]); |
---|
1168 | } |
---|
1169 | else |
---|
1170 | { |
---|
1171 | for(@n=1;@n<=size(lres0)/2;@n++) |
---|
1172 | { |
---|
1173 | if(specialIdealsEqual(lres0[2*@n-1],lres0[2*@n])==1) |
---|
1174 | { |
---|
1175 | lres0[2*@n-1]=groebner(lres0[2*@n-1]); |
---|
1176 | lres0[2*@n]=lres0[2*@n-1]; |
---|
1177 | attrib(lres0[2*@n],"isSB",1); |
---|
1178 | } |
---|
1179 | else |
---|
1180 | { |
---|
1181 | lres0[2*@n-1]=groebner(lres0[2*@n-1]); |
---|
1182 | lres0[2*@n]=groebner(lres0[2*@n]); |
---|
1183 | } |
---|
1184 | } |
---|
1185 | } |
---|
1186 | primary=primary+lres0; |
---|
1187 | |
---|
1188 | //============================================================= |
---|
1189 | // if(npars(@P)>0) |
---|
1190 | // { |
---|
1191 | // @ri= "ring @Phelp =" |
---|
1192 | // +string(char(@P))+", |
---|
1193 | // ("+varstr(@P)+","+parstr(@P)+",@t),(C,dp);"; |
---|
1194 | // } |
---|
1195 | // else |
---|
1196 | // { |
---|
1197 | // @ri= "ring @Phelp =" |
---|
1198 | // +string(char(@P))+",("+varstr(@P)+",@t),(C,dp);"; |
---|
1199 | // } |
---|
1200 | // execute(@ri); |
---|
1201 | // list @lvec; |
---|
1202 | // list @lr=imap(@P,lres0); |
---|
1203 | // ideal @lr1; |
---|
1204 | // |
---|
1205 | // if(size(@lr)==2) |
---|
1206 | // { |
---|
1207 | // @lr[2]=homog(@lr[2],@t); |
---|
1208 | // @lr1=std(@lr[2]); |
---|
1209 | // @lvec[2]=hilb(@lr1,1); |
---|
1210 | // } |
---|
1211 | // else |
---|
1212 | // { |
---|
1213 | // for(@n=1;@n<=size(@lr)/2;@n++) |
---|
1214 | // { |
---|
1215 | // if(specialIdealsEqual(@lr[2*@n-1],@lr[2*@n])==1) |
---|
1216 | // { |
---|
1217 | // @lr[2*@n-1]=homog(@lr[2*@n-1],@t); |
---|
1218 | // @lr1=std(@lr[2*@n-1]); |
---|
1219 | // @lvec[2*@n-1]=hilb(@lr1,1); |
---|
1220 | // @lvec[2*@n]=@lvec[2*@n-1]; |
---|
1221 | // } |
---|
1222 | // else |
---|
1223 | // { |
---|
1224 | // @lr[2*@n-1]=homog(@lr[2*@n-1],@t); |
---|
1225 | // @lr1=std(@lr[2*@n-1]); |
---|
1226 | // @lvec[2*@n-1]=hilb(@lr1,1); |
---|
1227 | // @lr[2*@n]=homog(@lr[2*@n],@t); |
---|
1228 | // @lr1=std(@lr[2*@n]); |
---|
1229 | // @lvec[2*@n]=hilb(@lr1,1); |
---|
1230 | // |
---|
1231 | // } |
---|
1232 | // } |
---|
1233 | // } |
---|
1234 | // @ri= "ring @Phelp1 =" |
---|
1235 | // +string(char(@P))+",("+varstr(@Phelp)+"),(C,lp);"; |
---|
1236 | // execute(@ri); |
---|
1237 | // list @lr=imap(@Phelp,@lr); |
---|
1238 | // |
---|
1239 | // kill @Phelp; |
---|
1240 | // if(size(@lr)==2) |
---|
1241 | // { |
---|
1242 | // @lr[2]=std(@lr[2],@lvec[2]); |
---|
1243 | // @lr[2]=subst(@lr[2],@t,1); |
---|
1244 | // |
---|
1245 | // } |
---|
1246 | // else |
---|
1247 | // { |
---|
1248 | // for(@n=1;@n<=size(@lr)/2;@n++) |
---|
1249 | // { |
---|
1250 | // if(specialIdealsEqual(@lr[2*@n-1],@lr[2*@n])==1) |
---|
1251 | // { |
---|
1252 | // @lr[2*@n-1]=std(@lr[2*@n-1],@lvec[2*@n-1]); |
---|
1253 | // @lr[2*@n-1]=subst(@lr[2*@n-1],@t,1); |
---|
1254 | // @lr[2*@n]=@lr[2*@n-1]; |
---|
1255 | // attrib(@lr[2*@n],"isSB",1); |
---|
1256 | // } |
---|
1257 | // else |
---|
1258 | // { |
---|
1259 | // @lr[2*@n-1]=std(@lr[2*@n-1],@lvec[2*@n-1]); |
---|
1260 | // @lr[2*@n-1]=subst(@lr[2*@n-1],@t,1); |
---|
1261 | // @lr[2*@n]=std(@lr[2*@n],@lvec[2*@n]); |
---|
1262 | // @lr[2*@n]=subst(@lr[2*@n],@t,1); |
---|
1263 | // } |
---|
1264 | // } |
---|
1265 | // } |
---|
1266 | // kill @lvec; |
---|
1267 | // setring @P; |
---|
1268 | // lres0=imap(@Phelp1,@lr); |
---|
1269 | // kill @Phelp1; |
---|
1270 | // for(@n=1;@n<=size(lres0);@n++) |
---|
1271 | // { |
---|
1272 | // lres0[@n]=clearSB(lres0[@n]); |
---|
1273 | // attrib(lres0[@n],"isSB",1); |
---|
1274 | // } |
---|
1275 | // |
---|
1276 | // primary[2*@k-1]=lres0[1]; |
---|
1277 | // primary[2*@k]=lres0[2]; |
---|
1278 | // @s=size(primary)/2; |
---|
1279 | // for(@n=1;@n<=size(lres0)/2-1;@n++) |
---|
1280 | // { |
---|
1281 | // primary[2*@s+2*@n-1]=lres0[2*@n+1]; |
---|
1282 | // primary[2*@s+2*@n]=lres0[2*@n+2]; |
---|
1283 | // } |
---|
1284 | // @k--; |
---|
1285 | //============================================================= |
---|
1286 | } |
---|
1287 | } |
---|
1288 | return(primary); |
---|
1289 | } |
---|
1290 | example |
---|
1291 | { "EXAMPLE:"; echo = 2; |
---|
1292 | ring r = 0,(x,y,z),lp; |
---|
1293 | poly p = z2+1; |
---|
1294 | poly q = z4+2; |
---|
1295 | ideal i = p^2*q^3,(y-z3)^3,(x-yz+z4)^4; |
---|
1296 | i=std(i); |
---|
1297 | list pr= zero_decomp(i,ideal(0),0); |
---|
1298 | pr; |
---|
1299 | } |
---|
1300 | /////////////////////////////////////////////////////////////////////////////// |
---|
1301 | proc extF(list l,list #) |
---|
1302 | { |
---|
1303 | //zero_dimensional primary decomposition after finite field extension |
---|
1304 | def R=basering; |
---|
1305 | int p=char(R); |
---|
1306 | |
---|
1307 | if((p==0)||(p>13)||(npars(R)>0)){return(l);} |
---|
1308 | |
---|
1309 | int ex=3; |
---|
1310 | if(size(#)>0){ex=#[1];} |
---|
1311 | |
---|
1312 | list peek,peek1; |
---|
1313 | while(size(l)>0) |
---|
1314 | { |
---|
1315 | if(size(l[2])==0) |
---|
1316 | { |
---|
1317 | peek[size(peek)+1]=l[1]; |
---|
1318 | } |
---|
1319 | else |
---|
1320 | { |
---|
1321 | peek1[size(peek1)+1]=l[1]; |
---|
1322 | peek1[size(peek1)+1]=l[2]; |
---|
1323 | } |
---|
1324 | l=delete(l,1); |
---|
1325 | l=delete(l,1); |
---|
1326 | } |
---|
1327 | if(size(peek)==0){return(peek1);} |
---|
1328 | |
---|
1329 | string gnir="ring RH=("+string(p)+"^"+string(ex)+",a),("+varstr(R)+"),lp;"; |
---|
1330 | execute(gnir); |
---|
1331 | string mp="minpoly="+string(minpoly)+";"; |
---|
1332 | gnir="ring RL=("+string(p)+",a),("+varstr(R)+"),lp;"; |
---|
1333 | execute(gnir); |
---|
1334 | execute(mp); |
---|
1335 | list L=imap(R,peek); |
---|
1336 | list pr, keep; |
---|
1337 | int i; |
---|
1338 | for(i=1;i<=size(L);i++) |
---|
1339 | { |
---|
1340 | attrib(L[i],"isSB",1); |
---|
1341 | pr=zero_decomp(L[i],0,0); |
---|
1342 | keep=keep+pr; |
---|
1343 | } |
---|
1344 | for(i=1;i<=size(keep);i++) |
---|
1345 | { |
---|
1346 | keep[i]=simplify(keep[i],1); |
---|
1347 | } |
---|
1348 | mp="poly pp="+string(minpoly)+";"; |
---|
1349 | |
---|
1350 | string gnir1="ring RS="+string(p)+",("+varstr(R)+",a),lp;"; |
---|
1351 | execute(gnir1); |
---|
1352 | execute(mp); |
---|
1353 | list L=imap(RL,keep); |
---|
1354 | |
---|
1355 | for(i=1;i<=size(L);i++) |
---|
1356 | { |
---|
1357 | L[i]=eliminate(L[i]+ideal(pp),a); |
---|
1358 | } |
---|
1359 | i=0; |
---|
1360 | int j; |
---|
1361 | while(i<size(L)/2-1) |
---|
1362 | { |
---|
1363 | i++; |
---|
1364 | j=i; |
---|
1365 | while(j<size(L)/2) |
---|
1366 | { |
---|
1367 | j++; |
---|
1368 | if(idealsEqual(L[2*i-1],L[2*j-1])) |
---|
1369 | { |
---|
1370 | L=delete(L,2*j-1); |
---|
1371 | L=delete(L,2*j-1); |
---|
1372 | j--; |
---|
1373 | } |
---|
1374 | } |
---|
1375 | } |
---|
1376 | setring R; |
---|
1377 | list re=imap(RS,L); |
---|
1378 | re=re+peek1; |
---|
1379 | |
---|
1380 | return(extF(re,ex+1)); |
---|
1381 | } |
---|
1382 | |
---|
1383 | /////////////////////////////////////////////////////////////////////////////// |
---|
1384 | proc zeroSp(ideal i) |
---|
1385 | { |
---|
1386 | //preparation for the separable closure |
---|
1387 | //decomposition into ideals of special type |
---|
1388 | //i.e. the minimal polynomials of every variable mod i are irreducible |
---|
1389 | //returns a list of 2 lists: rr=pe,qe |
---|
1390 | //the ideals in pe[l] are special, their special elements are in qe[l] |
---|
1391 | //pe[l] is a dp-Groebnerbasis |
---|
1392 | //the radical of the intersection of the pe[l] is equal to the radical of i |
---|
1393 | |
---|
1394 | def R=basering; |
---|
1395 | |
---|
1396 | //i has to be a reduced groebner basis |
---|
1397 | ideal F=finduni(i); |
---|
1398 | |
---|
1399 | int j,k,l,ready; |
---|
1400 | list fa; |
---|
1401 | fa[1]=factorize(F[1],1); |
---|
1402 | poly te,ti; |
---|
1403 | ideal tj; |
---|
1404 | //avoid factorization of the same polynomial |
---|
1405 | for(j=2;j<=size(F);j++) |
---|
1406 | { |
---|
1407 | for(k=1;k<=j-1;k++) |
---|
1408 | { |
---|
1409 | ti=F[k]; |
---|
1410 | te=subst(ti,var(k),var(j)); |
---|
1411 | if(te==F[j]) |
---|
1412 | { |
---|
1413 | tj=fa[k]; |
---|
1414 | fa[j]=subst(tj,var(k),var(j)); |
---|
1415 | ready=1; |
---|
1416 | break; |
---|
1417 | } |
---|
1418 | } |
---|
1419 | if(!ready) |
---|
1420 | { |
---|
1421 | fa[j]=factorize(F[j],1); |
---|
1422 | } |
---|
1423 | ready=0; |
---|
1424 | } |
---|
1425 | execute( "ring P=("+charstr(R)+"),("+varstr(R)+"),(C,dp);"); |
---|
1426 | ideal i=imap(R,i); |
---|
1427 | if(npars(basering)==0) |
---|
1428 | { |
---|
1429 | ideal J=fglm(R,i); |
---|
1430 | } |
---|
1431 | else |
---|
1432 | { |
---|
1433 | ideal J=groebner(i); |
---|
1434 | } |
---|
1435 | list fa=imap(R,fa); |
---|
1436 | list qe=J; //collects a dp-Groebnerbasis of the special ideals |
---|
1437 | list keep=ideal(0); //collects the special elements |
---|
1438 | |
---|
1439 | list re,em,ke; |
---|
1440 | ideal K,L; |
---|
1441 | |
---|
1442 | for(j=1;j<=nvars(basering);j++) |
---|
1443 | { |
---|
1444 | for(l=1;l<=size(qe);l++) |
---|
1445 | { |
---|
1446 | for(k=1;k<=size(fa[j]);k++) |
---|
1447 | { |
---|
1448 | L=std(qe[l],fa[j][k]); |
---|
1449 | K=keep[l],fa[j][k]; |
---|
1450 | if(deg(L[1])>0) |
---|
1451 | { |
---|
1452 | re[size(re)+1]=L; |
---|
1453 | ke[size(ke)+1]=K; |
---|
1454 | } |
---|
1455 | } |
---|
1456 | } |
---|
1457 | qe=re; |
---|
1458 | re=em; |
---|
1459 | keep=ke; |
---|
1460 | ke=em; |
---|
1461 | } |
---|
1462 | |
---|
1463 | setring R; |
---|
1464 | list qe=imap(P,keep); |
---|
1465 | list pe=imap(P,qe); |
---|
1466 | for(l=1;l<=size(qe);l++) |
---|
1467 | { |
---|
1468 | qe[l]=simplify(qe[l],2); |
---|
1469 | } |
---|
1470 | list rr=pe,qe; |
---|
1471 | return(rr); |
---|
1472 | } |
---|
1473 | /////////////////////////////////////////////////////////////////////////////// |
---|
1474 | |
---|
1475 | proc zeroSepClos(ideal I,ideal F) |
---|
1476 | { |
---|
1477 | //computes the separable closure of the special ideal I |
---|
1478 | //F is the set of special elements of I |
---|
1479 | //returns the separable closure sc(I) of I and an intvec v |
---|
1480 | //such that sc(I)=preimage(frobenius definde by v) |
---|
1481 | //i.e. var(i)----->var(i)^(p^v[i]) |
---|
1482 | |
---|
1483 | if(homog(I)==1){return(maxideal(1));} |
---|
1484 | |
---|
1485 | //assume F[i] irreducible in I and depending only on var(i) |
---|
1486 | |
---|
1487 | def R=basering; |
---|
1488 | int n=nvars(R); |
---|
1489 | int p=char(R); |
---|
1490 | intvec v; |
---|
1491 | v[n]=0; |
---|
1492 | int i,k; |
---|
1493 | list l; |
---|
1494 | |
---|
1495 | for(i=1;i<=n;i++) |
---|
1496 | { |
---|
1497 | l[i]=sep(F[i],i); |
---|
1498 | F[i]=l[i][1]; |
---|
1499 | if(l[i][2]>k){k=l[i][2];} |
---|
1500 | } |
---|
1501 | |
---|
1502 | if(k==0){return(list(I,v));} //the separable case |
---|
1503 | ideal m; |
---|
1504 | |
---|
1505 | for(i=1;i<=n;i++) |
---|
1506 | { |
---|
1507 | m[i]=var(i)^(p^l[i][2]); |
---|
1508 | v[i]=l[i][2]; |
---|
1509 | } |
---|
1510 | map phi=R,m; |
---|
1511 | ideal J=preimage(R,phi,I); |
---|
1512 | return(list(J,v)); |
---|
1513 | } |
---|
1514 | /////////////////////////////////////////////////////////////////////////////// |
---|
1515 | |
---|
1516 | proc insepDecomp(ideal i) |
---|
1517 | { |
---|
1518 | //decomposes i into special ideals |
---|
1519 | //computes the prime decomposition of the special ideals |
---|
1520 | //and transforms it back to a decomposition of i |
---|
1521 | |
---|
1522 | def R=basering; |
---|
1523 | list pr=zeroSp(i); |
---|
1524 | int l,k; |
---|
1525 | list re,wo,qr; |
---|
1526 | ideal m=maxideal(1); |
---|
1527 | ideal K; |
---|
1528 | map phi=R,m; |
---|
1529 | int p=char(R); |
---|
1530 | intvec op=option(get); |
---|
1531 | |
---|
1532 | for(l=1;l<=size(pr[1]);l++) |
---|
1533 | { |
---|
1534 | wo=zeroSepClos(pr[1][l],pr[2][l]); |
---|
1535 | for(k=1;k<=nvars(basering);k++) |
---|
1536 | { |
---|
1537 | m[k]=var(k)^(p^wo[2][k]); |
---|
1538 | } |
---|
1539 | phi=R,m; |
---|
1540 | qr=decomp(wo[1],2); |
---|
1541 | |
---|
1542 | option(redSB); |
---|
1543 | for(k=1;k<=size(qr)/2;k++) |
---|
1544 | { |
---|
1545 | K=qr[2*k]; |
---|
1546 | K=phi(K); |
---|
1547 | K=groebner(K); |
---|
1548 | re[size(re)+1]=zeroRad(K); |
---|
1549 | } |
---|
1550 | option(noredSB); |
---|
1551 | } |
---|
1552 | option(set,op); |
---|
1553 | return(re); |
---|
1554 | } |
---|
1555 | |
---|
1556 | |
---|
1557 | /////////////////////////////////////////////////////////////////////////////// |
---|
1558 | |
---|
1559 | static proc clearSB (ideal i,list #) |
---|
1560 | "USAGE: clearSB(i); i ideal which is SB ordered by monomial ordering |
---|
1561 | RETURN: ideal = minimal SB |
---|
1562 | NOTE: |
---|
1563 | EXAMPLE: example clearSB; shows an example |
---|
1564 | " |
---|
1565 | { |
---|
1566 | int k,j; |
---|
1567 | poly m; |
---|
1568 | int c=size(i); |
---|
1569 | |
---|
1570 | if(size(#)==0) |
---|
1571 | { |
---|
1572 | for(j=1;j<c;j++) |
---|
1573 | { |
---|
1574 | if(deg(i[j])==0) |
---|
1575 | { |
---|
1576 | i=ideal(1); |
---|
1577 | return(i); |
---|
1578 | } |
---|
1579 | if(deg(i[j])>0) |
---|
1580 | { |
---|
1581 | m=lead(i[j]); |
---|
1582 | for(k=j+1;k<=c;k++) |
---|
1583 | { |
---|
1584 | if(size(lead(i[k])/m)>0) |
---|
1585 | { |
---|
1586 | i[k]=0; |
---|
1587 | } |
---|
1588 | } |
---|
1589 | } |
---|
1590 | } |
---|
1591 | } |
---|
1592 | else |
---|
1593 | { |
---|
1594 | j=0; |
---|
1595 | while(j<c-1) |
---|
1596 | { |
---|
1597 | j++; |
---|
1598 | if(deg(i[j])==0) |
---|
1599 | { |
---|
1600 | i=ideal(1); |
---|
1601 | return(i); |
---|
1602 | } |
---|
1603 | if(deg(i[j])>0) |
---|
1604 | { |
---|
1605 | m=lead(i[j]); |
---|
1606 | for(k=j+1;k<=c;k++) |
---|
1607 | { |
---|
1608 | if(size(lead(i[k])/m)>0) |
---|
1609 | { |
---|
1610 | if((leadexp(m)!=leadexp(i[k]))||(#[j]<=#[k])) |
---|
1611 | { |
---|
1612 | i[k]=0; |
---|
1613 | } |
---|
1614 | else |
---|
1615 | { |
---|
1616 | i[j]=0; |
---|
1617 | break; |
---|
1618 | } |
---|
1619 | } |
---|
1620 | } |
---|
1621 | } |
---|
1622 | } |
---|
1623 | } |
---|
1624 | return(simplify(i,2)); |
---|
1625 | } |
---|
1626 | example |
---|
1627 | { "EXAMPLE:"; echo = 2; |
---|
1628 | ring r = (0,a,b),(x,y,z),dp; |
---|
1629 | ideal i=ax2+y,a2x+y,bx; |
---|
1630 | list l=1,2,1; |
---|
1631 | ideal j=clearSB(i,l); |
---|
1632 | j; |
---|
1633 | } |
---|
1634 | |
---|
1635 | /////////////////////////////////////////////////////////////////////////////// |
---|
1636 | |
---|
1637 | static proc independSet (ideal j) |
---|
1638 | "USAGE: independentSet(i); i ideal |
---|
1639 | RETURN: list = new varstring with the independent set at the end, |
---|
1640 | ordstring with the corresponding block ordering, |
---|
1641 | the integer where the independent set starts in the varstring |
---|
1642 | NOTE: |
---|
1643 | EXAMPLE: example independentSet; shows an example |
---|
1644 | " |
---|
1645 | { |
---|
1646 | int n,k,di; |
---|
1647 | list resu,hilf; |
---|
1648 | string var1,var2; |
---|
1649 | list v=indepSet(j,1); |
---|
1650 | |
---|
1651 | for(n=1;n<=size(v);n++) |
---|
1652 | { |
---|
1653 | di=0; |
---|
1654 | var1=""; |
---|
1655 | var2=""; |
---|
1656 | for(k=1;k<=size(v[n]);k++) |
---|
1657 | { |
---|
1658 | if(v[n][k]!=0) |
---|
1659 | { |
---|
1660 | di++; |
---|
1661 | var2=var2+"var("+string(k)+"),"; |
---|
1662 | } |
---|
1663 | else |
---|
1664 | { |
---|
1665 | var1=var1+"var("+string(k)+"),"; |
---|
1666 | } |
---|
1667 | } |
---|
1668 | if(di>0) |
---|
1669 | { |
---|
1670 | var1=var1+var2; |
---|
1671 | var1=var1[1..size(var1)-1]; |
---|
1672 | hilf[1]=var1; |
---|
1673 | hilf[2]="lp"; |
---|
1674 | //"lp("+string(nvars(basering)-di)+"),dp("+string(di)+")"; |
---|
1675 | hilf[3]=di; |
---|
1676 | resu[n]=hilf; |
---|
1677 | } |
---|
1678 | else |
---|
1679 | { |
---|
1680 | resu[n]=varstr(basering),ordstr(basering),0; |
---|
1681 | } |
---|
1682 | } |
---|
1683 | return(resu); |
---|
1684 | } |
---|
1685 | example |
---|
1686 | { "EXAMPLE:"; echo = 2; |
---|
1687 | ring s1=(0,x,y),(a,b,c,d,e,f,g),lp; |
---|
1688 | ideal i=ea-fbg,fa+be,ec-fdg,fc+de; |
---|
1689 | i=std(i); |
---|
1690 | list l=independSet(i); |
---|
1691 | l; |
---|
1692 | i=i,g; |
---|
1693 | l=independSet(i); |
---|
1694 | l; |
---|
1695 | |
---|
1696 | ring s=0,(x,y,z),lp; |
---|
1697 | ideal i=z,yx; |
---|
1698 | list l=independSet(i); |
---|
1699 | l; |
---|
1700 | |
---|
1701 | |
---|
1702 | } |
---|
1703 | /////////////////////////////////////////////////////////////////////////////// |
---|
1704 | |
---|
1705 | static proc maxIndependSet (ideal j) |
---|
1706 | "USAGE: maxIndependentSet(i); i ideal |
---|
1707 | RETURN: list = new varstring with the maximal independent set at the end, |
---|
1708 | ordstring with the corresponding block ordering, |
---|
1709 | the integer where the independent set starts in the varstring |
---|
1710 | NOTE: |
---|
1711 | EXAMPLE: example maxIndependentSet; shows an example |
---|
1712 | " |
---|
1713 | { |
---|
1714 | int n,k,di; |
---|
1715 | list resu,hilf; |
---|
1716 | string var1,var2; |
---|
1717 | list v=indepSet(j,0); |
---|
1718 | |
---|
1719 | for(n=1;n<=size(v);n++) |
---|
1720 | { |
---|
1721 | di=0; |
---|
1722 | var1=""; |
---|
1723 | var2=""; |
---|
1724 | for(k=1;k<=size(v[n]);k++) |
---|
1725 | { |
---|
1726 | if(v[n][k]!=0) |
---|
1727 | { |
---|
1728 | di++; |
---|
1729 | var2=var2+"var("+string(k)+"),"; |
---|
1730 | } |
---|
1731 | else |
---|
1732 | { |
---|
1733 | var1=var1+"var("+string(k)+"),"; |
---|
1734 | } |
---|
1735 | } |
---|
1736 | if(di>0) |
---|
1737 | { |
---|
1738 | var1=var1+var2; |
---|
1739 | var1=var1[1..size(var1)-1]; |
---|
1740 | hilf[1]=var1; |
---|
1741 | hilf[2]="lp"; |
---|
1742 | hilf[3]=di; |
---|
1743 | resu[n]=hilf; |
---|
1744 | } |
---|
1745 | else |
---|
1746 | { |
---|
1747 | resu[n]=varstr(basering),ordstr(basering),0; |
---|
1748 | } |
---|
1749 | } |
---|
1750 | return(resu); |
---|
1751 | } |
---|
1752 | example |
---|
1753 | { "EXAMPLE:"; echo = 2; |
---|
1754 | ring s1=(0,x,y),(a,b,c,d,e,f,g),lp; |
---|
1755 | ideal i=ea-fbg,fa+be,ec-fdg,fc+de; |
---|
1756 | i=std(i); |
---|
1757 | list l=maxIndependSet(i); |
---|
1758 | l; |
---|
1759 | i=i,g; |
---|
1760 | l=maxIndependSet(i); |
---|
1761 | l; |
---|
1762 | |
---|
1763 | ring s=0,(x,y,z),lp; |
---|
1764 | ideal i=z,yx; |
---|
1765 | list l=maxIndependSet(i); |
---|
1766 | l; |
---|
1767 | |
---|
1768 | |
---|
1769 | } |
---|
1770 | |
---|
1771 | /////////////////////////////////////////////////////////////////////////////// |
---|
1772 | |
---|
1773 | static proc prepareQuotientring (int nnp) |
---|
1774 | "USAGE: prepareQuotientring(nnp); nnp int |
---|
1775 | RETURN: string = to define Kvar(nnp+1),...,var(nvars)[..rest ] |
---|
1776 | NOTE: |
---|
1777 | EXAMPLE: example independentSet; shows an example |
---|
1778 | " |
---|
1779 | { |
---|
1780 | ideal @ih,@jh; |
---|
1781 | int npar=npars(basering); |
---|
1782 | int @n; |
---|
1783 | |
---|
1784 | string quotring= "ring quring = ("+charstr(basering); |
---|
1785 | for(@n=nnp+1;@n<=nvars(basering);@n++) |
---|
1786 | { |
---|
1787 | quotring=quotring+",var("+string(@n)+")"; |
---|
1788 | @ih=@ih+var(@n); |
---|
1789 | } |
---|
1790 | |
---|
1791 | quotring=quotring+"),(var(1)"; |
---|
1792 | @jh=@jh+var(1); |
---|
1793 | for(@n=2;@n<=nnp;@n++) |
---|
1794 | { |
---|
1795 | quotring=quotring+",var("+string(@n)+")"; |
---|
1796 | @jh=@jh+var(@n); |
---|
1797 | } |
---|
1798 | quotring=quotring+"),(C,lp);"; |
---|
1799 | |
---|
1800 | return(quotring); |
---|
1801 | |
---|
1802 | } |
---|
1803 | example |
---|
1804 | { "EXAMPLE:"; echo = 2; |
---|
1805 | ring s1=(0,x),(a,b,c,d,e,f,g),lp; |
---|
1806 | def @Q=basering; |
---|
1807 | list l= prepareQuotientring(3); |
---|
1808 | l; |
---|
1809 | execute(l[1]); |
---|
1810 | execute(l[2]); |
---|
1811 | basering; |
---|
1812 | phi; |
---|
1813 | setring @Q; |
---|
1814 | |
---|
1815 | } |
---|
1816 | |
---|
1817 | /////////////////////////////////////////////////////////////////////////////// |
---|
1818 | static proc cleanPrimary(list l) |
---|
1819 | { |
---|
1820 | int i,j; |
---|
1821 | list lh; |
---|
1822 | for(i=1;i<=size(l)/2;i++) |
---|
1823 | { |
---|
1824 | if(deg(l[2*i-1][1])>0) |
---|
1825 | { |
---|
1826 | j++; |
---|
1827 | lh[j]=l[2*i-1]; |
---|
1828 | j++; |
---|
1829 | lh[j]=l[2*i]; |
---|
1830 | } |
---|
1831 | } |
---|
1832 | return(lh); |
---|
1833 | } |
---|
1834 | /////////////////////////////////////////////////////////////////////////////// |
---|
1835 | |
---|
1836 | |
---|
1837 | proc minAssPrimesold(ideal i, list #) |
---|
1838 | "USAGE: minAssPrimes(i); i ideal |
---|
1839 | minAssPrimes(i,1); i ideal (to use also the factorizing Groebner) |
---|
1840 | RETURN: list = the minimal associated prime ideals of i |
---|
1841 | EXAMPLE: example minAssPrimes; shows an example |
---|
1842 | " |
---|
1843 | { |
---|
1844 | def @P=basering; |
---|
1845 | if(size(i)==0){return(list(ideal(0)));} |
---|
1846 | list qr=simplifyIdeal(i); |
---|
1847 | map phi=@P,qr[2]; |
---|
1848 | i=qr[1]; |
---|
1849 | |
---|
1850 | execute ("ring gnir = ("+charstr(basering)+"),("+varstr(basering)+"),(" |
---|
1851 | +ordstr(basering)+");"); |
---|
1852 | |
---|
1853 | |
---|
1854 | ideal i=fetch(@P,i); |
---|
1855 | if(size(#)==0) |
---|
1856 | { |
---|
1857 | int @wr; |
---|
1858 | list tluser,@res; |
---|
1859 | list primary=decomp(i,2); |
---|
1860 | |
---|
1861 | @res[1]=primary; |
---|
1862 | |
---|
1863 | tluser=union(@res); |
---|
1864 | setring @P; |
---|
1865 | list @res=imap(gnir,tluser); |
---|
1866 | return(phi(@res)); |
---|
1867 | } |
---|
1868 | list @res,empty; |
---|
1869 | ideal ser; |
---|
1870 | option(redSB); |
---|
1871 | list @pr=facstd(i); |
---|
1872 | if(size(@pr)==1) |
---|
1873 | { |
---|
1874 | attrib(@pr[1],"isSB",1); |
---|
1875 | if((dim(@pr[1])==0)&&(homog(@pr[1])==1)) |
---|
1876 | { |
---|
1877 | setring @P; |
---|
1878 | list @res=maxideal(1); |
---|
1879 | return(phi(@res)); |
---|
1880 | } |
---|
1881 | if(dim(@pr[1])>1) |
---|
1882 | { |
---|
1883 | setring @P; |
---|
1884 | // kill gnir; |
---|
1885 | execute ("ring gnir1 = ("+charstr(basering)+"), |
---|
1886 | ("+varstr(basering)+"),(C,lp);"); |
---|
1887 | ideal i=fetch(@P,i); |
---|
1888 | list @pr=facstd(i); |
---|
1889 | // ideal ser; |
---|
1890 | setring gnir; |
---|
1891 | @pr=fetch(gnir1,@pr); |
---|
1892 | kill gnir1; |
---|
1893 | } |
---|
1894 | } |
---|
1895 | option(noredSB); |
---|
1896 | int j,k,odim,ndim,count; |
---|
1897 | attrib(@pr[1],"isSB",1); |
---|
1898 | if(#[1]==77) |
---|
1899 | { |
---|
1900 | odim=dim(@pr[1]); |
---|
1901 | count=1; |
---|
1902 | intvec pos; |
---|
1903 | pos[size(@pr)]=0; |
---|
1904 | for(j=2;j<=size(@pr);j++) |
---|
1905 | { |
---|
1906 | attrib(@pr[j],"isSB",1); |
---|
1907 | ndim=dim(@pr[j]); |
---|
1908 | if(ndim>odim) |
---|
1909 | { |
---|
1910 | for(k=count;k<=j-1;k++) |
---|
1911 | { |
---|
1912 | pos[k]=1; |
---|
1913 | } |
---|
1914 | count=j; |
---|
1915 | odim=ndim; |
---|
1916 | } |
---|
1917 | if(ndim<odim) |
---|
1918 | { |
---|
1919 | pos[j]=1; |
---|
1920 | } |
---|
1921 | } |
---|
1922 | for(j=1;j<=size(@pr);j++) |
---|
1923 | { |
---|
1924 | if(pos[j]!=1) |
---|
1925 | { |
---|
1926 | @res[j]=decomp(@pr[j],2); |
---|
1927 | } |
---|
1928 | else |
---|
1929 | { |
---|
1930 | @res[j]=empty; |
---|
1931 | } |
---|
1932 | } |
---|
1933 | } |
---|
1934 | else |
---|
1935 | { |
---|
1936 | ser=ideal(1); |
---|
1937 | for(j=1;j<=size(@pr);j++) |
---|
1938 | { |
---|
1939 | //@pr[j]; |
---|
1940 | //pause(); |
---|
1941 | @res[j]=decomp(@pr[j],2); |
---|
1942 | // @res[j]=decomp(@pr[j],2,@pr[j],ser); |
---|
1943 | // for(k=1;k<=size(@res[j]);k++) |
---|
1944 | // { |
---|
1945 | // ser=intersect(ser,@res[j][k]); |
---|
1946 | // } |
---|
1947 | } |
---|
1948 | } |
---|
1949 | |
---|
1950 | @res=union(@res); |
---|
1951 | setring @P; |
---|
1952 | list @res=imap(gnir,@res); |
---|
1953 | return(phi(@res)); |
---|
1954 | } |
---|
1955 | example |
---|
1956 | { "EXAMPLE:"; echo = 2; |
---|
1957 | ring r = 32003,(x,y,z),lp; |
---|
1958 | poly p = z2+1; |
---|
1959 | poly q = z4+2; |
---|
1960 | ideal i = p^2*q^3,(y-z3)^3,(x-yz+z4)^4; |
---|
1961 | list pr= minAssPrimes(i); pr; |
---|
1962 | |
---|
1963 | minAssPrimes(i,1); |
---|
1964 | } |
---|
1965 | |
---|
1966 | static proc primT(ideal i) |
---|
1967 | { |
---|
1968 | //assumes that all generators of i are irreducible |
---|
1969 | //i is standard basis |
---|
1970 | |
---|
1971 | attrib(i,"isSB",1); |
---|
1972 | int j=size(i); |
---|
1973 | int k; |
---|
1974 | while(j>0) |
---|
1975 | { |
---|
1976 | if(deg(i[j])>1){break;} |
---|
1977 | j--; |
---|
1978 | } |
---|
1979 | if(j==0){return(1);} |
---|
1980 | if(deg(i[j])==vdim(i)){return(1);} |
---|
1981 | return(0); |
---|
1982 | } |
---|
1983 | |
---|
1984 | |
---|
1985 | static proc minAssPrimes(ideal i, list #) |
---|
1986 | "USAGE: minAssPrimes(i); i ideal |
---|
1987 | minAssPrimes(i,1); i ideal (to use also the factorizing Groebner) |
---|
1988 | RETURN: list = the minimal associated prime ideals of i |
---|
1989 | EXAMPLE: example minAssPrimes; shows an example |
---|
1990 | " |
---|
1991 | { |
---|
1992 | def P=basering; |
---|
1993 | if(size(i)==0){return(list(ideal(0)));} |
---|
1994 | list q=simplifyIdeal(i); |
---|
1995 | list re=maxideal(1); |
---|
1996 | int j,a,k; |
---|
1997 | intvec op=option(get); |
---|
1998 | map phi=P,q[2]; |
---|
1999 | |
---|
2000 | if(npars(P)==0){option(redSB);} |
---|
2001 | |
---|
2002 | i=std(q[1]); |
---|
2003 | if(dim(i)==-1){re=ideal(1);return(re);} |
---|
2004 | if((dim(i)==0)&&(npars(P)==0)) |
---|
2005 | { |
---|
2006 | int di=vdim(i); |
---|
2007 | execute ("ring gnir=("+charstr(P)+"),("+varstr(P)+"),lp;"); |
---|
2008 | ideal J=interred(imap(P,i)); |
---|
2009 | attrib(J,"isSB",1); |
---|
2010 | if(vdim(J)!=di) |
---|
2011 | { |
---|
2012 | J=fglm(P,i); |
---|
2013 | } |
---|
2014 | list pr=triangMH(J,2); |
---|
2015 | list qr,re; |
---|
2016 | |
---|
2017 | for(k=1;k<=size(pr);k++) |
---|
2018 | { |
---|
2019 | if(primT(pr[k])) |
---|
2020 | { |
---|
2021 | re[size(re)+1]=pr[k]; |
---|
2022 | } |
---|
2023 | else |
---|
2024 | { |
---|
2025 | attrib(pr[k],"isSB",1); |
---|
2026 | qr=decomp(pr[k],2); |
---|
2027 | for(j=1;j<=size(qr)/2;j++) |
---|
2028 | { |
---|
2029 | re[size(re)+1]=qr[2*j]; |
---|
2030 | } |
---|
2031 | } |
---|
2032 | } |
---|
2033 | setring P; |
---|
2034 | re=imap(gnir,re); |
---|
2035 | option(set,op); |
---|
2036 | return(phi(re)); |
---|
2037 | } |
---|
2038 | |
---|
2039 | if((size(#)==0)||(dim(i)==0)) |
---|
2040 | { |
---|
2041 | re[1]=decomp(i,2); |
---|
2042 | re=union(re); |
---|
2043 | option(set,op); |
---|
2044 | return(phi(re)); |
---|
2045 | } |
---|
2046 | |
---|
2047 | q=facstd(i); |
---|
2048 | |
---|
2049 | if((size(q)==1)&&(dim(i)>1)) |
---|
2050 | { |
---|
2051 | execute ("ring gnir=("+charstr(P)+"),("+varstr(P)+"),lp;"); |
---|
2052 | |
---|
2053 | list p=facstd(fetch(P,i)); |
---|
2054 | if(size(p)>1) |
---|
2055 | { |
---|
2056 | a=1; |
---|
2057 | setring P; |
---|
2058 | q=fetch(gnir,p); |
---|
2059 | } |
---|
2060 | else |
---|
2061 | { |
---|
2062 | setring P; |
---|
2063 | } |
---|
2064 | kill gnir; |
---|
2065 | } |
---|
2066 | |
---|
2067 | option(set,op); |
---|
2068 | for(j=1;j<=size(q);j++) |
---|
2069 | { |
---|
2070 | if(a==0){attrib(q[j],"isSB",1);} |
---|
2071 | re[j]=decomp(q[j],2); |
---|
2072 | } |
---|
2073 | re=union(re); |
---|
2074 | return(phi(re)); |
---|
2075 | } |
---|
2076 | example |
---|
2077 | { "EXAMPLE:"; echo = 2; |
---|
2078 | ring r = 32003,(x,y,z),lp; |
---|
2079 | poly p = z2+1; |
---|
2080 | poly q = z4+2; |
---|
2081 | ideal i = p^2*q^3,(y-z3)^3,(x-yz+z4)^4; |
---|
2082 | list pr= minAssPrimes(i); pr; |
---|
2083 | |
---|
2084 | minAssPrimes(i,1); |
---|
2085 | } |
---|
2086 | |
---|
2087 | static proc union(list li) |
---|
2088 | { |
---|
2089 | int i,j,k; |
---|
2090 | |
---|
2091 | def P=basering; |
---|
2092 | |
---|
2093 | execute("ring ir = ("+charstr(basering)+"),("+varstr(basering)+"),(C,lp);"); |
---|
2094 | list l=fetch(P,li); |
---|
2095 | list @erg; |
---|
2096 | |
---|
2097 | for(k=1;k<=size(l);k++) |
---|
2098 | { |
---|
2099 | for(j=1;j<=size(l[k])/2;j++) |
---|
2100 | { |
---|
2101 | if(deg(l[k][2*j][1])!=0) |
---|
2102 | { |
---|
2103 | i++; |
---|
2104 | @erg[i]=l[k][2*j]; |
---|
2105 | } |
---|
2106 | } |
---|
2107 | } |
---|
2108 | |
---|
2109 | list @wos; |
---|
2110 | i=0; |
---|
2111 | ideal i1,i2; |
---|
2112 | while(i<size(@erg)-1) |
---|
2113 | { |
---|
2114 | i++; |
---|
2115 | k=i+1; |
---|
2116 | i1=lead(@erg[i]); |
---|
2117 | attrib(i1,"isSB",1); |
---|
2118 | attrib(@erg[i],"isSB",1); |
---|
2119 | |
---|
2120 | while(k<=size(@erg)) |
---|
2121 | { |
---|
2122 | if(deg(@erg[i][1])==0) |
---|
2123 | { |
---|
2124 | break; |
---|
2125 | } |
---|
2126 | i2=lead(@erg[k]); |
---|
2127 | attrib(@erg[k],"isSB",1); |
---|
2128 | attrib(i2,"isSB",1); |
---|
2129 | |
---|
2130 | if(size(reduce(i1,i2,1))==0) |
---|
2131 | { |
---|
2132 | if(size(reduce(@erg[i],@erg[k],1))==0) |
---|
2133 | { |
---|
2134 | @erg[k]=ideal(1); |
---|
2135 | i2=ideal(1); |
---|
2136 | } |
---|
2137 | } |
---|
2138 | if(size(reduce(i2,i1,1))==0) |
---|
2139 | { |
---|
2140 | if(size(reduce(@erg[k],@erg[i],1))==0) |
---|
2141 | { |
---|
2142 | break; |
---|
2143 | } |
---|
2144 | } |
---|
2145 | k++; |
---|
2146 | if(k>size(@erg)) |
---|
2147 | { |
---|
2148 | @wos[size(@wos)+1]=@erg[i]; |
---|
2149 | } |
---|
2150 | } |
---|
2151 | } |
---|
2152 | if(deg(@erg[size(@erg)][1])!=0) |
---|
2153 | { |
---|
2154 | @wos[size(@wos)+1]=@erg[size(@erg)]; |
---|
2155 | } |
---|
2156 | setring P; |
---|
2157 | list @ser=fetch(ir,@wos); |
---|
2158 | return(@ser); |
---|
2159 | } |
---|
2160 | /////////////////////////////////////////////////////////////////////////////// |
---|
2161 | proc equidim(ideal i,list #) |
---|
2162 | "USAGE: equidim(i) or equidim(i,1) ; i ideal |
---|
2163 | RETURN: list of equidimensional ideals a[1],...,a[s] with: |
---|
2164 | - a[s] the equidimensional locus of i, i.e. the intersection |
---|
2165 | of the primary ideals of dimension of i |
---|
2166 | - a[1],...,a[s-1] the lower dimensional equidimensional loci. |
---|
2167 | NOTE: An embedded component q (primary ideal) of i can be replaced in the |
---|
2168 | decomposition by a primary ideal q1 with the same radical as q. @* |
---|
2169 | @code{equidim(i,1)} uses the algorithm of Eisenbud/Huneke/Vasconcelos. |
---|
2170 | |
---|
2171 | EXAMPLE:example equidim; shows an example |
---|
2172 | " |
---|
2173 | { |
---|
2174 | if(ord_test(basering)!=1) |
---|
2175 | { |
---|
2176 | ERROR( |
---|
2177 | "// Not implemented for this ordering, please change to global ordering." |
---|
2178 | ); |
---|
2179 | } |
---|
2180 | intvec op ; |
---|
2181 | def P = basering; |
---|
2182 | list eq; |
---|
2183 | intvec w; |
---|
2184 | int n,m; |
---|
2185 | int g=size(i); |
---|
2186 | int a=attrib(i,"isSB"); |
---|
2187 | int homo=homog(i); |
---|
2188 | if(size(#)!=0) |
---|
2189 | { |
---|
2190 | m=1; |
---|
2191 | } |
---|
2192 | |
---|
2193 | if(((homo==1)||(a==1))&&(find(ordstr(basering),"l")==0) |
---|
2194 | &&(find(ordstr(basering),"s")==0)) |
---|
2195 | { |
---|
2196 | execute("ring gnir = ("+charstr(basering)+"),("+varstr(basering)+"),(" |
---|
2197 | +ordstr(basering)+");"); |
---|
2198 | ideal i=imap(P,i); |
---|
2199 | ideal j=i; |
---|
2200 | if(a==1) |
---|
2201 | { |
---|
2202 | attrib(j,"isSB",1); |
---|
2203 | } |
---|
2204 | else |
---|
2205 | { |
---|
2206 | j=groebner(i); |
---|
2207 | } |
---|
2208 | } |
---|
2209 | else |
---|
2210 | { |
---|
2211 | execute("ring gnir = ("+charstr(basering)+"),("+varstr(basering)+"),dp;"); |
---|
2212 | ideal i=imap(P,i); |
---|
2213 | ideal j=groebner(i); |
---|
2214 | } |
---|
2215 | if(homo==1) |
---|
2216 | { |
---|
2217 | for(n=1;n<=nvars(basering);n++) |
---|
2218 | { |
---|
2219 | w[n]=ord(var(n)); |
---|
2220 | } |
---|
2221 | intvec hil=hilb(j,1,w); |
---|
2222 | } |
---|
2223 | |
---|
2224 | if ((dim(j)==-1)||(size(j)==0)||(nvars(basering)==1) |
---|
2225 | ||(dim(j)==0)||(dim(j)+g==nvars(basering))) |
---|
2226 | { |
---|
2227 | setring P; |
---|
2228 | eq[1]=i; |
---|
2229 | return(eq); |
---|
2230 | } |
---|
2231 | |
---|
2232 | if(m==0) |
---|
2233 | { |
---|
2234 | ideal k=equidimMax(j); |
---|
2235 | } |
---|
2236 | else |
---|
2237 | { |
---|
2238 | ideal k=equidimMaxEHV(j); |
---|
2239 | } |
---|
2240 | if(size(reduce(k,j,1))==0) |
---|
2241 | { |
---|
2242 | setring P; |
---|
2243 | eq[1]=i; |
---|
2244 | kill gnir; |
---|
2245 | return(eq); |
---|
2246 | } |
---|
2247 | op=option(get); |
---|
2248 | option(returnSB); |
---|
2249 | j=quotient(j,k); |
---|
2250 | option(set,op); |
---|
2251 | |
---|
2252 | list equi=equidim(j); |
---|
2253 | if(deg(equi[size(equi)][1])<=0) |
---|
2254 | { |
---|
2255 | equi[size(equi)]=k; |
---|
2256 | } |
---|
2257 | else |
---|
2258 | { |
---|
2259 | equi[size(equi)+1]=k; |
---|
2260 | } |
---|
2261 | setring P; |
---|
2262 | eq=imap(gnir,equi); |
---|
2263 | kill gnir; |
---|
2264 | return(eq); |
---|
2265 | } |
---|
2266 | example |
---|
2267 | { "EXAMPLE:"; echo = 2; |
---|
2268 | ring r = 32003,(x,y,z),dp; |
---|
2269 | ideal i = intersect(ideal(z),ideal(x,y),ideal(x2,z2),ideal(x5,y5,z5)); |
---|
2270 | equidim(i); |
---|
2271 | } |
---|
2272 | |
---|
2273 | /////////////////////////////////////////////////////////////////////////////// |
---|
2274 | proc equidimMax(ideal i) |
---|
2275 | "USAGE: equidimMax(i); i ideal |
---|
2276 | RETURN: ideal of equidimensional locus (of maximal dimension) of i. |
---|
2277 | EXAMPLE: example equidimMax; shows an example |
---|
2278 | " |
---|
2279 | { |
---|
2280 | if(ord_test(basering)!=1) |
---|
2281 | { |
---|
2282 | ERROR( |
---|
2283 | "// Not implemented for this ordering, please change to global ordering." |
---|
2284 | ); |
---|
2285 | } |
---|
2286 | def P = basering; |
---|
2287 | ideal eq; |
---|
2288 | intvec w; |
---|
2289 | int n; |
---|
2290 | int g=size(i); |
---|
2291 | int a=attrib(i,"isSB"); |
---|
2292 | int homo=homog(i); |
---|
2293 | |
---|
2294 | if(((homo==1)||(a==1))&&(find(ordstr(basering),"l")==0) |
---|
2295 | &&(find(ordstr(basering),"s")==0)) |
---|
2296 | { |
---|
2297 | execute("ring gnir = ("+charstr(basering)+"),("+varstr(basering)+"),(" |
---|
2298 | +ordstr(basering)+");"); |
---|
2299 | ideal i=imap(P,i); |
---|
2300 | ideal j=i; |
---|
2301 | if(a==1) |
---|
2302 | { |
---|
2303 | attrib(j,"isSB",1); |
---|
2304 | } |
---|
2305 | else |
---|
2306 | { |
---|
2307 | j=groebner(i); |
---|
2308 | } |
---|
2309 | } |
---|
2310 | else |
---|
2311 | { |
---|
2312 | execute("ring gnir = ("+charstr(basering)+"),("+varstr(basering)+"),dp;"); |
---|
2313 | ideal i=imap(P,i); |
---|
2314 | ideal j=groebner(i); |
---|
2315 | } |
---|
2316 | list indep; |
---|
2317 | ideal equ,equi; |
---|
2318 | if(homo==1) |
---|
2319 | { |
---|
2320 | for(n=1;n<=nvars(basering);n++) |
---|
2321 | { |
---|
2322 | w[n]=ord(var(n)); |
---|
2323 | } |
---|
2324 | intvec hil=hilb(j,1,w); |
---|
2325 | } |
---|
2326 | if ((dim(j)==-1)||(size(j)==0)||(nvars(basering)==1) |
---|
2327 | ||(dim(j)==0)||(dim(j)+g==nvars(basering))) |
---|
2328 | { |
---|
2329 | setring P; |
---|
2330 | return(i); |
---|
2331 | } |
---|
2332 | |
---|
2333 | indep=maxIndependSet(j); |
---|
2334 | |
---|
2335 | execute("ring gnir1 = ("+charstr(basering)+"),("+indep[1][1]+"),(" |
---|
2336 | +indep[1][2]+");"); |
---|
2337 | if(homo==1) |
---|
2338 | { |
---|
2339 | ideal j=std(imap(gnir,j),hil,w); |
---|
2340 | } |
---|
2341 | else |
---|
2342 | { |
---|
2343 | ideal j=groebner(imap(gnir,j)); |
---|
2344 | } |
---|
2345 | string quotring=prepareQuotientring(nvars(basering)-indep[1][3]); |
---|
2346 | execute(quotring); |
---|
2347 | ideal j=imap(gnir1,j); |
---|
2348 | kill gnir1; |
---|
2349 | j=clearSB(j); |
---|
2350 | ideal h; |
---|
2351 | for(n=1;n<=size(j);n++) |
---|
2352 | { |
---|
2353 | h[n]=leadcoef(j[n]); |
---|
2354 | } |
---|
2355 | setring gnir; |
---|
2356 | ideal h=imap(quring,h); |
---|
2357 | kill quring; |
---|
2358 | |
---|
2359 | list l=minSat(j,h); |
---|
2360 | |
---|
2361 | if(deg(l[2])>0) |
---|
2362 | { |
---|
2363 | equ=l[1]; |
---|
2364 | attrib(equ,"isSB",1); |
---|
2365 | j=std(j,l[2]); |
---|
2366 | |
---|
2367 | if(dim(equ)==dim(j)) |
---|
2368 | { |
---|
2369 | equi=equidimMax(j); |
---|
2370 | equ=interred(intersect(equ,equi)); |
---|
2371 | } |
---|
2372 | } |
---|
2373 | else |
---|
2374 | { |
---|
2375 | equ=i; |
---|
2376 | } |
---|
2377 | |
---|
2378 | setring P; |
---|
2379 | eq=imap(gnir,equ); |
---|
2380 | kill gnir; |
---|
2381 | return(eq); |
---|
2382 | } |
---|
2383 | example |
---|
2384 | { "EXAMPLE:"; echo = 2; |
---|
2385 | ring r = 32003,(x,y,z),dp; |
---|
2386 | ideal i = intersect(ideal(z),ideal(x,y),ideal(x2,z2),ideal(x5,y5,z5)); |
---|
2387 | equidimMax(i); |
---|
2388 | } |
---|
2389 | /////////////////////////////////////////////////////////////////////////////// |
---|
2390 | static proc islp() |
---|
2391 | { |
---|
2392 | string s=ordstr(basering); |
---|
2393 | int n=find(s,"lp"); |
---|
2394 | if(!n){return(0);} |
---|
2395 | int k=find(s,","); |
---|
2396 | string t=s[k+1..size(s)]; |
---|
2397 | int l=find(t,","); |
---|
2398 | t=s[1..k-1]; |
---|
2399 | int m=find(t,","); |
---|
2400 | if(l+m){return(0);} |
---|
2401 | return(1); |
---|
2402 | } |
---|
2403 | /////////////////////////////////////////////////////////////////////////////// |
---|
2404 | |
---|
2405 | proc algeDeco(ideal i, int w) |
---|
2406 | { |
---|
2407 | //reduces primery decomposition over algebraic extensions to |
---|
2408 | //the other cases |
---|
2409 | def R=basering; |
---|
2410 | int n=nvars(R); |
---|
2411 | |
---|
2412 | //---Anfang Provisorium |
---|
2413 | if(size(i)==2) |
---|
2414 | { |
---|
2415 | option(redSB); |
---|
2416 | ideal J=std(i); |
---|
2417 | option(noredSB); |
---|
2418 | if((size(J)==2)&&(deg(J[1])==1)) |
---|
2419 | { |
---|
2420 | ideal keep; |
---|
2421 | poly f; |
---|
2422 | int j; |
---|
2423 | for(j=1;j<=nvars(basering);j++) |
---|
2424 | { |
---|
2425 | f=J[2]; |
---|
2426 | while((f/var(j))*var(j)-f==0) |
---|
2427 | { |
---|
2428 | f=f/var(j); |
---|
2429 | keep=keep,var(j); |
---|
2430 | } |
---|
2431 | J[2]=f; |
---|
2432 | } |
---|
2433 | ideal K=factorize(J[2],1); |
---|
2434 | if(deg(K[1])==0){K=0;} |
---|
2435 | K=K+std(keep); |
---|
2436 | ideal L; |
---|
2437 | list resu; |
---|
2438 | for(j=1;j<=size(K);j++) |
---|
2439 | { |
---|
2440 | L=J[1],K[j]; |
---|
2441 | resu[j]=L; |
---|
2442 | } |
---|
2443 | return(resu); |
---|
2444 | } |
---|
2445 | } |
---|
2446 | //---Ende Provisorium |
---|
2447 | string mp="poly p="+string(minpoly)+";"; |
---|
2448 | string gnir="ring RH="+string(char(R))+",("+varstr(R)+","+string(par(1)) |
---|
2449 | +"),dp;"; |
---|
2450 | execute(gnir); |
---|
2451 | execute(mp); |
---|
2452 | ideal i=imap(R,i); |
---|
2453 | ideal I=subst(i,var(nvars(basering)),0); |
---|
2454 | int j; |
---|
2455 | for(j=1;j<=ncols(i);j++) |
---|
2456 | { |
---|
2457 | if(i[j]!=I[j]){break;} |
---|
2458 | } |
---|
2459 | if((j>ncols(i))&&(deg(p)==1)) |
---|
2460 | { |
---|
2461 | setring R; |
---|
2462 | kill RH; |
---|
2463 | kill gnir; |
---|
2464 | string gnir="ring RH="+string(char(R))+",("+varstr(R)+"),dp;"; |
---|
2465 | execute(gnir); |
---|
2466 | ideal i=imap(R,i); |
---|
2467 | ideal J; |
---|
2468 | } |
---|
2469 | else |
---|
2470 | { |
---|
2471 | i=i,p; |
---|
2472 | } |
---|
2473 | list pr; |
---|
2474 | |
---|
2475 | if(w==0) |
---|
2476 | { |
---|
2477 | pr=decomp(i); |
---|
2478 | } |
---|
2479 | if(w==1) |
---|
2480 | { |
---|
2481 | pr=prim_dec(i,1); |
---|
2482 | pr=reconvList(pr); |
---|
2483 | } |
---|
2484 | if(w==2) |
---|
2485 | { |
---|
2486 | pr=minAssPrimes(i); |
---|
2487 | } |
---|
2488 | if(n<nvars(basering)) |
---|
2489 | { |
---|
2490 | gnir="ring RS="+string(char(R))+",("+varstr(RH) |
---|
2491 | +"),(dp("+string(n)+"),lp);"; |
---|
2492 | execute(gnir); |
---|
2493 | list pr=imap(RH,pr); |
---|
2494 | ideal K; |
---|
2495 | for(j=1;j<=size(pr);j++) |
---|
2496 | { |
---|
2497 | K=groebner(pr[j]); |
---|
2498 | K=K[2..size(K)]; |
---|
2499 | pr[j]=K; |
---|
2500 | } |
---|
2501 | setring R; |
---|
2502 | list pr=imap(RS,pr); |
---|
2503 | } |
---|
2504 | else |
---|
2505 | { |
---|
2506 | setring R; |
---|
2507 | list pr=imap(RH,pr); |
---|
2508 | } |
---|
2509 | list re; |
---|
2510 | if(w==2) |
---|
2511 | { |
---|
2512 | re=pr; |
---|
2513 | } |
---|
2514 | else |
---|
2515 | { |
---|
2516 | re=convList(pr); |
---|
2517 | } |
---|
2518 | return(re); |
---|
2519 | } |
---|
2520 | |
---|
2521 | /////////////////////////////////////////////////////////////////////////////// |
---|
2522 | static proc decomp(ideal i,list #) |
---|
2523 | "USAGE: decomp(i); i ideal (for primary decomposition) (resp. |
---|
2524 | decomp(i,1); (for the associated primes of dimension of i) ) |
---|
2525 | decomp(i,2); (for the minimal associated primes) ) |
---|
2526 | RETURN: list = list of primary ideals and their associated primes |
---|
2527 | (at even positions in the list) |
---|
2528 | (resp. a list of the minimal associated primes) |
---|
2529 | NOTE: Algorithm of Gianni/Trager/Zacharias |
---|
2530 | EXAMPLE: example decomp; shows an example |
---|
2531 | " |
---|
2532 | { |
---|
2533 | intvec op; |
---|
2534 | def @P = basering; |
---|
2535 | list primary,indep,ltras; |
---|
2536 | intvec @vh,isat,@w; |
---|
2537 | int @wr,@k,@n,@m,@n1,@n2,@n3,homo,seri,keepdi; |
---|
2538 | ideal peek=i; |
---|
2539 | ideal ser,tras; |
---|
2540 | int isS=(attrib(i,"isSB")==1); |
---|
2541 | |
---|
2542 | if(size(#)>0) |
---|
2543 | { |
---|
2544 | if((#[1]==1)||(#[1]==2)) |
---|
2545 | { |
---|
2546 | @wr=#[1]; |
---|
2547 | if(size(#)>1) |
---|
2548 | { |
---|
2549 | seri=1; |
---|
2550 | peek=#[2]; |
---|
2551 | ser=#[3]; |
---|
2552 | } |
---|
2553 | } |
---|
2554 | else |
---|
2555 | { |
---|
2556 | seri=1; |
---|
2557 | peek=#[1]; |
---|
2558 | ser=#[2]; |
---|
2559 | } |
---|
2560 | } |
---|
2561 | |
---|
2562 | homo=homog(i); |
---|
2563 | if(homo==1) |
---|
2564 | { |
---|
2565 | if(attrib(i,"isSB")!=1) |
---|
2566 | { |
---|
2567 | ltras=mstd(i); |
---|
2568 | attrib(ltras[1],"isSB",1); |
---|
2569 | } |
---|
2570 | else |
---|
2571 | { |
---|
2572 | ltras=i,i; |
---|
2573 | attrib(ltras[1],"isSB",1); |
---|
2574 | } |
---|
2575 | tras=ltras[1]; |
---|
2576 | attrib(tras,"isSB",1); |
---|
2577 | if(dim(tras)==0) |
---|
2578 | { |
---|
2579 | primary[1]=ltras[2]; |
---|
2580 | primary[2]=maxideal(1); |
---|
2581 | if(@wr>0) |
---|
2582 | { |
---|
2583 | list l; |
---|
2584 | l[1]=maxideal(1); |
---|
2585 | l[2]=maxideal(1); |
---|
2586 | return(l); |
---|
2587 | } |
---|
2588 | return(primary); |
---|
2589 | } |
---|
2590 | for(@n=1;@n<=nvars(basering);@n++) |
---|
2591 | { |
---|
2592 | @w[@n]=ord(var(@n)); |
---|
2593 | } |
---|
2594 | intvec @hilb=hilb(tras,1,@w); |
---|
2595 | intvec keephilb=@hilb; |
---|
2596 | } |
---|
2597 | |
---|
2598 | //---------------------------------------------------------------- |
---|
2599 | //i is the zero-ideal |
---|
2600 | //---------------------------------------------------------------- |
---|
2601 | |
---|
2602 | if(size(i)==0) |
---|
2603 | { |
---|
2604 | primary=i,i; |
---|
2605 | return(primary); |
---|
2606 | } |
---|
2607 | |
---|
2608 | //---------------------------------------------------------------- |
---|
2609 | //pass to the lexicographical ordering and compute a standardbasis |
---|
2610 | //---------------------------------------------------------------- |
---|
2611 | |
---|
2612 | int lp=islp(); |
---|
2613 | |
---|
2614 | execute("ring gnir = ("+charstr(basering)+"),("+varstr(basering)+"),(C,lp);"); |
---|
2615 | op=option(get); |
---|
2616 | option(redSB); |
---|
2617 | |
---|
2618 | ideal ser=fetch(@P,ser); |
---|
2619 | |
---|
2620 | if(homo==1) |
---|
2621 | { |
---|
2622 | if(!lp) |
---|
2623 | { |
---|
2624 | ideal @j=std(fetch(@P,i),@hilb,@w); |
---|
2625 | } |
---|
2626 | else |
---|
2627 | { |
---|
2628 | ideal @j=fetch(@P,tras); |
---|
2629 | attrib(@j,"isSB",1); |
---|
2630 | } |
---|
2631 | } |
---|
2632 | else |
---|
2633 | { |
---|
2634 | if(lp&&isS) |
---|
2635 | { |
---|
2636 | ideal @j=fetch(@P,i); |
---|
2637 | attrib(@j,"isSB",1); |
---|
2638 | } |
---|
2639 | else |
---|
2640 | { |
---|
2641 | ideal @j=groebner(fetch(@P,i)); |
---|
2642 | } |
---|
2643 | } |
---|
2644 | option(set,op); |
---|
2645 | if(seri==1) |
---|
2646 | { |
---|
2647 | ideal peek=fetch(@P,peek); |
---|
2648 | attrib(peek,"isSB",1); |
---|
2649 | } |
---|
2650 | else |
---|
2651 | { |
---|
2652 | ideal peek=@j; |
---|
2653 | } |
---|
2654 | if(size(ser)==0) |
---|
2655 | { |
---|
2656 | ideal fried; |
---|
2657 | @n=size(@j); |
---|
2658 | for(@k=1;@k<=@n;@k++) |
---|
2659 | { |
---|
2660 | if(deg(lead(@j[@k]))==1) |
---|
2661 | { |
---|
2662 | fried[size(fried)+1]=@j[@k]; |
---|
2663 | @j[@k]=0; |
---|
2664 | } |
---|
2665 | } |
---|
2666 | if(size(fried)==nvars(basering)) |
---|
2667 | { |
---|
2668 | setring @P; |
---|
2669 | primary[1]=i; |
---|
2670 | primary[2]=i; |
---|
2671 | return(primary); |
---|
2672 | } |
---|
2673 | if(size(fried)>0) |
---|
2674 | { |
---|
2675 | string newva; |
---|
2676 | string newma; |
---|
2677 | for(@k=1;@k<=nvars(basering);@k++) |
---|
2678 | { |
---|
2679 | @n1=0; |
---|
2680 | for(@n=1;@n<=size(fried);@n++) |
---|
2681 | { |
---|
2682 | if(leadmonom(fried[@n])==var(@k)) |
---|
2683 | { |
---|
2684 | @n1=1; |
---|
2685 | break; |
---|
2686 | } |
---|
2687 | } |
---|
2688 | if(@n1==0) |
---|
2689 | { |
---|
2690 | newva=newva+string(var(@k))+","; |
---|
2691 | newma=newma+string(var(@k))+","; |
---|
2692 | } |
---|
2693 | else |
---|
2694 | { |
---|
2695 | newma=newma+string(0)+","; |
---|
2696 | } |
---|
2697 | } |
---|
2698 | newva[size(newva)]=")"; |
---|
2699 | newma[size(newma)]=";"; |
---|
2700 | execute("ring @deirf=("+charstr(gnir)+"),("+newva+",lp;"); |
---|
2701 | execute("map @kappa=gnir,"+newma); |
---|
2702 | ideal @j= @kappa(@j); |
---|
2703 | @j=simplify(@j,2); |
---|
2704 | attrib(@j,"isSB",1); |
---|
2705 | list pr=decomp(@j); |
---|
2706 | setring gnir; |
---|
2707 | list pr=imap(@deirf,pr); |
---|
2708 | for(@k=1;@k<=size(pr);@k++) |
---|
2709 | { |
---|
2710 | @j=pr[@k]+fried; |
---|
2711 | pr[@k]=@j; |
---|
2712 | } |
---|
2713 | setring @P; |
---|
2714 | return(imap(gnir,pr)); |
---|
2715 | } |
---|
2716 | } |
---|
2717 | //---------------------------------------------------------------- |
---|
2718 | //j is the ring |
---|
2719 | //---------------------------------------------------------------- |
---|
2720 | |
---|
2721 | if (dim(@j)==-1) |
---|
2722 | { |
---|
2723 | setring @P; |
---|
2724 | primary=ideal(1),ideal(1); |
---|
2725 | return(primary); |
---|
2726 | } |
---|
2727 | |
---|
2728 | //---------------------------------------------------------------- |
---|
2729 | // the case of one variable |
---|
2730 | //---------------------------------------------------------------- |
---|
2731 | |
---|
2732 | if(nvars(basering)==1) |
---|
2733 | { |
---|
2734 | |
---|
2735 | list fac=factor(@j[1]); |
---|
2736 | list gprimary; |
---|
2737 | for(@k=1;@k<=size(fac[1]);@k++) |
---|
2738 | { |
---|
2739 | if(@wr==0) |
---|
2740 | { |
---|
2741 | gprimary[2*@k-1]=ideal(fac[1][@k]^fac[2][@k]); |
---|
2742 | gprimary[2*@k]=ideal(fac[1][@k]); |
---|
2743 | } |
---|
2744 | else |
---|
2745 | { |
---|
2746 | gprimary[2*@k-1]=ideal(fac[1][@k]); |
---|
2747 | gprimary[2*@k]=ideal(fac[1][@k]); |
---|
2748 | } |
---|
2749 | } |
---|
2750 | setring @P; |
---|
2751 | primary=fetch(gnir,gprimary); |
---|
2752 | |
---|
2753 | return(primary); |
---|
2754 | } |
---|
2755 | |
---|
2756 | //------------------------------------------------------------------ |
---|
2757 | //the zero-dimensional case |
---|
2758 | //------------------------------------------------------------------ |
---|
2759 | if (dim(@j)==0) |
---|
2760 | { |
---|
2761 | op=option(get); |
---|
2762 | option(redSB); |
---|
2763 | |
---|
2764 | list gprimary= zero_decomp(@j,ser,@wr); |
---|
2765 | option(set,op); |
---|
2766 | setring @P; |
---|
2767 | primary=fetch(gnir,gprimary); |
---|
2768 | if(size(ser)>0) |
---|
2769 | { |
---|
2770 | primary=cleanPrimary(primary); |
---|
2771 | } |
---|
2772 | return(primary); |
---|
2773 | } |
---|
2774 | |
---|
2775 | poly @gs,@gh,@p; |
---|
2776 | string @va,quotring; |
---|
2777 | list quprimary,htprimary,collectprimary,lsau,lnew,allindep,restindep; |
---|
2778 | ideal @h; |
---|
2779 | int jdim=dim(@j); |
---|
2780 | list fett; |
---|
2781 | int lauf,di,newtest; |
---|
2782 | //------------------------------------------------------------------ |
---|
2783 | //search for a maximal independent set indep,i.e. |
---|
2784 | //look for subring such that the intersection with the ideal is zero |
---|
2785 | //j intersected with K[var(indep[3]+1),...,var(nvar] is zero, |
---|
2786 | //indep[1] is the new varstring and indep[2] the string for block-ordering |
---|
2787 | //------------------------------------------------------------------ |
---|
2788 | if(@wr!=1) |
---|
2789 | { |
---|
2790 | allindep=independSet(@j); |
---|
2791 | for(@m=1;@m<=size(allindep);@m++) |
---|
2792 | { |
---|
2793 | if(allindep[@m][3]==jdim) |
---|
2794 | { |
---|
2795 | di++; |
---|
2796 | indep[di]=allindep[@m]; |
---|
2797 | } |
---|
2798 | else |
---|
2799 | { |
---|
2800 | lauf++; |
---|
2801 | restindep[lauf]=allindep[@m]; |
---|
2802 | } |
---|
2803 | } |
---|
2804 | } |
---|
2805 | else |
---|
2806 | { |
---|
2807 | indep=maxIndependSet(@j); |
---|
2808 | } |
---|
2809 | |
---|
2810 | ideal jkeep=@j; |
---|
2811 | if(ordstr(@P)[1]=="w") |
---|
2812 | { |
---|
2813 | execute("ring @Phelp=("+charstr(gnir)+"),("+varstr(gnir)+"),("+ordstr(@P)+");"); |
---|
2814 | } |
---|
2815 | else |
---|
2816 | { |
---|
2817 | execute( "ring @Phelp=("+charstr(gnir)+"),("+varstr(gnir)+"),(C,dp);"); |
---|
2818 | } |
---|
2819 | |
---|
2820 | if(homo==1) |
---|
2821 | { |
---|
2822 | if((ordstr(@P)[3]=="d")||(ordstr(@P)[1]=="d")||(ordstr(@P)[1]=="w") |
---|
2823 | ||(ordstr(@P)[3]=="w")) |
---|
2824 | { |
---|
2825 | ideal jwork=imap(@P,tras); |
---|
2826 | attrib(jwork,"isSB",1); |
---|
2827 | } |
---|
2828 | else |
---|
2829 | { |
---|
2830 | ideal jwork=std(imap(gnir,@j),@hilb,@w); |
---|
2831 | } |
---|
2832 | |
---|
2833 | } |
---|
2834 | else |
---|
2835 | { |
---|
2836 | ideal jwork=groebner(imap(gnir,@j)); |
---|
2837 | } |
---|
2838 | list hquprimary; |
---|
2839 | poly @p,@q; |
---|
2840 | ideal @h,fac,ser; |
---|
2841 | di=dim(jwork); |
---|
2842 | keepdi=di; |
---|
2843 | |
---|
2844 | setring gnir; |
---|
2845 | for(@m=1;@m<=size(indep);@m++) |
---|
2846 | { |
---|
2847 | isat=0; |
---|
2848 | @n2=0; |
---|
2849 | if((indep[@m][1]==varstr(basering))&&(@m==1)) |
---|
2850 | //this is the good case, nothing to do, just to have the same notations |
---|
2851 | //change the ring |
---|
2852 | { |
---|
2853 | execute("ring gnir1 = ("+charstr(basering)+"),("+varstr(basering)+"),(" |
---|
2854 | +ordstr(basering)+");"); |
---|
2855 | ideal @j=fetch(gnir,@j); |
---|
2856 | attrib(@j,"isSB",1); |
---|
2857 | ideal ser=fetch(gnir,ser); |
---|
2858 | |
---|
2859 | } |
---|
2860 | else |
---|
2861 | { |
---|
2862 | @va=string(maxideal(1)); |
---|
2863 | execute("ring gnir1 = ("+charstr(basering)+"),("+indep[@m][1]+"),(" |
---|
2864 | +indep[@m][2]+");"); |
---|
2865 | execute("map phi=gnir,"+@va+";"); |
---|
2866 | op=option(get); |
---|
2867 | option(redSB); |
---|
2868 | if(homo==1) |
---|
2869 | { |
---|
2870 | ideal @j=std(phi(@j),@hilb,@w); |
---|
2871 | } |
---|
2872 | else |
---|
2873 | { |
---|
2874 | ideal @j=groebner(phi(@j)); |
---|
2875 | } |
---|
2876 | ideal ser=phi(ser); |
---|
2877 | |
---|
2878 | option(set,op); |
---|
2879 | } |
---|
2880 | if((deg(@j[1])==0)||(dim(@j)<jdim)) |
---|
2881 | { |
---|
2882 | setring gnir; |
---|
2883 | break; |
---|
2884 | } |
---|
2885 | for (lauf=1;lauf<=size(@j);lauf++) |
---|
2886 | { |
---|
2887 | fett[lauf]=size(@j[lauf]); |
---|
2888 | } |
---|
2889 | //------------------------------------------------------------------------ |
---|
2890 | //we have now the following situation: |
---|
2891 | //j intersected with K[var(nnp+1),..,var(nva)] is zero so we may pass |
---|
2892 | //to this quotientring, j is their still a standardbasis, the |
---|
2893 | //leading coefficients of the polynomials there (polynomials in |
---|
2894 | //K[var(nnp+1),..,var(nva)]) are collected in the list h, |
---|
2895 | //we need their ggt, gh, because of the following: let |
---|
2896 | //(j:gh^n)=(j:gh^infinity) then j*K(var(nnp+1),..,var(nva))[..the rest..] |
---|
2897 | //intersected with K[var(1),...,var(nva)] is (j:gh^n) |
---|
2898 | //on the other hand j=(j,gh^n) intersected with (j:gh^n) |
---|
2899 | |
---|
2900 | //------------------------------------------------------------------------ |
---|
2901 | |
---|
2902 | //arrangement for quotientring K(var(nnp+1),..,var(nva))[..the rest..] and |
---|
2903 | //map phi:K[var(1),...,var(nva)] --->K(var(nnpr+1),..,var(nva))[..rest..] |
---|
2904 | //------------------------------------------------------------------------ |
---|
2905 | |
---|
2906 | quotring=prepareQuotientring(nvars(basering)-indep[@m][3]); |
---|
2907 | |
---|
2908 | //--------------------------------------------------------------------- |
---|
2909 | //we pass to the quotientring K(var(nnp+1),..,var(nva))[..the rest..] |
---|
2910 | //--------------------------------------------------------------------- |
---|
2911 | |
---|
2912 | execute(quotring); |
---|
2913 | |
---|
2914 | // @j considered in the quotientring |
---|
2915 | ideal @j=imap(gnir1,@j); |
---|
2916 | ideal ser=imap(gnir1,ser); |
---|
2917 | |
---|
2918 | kill gnir1; |
---|
2919 | |
---|
2920 | //j is a standardbasis in the quotientring but usually not minimal |
---|
2921 | //here it becomes minimal |
---|
2922 | |
---|
2923 | @j=clearSB(@j,fett); |
---|
2924 | attrib(@j,"isSB",1); |
---|
2925 | |
---|
2926 | //we need later ggt(h[1],...)=gh for saturation |
---|
2927 | ideal @h; |
---|
2928 | if(deg(@j[1])>0) |
---|
2929 | { |
---|
2930 | for(@n=1;@n<=size(@j);@n++) |
---|
2931 | { |
---|
2932 | @h[@n]=leadcoef(@j[@n]); |
---|
2933 | } |
---|
2934 | //the primary decomposition of j*K(var(nnp+1),..,var(nva))[..the rest..] |
---|
2935 | op=option(get); |
---|
2936 | option(redSB); |
---|
2937 | |
---|
2938 | list uprimary= zero_decomp(@j,ser,@wr); |
---|
2939 | option(set,op); |
---|
2940 | } |
---|
2941 | else |
---|
2942 | { |
---|
2943 | list uprimary; |
---|
2944 | uprimary[1]=ideal(1); |
---|
2945 | uprimary[2]=ideal(1); |
---|
2946 | } |
---|
2947 | //we need the intersection of the ideals in the list quprimary with the |
---|
2948 | //polynomialring, i.e. let q=(f1,...,fr) in the quotientring such an ideal |
---|
2949 | //but fi polynomials, then the intersection of q with the polynomialring |
---|
2950 | //is the saturation of the ideal generated by f1,...,fr with respect to |
---|
2951 | //h which is the lcm of the leading coefficients of the fi considered in |
---|
2952 | //in the quotientring: this is coded in saturn |
---|
2953 | |
---|
2954 | list saturn; |
---|
2955 | ideal hpl; |
---|
2956 | |
---|
2957 | for(@n=1;@n<=size(uprimary);@n++) |
---|
2958 | { |
---|
2959 | uprimary[@n]=interred(uprimary[@n]); // temporary fix |
---|
2960 | hpl=0; |
---|
2961 | for(@n1=1;@n1<=size(uprimary[@n]);@n1++) |
---|
2962 | { |
---|
2963 | hpl=hpl,leadcoef(uprimary[@n][@n1]); |
---|
2964 | } |
---|
2965 | saturn[@n]=hpl; |
---|
2966 | } |
---|
2967 | |
---|
2968 | //-------------------------------------------------------------------- |
---|
2969 | //we leave the quotientring K(var(nnp+1),..,var(nva))[..the rest..] |
---|
2970 | //back to the polynomialring |
---|
2971 | //--------------------------------------------------------------------- |
---|
2972 | setring gnir; |
---|
2973 | |
---|
2974 | collectprimary=imap(quring,uprimary); |
---|
2975 | lsau=imap(quring,saturn); |
---|
2976 | @h=imap(quring,@h); |
---|
2977 | |
---|
2978 | kill quring; |
---|
2979 | |
---|
2980 | @n2=size(quprimary); |
---|
2981 | @n3=@n2; |
---|
2982 | |
---|
2983 | for(@n1=1;@n1<=size(collectprimary)/2;@n1++) |
---|
2984 | { |
---|
2985 | if(deg(collectprimary[2*@n1][1])>0) |
---|
2986 | { |
---|
2987 | @n2++; |
---|
2988 | quprimary[@n2]=collectprimary[2*@n1-1]; |
---|
2989 | lnew[@n2]=lsau[2*@n1-1]; |
---|
2990 | @n2++; |
---|
2991 | lnew[@n2]=lsau[2*@n1]; |
---|
2992 | quprimary[@n2]=collectprimary[2*@n1]; |
---|
2993 | } |
---|
2994 | } |
---|
2995 | |
---|
2996 | //here the intersection with the polynomialring |
---|
2997 | //mentioned above is really computed |
---|
2998 | for(@n=@n3/2+1;@n<=@n2/2;@n++) |
---|
2999 | { |
---|
3000 | if(specialIdealsEqual(quprimary[2*@n-1],quprimary[2*@n])) |
---|
3001 | { |
---|
3002 | quprimary[2*@n-1]=sat2(quprimary[2*@n-1],lnew[2*@n-1])[1]; |
---|
3003 | quprimary[2*@n]=quprimary[2*@n-1]; |
---|
3004 | } |
---|
3005 | else |
---|
3006 | { |
---|
3007 | if(@wr==0) |
---|
3008 | { |
---|
3009 | quprimary[2*@n-1]=sat2(quprimary[2*@n-1],lnew[2*@n-1])[1]; |
---|
3010 | } |
---|
3011 | quprimary[2*@n]=sat2(quprimary[2*@n],lnew[2*@n])[1]; |
---|
3012 | } |
---|
3013 | } |
---|
3014 | |
---|
3015 | if(size(@h)>0) |
---|
3016 | { |
---|
3017 | //--------------------------------------------------------------- |
---|
3018 | //we change to @Phelp to have the ordering dp for saturation |
---|
3019 | //--------------------------------------------------------------- |
---|
3020 | setring @Phelp; |
---|
3021 | @h=imap(gnir,@h); |
---|
3022 | if(@wr!=1) |
---|
3023 | { |
---|
3024 | @q=minSat(jwork,@h)[2]; |
---|
3025 | } |
---|
3026 | else |
---|
3027 | { |
---|
3028 | fac=ideal(0); |
---|
3029 | for(lauf=1;lauf<=ncols(@h);lauf++) |
---|
3030 | { |
---|
3031 | if(deg(@h[lauf])>0) |
---|
3032 | { |
---|
3033 | fac=fac+factorize(@h[lauf],1); |
---|
3034 | } |
---|
3035 | } |
---|
3036 | fac=simplify(fac,4); |
---|
3037 | @q=1; |
---|
3038 | for(lauf=1;lauf<=size(fac);lauf++) |
---|
3039 | { |
---|
3040 | @q=@q*fac[lauf]; |
---|
3041 | } |
---|
3042 | } |
---|
3043 | jwork=std(jwork,@q); |
---|
3044 | keepdi=dim(jwork); |
---|
3045 | if(keepdi<di) |
---|
3046 | { |
---|
3047 | setring gnir; |
---|
3048 | @j=imap(@Phelp,jwork); |
---|
3049 | break; |
---|
3050 | } |
---|
3051 | if(homo==1) |
---|
3052 | { |
---|
3053 | @hilb=hilb(jwork,1,@w); |
---|
3054 | } |
---|
3055 | |
---|
3056 | setring gnir; |
---|
3057 | @j=imap(@Phelp,jwork); |
---|
3058 | } |
---|
3059 | } |
---|
3060 | |
---|
3061 | if((size(quprimary)==0)&&(@wr==1)) |
---|
3062 | { |
---|
3063 | @j=ideal(1); |
---|
3064 | quprimary[1]=ideal(1); |
---|
3065 | quprimary[2]=ideal(1); |
---|
3066 | } |
---|
3067 | if((size(quprimary)==0)) |
---|
3068 | { |
---|
3069 | keepdi=di-1; |
---|
3070 | } |
---|
3071 | //--------------------------------------------------------------- |
---|
3072 | //notice that j=sat(j,gh) intersected with (j,gh^n) |
---|
3073 | //we finished with sat(j,gh) and have to start with (j,gh^n) |
---|
3074 | //--------------------------------------------------------------- |
---|
3075 | if((deg(@j[1])!=0)&&(@wr!=1)) |
---|
3076 | { |
---|
3077 | if(size(quprimary)>0) |
---|
3078 | { |
---|
3079 | setring @Phelp; |
---|
3080 | ser=imap(gnir,ser); |
---|
3081 | hquprimary=imap(gnir,quprimary); |
---|
3082 | if(@wr==0) |
---|
3083 | { |
---|
3084 | ideal htest=hquprimary[1]; |
---|
3085 | for (@n1=2;@n1<=size(hquprimary)/2;@n1++) |
---|
3086 | { |
---|
3087 | htest=intersect(htest,hquprimary[2*@n1-1]); |
---|
3088 | } |
---|
3089 | } |
---|
3090 | else |
---|
3091 | { |
---|
3092 | ideal htest=hquprimary[2]; |
---|
3093 | |
---|
3094 | for (@n1=2;@n1<=size(hquprimary)/2;@n1++) |
---|
3095 | { |
---|
3096 | htest=intersect(htest,hquprimary[2*@n1]); |
---|
3097 | } |
---|
3098 | } |
---|
3099 | |
---|
3100 | if(size(ser)>0) |
---|
3101 | { |
---|
3102 | ser=intersect(htest,ser); |
---|
3103 | } |
---|
3104 | else |
---|
3105 | { |
---|
3106 | ser=htest; |
---|
3107 | } |
---|
3108 | setring gnir; |
---|
3109 | ser=imap(@Phelp,ser); |
---|
3110 | } |
---|
3111 | if(size(reduce(ser,peek,1))!=0) |
---|
3112 | { |
---|
3113 | for(@m=1;@m<=size(restindep);@m++) |
---|
3114 | { |
---|
3115 | // if(restindep[@m][3]>=keepdi) |
---|
3116 | // { |
---|
3117 | isat=0; |
---|
3118 | @n2=0; |
---|
3119 | |
---|
3120 | if(restindep[@m][1]==varstr(basering)) |
---|
3121 | //the good case, nothing to do, just to have the same notations |
---|
3122 | //change the ring |
---|
3123 | { |
---|
3124 | execute("ring gnir1 = ("+charstr(basering)+"),("+ |
---|
3125 | varstr(basering)+"),("+ordstr(basering)+");"); |
---|
3126 | ideal @j=fetch(gnir,jkeep); |
---|
3127 | attrib(@j,"isSB",1); |
---|
3128 | } |
---|
3129 | else |
---|
3130 | { |
---|
3131 | @va=string(maxideal(1)); |
---|
3132 | execute("ring gnir1 = ("+charstr(basering)+"),("+ |
---|
3133 | restindep[@m][1]+"),(" +restindep[@m][2]+");"); |
---|
3134 | execute("map phi=gnir,"+@va+";"); |
---|
3135 | op=option(get); |
---|
3136 | option(redSB); |
---|
3137 | if(homo==1) |
---|
3138 | { |
---|
3139 | ideal @j=std(phi(jkeep),keephilb,@w); |
---|
3140 | } |
---|
3141 | else |
---|
3142 | { |
---|
3143 | ideal @j=groebner(phi(jkeep)); |
---|
3144 | } |
---|
3145 | ideal ser=phi(ser); |
---|
3146 | option(set,op); |
---|
3147 | } |
---|
3148 | |
---|
3149 | for (lauf=1;lauf<=size(@j);lauf++) |
---|
3150 | { |
---|
3151 | fett[lauf]=size(@j[lauf]); |
---|
3152 | } |
---|
3153 | //------------------------------------------------------------------ |
---|
3154 | //we have now the following situation: |
---|
3155 | //j intersected with K[var(nnp+1),..,var(nva)] is zero so we may |
---|
3156 | //pass to this quotientring, j is their still a standardbasis, the |
---|
3157 | //leading coefficients of the polynomials there (polynomials in |
---|
3158 | //K[var(nnp+1),..,var(nva)]) are collected in the list h, |
---|
3159 | //we need their ggt, gh, because of the following: |
---|
3160 | //let (j:gh^n)=(j:gh^infinity) then |
---|
3161 | //j*K(var(nnp+1),..,var(nva))[..the rest..] |
---|
3162 | //intersected with K[var(1),...,var(nva)] is (j:gh^n) |
---|
3163 | //on the other hand j=(j,gh^n) intersected with (j:gh^n) |
---|
3164 | |
---|
3165 | //------------------------------------------------------------------ |
---|
3166 | |
---|
3167 | //the arrangement for the quotientring |
---|
3168 | // K(var(nnp+1),..,var(nva))[..the rest..] |
---|
3169 | //and the map phi:K[var(1),...,var(nva)] ----> |
---|
3170 | //--->K(var(nnpr+1),..,var(nva))[..the rest..] |
---|
3171 | //------------------------------------------------------------------ |
---|
3172 | |
---|
3173 | quotring=prepareQuotientring(nvars(basering)-restindep[@m][3]); |
---|
3174 | |
---|
3175 | //------------------------------------------------------------------ |
---|
3176 | //we pass to the quotientring K(var(nnp+1),..,var(nva))[..rest..] |
---|
3177 | //------------------------------------------------------------------ |
---|
3178 | |
---|
3179 | execute(quotring); |
---|
3180 | |
---|
3181 | // @j considered in the quotientring |
---|
3182 | ideal @j=imap(gnir1,@j); |
---|
3183 | ideal ser=imap(gnir1,ser); |
---|
3184 | |
---|
3185 | kill gnir1; |
---|
3186 | |
---|
3187 | //j is a standardbasis in the quotientring but usually not minimal |
---|
3188 | //here it becomes minimal |
---|
3189 | @j=clearSB(@j,fett); |
---|
3190 | attrib(@j,"isSB",1); |
---|
3191 | |
---|
3192 | //we need later ggt(h[1],...)=gh for saturation |
---|
3193 | ideal @h; |
---|
3194 | |
---|
3195 | for(@n=1;@n<=size(@j);@n++) |
---|
3196 | { |
---|
3197 | @h[@n]=leadcoef(@j[@n]); |
---|
3198 | } |
---|
3199 | //the primary decomposition of j*K(var(nnp+1),..,var(nva))[..rest..] |
---|
3200 | |
---|
3201 | op=option(get); |
---|
3202 | option(redSB); |
---|
3203 | list uprimary= zero_decomp(@j,ser,@wr); |
---|
3204 | option(set,op); |
---|
3205 | |
---|
3206 | //we need the intersection of the ideals in the list quprimary with |
---|
3207 | //the polynomialring, i.e. let q=(f1,...,fr) in the quotientring |
---|
3208 | //such an ideal but fi polynomials, then the intersection of q with |
---|
3209 | //the polynomialring is the saturation of the ideal generated by |
---|
3210 | //f1,...,fr with respect toh which is the lcm of the leading |
---|
3211 | //coefficients of the fi considered in the quotientring: |
---|
3212 | //this is coded in saturn |
---|
3213 | |
---|
3214 | list saturn; |
---|
3215 | ideal hpl; |
---|
3216 | |
---|
3217 | for(@n=1;@n<=size(uprimary);@n++) |
---|
3218 | { |
---|
3219 | hpl=0; |
---|
3220 | for(@n1=1;@n1<=size(uprimary[@n]);@n1++) |
---|
3221 | { |
---|
3222 | hpl=hpl,leadcoef(uprimary[@n][@n1]); |
---|
3223 | } |
---|
3224 | saturn[@n]=hpl; |
---|
3225 | } |
---|
3226 | //------------------------------------------------------------------ |
---|
3227 | //we leave the quotientring K(var(nnp+1),..,var(nva))[..rest..] |
---|
3228 | //back to the polynomialring |
---|
3229 | //------------------------------------------------------------------ |
---|
3230 | setring gnir; |
---|
3231 | |
---|
3232 | collectprimary=imap(quring,uprimary); |
---|
3233 | lsau=imap(quring,saturn); |
---|
3234 | @h=imap(quring,@h); |
---|
3235 | |
---|
3236 | kill quring; |
---|
3237 | |
---|
3238 | |
---|
3239 | @n2=size(quprimary); |
---|
3240 | @n3=@n2; |
---|
3241 | |
---|
3242 | for(@n1=1;@n1<=size(collectprimary)/2;@n1++) |
---|
3243 | { |
---|
3244 | if(deg(collectprimary[2*@n1][1])>0) |
---|
3245 | { |
---|
3246 | @n2++; |
---|
3247 | quprimary[@n2]=collectprimary[2*@n1-1]; |
---|
3248 | lnew[@n2]=lsau[2*@n1-1]; |
---|
3249 | @n2++; |
---|
3250 | lnew[@n2]=lsau[2*@n1]; |
---|
3251 | quprimary[@n2]=collectprimary[2*@n1]; |
---|
3252 | } |
---|
3253 | } |
---|
3254 | |
---|
3255 | |
---|
3256 | //here the intersection with the polynomialring |
---|
3257 | //mentioned above is really computed |
---|
3258 | |
---|
3259 | for(@n=@n3/2+1;@n<=@n2/2;@n++) |
---|
3260 | { |
---|
3261 | if(specialIdealsEqual(quprimary[2*@n-1],quprimary[2*@n])) |
---|
3262 | { |
---|
3263 | quprimary[2*@n-1]=sat2(quprimary[2*@n-1],lnew[2*@n-1])[1]; |
---|
3264 | quprimary[2*@n]=quprimary[2*@n-1]; |
---|
3265 | } |
---|
3266 | else |
---|
3267 | { |
---|
3268 | if(@wr==0) |
---|
3269 | { |
---|
3270 | quprimary[2*@n-1]=sat2(quprimary[2*@n-1],lnew[2*@n-1])[1]; |
---|
3271 | } |
---|
3272 | quprimary[2*@n]=sat2(quprimary[2*@n],lnew[2*@n])[1]; |
---|
3273 | } |
---|
3274 | } |
---|
3275 | if(@n2>=@n3+2) |
---|
3276 | { |
---|
3277 | setring @Phelp; |
---|
3278 | ser=imap(gnir,ser); |
---|
3279 | hquprimary=imap(gnir,quprimary); |
---|
3280 | for(@n=@n3/2+1;@n<=@n2/2;@n++) |
---|
3281 | { |
---|
3282 | if(@wr==0) |
---|
3283 | { |
---|
3284 | ser=intersect(ser,hquprimary[2*@n-1]); |
---|
3285 | } |
---|
3286 | else |
---|
3287 | { |
---|
3288 | ser=intersect(ser,hquprimary[2*@n]); |
---|
3289 | } |
---|
3290 | } |
---|
3291 | setring gnir; |
---|
3292 | ser=imap(@Phelp,ser); |
---|
3293 | } |
---|
3294 | |
---|
3295 | // } |
---|
3296 | } |
---|
3297 | if(size(reduce(ser,peek,1))!=0) |
---|
3298 | { |
---|
3299 | if(@wr>0) |
---|
3300 | { |
---|
3301 | htprimary=decomp(@j,@wr,peek,ser); |
---|
3302 | } |
---|
3303 | else |
---|
3304 | { |
---|
3305 | htprimary=decomp(@j,peek,ser); |
---|
3306 | } |
---|
3307 | // here we collect now both results primary(sat(j,gh)) |
---|
3308 | // and primary(j,gh^n) |
---|
3309 | @n=size(quprimary); |
---|
3310 | for (@k=1;@k<=size(htprimary);@k++) |
---|
3311 | { |
---|
3312 | quprimary[@n+@k]=htprimary[@k]; |
---|
3313 | } |
---|
3314 | } |
---|
3315 | } |
---|
3316 | |
---|
3317 | } |
---|
3318 | //--------------------------------------------------------------------------- |
---|
3319 | //back to the ring we started with |
---|
3320 | //the final result: primary |
---|
3321 | //--------------------------------------------------------------------------- |
---|
3322 | |
---|
3323 | setring @P; |
---|
3324 | primary=imap(gnir,quprimary); |
---|
3325 | return(primary); |
---|
3326 | } |
---|
3327 | |
---|
3328 | |
---|
3329 | example |
---|
3330 | { "EXAMPLE:"; echo = 2; |
---|
3331 | ring r = 32003,(x,y,z),lp; |
---|
3332 | poly p = z2+1; |
---|
3333 | poly q = z4+2; |
---|
3334 | ideal i = p^2*q^3,(y-z3)^3,(x-yz+z4)^4; |
---|
3335 | list pr= decomp(i); |
---|
3336 | pr; |
---|
3337 | testPrimary( pr, i); |
---|
3338 | } |
---|
3339 | |
---|
3340 | /////////////////////////////////////////////////////////////////////////////// |
---|
3341 | static proc powerCoeffs(poly f,int e) |
---|
3342 | //computes a polynomial with the same monomials as f but coefficients |
---|
3343 | //the p^e th power of the coefficients of f |
---|
3344 | { |
---|
3345 | int i; |
---|
3346 | poly g; |
---|
3347 | int ex=char(basering)^e; |
---|
3348 | for(i=1;i<=size(f);i++) |
---|
3349 | { |
---|
3350 | g=g+leadcoef(f[i])^ex*leadmonom(f[i]); |
---|
3351 | } |
---|
3352 | return(g); |
---|
3353 | } |
---|
3354 | /////////////////////////////////////////////////////////////////////////////// |
---|
3355 | |
---|
3356 | proc sep(poly f,int i, list #) |
---|
3357 | "USAGE: input: a polynomial f depending on the i-th variable and optional |
---|
3358 | an integer k considering the polynomial f defined over Fp(t1,...,tm) |
---|
3359 | as polynomial over Fp(t(1)^(p^-k),...,t(m)^(p^-k)) |
---|
3360 | RETURN: the separabel part of f as polynomial in Fp(t1,...,tm) |
---|
3361 | and an integer k to indicate that f should be considerd |
---|
3362 | as polynomial over Fp(t(1)^(p^-k),...,t(m)^(p^-k)) |
---|
3363 | EXAMPLE: example sep; shows an example |
---|
3364 | { |
---|
3365 | def R=basering; |
---|
3366 | int k; |
---|
3367 | if(size(#)>0){k=#[1];} |
---|
3368 | |
---|
3369 | |
---|
3370 | poly h=gcd(f,diff(f,var(i))); |
---|
3371 | if((reduce(f,std(h))!=0)||(reduce(diff(f,var(i)),std(h))!=0)) |
---|
3372 | { |
---|
3373 | ERROR("FEHLER IN GCD"); |
---|
3374 | } |
---|
3375 | poly g1=lift(h,f)[1][1]; // f/h |
---|
3376 | poly h1; |
---|
3377 | |
---|
3378 | while(h!=h1) |
---|
3379 | { |
---|
3380 | h1=h; |
---|
3381 | h=gcd(h,diff(h,var(i))); |
---|
3382 | } |
---|
3383 | |
---|
3384 | if(deg(h1)==0){return(list(g1,k));} //in characteristic 0 we return here |
---|
3385 | |
---|
3386 | k++; |
---|
3387 | |
---|
3388 | ideal ma=maxideal(1); |
---|
3389 | ma[i]=var(i)^char(R); |
---|
3390 | map phi=R,ma; |
---|
3391 | ideal hh=h; //this is technical because preimage works only for ideals |
---|
3392 | |
---|
3393 | poly u=preimage(R,phi,hh)[1]; //h=u(x(i)^p) |
---|
3394 | |
---|
3395 | list g2=sep(u,i,k); //we consider u(t(1)^(p^-1),...,t(m)^(p^-1)) |
---|
3396 | g1=powerCoeffs(g1,g2[2]-k+1); //to have g1 over the same field as g2[1] |
---|
3397 | |
---|
3398 | list g3=sep(g1*g2[1],i,g2[2]); |
---|
3399 | return(g3); |
---|
3400 | } |
---|
3401 | example |
---|
3402 | { "EXAMPLE:"; echo = 2; |
---|
3403 | ring R=(5,t,s),(x,y,z),dp; |
---|
3404 | poly f=(x^25-t*x^5+t)*(x^3+s); |
---|
3405 | sep(f,1); |
---|
3406 | } |
---|
3407 | |
---|
3408 | /////////////////////////////////////////////////////////////////////////////// |
---|
3409 | proc zeroRad(ideal I,list #) |
---|
3410 | "USAGE: zeroRad(I) , I a zero-dimensional ideal |
---|
3411 | RETURN: the radical of I |
---|
3412 | NOTE: Algorithm of Kemper |
---|
3413 | EXAMPLE: example zeroRad; shows an example |
---|
3414 | { |
---|
3415 | if(homog(I)==1){return(maxideal(1));} |
---|
3416 | //I needs to be a reduced standard basis |
---|
3417 | def R=basering; |
---|
3418 | int m=npars(R); |
---|
3419 | int n=nvars(R); |
---|
3420 | int p=char(R); |
---|
3421 | int d=vdim(I); |
---|
3422 | int i,k; |
---|
3423 | list l; |
---|
3424 | if(((p==0)||(p>d))&&(d==deg(I[1]))) |
---|
3425 | { |
---|
3426 | intvec e=leadexp(I[1]); |
---|
3427 | for(i=1;i<=nvars(basering);i++) |
---|
3428 | { |
---|
3429 | if(e[i]!=0) break; |
---|
3430 | } |
---|
3431 | I[1]=sep(I[1],i)[1]; |
---|
3432 | return(interred(I)); |
---|
3433 | } |
---|
3434 | intvec op=option(get); |
---|
3435 | |
---|
3436 | option(redSB); |
---|
3437 | ideal F=finduni(I);//F[i] generates I intersected with K[var(i)] |
---|
3438 | |
---|
3439 | option(set,op); |
---|
3440 | if(size(#)>0){I=#[1];} |
---|
3441 | |
---|
3442 | for(i=1;i<=n;i++) |
---|
3443 | { |
---|
3444 | l[i]=sep(F[i],i); |
---|
3445 | F[i]=l[i][1]; |
---|
3446 | if(l[i][2]>k){k=l[i][2];} //computation of the maximal k |
---|
3447 | } |
---|
3448 | |
---|
3449 | if((k==0)||(m==0)){return(interred(I+F));} //the separable case |
---|
3450 | |
---|
3451 | for(i=1;i<=n;i++) //consider all polynomials over |
---|
3452 | { //Fp(t(1)^(p^-k),...,t(m)^(p^-k)) |
---|
3453 | F[i]=powerCoeffs(F[i],k-l[i][2]); |
---|
3454 | } |
---|
3455 | |
---|
3456 | string cR="ring @R="+string(p)+",("+parstr(R)+","+varstr(R)+"),dp;"; |
---|
3457 | execute(cR); |
---|
3458 | ideal F=imap(R,F); |
---|
3459 | |
---|
3460 | string nR="ring @S="+string(p)+",(y(1..m),"+varstr(R)+","+parstr(R)+"),dp;"; |
---|
3461 | execute(nR); |
---|
3462 | |
---|
3463 | ideal G=fetch(@R,F); //G[i](t(1)^(p^-k),...,t(m)^(p^-k),x(i))=sep(F[i]) |
---|
3464 | |
---|
3465 | ideal I=imap(R,I); |
---|
3466 | ideal J=I+G; |
---|
3467 | poly el=1; |
---|
3468 | k=p^k; |
---|
3469 | for(i=1;i<=m;i++) |
---|
3470 | { |
---|
3471 | J=J,var(i)^k-var(m+n+i); |
---|
3472 | el=el*y(i); |
---|
3473 | } |
---|
3474 | |
---|
3475 | J=eliminate(J,el); |
---|
3476 | setring R; |
---|
3477 | ideal J=imap(@S,J); |
---|
3478 | return(J); |
---|
3479 | } |
---|
3480 | example |
---|
3481 | { "EXAMPLE:"; echo = 2; |
---|
3482 | ring R=(5,t),(x,y),dp; |
---|
3483 | ideal I=x^5-t,y^5-t; |
---|
3484 | zeroRad(I); |
---|
3485 | } |
---|
3486 | |
---|
3487 | /////////////////////////////////////////////////////////////////////////////// |
---|
3488 | static proc radicalKL (ideal i,ideal ser,list #) |
---|
3489 | { |
---|
3490 | attrib(i,"isSB",1); // i needs to be a reduced standard basis |
---|
3491 | list indep,fett; |
---|
3492 | intvec @w,@hilb,op; |
---|
3493 | int @wr,@n,@m,lauf,di; |
---|
3494 | ideal fac,@h,collectrad,lsau; |
---|
3495 | poly @q; |
---|
3496 | string @va,quotring; |
---|
3497 | |
---|
3498 | def @P = basering; |
---|
3499 | int jdim=dim(i); |
---|
3500 | int homo=homog(i); |
---|
3501 | ideal rad=ideal(1); |
---|
3502 | ideal te=ser; |
---|
3503 | if(size(#)>0) |
---|
3504 | { |
---|
3505 | @wr=#[1]; |
---|
3506 | } |
---|
3507 | if(homo==1) |
---|
3508 | { |
---|
3509 | for(@n=1;@n<=nvars(basering);@n++) |
---|
3510 | { |
---|
3511 | @w[@n]=ord(var(@n)); |
---|
3512 | } |
---|
3513 | @hilb=hilb(i,1,@w); |
---|
3514 | } |
---|
3515 | |
---|
3516 | |
---|
3517 | //--------------------------------------------------------------------------- |
---|
3518 | //j is the ring |
---|
3519 | //--------------------------------------------------------------------------- |
---|
3520 | |
---|
3521 | if (jdim==-1) |
---|
3522 | { |
---|
3523 | |
---|
3524 | return(ideal(1)); |
---|
3525 | } |
---|
3526 | |
---|
3527 | //--------------------------------------------------------------------------- |
---|
3528 | //the zero-dimensional case |
---|
3529 | //--------------------------------------------------------------------------- |
---|
3530 | |
---|
3531 | if (jdim==0) |
---|
3532 | { |
---|
3533 | return(zeroRad(i)); |
---|
3534 | } |
---|
3535 | //------------------------------------------------------------------------- |
---|
3536 | //search for a maximal independent set indep,i.e. |
---|
3537 | //look for subring such that the intersection with the ideal is zero |
---|
3538 | //j intersected with K[var(indep[3]+1),...,var(nvar] is zero, |
---|
3539 | //indep[1] is the new varstring, indep[2] the string for the block-ordering |
---|
3540 | //------------------------------------------------------------------------- |
---|
3541 | |
---|
3542 | indep=maxIndependSet(i); |
---|
3543 | |
---|
3544 | for(@m=1;@m<=size(indep);@m++) |
---|
3545 | { |
---|
3546 | if((indep[@m][1]==varstr(basering))&&(@m==1)) |
---|
3547 | //this is the good case, nothing to do, just to have the same notations |
---|
3548 | //change the ring |
---|
3549 | { |
---|
3550 | execute("ring gnir1 = ("+charstr(basering)+"),("+varstr(basering)+"),(" |
---|
3551 | +ordstr(basering)+");"); |
---|
3552 | ideal @j=fetch(@P,i); |
---|
3553 | attrib(@j,"isSB",1); |
---|
3554 | } |
---|
3555 | else |
---|
3556 | { |
---|
3557 | @va=string(maxideal(1)); |
---|
3558 | execute("ring gnir1 = ("+charstr(basering)+"),("+indep[@m][1]+"),(" |
---|
3559 | +indep[@m][2]+");"); |
---|
3560 | execute("map phi=@P,"+@va+";"); |
---|
3561 | if(homo==1) |
---|
3562 | { |
---|
3563 | ideal @j=std(phi(i),@hilb,@w); |
---|
3564 | } |
---|
3565 | else |
---|
3566 | { |
---|
3567 | ideal @j=groebner(phi(i)); |
---|
3568 | } |
---|
3569 | } |
---|
3570 | if((deg(@j[1])==0)||(dim(@j)<jdim)) |
---|
3571 | { |
---|
3572 | setring @P; |
---|
3573 | break; |
---|
3574 | } |
---|
3575 | for (lauf=1;lauf<=size(@j);lauf++) |
---|
3576 | { |
---|
3577 | fett[lauf]=size(@j[lauf]); |
---|
3578 | } |
---|
3579 | //------------------------------------------------------------------------ |
---|
3580 | //we have now the following situation: |
---|
3581 | //j intersected with K[var(nnp+1),..,var(nva)] is zero so we may pass |
---|
3582 | //to this quotientring, j is their still a standardbasis, the |
---|
3583 | //leading coefficients of the polynomials there (polynomials in |
---|
3584 | //K[var(nnp+1),..,var(nva)]) are collected in the list h, |
---|
3585 | //we need their ggt, gh, because of the following: |
---|
3586 | //let (j:gh^n)=(j:gh^infinity) then j*K(var(nnp+1),..,var(nva))[..rest..] |
---|
3587 | //intersected with K[var(1),...,var(nva)] is (j:gh^n) |
---|
3588 | //on the other hand j=(j,gh^n) intersected with (j:gh^n) |
---|
3589 | |
---|
3590 | //------------------------------------------------------------------------ |
---|
3591 | |
---|
3592 | //the arrangement for the quotientring K(var(nnp+1),..,var(nva))[..rest..] |
---|
3593 | //and the map phi:K[var(1),...,var(nva)] -----> |
---|
3594 | //K(var(nnpr+1),..,var(nva))[..the rest..] |
---|
3595 | //------------------------------------------------------------------------ |
---|
3596 | quotring=prepareQuotientring(nvars(basering)-indep[@m][3]); |
---|
3597 | |
---|
3598 | //------------------------------------------------------------------------ |
---|
3599 | //we pass to the quotientring K(var(nnp+1),..,var(nva))[..the rest..] |
---|
3600 | //------------------------------------------------------------------------ |
---|
3601 | |
---|
3602 | execute(quotring); |
---|
3603 | |
---|
3604 | // @j considered in the quotientring |
---|
3605 | ideal @j=imap(gnir1,@j); |
---|
3606 | |
---|
3607 | kill gnir1; |
---|
3608 | |
---|
3609 | //j is a standardbasis in the quotientring but usually not minimal |
---|
3610 | //here it becomes minimal |
---|
3611 | |
---|
3612 | @j=clearSB(@j,fett); |
---|
3613 | |
---|
3614 | //we need later ggt(h[1],...)=gh for saturation |
---|
3615 | ideal @h; |
---|
3616 | if(deg(@j[1])>0) |
---|
3617 | { |
---|
3618 | for(@n=1;@n<=size(@j);@n++) |
---|
3619 | { |
---|
3620 | @h[@n]=leadcoef(@j[@n]); |
---|
3621 | } |
---|
3622 | op=option(get); |
---|
3623 | option(redSB); |
---|
3624 | @j=interred(@j); //to obtain a reduced standardbasis |
---|
3625 | attrib(@j,"isSB",1); |
---|
3626 | option(set,op); |
---|
3627 | |
---|
3628 | ideal zero_rad= zeroRad(@j); |
---|
3629 | } |
---|
3630 | else |
---|
3631 | { |
---|
3632 | ideal zero_rad=ideal(1); |
---|
3633 | } |
---|
3634 | |
---|
3635 | //we need the intersection of the ideals in the list quprimary with the |
---|
3636 | //polynomialring, i.e. let q=(f1,...,fr) in the quotientring such an ideal |
---|
3637 | //but fi polynomials, then the intersection of q with the polynomialring |
---|
3638 | //is the saturation of the ideal generated by f1,...,fr with respect to |
---|
3639 | //h which is the lcm of the leading coefficients of the fi considered in |
---|
3640 | //the quotientring: this is coded in saturn |
---|
3641 | |
---|
3642 | zero_rad=std(zero_rad); |
---|
3643 | |
---|
3644 | ideal hpl; |
---|
3645 | |
---|
3646 | for(@n=1;@n<=size(zero_rad);@n++) |
---|
3647 | { |
---|
3648 | hpl=hpl,leadcoef(zero_rad[@n]); |
---|
3649 | } |
---|
3650 | |
---|
3651 | //------------------------------------------------------------------------ |
---|
3652 | //we leave the quotientring K(var(nnp+1),..,var(nva))[..the rest..] |
---|
3653 | //back to the polynomialring |
---|
3654 | //------------------------------------------------------------------------ |
---|
3655 | setring @P; |
---|
3656 | |
---|
3657 | collectrad=imap(quring,zero_rad); |
---|
3658 | lsau=simplify(imap(quring,hpl),2); |
---|
3659 | @h=imap(quring,@h); |
---|
3660 | |
---|
3661 | kill quring; |
---|
3662 | |
---|
3663 | |
---|
3664 | //here the intersection with the polynomialring |
---|
3665 | //mentioned above is really computed |
---|
3666 | |
---|
3667 | collectrad=sat2(collectrad,lsau)[1]; |
---|
3668 | if(deg(@h[1])>=0) |
---|
3669 | { |
---|
3670 | fac=ideal(0); |
---|
3671 | for(lauf=1;lauf<=ncols(@h);lauf++) |
---|
3672 | { |
---|
3673 | if(deg(@h[lauf])>0) |
---|
3674 | { |
---|
3675 | fac=fac+factorize(@h[lauf],1); |
---|
3676 | } |
---|
3677 | } |
---|
3678 | fac=simplify(fac,4); |
---|
3679 | @q=1; |
---|
3680 | for(lauf=1;lauf<=size(fac);lauf++) |
---|
3681 | { |
---|
3682 | @q=@q*fac[lauf]; |
---|
3683 | } |
---|
3684 | op=option(get); |
---|
3685 | option(returnSB); |
---|
3686 | option(redSB); |
---|
3687 | i=quotient(i+ideal(@q),rad); |
---|
3688 | attrib(i,"isSB",1); |
---|
3689 | option(set,op); |
---|
3690 | |
---|
3691 | } |
---|
3692 | if((deg(rad[1])>0)&&(deg(collectrad[1])>0)) |
---|
3693 | { |
---|
3694 | rad=intersect(rad,collectrad); |
---|
3695 | te=intersect(te,collectrad); |
---|
3696 | te=simplify(reduce(te,i,1),2); |
---|
3697 | } |
---|
3698 | else |
---|
3699 | { |
---|
3700 | if(deg(collectrad[1])>0) |
---|
3701 | { |
---|
3702 | rad=collectrad; |
---|
3703 | te=intersect(te,collectrad); |
---|
3704 | te=simplify(reduce(te,i,1),2); |
---|
3705 | } |
---|
3706 | } |
---|
3707 | |
---|
3708 | if((dim(i)<jdim)||(size(te)==0)) |
---|
3709 | { |
---|
3710 | break; |
---|
3711 | } |
---|
3712 | if(homo==1) |
---|
3713 | { |
---|
3714 | @hilb=hilb(i,1,@w); |
---|
3715 | } |
---|
3716 | } |
---|
3717 | if(((@wr==1)&&(dim(i)<jdim))||(deg(i[1])==0)||(size(te)==0)) |
---|
3718 | { |
---|
3719 | return(rad); |
---|
3720 | } |
---|
3721 | rad=intersect(rad,radicalKL(i,ideal(1),@wr)); |
---|
3722 | return(rad); |
---|
3723 | } |
---|
3724 | |
---|
3725 | /////////////////////////////////////////////////////////////////////////////// |
---|
3726 | |
---|
3727 | proc radicalEHV(ideal i) |
---|
3728 | "USAGE: radicalEHV(i); i ideal. |
---|
3729 | RETURN: ideal, the radical of i. |
---|
3730 | NOTE: Uses the algorithm of Eisenbud/Huneke/Vasconcelos, which |
---|
3731 | reduces the computation to the complete intersection case, |
---|
3732 | by taking, in the general case, a generic linear combination |
---|
3733 | of the input. |
---|
3734 | Works only in characteristic 0 or p large. |
---|
3735 | EXAMPLE: example radicalEHV; shows an example |
---|
3736 | " |
---|
3737 | { |
---|
3738 | if(ord_test(basering)!=1) |
---|
3739 | { |
---|
3740 | ERROR( |
---|
3741 | "// Not implemented for this ordering, please change to global ordering." |
---|
3742 | ); |
---|
3743 | } |
---|
3744 | if((char(basering)<100)&&(char(basering)!=0)) |
---|
3745 | { |
---|
3746 | "WARNING: The characteristic is too small, the result may be wrong"; |
---|
3747 | } |
---|
3748 | ideal J,I,I0,radI0,L,radI1,I2,radI2; |
---|
3749 | int l,n; |
---|
3750 | intvec op=option(get); |
---|
3751 | matrix M; |
---|
3752 | |
---|
3753 | option(redSB); |
---|
3754 | list m=mstd(i); |
---|
3755 | I=m[2]; |
---|
3756 | option(set,op); |
---|
3757 | |
---|
3758 | int cod=nvars(basering)-dim(m[1]); |
---|
3759 | //-------------------complete intersection case:---------------------- |
---|
3760 | if(cod==size(m[2])) |
---|
3761 | { |
---|
3762 | J=minor(jacob(I),cod); |
---|
3763 | return(quotient(I,J)); |
---|
3764 | } |
---|
3765 | //-----first codim elements of I are a complete intersection:--------- |
---|
3766 | for(l=1;l<=cod;l++) |
---|
3767 | { |
---|
3768 | I0[l]=I[l]; |
---|
3769 | } |
---|
3770 | n=dim(std(I0))+cod-nvars(basering); |
---|
3771 | //-----last codim elements of I are a complete intersection:---------- |
---|
3772 | if(n!=0) |
---|
3773 | { |
---|
3774 | for(l=1;l<=cod;l++) |
---|
3775 | { |
---|
3776 | I0[l]=I[size(I)-l+1]; |
---|
3777 | } |
---|
3778 | n=dim(std(I0))+cod-nvars(basering); |
---|
3779 | } |
---|
3780 | //-----taking a generic linear combination of the input:-------------- |
---|
3781 | if(n!=0) |
---|
3782 | { |
---|
3783 | M=transpose(sparsetriag(size(m[2]),cod,95,1)); |
---|
3784 | I0=ideal(M*transpose(I)); |
---|
3785 | n=dim(std(I0))+cod-nvars(basering); |
---|
3786 | } |
---|
3787 | //-----taking a more generic linear combination of the input:--------- |
---|
3788 | if(n!=0) |
---|
3789 | { |
---|
3790 | M=transpose(sparsetriag(size(m[2]),cod,0,100)); |
---|
3791 | I0=ideal(M*transpose(I)); |
---|
3792 | n=dim(std(I0))+cod-nvars(basering); |
---|
3793 | } |
---|
3794 | if(n==0) |
---|
3795 | { |
---|
3796 | J=minor(jacob(I0),cod); |
---|
3797 | radI0=quotient(I0,J); |
---|
3798 | L=quotient(radI0,I); |
---|
3799 | radI1=quotient(radI0,L); |
---|
3800 | |
---|
3801 | if(size(reduce(radI1,m[1],1))==0) |
---|
3802 | { |
---|
3803 | return(I); |
---|
3804 | } |
---|
3805 | |
---|
3806 | I2=sat(I,radI1)[1]; |
---|
3807 | |
---|
3808 | if(deg(I2[1])<=0) |
---|
3809 | { |
---|
3810 | return(radI1); |
---|
3811 | } |
---|
3812 | return(intersect(radI1,radicalEHV(I2))); |
---|
3813 | } |
---|
3814 | //---------------------general case------------------------------------- |
---|
3815 | return(radical(I)); |
---|
3816 | } |
---|
3817 | example |
---|
3818 | { "EXAMPLE:"; echo = 2; |
---|
3819 | ring r = 0,(x,y,z),dp; |
---|
3820 | poly p = z2+1; |
---|
3821 | poly q = z3+2; |
---|
3822 | ideal i = p*q^2,y-z2; |
---|
3823 | ideal pr= radicalEHV(i); |
---|
3824 | pr; |
---|
3825 | } |
---|
3826 | |
---|
3827 | /////////////////////////////////////////////////////////////////////////////// |
---|
3828 | |
---|
3829 | proc Ann(module M) |
---|
3830 | "USAGE: Ann(M); M module |
---|
3831 | RETURN: ideal, the annihilator of coker(M) |
---|
3832 | NOTE: The output is the ideal of all elements a of the basering R such that |
---|
3833 | a * R^m is contained in M (m=number of rows of M). |
---|
3834 | EXAMPLE: example Ann; shows an example |
---|
3835 | " |
---|
3836 | { |
---|
3837 | M=prune(M); //to obtain a small embedding |
---|
3838 | ideal ann=quotient1(M,freemodule(nrows(M))); |
---|
3839 | return(ann); |
---|
3840 | } |
---|
3841 | example |
---|
3842 | { "EXAMPLE:"; echo = 2; |
---|
3843 | ring r = 0,(x,y,z),lp; |
---|
3844 | module M = x2-y2,z3; |
---|
3845 | Ann(M); |
---|
3846 | M = [1,x2],[y,x]; |
---|
3847 | Ann(M); |
---|
3848 | qring Q=std(xy-1); |
---|
3849 | module M=imap(r,M); |
---|
3850 | Ann(M); |
---|
3851 | } |
---|
3852 | |
---|
3853 | /////////////////////////////////////////////////////////////////////////////// |
---|
3854 | |
---|
3855 | //computes the equidimensional part of the ideal i of codimension e |
---|
3856 | static proc int_ass_primary_e(ideal i, int e) |
---|
3857 | { |
---|
3858 | if(homog(i)!=1) |
---|
3859 | { |
---|
3860 | i=std(i); |
---|
3861 | } |
---|
3862 | list re=sres(i,0); //the resolution |
---|
3863 | re=minres(re); //minimized resolution |
---|
3864 | ideal ann=AnnExt_R(e,re); |
---|
3865 | if(nvars(basering)-dim(std(ann))!=e) |
---|
3866 | { |
---|
3867 | return(ideal(1)); |
---|
3868 | } |
---|
3869 | return(ann); |
---|
3870 | } |
---|
3871 | |
---|
3872 | /////////////////////////////////////////////////////////////////////////////// |
---|
3873 | |
---|
3874 | //computes the annihilator of Ext^n(R/i,R) with given resolution re |
---|
3875 | //n is not necessarily the number of variables |
---|
3876 | static proc AnnExt_R(int n,list re) |
---|
3877 | { |
---|
3878 | if(n<nvars(basering)) |
---|
3879 | { |
---|
3880 | matrix f=transpose(re[n+1]); //Hom(_,R) |
---|
3881 | module k=nres(f,2)[2]; //the kernel |
---|
3882 | matrix g=transpose(re[n]); //the image of Hom(_,R) |
---|
3883 | |
---|
3884 | ideal ann=quotient1(g,k); //the anihilator |
---|
3885 | } |
---|
3886 | else |
---|
3887 | { |
---|
3888 | ideal ann=Ann(transpose(re[n])); |
---|
3889 | } |
---|
3890 | return(ann); |
---|
3891 | } |
---|
3892 | /////////////////////////////////////////////////////////////////////////////// |
---|
3893 | |
---|
3894 | static proc analyze(list pr) |
---|
3895 | { |
---|
3896 | int ii,jj; |
---|
3897 | for(ii=1;ii<=size(pr)/2;ii++) |
---|
3898 | { |
---|
3899 | dim(std(pr[2*ii])); |
---|
3900 | idealsEqual(pr[2*ii-1],pr[2*ii]); |
---|
3901 | "==========================="; |
---|
3902 | } |
---|
3903 | |
---|
3904 | for(ii=size(pr)/2;ii>1;ii--) |
---|
3905 | { |
---|
3906 | for(jj=1;jj<ii;jj++) |
---|
3907 | { |
---|
3908 | if(size(reduce(pr[2*jj],std(pr[2*ii],1)))==0) |
---|
3909 | { |
---|
3910 | "eingebette Komponente"; |
---|
3911 | jj; |
---|
3912 | ii; |
---|
3913 | } |
---|
3914 | } |
---|
3915 | } |
---|
3916 | } |
---|
3917 | |
---|
3918 | /////////////////////////////////////////////////////////////////////////////// |
---|
3919 | // |
---|
3920 | // Shimoyama-Yokoyama |
---|
3921 | // |
---|
3922 | /////////////////////////////////////////////////////////////////////////////// |
---|
3923 | |
---|
3924 | static proc simplifyIdeal(ideal i) |
---|
3925 | { |
---|
3926 | def r=basering; |
---|
3927 | |
---|
3928 | int j,k; |
---|
3929 | map phi; |
---|
3930 | poly p; |
---|
3931 | |
---|
3932 | ideal iwork=i; |
---|
3933 | ideal imap1=maxideal(1); |
---|
3934 | ideal imap2=maxideal(1); |
---|
3935 | |
---|
3936 | |
---|
3937 | for(j=1;j<=nvars(basering);j++) |
---|
3938 | { |
---|
3939 | for(k=1;k<=size(i);k++) |
---|
3940 | { |
---|
3941 | if(deg(iwork[k]/var(j))==0) |
---|
3942 | { |
---|
3943 | p=-1/leadcoef(iwork[k]/var(j))*iwork[k]; |
---|
3944 | imap1[j]=p+2*var(j); |
---|
3945 | phi=r,imap1; |
---|
3946 | iwork=phi(iwork); |
---|
3947 | iwork=subst(iwork,var(j),0); |
---|
3948 | iwork[k]=var(j); |
---|
3949 | imap1=maxideal(1); |
---|
3950 | imap2[j]=-p; |
---|
3951 | break; |
---|
3952 | } |
---|
3953 | } |
---|
3954 | } |
---|
3955 | return(iwork,imap2); |
---|
3956 | } |
---|
3957 | |
---|
3958 | |
---|
3959 | /////////////////////////////////////////////////////// |
---|
3960 | // ini_mod |
---|
3961 | // input: a polynomial p |
---|
3962 | // output: the initial term of p as needed |
---|
3963 | // in the context of characteristic sets |
---|
3964 | ////////////////////////////////////////////////////// |
---|
3965 | |
---|
3966 | static proc ini_mod(poly p) |
---|
3967 | { |
---|
3968 | if (p==0) |
---|
3969 | { |
---|
3970 | return(0); |
---|
3971 | } |
---|
3972 | int n; matrix m; |
---|
3973 | for( n=nvars(basering); n>0; n=n-1) |
---|
3974 | { |
---|
3975 | m=coef(p,var(n)); |
---|
3976 | if(m[1,1]!=1) |
---|
3977 | { |
---|
3978 | p=m[2,1]; |
---|
3979 | break; |
---|
3980 | } |
---|
3981 | } |
---|
3982 | if(deg(p)==0) |
---|
3983 | { |
---|
3984 | p=0; |
---|
3985 | } |
---|
3986 | return(p); |
---|
3987 | } |
---|
3988 | /////////////////////////////////////////////////////// |
---|
3989 | // min_ass_prim_charsets |
---|
3990 | // input: generators of an ideal PS and an integer cho |
---|
3991 | // If cho=0, the given ordering of the variables is used. |
---|
3992 | // Otherwise, the system tries to find an "optimal ordering", |
---|
3993 | // which in some cases may considerably speed up the algorithm |
---|
3994 | // output: the minimal associated primes of PS |
---|
3995 | // algorithm: via characteriostic sets |
---|
3996 | ////////////////////////////////////////////////////// |
---|
3997 | |
---|
3998 | |
---|
3999 | static proc min_ass_prim_charsets (ideal PS, int cho) |
---|
4000 | { |
---|
4001 | if((cho<0) and (cho>1)) |
---|
4002 | { |
---|
4003 | "ERROR: <int> must be 0 or 1" |
---|
4004 | return(); |
---|
4005 | } |
---|
4006 | if(system("version")>933) |
---|
4007 | { |
---|
4008 | option(notWarnSB); |
---|
4009 | } |
---|
4010 | if(cho==0) |
---|
4011 | { |
---|
4012 | return(min_ass_prim_charsets0(PS)); |
---|
4013 | } |
---|
4014 | else |
---|
4015 | { |
---|
4016 | return(min_ass_prim_charsets1(PS)); |
---|
4017 | } |
---|
4018 | } |
---|
4019 | /////////////////////////////////////////////////////// |
---|
4020 | // min_ass_prim_charsets0 |
---|
4021 | // input: generators of an ideal PS |
---|
4022 | // output: the minimal associated primes of PS |
---|
4023 | // algorithm: via characteristic sets |
---|
4024 | // the given ordering of the variables is used |
---|
4025 | ////////////////////////////////////////////////////// |
---|
4026 | |
---|
4027 | |
---|
4028 | static proc min_ass_prim_charsets0 (ideal PS) |
---|
4029 | { |
---|
4030 | intvec op; |
---|
4031 | matrix m=char_series(PS); // We compute an irreducible |
---|
4032 | // characteristic series |
---|
4033 | int i,j,k; |
---|
4034 | list PSI; |
---|
4035 | list PHI; // the ideals given by the characteristic series |
---|
4036 | for(i=nrows(m);i>=1; i--) |
---|
4037 | { |
---|
4038 | PHI[i]=ideal(m[i,1..ncols(m)]); |
---|
4039 | } |
---|
4040 | // We compute the radical of each ideal in PHI |
---|
4041 | ideal I,JS,II; |
---|
4042 | int sizeJS, sizeII; |
---|
4043 | for(i=size(PHI);i>=1; i--) |
---|
4044 | { |
---|
4045 | I=0; |
---|
4046 | for(j=size(PHI[i]);j>0;j=j-1) |
---|
4047 | { |
---|
4048 | I=I+ini_mod(PHI[i][j]); |
---|
4049 | } |
---|
4050 | JS=std(PHI[i]); |
---|
4051 | sizeJS=size(JS); |
---|
4052 | for(j=size(I);j>0;j=j-1) |
---|
4053 | { |
---|
4054 | II=0; |
---|
4055 | sizeII=0; |
---|
4056 | k=0; |
---|
4057 | while(k<=sizeII) // successive saturation |
---|
4058 | { |
---|
4059 | op=option(get); |
---|
4060 | option(returnSB); |
---|
4061 | II=quotient(JS,I[j]); |
---|
4062 | option(set,op); |
---|
4063 | sizeII=size(II); |
---|
4064 | if(sizeII==sizeJS) |
---|
4065 | { |
---|
4066 | for(k=1;k<=sizeII;k++) |
---|
4067 | { |
---|
4068 | if(leadexp(II[k])!=leadexp(JS[k])) break; |
---|
4069 | } |
---|
4070 | } |
---|
4071 | JS=II; |
---|
4072 | sizeJS=sizeII; |
---|
4073 | } |
---|
4074 | } |
---|
4075 | PSI=insert(PSI,JS); |
---|
4076 | } |
---|
4077 | int sizePSI=size(PSI); |
---|
4078 | // We eliminate redundant ideals |
---|
4079 | for(i=1;i<sizePSI;i++) |
---|
4080 | { |
---|
4081 | for(j=i+1;j<=sizePSI;j++) |
---|
4082 | { |
---|
4083 | if(size(PSI[i])!=0) |
---|
4084 | { |
---|
4085 | if(size(PSI[j])!=0) |
---|
4086 | { |
---|
4087 | if(size(NF(PSI[i],PSI[j],1))==0) |
---|
4088 | { |
---|
4089 | PSI[j]=ideal(0); |
---|
4090 | } |
---|
4091 | else |
---|
4092 | { |
---|
4093 | if(size(NF(PSI[j],PSI[i],1))==0) |
---|
4094 | { |
---|
4095 | PSI[i]=ideal(0); |
---|
4096 | } |
---|
4097 | } |
---|
4098 | } |
---|
4099 | } |
---|
4100 | } |
---|
4101 | } |
---|
4102 | for(i=sizePSI;i>=1;i--) |
---|
4103 | { |
---|
4104 | if(size(PSI[i])==0) |
---|
4105 | { |
---|
4106 | PSI=delete(PSI,i); |
---|
4107 | } |
---|
4108 | } |
---|
4109 | return (PSI); |
---|
4110 | } |
---|
4111 | |
---|
4112 | /////////////////////////////////////////////////////// |
---|
4113 | // min_ass_prim_charsets1 |
---|
4114 | // input: generators of an ideal PS |
---|
4115 | // output: the minimal associated primes of PS |
---|
4116 | // algorithm: via characteristic sets |
---|
4117 | // input: generators of an ideal PS and an integer i |
---|
4118 | // The system tries to find an "optimal ordering" of |
---|
4119 | // the variables |
---|
4120 | ////////////////////////////////////////////////////// |
---|
4121 | |
---|
4122 | |
---|
4123 | static proc min_ass_prim_charsets1 (ideal PS) |
---|
4124 | { |
---|
4125 | intvec op; |
---|
4126 | def oldring=basering; |
---|
4127 | string n=system("neworder",PS); |
---|
4128 | execute("ring r=("+charstr(oldring)+"),("+n+"),dp;"); |
---|
4129 | ideal PS=imap(oldring,PS); |
---|
4130 | matrix m=char_series(PS); // We compute an irreducible |
---|
4131 | // characteristic series |
---|
4132 | int i,j,k; |
---|
4133 | ideal I; |
---|
4134 | list PSI; |
---|
4135 | list PHI; // the ideals given by the characteristic series |
---|
4136 | list ITPHI; // their initial terms |
---|
4137 | for(i=nrows(m);i>=1; i--) |
---|
4138 | { |
---|
4139 | PHI[i]=ideal(m[i,1..ncols(m)]); |
---|
4140 | I=0; |
---|
4141 | for(j=size(PHI[i]);j>0;j=j-1) |
---|
4142 | { |
---|
4143 | I=I,ini_mod(PHI[i][j]); |
---|
4144 | } |
---|
4145 | I=I[2..ncols(I)]; |
---|
4146 | ITPHI[i]=I; |
---|
4147 | } |
---|
4148 | setring oldring; |
---|
4149 | matrix m=imap(r,m); |
---|
4150 | list PHI=imap(r,PHI); |
---|
4151 | list ITPHI=imap(r,ITPHI); |
---|
4152 | // We compute the radical of each ideal in PHI |
---|
4153 | ideal I,JS,II; |
---|
4154 | int sizeJS, sizeII; |
---|
4155 | for(i=size(PHI);i>=1; i--) |
---|
4156 | { |
---|
4157 | I=0; |
---|
4158 | for(j=size(PHI[i]);j>0;j=j-1) |
---|
4159 | { |
---|
4160 | I=I+ITPHI[i][j]; |
---|
4161 | } |
---|
4162 | JS=std(PHI[i]); |
---|
4163 | sizeJS=size(JS); |
---|
4164 | for(j=size(I);j>0;j=j-1) |
---|
4165 | { |
---|
4166 | II=0; |
---|
4167 | sizeII=0; |
---|
4168 | k=0; |
---|
4169 | while(k<=sizeII) // successive iteration |
---|
4170 | { |
---|
4171 | op=option(get); |
---|
4172 | option(returnSB); |
---|
4173 | II=quotient(JS,I[j]); |
---|
4174 | option(set,op); |
---|
4175 | //std |
---|
4176 | // II=std(II); |
---|
4177 | sizeII=size(II); |
---|
4178 | if(sizeII==sizeJS) |
---|
4179 | { |
---|
4180 | for(k=1;k<=sizeII;k++) |
---|
4181 | { |
---|
4182 | if(leadexp(II[k])!=leadexp(JS[k])) break; |
---|
4183 | } |
---|
4184 | } |
---|
4185 | JS=II; |
---|
4186 | sizeJS=sizeII; |
---|
4187 | } |
---|
4188 | } |
---|
4189 | PSI=insert(PSI,JS); |
---|
4190 | } |
---|
4191 | int sizePSI=size(PSI); |
---|
4192 | // We eliminate redundant ideals |
---|
4193 | for(i=1;i<sizePSI;i++) |
---|
4194 | { |
---|
4195 | for(j=i+1;j<=sizePSI;j++) |
---|
4196 | { |
---|
4197 | if(size(PSI[i])!=0) |
---|
4198 | { |
---|
4199 | if(size(PSI[j])!=0) |
---|
4200 | { |
---|
4201 | if(size(NF(PSI[i],PSI[j],1))==0) |
---|
4202 | { |
---|
4203 | PSI[j]=ideal(0); |
---|
4204 | } |
---|
4205 | else |
---|
4206 | { |
---|
4207 | if(size(NF(PSI[j],PSI[i],1))==0) |
---|
4208 | { |
---|
4209 | PSI[i]=ideal(0); |
---|
4210 | } |
---|
4211 | } |
---|
4212 | } |
---|
4213 | } |
---|
4214 | } |
---|
4215 | } |
---|
4216 | for(i=sizePSI;i>=1;i--) |
---|
4217 | { |
---|
4218 | if(size(PSI[i])==0) |
---|
4219 | { |
---|
4220 | PSI=delete(PSI,i); |
---|
4221 | } |
---|
4222 | } |
---|
4223 | return (PSI); |
---|
4224 | } |
---|
4225 | |
---|
4226 | |
---|
4227 | ///////////////////////////////////////////////////// |
---|
4228 | // proc prim_dec |
---|
4229 | // input: generators of an ideal I and an integer choose |
---|
4230 | // If choose=0, min_ass_prim_charsets with the given |
---|
4231 | // ordering of the variables is used. |
---|
4232 | // If choose=1, min_ass_prim_charsets with the "optimized" |
---|
4233 | // ordering of the variables is used. |
---|
4234 | // If choose=2, minAssPrimes from primdec.lib is used |
---|
4235 | // If choose=3, minAssPrimes+factorizing Buchberger from primdec.lib is used |
---|
4236 | // output: a primary decomposition of I, i.e., a list |
---|
4237 | // of pairs consisting of a standard basis of a primary component |
---|
4238 | // of I and a standard basis of the corresponding associated prime. |
---|
4239 | // To compute the minimal associated primes of a given ideal |
---|
4240 | // min_ass_prim_l is called, i.e., the minimal associated primes |
---|
4241 | // are computed via characteristic sets. |
---|
4242 | // In the homogeneous case, the performance of the procedure |
---|
4243 | // will be improved if I is already given by a minimal set of |
---|
4244 | // generators. Apply minbase if necessary. |
---|
4245 | ////////////////////////////////////////////////////////// |
---|
4246 | |
---|
4247 | |
---|
4248 | static proc prim_dec(ideal I, int choose) |
---|
4249 | { |
---|
4250 | if((choose<0) or (choose>3)) |
---|
4251 | { |
---|
4252 | "ERROR: <int> must be 0 or 1 or 2 or 3"; |
---|
4253 | return(); |
---|
4254 | } |
---|
4255 | if(system("version")>933) |
---|
4256 | { |
---|
4257 | option(notWarnSB); |
---|
4258 | } |
---|
4259 | ideal H=1; // The intersection of the primary components |
---|
4260 | list U; // the leaves of the decomposition tree, i.e., |
---|
4261 | // pairs consisting of a primary component of I |
---|
4262 | // and the corresponding associated prime |
---|
4263 | list W; // the non-leaf vertices in the decomposition tree. |
---|
4264 | // every entry has 6 components: |
---|
4265 | // 1- the vertex itself , i.e., a standard bais of the |
---|
4266 | // given ideal I (type 1), or a standard basis of a |
---|
4267 | // pseudo-primary component arising from |
---|
4268 | // pseudo-primary decomposition (type 2), or a |
---|
4269 | // standard basis of a remaining component arising from |
---|
4270 | // pseudo-primary decomposition or extraction (type 3) |
---|
4271 | // 2- the type of the vertex as indicated above |
---|
4272 | // 3- the weighted_tree_depth of the vertex |
---|
4273 | // 4- the tester of the vertex |
---|
4274 | // 5- a standard basis of the associated prime |
---|
4275 | // of a vertex of type 2, or 0 otherwise |
---|
4276 | // 6- a list of pairs consisting of a standard |
---|
4277 | // basis of a minimal associated prime ideal |
---|
4278 | // of the father of the vertex and the |
---|
4279 | // irreducible factors of the "minimal |
---|
4280 | // divisor" of the seperator or extractor |
---|
4281 | // corresponding to the prime ideal |
---|
4282 | // as computed by the procedure minsat, |
---|
4283 | // if the vertex is of type 3, or |
---|
4284 | // the empty list otherwise |
---|
4285 | ideal SI=std(I); |
---|
4286 | if(SI[1]==1) // primdecSY(ideal(1)) |
---|
4287 | { |
---|
4288 | return(list()); |
---|
4289 | } |
---|
4290 | int ncolsSI=ncols(SI); |
---|
4291 | int ncolsH=1; |
---|
4292 | W[1]=list(I,1,0,poly(1),ideal(0),list()); // The root of the tree |
---|
4293 | int weighted_tree_depth; |
---|
4294 | int i,j; |
---|
4295 | int check; |
---|
4296 | list V; // current vertex |
---|
4297 | list VV; // new vertex |
---|
4298 | list QQ; |
---|
4299 | list WI; |
---|
4300 | ideal Qi,SQ,SRest,fac; |
---|
4301 | poly tester; |
---|
4302 | |
---|
4303 | while(1) |
---|
4304 | { |
---|
4305 | i=1; |
---|
4306 | while(1) |
---|
4307 | { |
---|
4308 | while(i<=size(W)) // find vertex V of smallest weighted tree-depth |
---|
4309 | { |
---|
4310 | if (W[i][3]<=weighted_tree_depth) break; |
---|
4311 | i++; |
---|
4312 | } |
---|
4313 | if (i<=size(W)) break; |
---|
4314 | i=1; |
---|
4315 | weighted_tree_depth++; |
---|
4316 | } |
---|
4317 | V=W[i]; |
---|
4318 | W=delete(W,i); // delete V from W |
---|
4319 | |
---|
4320 | // now proceed by type of vertex V |
---|
4321 | |
---|
4322 | if (V[2]==2) // extraction needed |
---|
4323 | { |
---|
4324 | SQ,SRest,fac=extraction(V[1],V[5]); |
---|
4325 | // standard basis of primary component, |
---|
4326 | // standard basis of remaining component, |
---|
4327 | // irreducible factors of |
---|
4328 | // the "minimal divisor" of the extractor |
---|
4329 | // as computed by the procedure minsat, |
---|
4330 | check=0; |
---|
4331 | for(j=1;j<=ncolsH;j++) |
---|
4332 | { |
---|
4333 | if (NF(H[j],SQ,1)!=0) // Q is not redundant |
---|
4334 | { |
---|
4335 | check=1; |
---|
4336 | break; |
---|
4337 | } |
---|
4338 | } |
---|
4339 | if(check==1) // Q is not redundant |
---|
4340 | { |
---|
4341 | QQ=list(); |
---|
4342 | QQ[1]=list(SQ,V[5]); // primary component, associated prime, |
---|
4343 | // i.e., standard bases thereof |
---|
4344 | U=U+QQ; |
---|
4345 | H=intersect(H,SQ); |
---|
4346 | H=std(H); |
---|
4347 | ncolsH=ncols(H); |
---|
4348 | check=0; |
---|
4349 | if(ncolsH==ncolsSI) |
---|
4350 | { |
---|
4351 | for(j=1;j<=ncolsSI;j++) |
---|
4352 | { |
---|
4353 | if(leadexp(H[j])!=leadexp(SI[j])) |
---|
4354 | { |
---|
4355 | check=1; |
---|
4356 | break; |
---|
4357 | } |
---|
4358 | } |
---|
4359 | } |
---|
4360 | else |
---|
4361 | { |
---|
4362 | check=1; |
---|
4363 | } |
---|
4364 | if(check==0) // H==I => U is a primary decomposition |
---|
4365 | { |
---|
4366 | return(U); |
---|
4367 | } |
---|
4368 | } |
---|
4369 | if (SRest[1]!=1) // the remaining component is not |
---|
4370 | // the whole ring |
---|
4371 | { |
---|
4372 | if (rad_con(V[4],SRest)==0) // the new vertex is not the |
---|
4373 | // root of a redundant subtree |
---|
4374 | { |
---|
4375 | VV[1]=SRest; // remaining component |
---|
4376 | VV[2]=3; // pseudoprimdec_special |
---|
4377 | VV[3]=V[3]+1; // weighted depth |
---|
4378 | VV[4]=V[4]; // the tester did not change |
---|
4379 | VV[5]=ideal(0); |
---|
4380 | VV[6]=list(list(V[5],fac)); |
---|
4381 | W=insert(W,VV,size(W)); |
---|
4382 | } |
---|
4383 | } |
---|
4384 | } |
---|
4385 | else |
---|
4386 | { |
---|
4387 | if (V[2]==3) // pseudo_prim_dec_special is needed |
---|
4388 | { |
---|
4389 | QQ,SRest=pseudo_prim_dec_special_charsets(V[1],V[6],choose); |
---|
4390 | // QQ = quadruples: |
---|
4391 | // standard basis of pseudo-primary component, |
---|
4392 | // standard basis of corresponding prime, |
---|
4393 | // seperator, irreducible factors of |
---|
4394 | // the "minimal divisor" of the seperator |
---|
4395 | // as computed by the procedure minsat, |
---|
4396 | // SRest=standard basis of remaining component |
---|
4397 | } |
---|
4398 | else // V is the root, pseudo_prim_dec is needed |
---|
4399 | { |
---|
4400 | QQ,SRest=pseudo_prim_dec_charsets(I,SI,choose); |
---|
4401 | // QQ = quadruples: |
---|
4402 | // standard basis of pseudo-primary component, |
---|
4403 | // standard basis of corresponding prime, |
---|
4404 | // seperator, irreducible factors of |
---|
4405 | // the "minimal divisor" of the seperator |
---|
4406 | // as computed by the procedure minsat, |
---|
4407 | // SRest=standard basis of remaining component |
---|
4408 | |
---|
4409 | } |
---|
4410 | //check |
---|
4411 | for(i=size(QQ);i>=1;i--) |
---|
4412 | //for(i=1;i<=size(QQ);i++) |
---|
4413 | { |
---|
4414 | tester=QQ[i][3]*V[4]; |
---|
4415 | Qi=QQ[i][2]; |
---|
4416 | if(NF(tester,Qi,1)!=0) // the new vertex is not the |
---|
4417 | // root of a redundant subtree |
---|
4418 | { |
---|
4419 | VV[1]=QQ[i][1]; |
---|
4420 | VV[2]=2; |
---|
4421 | VV[3]=V[3]+1; |
---|
4422 | VV[4]=tester; // the new tester as computed above |
---|
4423 | VV[5]=Qi; // QQ[i][2]; |
---|
4424 | VV[6]=list(); |
---|
4425 | W=insert(W,VV,size(W)); |
---|
4426 | } |
---|
4427 | } |
---|
4428 | if (SRest[1]!=1) // the remaining component is not |
---|
4429 | // the whole ring |
---|
4430 | { |
---|
4431 | if (rad_con(V[4],SRest)==0) // the vertex is not the root |
---|
4432 | // of a redundant subtree |
---|
4433 | { |
---|
4434 | VV[1]=SRest; |
---|
4435 | VV[2]=3; |
---|
4436 | VV[3]=V[3]+2; |
---|
4437 | VV[4]=V[4]; // the tester did not change |
---|
4438 | VV[5]=ideal(0); |
---|
4439 | WI=list(); |
---|
4440 | for(i=1;i<=size(QQ);i++) |
---|
4441 | { |
---|
4442 | WI=insert(WI,list(QQ[i][2],QQ[i][4])); |
---|
4443 | } |
---|
4444 | VV[6]=WI; |
---|
4445 | W=insert(W,VV,size(W)); |
---|
4446 | } |
---|
4447 | } |
---|
4448 | } |
---|
4449 | } |
---|
4450 | } |
---|
4451 | |
---|
4452 | ////////////////////////////////////////////////////////////////////////// |
---|
4453 | // proc pseudo_prim_dec_charsets |
---|
4454 | // input: Generators of an arbitrary ideal I, a standard basis SI of I, |
---|
4455 | // and an integer choo |
---|
4456 | // If choo=0, min_ass_prim_charsets with the given |
---|
4457 | // ordering of the variables is used. |
---|
4458 | // If choo=1, min_ass_prim_charsets with the "optimized" |
---|
4459 | // ordering of the variables is used. |
---|
4460 | // If choo=2, minAssPrimes from primdec.lib is used |
---|
4461 | // If choo=3, minAssPrimes+factorizing Buchberger from primdec.lib is used |
---|
4462 | // output: a pseudo primary decomposition of I, i.e., a list |
---|
4463 | // of pseudo primary components together with a standard basis of the |
---|
4464 | // remaining component. Each pseudo primary component is |
---|
4465 | // represented by a quadrupel: A standard basis of the component, |
---|
4466 | // a standard basis of the corresponding associated prime, the |
---|
4467 | // seperator of the component, and the irreducible factors of the |
---|
4468 | // "minimal divisor" of the seperator as computed by the procedure minsat, |
---|
4469 | // calls proc pseudo_prim_dec_i |
---|
4470 | ////////////////////////////////////////////////////////////////////////// |
---|
4471 | |
---|
4472 | |
---|
4473 | static proc pseudo_prim_dec_charsets (ideal I, ideal SI, int choo) |
---|
4474 | { |
---|
4475 | list L; // The list of minimal associated primes, |
---|
4476 | // each one given by a standard basis |
---|
4477 | if((choo==0) or (choo==1)) |
---|
4478 | { |
---|
4479 | L=min_ass_prim_charsets(I,choo); |
---|
4480 | } |
---|
4481 | else |
---|
4482 | { |
---|
4483 | if(choo==2) |
---|
4484 | { |
---|
4485 | L=minAssPrimes(I); |
---|
4486 | } |
---|
4487 | else |
---|
4488 | { |
---|
4489 | L=minAssPrimes(I,1); |
---|
4490 | } |
---|
4491 | for(int i=size(L);i>=1;i=i-1) |
---|
4492 | { |
---|
4493 | L[i]=std(L[i]); |
---|
4494 | } |
---|
4495 | } |
---|
4496 | return (pseudo_prim_dec_i(SI,L)); |
---|
4497 | } |
---|
4498 | |
---|
4499 | //////////////////////////////////////////////////////////////// |
---|
4500 | // proc pseudo_prim_dec_special_charsets |
---|
4501 | // input: a standard basis of an ideal I whose radical is the |
---|
4502 | // intersection of the radicals of ideals generated by one prime ideal |
---|
4503 | // P_i together with one polynomial f_i, the list V6 must be the list of |
---|
4504 | // pairs (standard basis of P_i, irreducible factors of f_i), |
---|
4505 | // and an integer choo |
---|
4506 | // If choo=0, min_ass_prim_charsets with the given |
---|
4507 | // ordering of the variables is used. |
---|
4508 | // If choo=1, min_ass_prim_charsets with the "optimized" |
---|
4509 | // ordering of the variables is used. |
---|
4510 | // If choo=2, minAssPrimes from primdec.lib is used |
---|
4511 | // If choo=3, minAssPrimes+factorizing Buchberger from primdec.lib is used |
---|
4512 | // output: a pseudo primary decomposition of I, i.e., a list |
---|
4513 | // of pseudo primary components together with a standard basis of the |
---|
4514 | // remaining component. Each pseudo primary component is |
---|
4515 | // represented by a quadrupel: A standard basis of the component, |
---|
4516 | // a standard basis of the corresponding associated prime, the |
---|
4517 | // seperator of the component, and the irreducible factors of the |
---|
4518 | // "minimal divisor" of the seperator as computed by the procedure minsat, |
---|
4519 | // calls proc pseudo_prim_dec_i |
---|
4520 | //////////////////////////////////////////////////////////////// |
---|
4521 | |
---|
4522 | |
---|
4523 | static proc pseudo_prim_dec_special_charsets (ideal SI,list V6, int choo) |
---|
4524 | { |
---|
4525 | int i,j,l; |
---|
4526 | list m; |
---|
4527 | list L; |
---|
4528 | int sizeL; |
---|
4529 | ideal P,SP; ideal fac; |
---|
4530 | int dimSP; |
---|
4531 | for(l=size(V6);l>=1;l--) // creates a list of associated primes |
---|
4532 | // of I, possibly redundant |
---|
4533 | { |
---|
4534 | P=V6[l][1]; |
---|
4535 | fac=V6[l][2]; |
---|
4536 | for(i=ncols(fac);i>=1;i--) |
---|
4537 | { |
---|
4538 | SP=P+fac[i]; |
---|
4539 | SP=std(SP); |
---|
4540 | if(SP[1]!=1) |
---|
4541 | { |
---|
4542 | if((choo==0) or (choo==1)) |
---|
4543 | { |
---|
4544 | m=min_ass_prim_charsets(SP,choo); // a list of SB |
---|
4545 | } |
---|
4546 | else |
---|
4547 | { |
---|
4548 | if(choo==2) |
---|
4549 | { |
---|
4550 | m=minAssPrimes(SP); |
---|
4551 | } |
---|
4552 | else |
---|
4553 | { |
---|
4554 | m=minAssPrimes(SP,1); |
---|
4555 | } |
---|
4556 | for(j=size(m);j>=1;j=j-1) |
---|
4557 | { |
---|
4558 | m[j]=std(m[j]); |
---|
4559 | } |
---|
4560 | } |
---|
4561 | dimSP=dim(SP); |
---|
4562 | for(j=size(m);j>=1; j--) |
---|
4563 | { |
---|
4564 | if(dim(m[j])==dimSP) |
---|
4565 | { |
---|
4566 | L=insert(L,m[j],size(L)); |
---|
4567 | } |
---|
4568 | } |
---|
4569 | } |
---|
4570 | } |
---|
4571 | } |
---|
4572 | sizeL=size(L); |
---|
4573 | for(i=1;i<sizeL;i++) // get rid of redundant primes |
---|
4574 | { |
---|
4575 | for(j=i+1;j<=sizeL;j++) |
---|
4576 | { |
---|
4577 | if(size(L[i])!=0) |
---|
4578 | { |
---|
4579 | if(size(L[j])!=0) |
---|
4580 | { |
---|
4581 | if(size(NF(L[i],L[j],1))==0) |
---|
4582 | { |
---|
4583 | L[j]=ideal(0); |
---|
4584 | } |
---|
4585 | else |
---|
4586 | { |
---|
4587 | if(size(NF(L[j],L[i],1))==0) |
---|
4588 | { |
---|
4589 | L[i]=ideal(0); |
---|
4590 | } |
---|
4591 | } |
---|
4592 | } |
---|
4593 | } |
---|
4594 | } |
---|
4595 | } |
---|
4596 | for(i=sizeL;i>=1;i--) |
---|
4597 | { |
---|
4598 | if(size(L[i])==0) |
---|
4599 | { |
---|
4600 | L=delete(L,i); |
---|
4601 | } |
---|
4602 | } |
---|
4603 | return (pseudo_prim_dec_i(SI,L)); |
---|
4604 | } |
---|
4605 | |
---|
4606 | |
---|
4607 | //////////////////////////////////////////////////////////////// |
---|
4608 | // proc pseudo_prim_dec_i |
---|
4609 | // input: A standard basis of an arbitrary ideal I, and standard bases |
---|
4610 | // of the minimal associated primes of I |
---|
4611 | // output: a pseudo primary decomposition of I, i.e., a list |
---|
4612 | // of pseudo primary components together with a standard basis of the |
---|
4613 | // remaining component. Each pseudo primary component is |
---|
4614 | // represented by a quadrupel: A standard basis of the component Q_i, |
---|
4615 | // a standard basis of the corresponding associated prime P_i, the |
---|
4616 | // seperator of the component, and the irreducible factors of the |
---|
4617 | // "minimal divisor" of the seperator as computed by the procedure minsat, |
---|
4618 | //////////////////////////////////////////////////////////////// |
---|
4619 | |
---|
4620 | |
---|
4621 | static proc pseudo_prim_dec_i (ideal SI, list L) |
---|
4622 | { |
---|
4623 | list Q; |
---|
4624 | if (size(L)==1) // one minimal associated prime only |
---|
4625 | // the ideal is already pseudo primary |
---|
4626 | { |
---|
4627 | Q=SI,L[1],1; |
---|
4628 | list QQ; |
---|
4629 | QQ[1]=Q; |
---|
4630 | return (QQ,ideal(1)); |
---|
4631 | } |
---|
4632 | |
---|
4633 | poly f0,f,g; |
---|
4634 | ideal fac; |
---|
4635 | int i,j,k,l; |
---|
4636 | ideal SQi; |
---|
4637 | ideal I'=SI; |
---|
4638 | list QP; |
---|
4639 | int sizeL=size(L); |
---|
4640 | for(i=1;i<=sizeL;i++) |
---|
4641 | { |
---|
4642 | fac=0; |
---|
4643 | for(j=1;j<=sizeL;j++) // compute the seperator sep_i |
---|
4644 | // of the i-th component |
---|
4645 | { |
---|
4646 | if (i!=j) // search g not in L[i], but L[j] |
---|
4647 | { |
---|
4648 | for(k=1;k<=ncols(L[j]);k++) |
---|
4649 | { |
---|
4650 | if(NF(L[j][k],L[i],1)!=0) |
---|
4651 | { |
---|
4652 | break; |
---|
4653 | } |
---|
4654 | } |
---|
4655 | fac=fac+L[j][k]; |
---|
4656 | } |
---|
4657 | } |
---|
4658 | // delete superfluous polynomials |
---|
4659 | fac=simplify(fac,8); |
---|
4660 | // saturation |
---|
4661 | SQi,f0,f,fac=minsat_ppd(SI,fac); |
---|
4662 | I'=I',f; |
---|
4663 | QP=SQi,L[i],f0,fac; |
---|
4664 | // the quadrupel: |
---|
4665 | // a standard basis of Q_i, |
---|
4666 | // a standard basis of P_i, |
---|
4667 | // sep_i, |
---|
4668 | // irreducible factors of |
---|
4669 | // the "minimal divisor" of the seperator |
---|
4670 | // as computed by the procedure minsat, |
---|
4671 | Q[i]=QP; |
---|
4672 | } |
---|
4673 | I'=std(I'); |
---|
4674 | return (Q, I'); |
---|
4675 | // I' = remaining component |
---|
4676 | } |
---|
4677 | |
---|
4678 | |
---|
4679 | //////////////////////////////////////////////////////////////// |
---|
4680 | // proc extraction |
---|
4681 | // input: A standard basis of a pseudo primary ideal I, and a standard |
---|
4682 | // basis of the unique minimal associated prime P of I |
---|
4683 | // output: an extraction of I, i.e., a standard basis of the primary |
---|
4684 | // component Q of I with associated prime P, a standard basis of the |
---|
4685 | // remaining component, and the irreducible factors of the |
---|
4686 | // "minimal divisor" of the extractor as computed by the procedure minsat |
---|
4687 | //////////////////////////////////////////////////////////////// |
---|
4688 | |
---|
4689 | |
---|
4690 | static proc extraction (ideal SI, ideal SP) |
---|
4691 | { |
---|
4692 | list indsets=indepSet(SP,0); |
---|
4693 | poly f; |
---|
4694 | if(size(indsets)!=0) //check, whether dim P != 0 |
---|
4695 | { |
---|
4696 | intvec v; // a maximal independent set of variables |
---|
4697 | // modulo P |
---|
4698 | string U; // the independent variables |
---|
4699 | string A; // the dependent variables |
---|
4700 | int j,k; |
---|
4701 | int a; // the size of A |
---|
4702 | int degf; |
---|
4703 | ideal g; |
---|
4704 | list polys; |
---|
4705 | int sizepolys; |
---|
4706 | list newpoly; |
---|
4707 | def R=basering; |
---|
4708 | //intvec hv=hilb(SI,1); |
---|
4709 | for (k=1;k<=size(indsets);k++) |
---|
4710 | { |
---|
4711 | v=indsets[k]; |
---|
4712 | for (j=1;j<=nvars(R);j++) |
---|
4713 | { |
---|
4714 | if (v[j]==1) |
---|
4715 | { |
---|
4716 | U=U+varstr(j)+","; |
---|
4717 | } |
---|
4718 | else |
---|
4719 | { |
---|
4720 | A=A+varstr(j)+","; |
---|
4721 | a++; |
---|
4722 | } |
---|
4723 | } |
---|
4724 | |
---|
4725 | U[size(U)]=")"; // we compute the extractor of I (w.r.t. U) |
---|
4726 | execute("ring RAU=("+charstr(basering)+"),("+A+U+",(dp("+string(a)+"),dp);"); |
---|
4727 | ideal I=imap(R,SI); |
---|
4728 | //I=std(I,hv); // the standard basis in (R[U])[A] |
---|
4729 | I=std(I); // the standard basis in (R[U])[A] |
---|
4730 | A[size(A)]=")"; |
---|
4731 | execute("ring Rloc=("+charstr(basering)+","+U+",("+A+",dp;"); |
---|
4732 | ideal I=imap(RAU,I); |
---|
4733 | //"std in lokalisierung:"+newline,I; |
---|
4734 | ideal h; |
---|
4735 | for(j=ncols(I);j>=1;j--) |
---|
4736 | { |
---|
4737 | h[j]=leadcoef(I[j]); // consider I in (R(U))[A] |
---|
4738 | } |
---|
4739 | setring R; |
---|
4740 | g=imap(Rloc,h); |
---|
4741 | kill RAU,Rloc; |
---|
4742 | U=""; |
---|
4743 | A=""; |
---|
4744 | a=0; |
---|
4745 | f=lcm(g); |
---|
4746 | newpoly[1]=f; |
---|
4747 | polys=polys+newpoly; |
---|
4748 | newpoly=list(); |
---|
4749 | } |
---|
4750 | f=polys[1]; |
---|
4751 | degf=deg(f); |
---|
4752 | sizepolys=size(polys); |
---|
4753 | for (k=2;k<=sizepolys;k++) |
---|
4754 | { |
---|
4755 | if (deg(polys[k])<degf) |
---|
4756 | { |
---|
4757 | f=polys[k]; |
---|
4758 | degf=deg(f); |
---|
4759 | } |
---|
4760 | } |
---|
4761 | } |
---|
4762 | else |
---|
4763 | { |
---|
4764 | f=1; |
---|
4765 | } |
---|
4766 | poly f0,h0; ideal SQ; ideal fac; |
---|
4767 | if(f!=1) |
---|
4768 | { |
---|
4769 | SQ,f0,h0,fac=minsat(SI,f); |
---|
4770 | return(SQ,std(SI+h0),fac); |
---|
4771 | // the tripel |
---|
4772 | // a standard basis of Q, |
---|
4773 | // a standard basis of remaining component, |
---|
4774 | // irreducible factors of |
---|
4775 | // the "minimal divisor" of the extractor |
---|
4776 | // as computed by the procedure minsat |
---|
4777 | } |
---|
4778 | else |
---|
4779 | { |
---|
4780 | return(SI,ideal(1),ideal(1)); |
---|
4781 | } |
---|
4782 | } |
---|
4783 | |
---|
4784 | ///////////////////////////////////////////////////// |
---|
4785 | // proc minsat |
---|
4786 | // input: a standard basis of an ideal I and a polynomial p |
---|
4787 | // output: a standard basis IS of the saturation of I w.r. to p, |
---|
4788 | // the maximal squarefree factor f0 of p, |
---|
4789 | // the "minimal divisor" f of f0 such that the saturation of |
---|
4790 | // I w.r. to f equals the saturation of I w.r. to f0 (which is IS), |
---|
4791 | // the irreducible factors of f |
---|
4792 | ////////////////////////////////////////////////////////// |
---|
4793 | |
---|
4794 | |
---|
4795 | static proc minsat(ideal SI, poly p) |
---|
4796 | { |
---|
4797 | ideal fac=factorize(p,1); //the irreducible factors of p |
---|
4798 | fac=sort(fac)[1]; |
---|
4799 | int i,k; |
---|
4800 | poly f0=1; |
---|
4801 | for(i=ncols(fac);i>=1;i--) |
---|
4802 | { |
---|
4803 | f0=f0*fac[i]; |
---|
4804 | } |
---|
4805 | poly f=1; |
---|
4806 | ideal iold; |
---|
4807 | list quotM; |
---|
4808 | quotM[1]=SI; |
---|
4809 | quotM[2]=fac; |
---|
4810 | quotM[3]=f0; |
---|
4811 | // we deal seperately with the first quotient; |
---|
4812 | // factors, which do not contribute to this one, |
---|
4813 | // are omitted |
---|
4814 | iold=quotM[1]; |
---|
4815 | quotM=minquot(quotM); |
---|
4816 | fac=quotM[2]; |
---|
4817 | if(quotM[3]==1) |
---|
4818 | { |
---|
4819 | return(quotM[1],f0,f,fac); |
---|
4820 | } |
---|
4821 | while(special_ideals_equal(iold,quotM[1])==0) |
---|
4822 | { |
---|
4823 | f=f*quotM[3]; |
---|
4824 | iold=quotM[1]; |
---|
4825 | quotM=minquot(quotM); |
---|
4826 | } |
---|
4827 | return(quotM[1],f0,f,fac); // the quadrupel ((I:p),f0,f, irr. factors of f) |
---|
4828 | } |
---|
4829 | |
---|
4830 | ///////////////////////////////////////////////////// |
---|
4831 | // proc minsat_ppd |
---|
4832 | // input: a standard basis of an ideal I and a polynomial p |
---|
4833 | // output: a standard basis IS of the saturation of I w.r. to p, |
---|
4834 | // the maximal squarefree factor f0 of p, |
---|
4835 | // the "minimal divisor" f of f0 such that the saturation of |
---|
4836 | // I w.r. to f equals the saturation of I w.r. to f0 (which is IS), |
---|
4837 | // the irreducible factors of f |
---|
4838 | ////////////////////////////////////////////////////////// |
---|
4839 | |
---|
4840 | |
---|
4841 | static proc minsat_ppd(ideal SI, ideal fac) |
---|
4842 | { |
---|
4843 | fac=sort(fac)[1]; |
---|
4844 | int i,k; |
---|
4845 | poly f0=1; |
---|
4846 | for(i=ncols(fac);i>=1;i--) |
---|
4847 | { |
---|
4848 | f0=f0*fac[i]; |
---|
4849 | } |
---|
4850 | poly f=1; |
---|
4851 | ideal iold; |
---|
4852 | list quotM; |
---|
4853 | quotM[1]=SI; |
---|
4854 | quotM[2]=fac; |
---|
4855 | quotM[3]=f0; |
---|
4856 | // we deal seperately with the first quotient; |
---|
4857 | // factors, which do not contribute to this one, |
---|
4858 | // are omitted |
---|
4859 | iold=quotM[1]; |
---|
4860 | quotM=minquot(quotM); |
---|
4861 | fac=quotM[2]; |
---|
4862 | if(quotM[3]==1) |
---|
4863 | { |
---|
4864 | return(quotM[1],f0,f,fac); |
---|
4865 | } |
---|
4866 | while(special_ideals_equal(iold,quotM[1])==0) |
---|
4867 | { |
---|
4868 | f=f*quotM[3]; |
---|
4869 | iold=quotM[1]; |
---|
4870 | quotM=minquot(quotM); |
---|
4871 | k++; |
---|
4872 | } |
---|
4873 | return(quotM[1],f0,f,fac); // the quadrupel ((I:p),f0,f, irr. factors of f) |
---|
4874 | } |
---|
4875 | ///////////////////////////////////////////////////////////////// |
---|
4876 | // proc minquot |
---|
4877 | // input: a list with 3 components: a standard basis |
---|
4878 | // of an ideal I, a set of irreducible polynomials, and |
---|
4879 | // there product f0 |
---|
4880 | // output: a standard basis of the ideal (I:f0), the irreducible |
---|
4881 | // factors of the "minimal divisor" f of f0 with (I:f0) = (I:f), |
---|
4882 | // the "minimal divisor" f |
---|
4883 | ///////////////////////////////////////////////////////////////// |
---|
4884 | |
---|
4885 | static proc minquot(list tsil) |
---|
4886 | { |
---|
4887 | intvec op; |
---|
4888 | int i,j,k,action; |
---|
4889 | ideal verg; |
---|
4890 | list l; |
---|
4891 | poly g; |
---|
4892 | ideal laedi=tsil[1]; |
---|
4893 | ideal fac=tsil[2]; |
---|
4894 | poly f=tsil[3]; |
---|
4895 | |
---|
4896 | //std |
---|
4897 | // ideal star=quotient(laedi,f); |
---|
4898 | // star=std(star); |
---|
4899 | op=option(get); |
---|
4900 | option(returnSB); |
---|
4901 | ideal star=quotient(laedi,f); |
---|
4902 | option(set,op); |
---|
4903 | if(special_ideals_equal(laedi,star)==1) |
---|
4904 | { |
---|
4905 | return(laedi,ideal(1),1); |
---|
4906 | } |
---|
4907 | action=1; |
---|
4908 | while(action==1) |
---|
4909 | { |
---|
4910 | if(size(fac)==1) |
---|
4911 | { |
---|
4912 | action=0; |
---|
4913 | break; |
---|
4914 | } |
---|
4915 | for(i=1;i<=size(fac);i++) |
---|
4916 | { |
---|
4917 | g=1; |
---|
4918 | for(j=1;j<=size(fac);j++) |
---|
4919 | { |
---|
4920 | if(i!=j) |
---|
4921 | { |
---|
4922 | g=g*fac[j]; |
---|
4923 | } |
---|
4924 | } |
---|
4925 | //std |
---|
4926 | // verg=quotient(laedi,g); |
---|
4927 | // verg=std(verg); |
---|
4928 | op=option(get); |
---|
4929 | option(returnSB); |
---|
4930 | verg=quotient(laedi,g); |
---|
4931 | option(set,op); |
---|
4932 | if(special_ideals_equal(verg,star)==1) |
---|
4933 | { |
---|
4934 | f=g; |
---|
4935 | fac[i]=0; |
---|
4936 | fac=simplify(fac,2); |
---|
4937 | break; |
---|
4938 | } |
---|
4939 | if(i==size(fac)) |
---|
4940 | { |
---|
4941 | action=0; |
---|
4942 | } |
---|
4943 | } |
---|
4944 | } |
---|
4945 | l=star,fac,f; |
---|
4946 | return(l); |
---|
4947 | } |
---|
4948 | ///////////////////////////////////////////////// |
---|
4949 | // proc special_ideals_equal |
---|
4950 | // input: standard bases of ideal k1 and k2 such that |
---|
4951 | // k1 is contained in k2, or k2 is contained ink1 |
---|
4952 | // output: 1, if k1 equals k2, 0 otherwise |
---|
4953 | ////////////////////////////////////////////////// |
---|
4954 | |
---|
4955 | static proc special_ideals_equal( ideal k1, ideal k2) |
---|
4956 | { |
---|
4957 | int j; |
---|
4958 | if(size(k1)==size(k2)) |
---|
4959 | { |
---|
4960 | for(j=1;j<=size(k1);j++) |
---|
4961 | { |
---|
4962 | if(leadexp(k1[j])!=leadexp(k2[j])) |
---|
4963 | { |
---|
4964 | return(0); |
---|
4965 | } |
---|
4966 | } |
---|
4967 | return(1); |
---|
4968 | } |
---|
4969 | return(0); |
---|
4970 | } |
---|
4971 | |
---|
4972 | |
---|
4973 | /////////////////////////////////////////////////////////////////////////////// |
---|
4974 | |
---|
4975 | static proc convList(list l) |
---|
4976 | { |
---|
4977 | int i; |
---|
4978 | list re,he; |
---|
4979 | for(i=1;i<=size(l)/2;i++) |
---|
4980 | { |
---|
4981 | he=l[2*i-1],l[2*i]; |
---|
4982 | re[i]=he; |
---|
4983 | } |
---|
4984 | return(re); |
---|
4985 | } |
---|
4986 | /////////////////////////////////////////////////////////////////////////////// |
---|
4987 | |
---|
4988 | static proc reconvList(list l) |
---|
4989 | { |
---|
4990 | int i; |
---|
4991 | list re; |
---|
4992 | for(i=1;i<=size(l);i++) |
---|
4993 | { |
---|
4994 | re[2*i-1]=l[i][1]; |
---|
4995 | re[2*i]=l[i][2]; |
---|
4996 | } |
---|
4997 | return(re); |
---|
4998 | } |
---|
4999 | |
---|
5000 | /////////////////////////////////////////////////////////////////////////////// |
---|
5001 | // |
---|
5002 | // The main procedures |
---|
5003 | // |
---|
5004 | /////////////////////////////////////////////////////////////////////////////// |
---|
5005 | |
---|
5006 | proc primdecGTZ(ideal i) |
---|
5007 | "USAGE: primdecGTZ(i); i ideal |
---|
5008 | RETURN: a list pr of primary ideals and their associated primes: |
---|
5009 | @format |
---|
5010 | pr[i][1] the i-th primary component, |
---|
5011 | pr[i][2] the i-th prime component. |
---|
5012 | @end format |
---|
5013 | NOTE: Algorithm of Gianni/Trager/Zacharias. |
---|
5014 | Designed for characteristic 0, works also in char k > 0, if it |
---|
5015 | terminates (may result in an infinite loop in small characteristic!) |
---|
5016 | EXAMPLE: example primdecGTZ; shows an example |
---|
5017 | " |
---|
5018 | { |
---|
5019 | if(ord_test(basering)!=1) |
---|
5020 | { |
---|
5021 | ERROR( |
---|
5022 | "// Not implemented for this ordering, please change to global ordering." |
---|
5023 | ); |
---|
5024 | } |
---|
5025 | if(minpoly!=0) |
---|
5026 | { |
---|
5027 | return(algeDeco(i,0)); |
---|
5028 | ERROR( |
---|
5029 | "// Not implemented yet for algebraic extensions.Simulate the ring extension by adding the minpoly to the ideal" |
---|
5030 | ); |
---|
5031 | } |
---|
5032 | return(convList(decomp(i))); |
---|
5033 | } |
---|
5034 | example |
---|
5035 | { "EXAMPLE:"; echo = 2; |
---|
5036 | ring r = 0,(x,y,z),lp; |
---|
5037 | poly p = z2+1; |
---|
5038 | poly q = z3+2; |
---|
5039 | ideal i = p*q^2,y-z2; |
---|
5040 | list pr = primdecGTZ(i); |
---|
5041 | pr; |
---|
5042 | } |
---|
5043 | /////////////////////////////////////////////////////////////////////////////// |
---|
5044 | |
---|
5045 | proc primdecSY(ideal i, list #) |
---|
5046 | "USAGE: primdecSY(i); i ideal, c int |
---|
5047 | RETURN: a list pr of primary ideals and their associated primes: |
---|
5048 | @format |
---|
5049 | pr[i][1] the i-th primary component, |
---|
5050 | pr[i][2] the i-th prime component. |
---|
5051 | @end format |
---|
5052 | NOTE: Algorithm of Shimoyama/Yokoyama. |
---|
5053 | @format |
---|
5054 | if c=0, the given ordering of the variables is used, |
---|
5055 | if c=1, minAssChar tries to use an optimal ordering, |
---|
5056 | if c=2, minAssGTZ is used, |
---|
5057 | if c=3, minAssGTZ and facstd are used. |
---|
5058 | @end format |
---|
5059 | EXAMPLE: example primdecSY; shows an example |
---|
5060 | " |
---|
5061 | { |
---|
5062 | if(ord_test(basering)!=1) |
---|
5063 | { |
---|
5064 | ERROR( |
---|
5065 | "// Not implemented for this ordering, please change to global ordering." |
---|
5066 | ); |
---|
5067 | } |
---|
5068 | i=simplify(i,2); |
---|
5069 | if ((i[1]==0)||(i[1]==1)) |
---|
5070 | { |
---|
5071 | list L=list(ideal(i[1]),ideal(i[1])); |
---|
5072 | return(list(L)); |
---|
5073 | } |
---|
5074 | if(minpoly!=0) |
---|
5075 | { |
---|
5076 | return(algeDeco(i,1)); |
---|
5077 | } |
---|
5078 | if (size(#)==1) |
---|
5079 | { return(prim_dec(i,#[1])); } |
---|
5080 | else |
---|
5081 | { return(prim_dec(i,1)); } |
---|
5082 | } |
---|
5083 | example |
---|
5084 | { "EXAMPLE:"; echo = 2; |
---|
5085 | ring r = 0,(x,y,z),lp; |
---|
5086 | poly p = z2+1; |
---|
5087 | poly q = z3+2; |
---|
5088 | ideal i = p*q^2,y-z2; |
---|
5089 | list pr = primdecSY(i); |
---|
5090 | pr; |
---|
5091 | } |
---|
5092 | /////////////////////////////////////////////////////////////////////////////// |
---|
5093 | proc minAssGTZ(ideal i,list #) |
---|
5094 | "USAGE: minAssGTZ(i); i ideal |
---|
5095 | minAssGTZ(i,1); i ideal does not use the factorizing Groebner |
---|
5096 | RETURN: a list, the minimal associated prime ideals of i. |
---|
5097 | NOTE: Designed for characteristic 0, works also in char k > 0 based |
---|
5098 | on an algorithm of Yokoyama |
---|
5099 | EXAMPLE: example minAssGTZ; shows an example |
---|
5100 | " |
---|
5101 | { |
---|
5102 | if(ord_test(basering)!=1) |
---|
5103 | { |
---|
5104 | ERROR( |
---|
5105 | "// Not implemented for this ordering, please change to global ordering." |
---|
5106 | ); |
---|
5107 | } |
---|
5108 | if(minpoly!=0) |
---|
5109 | { |
---|
5110 | return(algeDeco(i,2)); |
---|
5111 | ERROR( |
---|
5112 | "// Not implemented for algebraic extensions. Simulate the ring extension by adding the minpoly to the ideal" |
---|
5113 | ); |
---|
5114 | } |
---|
5115 | if(size(#)==0) |
---|
5116 | { |
---|
5117 | return(minAssPrimes(i,1)); |
---|
5118 | } |
---|
5119 | return(minAssPrimes(i)); |
---|
5120 | } |
---|
5121 | example |
---|
5122 | { "EXAMPLE:"; echo = 2; |
---|
5123 | ring r = 0,(x,y,z),dp; |
---|
5124 | poly p = z2+1; |
---|
5125 | poly q = z3+2; |
---|
5126 | ideal i = p*q^2,y-z2; |
---|
5127 | list pr = minAssGTZ(i); |
---|
5128 | pr; |
---|
5129 | } |
---|
5130 | |
---|
5131 | /////////////////////////////////////////////////////////////////////////////// |
---|
5132 | proc minAssChar(ideal i, list #) |
---|
5133 | "USAGE: minAssChar(i[,c]); i ideal, c int. |
---|
5134 | RETURN: list, the minimal associated prime ideals of i. |
---|
5135 | NOTE: If c=0, the given ordering of the variables is used. @* |
---|
5136 | Otherwise, the system tries to find an optimal ordering, |
---|
5137 | which in some cases may considerably speed up the algorithm. @* |
---|
5138 | Due to a bug in the factorization, the result may be not completely |
---|
5139 | decomposed in small characteristic. |
---|
5140 | EXAMPLE: example minAssChar; shows an example |
---|
5141 | " |
---|
5142 | { |
---|
5143 | if(ord_test(basering)!=1) |
---|
5144 | { |
---|
5145 | ERROR( |
---|
5146 | "// Not implemented for this ordering, please change to global ordering." |
---|
5147 | ); |
---|
5148 | } |
---|
5149 | if (size(#)==1) |
---|
5150 | { return(min_ass_prim_charsets(i,#[1])); } |
---|
5151 | else |
---|
5152 | { return(min_ass_prim_charsets(i,1)); } |
---|
5153 | } |
---|
5154 | example |
---|
5155 | { "EXAMPLE:"; echo = 2; |
---|
5156 | ring r = 0,(x,y,z),dp; |
---|
5157 | poly p = z2+1; |
---|
5158 | poly q = z3+2; |
---|
5159 | ideal i = p*q^2,y-z2; |
---|
5160 | list pr = minAssChar(i); |
---|
5161 | pr; |
---|
5162 | } |
---|
5163 | /////////////////////////////////////////////////////////////////////////////// |
---|
5164 | proc equiRadical(ideal i) |
---|
5165 | "USAGE: equiRadical(i); i ideal |
---|
5166 | RETURN: ideal, intersection of associated primes of i of maximal dimension. |
---|
5167 | NOTE: A combination of the algorithms of Krick/Logar and Kemper is used. |
---|
5168 | Works also in positive characteristic (Kempers algorithm). |
---|
5169 | EXAMPLE: example equiRadical; shows an example |
---|
5170 | " |
---|
5171 | { |
---|
5172 | if(ord_test(basering)!=1) |
---|
5173 | { |
---|
5174 | ERROR( |
---|
5175 | "// Not implemented for this ordering, please change to global ordering." |
---|
5176 | ); |
---|
5177 | } |
---|
5178 | return(radical(i,1)); |
---|
5179 | } |
---|
5180 | example |
---|
5181 | { "EXAMPLE:"; echo = 2; |
---|
5182 | ring r = 0,(x,y,z),dp; |
---|
5183 | poly p = z2+1; |
---|
5184 | poly q = z3+2; |
---|
5185 | ideal i = p*q^2,y-z2; |
---|
5186 | ideal pr= equiRadical(i); |
---|
5187 | pr; |
---|
5188 | } |
---|
5189 | |
---|
5190 | /////////////////////////////////////////////////////////////////////////////// |
---|
5191 | proc radical(ideal i,list #) |
---|
5192 | "USAGE: radical(i); i ideal. |
---|
5193 | RETURN: ideal, the radical of i. |
---|
5194 | NOTE: A combination of the algorithms of Krick/Logar and Kemper is used. |
---|
5195 | Works also in positive characteristic (Kempers algorithm). |
---|
5196 | EXAMPLE: example radical; shows an example |
---|
5197 | " |
---|
5198 | { |
---|
5199 | if(ord_test(basering)!=1) |
---|
5200 | { |
---|
5201 | ERROR( |
---|
5202 | "// Not implemented for this ordering, please change to global ordering." |
---|
5203 | ); |
---|
5204 | } |
---|
5205 | def @P=basering; |
---|
5206 | int j,il; |
---|
5207 | if(size(#)>0){il=#[1];} |
---|
5208 | if(size(i)==0){return(ideal(0));} |
---|
5209 | ideal re=1; |
---|
5210 | intvec op = option(get); |
---|
5211 | list qr=simplifyIdeal(i); |
---|
5212 | ideal isave=i; |
---|
5213 | map phi=@P,qr[2]; |
---|
5214 | |
---|
5215 | option(redSB); |
---|
5216 | i=groebner(qr[1]); |
---|
5217 | option(set,op); |
---|
5218 | int di=dim(i); |
---|
5219 | |
---|
5220 | if(di==0) |
---|
5221 | { |
---|
5222 | i=zeroRad(i,qr[1]); |
---|
5223 | return(interred(phi(i))); |
---|
5224 | } |
---|
5225 | |
---|
5226 | option(redSB); |
---|
5227 | list pr=i; |
---|
5228 | if (!homog(i)) |
---|
5229 | { |
---|
5230 | pr=facstd(i); |
---|
5231 | } |
---|
5232 | option(set,op); |
---|
5233 | int s=size(pr); |
---|
5234 | |
---|
5235 | for(j=1;j<=s;j++) |
---|
5236 | { |
---|
5237 | attrib(pr[s+1-j],"isSB",1); |
---|
5238 | if((size(reduce(re,pr[s+1-j],1))!=0)&&((dim(pr[s+1-j])==di)||!il)) |
---|
5239 | { |
---|
5240 | re=intersect(re,radicalKL(pr[s+1-j],re,il)); |
---|
5241 | } |
---|
5242 | } |
---|
5243 | return(interred(phi(re))); |
---|
5244 | } |
---|
5245 | example |
---|
5246 | { "EXAMPLE:"; echo = 2; |
---|
5247 | ring r = 0,(x,y,z),dp; |
---|
5248 | poly p = z2+1; |
---|
5249 | poly q = z3+2; |
---|
5250 | ideal i = p*q^2,y-z2; |
---|
5251 | ideal pr= radical(i); |
---|
5252 | pr; |
---|
5253 | } |
---|
5254 | /////////////////////////////////////////////////////////////////////////////// |
---|
5255 | proc prepareAss(ideal i) |
---|
5256 | "USAGE: prepareAss(i); i ideal |
---|
5257 | RETURN: list, the radicals of the maximal dimensional components of i. |
---|
5258 | NOTE: Uses algorithm of Eisenbud/Huneke/Vasconcelos. |
---|
5259 | EXAMPLE: example prepareAss; shows an example |
---|
5260 | " |
---|
5261 | { |
---|
5262 | if(ord_test(basering)!=1) |
---|
5263 | { |
---|
5264 | ERROR( |
---|
5265 | "// Not implemented for this ordering, please change to global ordering." |
---|
5266 | ); |
---|
5267 | } |
---|
5268 | ideal j=std(i); |
---|
5269 | int cod=nvars(basering)-dim(j); |
---|
5270 | int e; |
---|
5271 | list er; |
---|
5272 | ideal ann; |
---|
5273 | if(homog(i)==1) |
---|
5274 | { |
---|
5275 | list re=sres(j,0); //the resolution |
---|
5276 | re=minres(re); //minimized resolution |
---|
5277 | } |
---|
5278 | else |
---|
5279 | { |
---|
5280 | list re=mres(i,0); |
---|
5281 | } |
---|
5282 | for(e=cod;e<=nvars(basering);e++) |
---|
5283 | { |
---|
5284 | ann=AnnExt_R(e,re); |
---|
5285 | |
---|
5286 | if(nvars(basering)-dim(std(ann))==e) |
---|
5287 | { |
---|
5288 | er[size(er)+1]=equiRadical(ann); |
---|
5289 | } |
---|
5290 | } |
---|
5291 | return(er); |
---|
5292 | } |
---|
5293 | example |
---|
5294 | { "EXAMPLE:"; echo = 2; |
---|
5295 | ring r = 0,(x,y,z),dp; |
---|
5296 | poly p = z2+1; |
---|
5297 | poly q = z3+2; |
---|
5298 | ideal i = p*q^2,y-z2; |
---|
5299 | list pr = prepareAss(i); |
---|
5300 | pr; |
---|
5301 | } |
---|
5302 | /////////////////////////////////////////////////////////////////////////////// |
---|
5303 | proc equidimMaxEHV(ideal i) |
---|
5304 | "USAGE: equidimMaxEHV(i); i ideal |
---|
5305 | RETURN: ideal, the equidimensional component (of maximal dimension) of i. |
---|
5306 | NOTE: Uses algorithm of Eisenbud, Huneke and Vasconcelos. |
---|
5307 | EXAMPLE: example equidimMaxEHV; shows an example |
---|
5308 | " |
---|
5309 | { |
---|
5310 | if(ord_test(basering)!=1) |
---|
5311 | { |
---|
5312 | ERROR( |
---|
5313 | "// Not implemented for this ordering, please change to global ordering." |
---|
5314 | ); |
---|
5315 | } |
---|
5316 | ideal j=groebner(i); |
---|
5317 | int cod=nvars(basering)-dim(j); |
---|
5318 | int e; |
---|
5319 | ideal ann; |
---|
5320 | if(homog(i)==1) |
---|
5321 | { |
---|
5322 | list re=sres(j,0); //the resolution |
---|
5323 | re=minres(re); //minimized resolution |
---|
5324 | } |
---|
5325 | else |
---|
5326 | { |
---|
5327 | list re=mres(i,0); |
---|
5328 | } |
---|
5329 | ann=AnnExt_R(cod,re); |
---|
5330 | return(ann); |
---|
5331 | } |
---|
5332 | example |
---|
5333 | { "EXAMPLE:"; echo = 2; |
---|
5334 | ring r = 0,(x,y,z),dp; |
---|
5335 | ideal i=intersect(ideal(z),ideal(x,y),ideal(x2,z2),ideal(x5,y5,z5)); |
---|
5336 | equidimMaxEHV(i); |
---|
5337 | } |
---|
5338 | |
---|
5339 | proc testPrimary(list pr, ideal k) |
---|
5340 | "USAGE: testPrimary(pr,k); pr a list, k an ideal. |
---|
5341 | ASSUME: pr is the result of primdecGTZ(k) or primdecSY(k). |
---|
5342 | RETURN: int, 1 if the intersection of the ideals in pr is k, 0 if not |
---|
5343 | EXAMPLE: example testPrimary; shows an example |
---|
5344 | " |
---|
5345 | { |
---|
5346 | int i; |
---|
5347 | pr=reconvList(pr); |
---|
5348 | ideal j=pr[1]; |
---|
5349 | for (i=2;i<=size(pr)/2;i++) |
---|
5350 | { |
---|
5351 | j=intersect(j,pr[2*i-1]); |
---|
5352 | } |
---|
5353 | return(idealsEqual(j,k)); |
---|
5354 | } |
---|
5355 | example |
---|
5356 | { "EXAMPLE:"; echo = 2; |
---|
5357 | ring r = 32003,(x,y,z),dp; |
---|
5358 | poly p = z2+1; |
---|
5359 | poly q = z4+2; |
---|
5360 | ideal i = p^2*q^3,(y-z3)^3,(x-yz+z4)^4; |
---|
5361 | list pr = primdecGTZ(i); |
---|
5362 | testPrimary(pr,i); |
---|
5363 | } |
---|
5364 | |
---|
5365 | /////////////////////////////////////////////////////////////////////////////// |
---|
5366 | proc zerodec(ideal I) |
---|
5367 | "USAGE: zerodec(I); I ideal |
---|
5368 | ASSUME: I is zero-dimensional, the characteristic of the ground field is 0 |
---|
5369 | RETURN: list of primary ideals, the zero-dimensional decomposition of I |
---|
5370 | NOTE: The algorithm (of Monico), works well only for a small total number |
---|
5371 | of solutions (@code{vdim(std(I))} should be < 100) and without |
---|
5372 | parameters. In practice, it works also in large characteristic p>0 |
---|
5373 | but may fail for small p. |
---|
5374 | @* If printlevel > 0 (default = 0) additional information is displayed. |
---|
5375 | EXAMPLE: example zerodec; shows an example |
---|
5376 | " |
---|
5377 | { |
---|
5378 | if(ord_test(basering)!=1) |
---|
5379 | { |
---|
5380 | ERROR( |
---|
5381 | "// Not implemented for this ordering, please change to global ordering." |
---|
5382 | ); |
---|
5383 | } |
---|
5384 | def R=basering; |
---|
5385 | poly q; |
---|
5386 | int j,time; |
---|
5387 | matrix m; |
---|
5388 | list re; |
---|
5389 | poly va=var(1); |
---|
5390 | ideal J=groebner(I); |
---|
5391 | ideal ba=kbase(J); |
---|
5392 | int d=vdim(J); |
---|
5393 | dbprint(printlevel-voice+2,"// multiplicity of ideal : "+ string(d)); |
---|
5394 | //------ compute matrix of multiplication on R/I with generic element p ----- |
---|
5395 | int e=nvars(basering); |
---|
5396 | poly p=randomLast(100)[e]+random(-50,50); //the generic element |
---|
5397 | matrix n[d][d]; |
---|
5398 | time = timer; |
---|
5399 | for(j=2;j<=e;j++) |
---|
5400 | { |
---|
5401 | va=va*var(j); |
---|
5402 | } |
---|
5403 | for(j=1;j<=d;j++) |
---|
5404 | { |
---|
5405 | q=reduce(p*ba[j],J); |
---|
5406 | m=coeffs(q,ba,va); |
---|
5407 | n[j,1..d]=m[1..d,1]; |
---|
5408 | } |
---|
5409 | dbprint(printlevel-voice+2, |
---|
5410 | "// time for computing multiplication matrix (with generic element) : "+ |
---|
5411 | string(timer-time)); |
---|
5412 | //---------------- compute characteristic polynomial of matrix -------------- |
---|
5413 | execute("ring P1=("+charstr(R)+"),T,dp;"); |
---|
5414 | matrix n=imap(R,n); |
---|
5415 | time = timer; |
---|
5416 | poly charpol=det(n-T*freemodule(d)); |
---|
5417 | dbprint(printlevel-voice+2,"// time for computing char poly: "+ |
---|
5418 | string(timer-time)); |
---|
5419 | //------------------- factorize characteristic polynomial ------------------- |
---|
5420 | //check first if constant term of charpoly is != 0 (which is true for |
---|
5421 | //sufficiently generic element) |
---|
5422 | if(charpol[size(charpol)]!=0) |
---|
5423 | { |
---|
5424 | time = timer; |
---|
5425 | list fac=factor(charpol); |
---|
5426 | testFactor(fac,charpol); |
---|
5427 | dbprint(printlevel-voice+2,"// time for factorizing char poly: "+ |
---|
5428 | string(timer-time)); |
---|
5429 | int f=size(fac[1]); |
---|
5430 | //--------------------------- the irreducible case -------------------------- |
---|
5431 | if(f==1) |
---|
5432 | { |
---|
5433 | setring R; |
---|
5434 | re=I; |
---|
5435 | return(re); |
---|
5436 | } |
---|
5437 | //---------------------------- the reducible case --------------------------- |
---|
5438 | //if f_i are the irreducible factors of charpoly, mult=ri, then <I,g_i^ri> |
---|
5439 | //are the primary components where g_i = f_i(p). However, substituting p in |
---|
5440 | //f_i may result in a huge object although the final result may be small. |
---|
5441 | //Hence it is better to simultaneously reduce with I. For this we need a new |
---|
5442 | //ring. |
---|
5443 | execute("ring P=("+charstr(R)+"),(T,"+varstr(R)+"),(dp(1),dp);"); |
---|
5444 | list rfac=imap(P1,fac); |
---|
5445 | intvec ov=option(get);; |
---|
5446 | option(redSB); |
---|
5447 | list re1; |
---|
5448 | ideal new = T-imap(R,p),imap(R,J); |
---|
5449 | attrib(new, "isSB",1); //we know that new is a standard basis |
---|
5450 | for(j=1;j<=f;j++) |
---|
5451 | { |
---|
5452 | re1[j]=reduce(rfac[1][j]^rfac[2][j],new); |
---|
5453 | } |
---|
5454 | setring R; |
---|
5455 | re = imap(P,re1); |
---|
5456 | for(j=1;j<=f;j++) |
---|
5457 | { |
---|
5458 | J=I,re[j]; |
---|
5459 | re[j]=interred(J); |
---|
5460 | } |
---|
5461 | option(set,ov); |
---|
5462 | return(re); |
---|
5463 | } |
---|
5464 | else |
---|
5465 | //------------------- choice of generic element failed ------------------- |
---|
5466 | { |
---|
5467 | dbprint(printlevel-voice+2,"// try new generic element!"); |
---|
5468 | setring R; |
---|
5469 | return(zerodec(I)); |
---|
5470 | } |
---|
5471 | } |
---|
5472 | example |
---|
5473 | { "EXAMPLE:"; echo = 2; |
---|
5474 | ring r = 0,(x,y),dp; |
---|
5475 | ideal i = x2-2,y2-2; |
---|
5476 | list pr = zerodec(i); |
---|
5477 | pr; |
---|
5478 | } |
---|
5479 | //////////////////////////////////////////////////////////////////////////// |
---|
5480 | /* |
---|
5481 | //Beispiele Wenk-Dipl (in ~/Texfiles/Diplom/Wenk/Examples/) |
---|
5482 | //Zeiten: Singular/Singular/Singular -r123456789 -v :wilde13 (PentiumPro200) |
---|
5483 | //Singular for HPUX-9 version 1-3-8 (2000060214) Jun 2 2000 15:31:26 |
---|
5484 | //(wilde13) |
---|
5485 | |
---|
5486 | //1. vdim=20, 3 Komponenten |
---|
5487 | //zerodec-time:2(1) (matrix:1 charpoly:0 factor:1) |
---|
5488 | //primdecGTZ-time: 1(0) |
---|
5489 | ring rs= 0,(a,b,c),dp; |
---|
5490 | poly f1= a^2*b*c + a*b^2*c + a*b*c^2 + a*b*c + a*b + a*c + b*c; |
---|
5491 | poly f2= a^2*b^2*c + a*b^2*c^2 + a^2*b*c + a*b*c + b*c + a + c; |
---|
5492 | poly f3= a^2*b^2*c^2 + a^2*b^2*c + a*b^2*c + a*b*c + a*c + c + 1; |
---|
5493 | ideal gls=f1,f2,f3; |
---|
5494 | int time=timer; |
---|
5495 | printlevel =1; |
---|
5496 | time=timer; list pr1=zerodec(gls); timer-time;size(pr1); |
---|
5497 | time=timer; list pr =primdecGTZ(gls); timer-time;size(pr); |
---|
5498 | time=timer; ideal ra =radical(gls); timer-time;size(pr); |
---|
5499 | |
---|
5500 | //2.cyclic5 vdim=70, 20 Komponenten |
---|
5501 | //zerodec-time:36(28) (matrix:1(0) charpoly:18(19) factor:17(9) |
---|
5502 | //primdecGTZ-time: 28(5) |
---|
5503 | //radical : 0 |
---|
5504 | ring rs= 0,(a,b,c,d,e),dp; |
---|
5505 | poly f0= a + b + c + d + e + 1; |
---|
5506 | poly f1= a + b + c + d + e; |
---|
5507 | poly f2= a*b + b*c + c*d + a*e + d*e; |
---|
5508 | poly f3= a*b*c + b*c*d + a*b*e + a*d*e + c*d*e; |
---|
5509 | poly f4= a*b*c*d + a*b*c*e + a*b*d*e + a*c*d*e + b*c*d*e; |
---|
5510 | poly f5= a*b*c*d*e - 1; |
---|
5511 | ideal gls= f1,f2,f3,f4,f5; |
---|
5512 | |
---|
5513 | //3. random vdim=40, 1 Komponente |
---|
5514 | //zerodec-time:126(304) (matrix:1 charpoly:115(298) factor:10(5)) |
---|
5515 | //primdecGTZ-time:17 (11) |
---|
5516 | ring rs=0,(x,y,z),dp; |
---|
5517 | poly f1=2*x^2 + 4*x + 3*y^2 + 7*x*z + 9*y*z + 5*z^2; |
---|
5518 | poly f2=7*x^3 + 8*x*y + 12*y^2 + 18*x*z + 3*y^4*z + 10*z^3 + 12; |
---|
5519 | poly f3=3*x^4 + 1*x*y*z + 6*y^3 + 3*x*z^2 + 2*y*z^2 + 4*z^2 + 5; |
---|
5520 | ideal gls=f1,f2,f3; |
---|
5521 | |
---|
5522 | //4. introduction into resultants, sturmfels, vdim=28, 1 Komponente |
---|
5523 | //zerodec-time:4 (matrix:0 charpoly:0 factor:4) |
---|
5524 | //primdecGTZ-time:1 |
---|
5525 | ring rs=0,(x,y),dp; |
---|
5526 | poly f1= x4+y4-1; |
---|
5527 | poly f2= x5y2-4x3y3+x2y5-1; |
---|
5528 | ideal gls=f1,f2; |
---|
5529 | |
---|
5530 | //5. 3 quadratic equations with random coeffs, vdim=8, 1 Komponente |
---|
5531 | //zerodec-time:0(0) (matrix:0 charpoly:0 factor:0) |
---|
5532 | //primdecGTZ-time:1(0) |
---|
5533 | ring rs=0,(x,y,z),dp; |
---|
5534 | poly f1=2*x^2 + 4*x*y + 3*y^2 + 7*x*z + 9*y*z + 5*z^2 + 2; |
---|
5535 | poly f2=7*x^2 + 8*x*y + 12*y^2 + 18*x*z + 3*y*z + 10*z^2 + 12; |
---|
5536 | poly f3=3*x^2 + 1*x*y + 6*y^2 + 3*x*z + 2*y*z + 4*z^2 + 5; |
---|
5537 | ideal gls=f1,f2,f3; |
---|
5538 | |
---|
5539 | //6. 3 polys vdim=24, 1 Komponente |
---|
5540 | // run("ex14",2); |
---|
5541 | //zerodec-time:5(4) (matrix:0 charpoly:3(3) factor:2(1)) |
---|
5542 | //primdecGTZ-time:4 (2) |
---|
5543 | ring rs=0,(x1,x2,x3,x4),dp; |
---|
5544 | poly f1=16*x1^2 + 3*x2^2 + 5*x3^4 - 1 - 4*x4 + x4^3; |
---|
5545 | poly f2=5*x1^3 + 3*x2^2 + 4*x3^2*x4 + 2*x1*x4 - 1 + x4 + 4*x1 + x2 + x3 + x4; |
---|
5546 | poly f3=-4*x1^2 + x2^2 + x3^2 - 3 + x4^2 + 4*x1 + x2 + x3 + x4; |
---|
5547 | poly f4=-4*x1 + x2 + x3 + x4; |
---|
5548 | ideal gls=f1,f2,f3,f4; |
---|
5549 | |
---|
5550 | //7. ex43, PoSSo, caprasse vdim=56, 16 Komponenten |
---|
5551 | //zerodec-time:23(15) (matrix:0 charpoly:16(13) factor:3(2)) |
---|
5552 | //primdecGTZ-time:3 (2) |
---|
5553 | ring rs= 0,(y,z,x,t),dp; |
---|
5554 | ideal gls=y^2*z+2*y*x*t-z-2*x, |
---|
5555 | 4*y^2*z*x-z*x^3+2*y^3*t+4*y*x^2*t-10*y^2+4*z*x+4*x^2-10*y*t+2, |
---|
5556 | 2*y*z*t+x*t^2-2*z-x, |
---|
5557 | -z^3*x+4*y*z^2*t+4*z*x*t^2+2*y*t^3+4*z^2+4*z*x-10*y*t-10*t^2+2; |
---|
5558 | |
---|
5559 | //8. Arnborg-System, n=6 (II), vdim=156, 90 Komponenten |
---|
5560 | //zerodec-time (char32003):127(45)(matrix:2(0) charpoly:106(37) factor:16(7)) |
---|
5561 | //primdecGTZ-time(char32003) :81 (18) |
---|
5562 | //ring rs= 0,(a,b,c,d,x,f),dp; |
---|
5563 | ring rs= 32003,(a,b,c,d,x,f),dp; |
---|
5564 | ideal gls=a+b+c+d+x+f, ab+bc+cd+dx+xf+af, abc+bcd+cdx+d*xf+axf+abf, |
---|
5565 | abcd+bcdx+cd*xf+ad*xf+abxf+abcf, abcdx+bcd*xf+acd*xf+abd*xf+abcxf+abcdf, |
---|
5566 | abcd*xf-1; |
---|
5567 | |
---|
5568 | //9. ex42, PoSSo, Methan6_1, vdim=27, 2 Komponenten |
---|
5569 | //zerodec-time:610 (matrix:10 charpoly:557 factor:26) |
---|
5570 | //primdecGTZ-time: 118 |
---|
5571 | //zerodec-time(char32003):2 |
---|
5572 | //primdecGTZ-time(char32003):4 |
---|
5573 | //ring rs= 0,(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10),dp; |
---|
5574 | ring rs= 32003,(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10),dp; |
---|
5575 | ideal gls=64*x2*x7-10*x1*x8+10*x7*x9+11*x7*x10-320000*x1, |
---|
5576 | -32*x2*x7-5*x2*x8-5*x2*x10+160000*x1-5000*x2, |
---|
5577 | -x3*x8+x6*x8+x9*x10+210*x6+1300000, |
---|
5578 | -x4*x8+700000, |
---|
5579 | x10^2-2*x5, |
---|
5580 | -x6*x8+x7*x9-210*x6, |
---|
5581 | -64*x2*x7-10*x7*x9-11*x7*x10+320000*x1-16*x7+7000000, |
---|
5582 | -10*x1*x8-10*x2*x8-10*x3*x8-10*x4*x8-10*x6*x8+10*x2*x10+11*x7*x10 |
---|
5583 | +20000*x2+14*x5, |
---|
5584 | x4*x8-x7*x9-x9*x10-410*x9, |
---|
5585 | 10*x2*x8+10*x3*x8+10*x6*x8+10*x7*x9-10*x2*x10-11*x7*x10-10*x9*x10 |
---|
5586 | -10*x10^2+1400*x6-4200*x10; |
---|
5587 | |
---|
5588 | //10. ex33, walk-s7, Diplomarbeit von Tim, vdim=114 |
---|
5589 | //zerfaellt in unterschiedlich viele Komponenten in versch. Charkteristiken: |
---|
5590 | //char32003:30, char0:3(2xdeg1,1xdeg112!), char181:4(2xdeg1,1xdeg28,1xdeg84) |
---|
5591 | //char 0: zerodec-time:10075 (ca 3h) (matrix:3 charpoly:9367, factor:680 |
---|
5592 | // + 24 sec fuer Normalform (anstatt einsetzen), total [29623k]) |
---|
5593 | // primdecGTZ-time: 214 |
---|
5594 | //char 32003:zerodec-time:197(68) (matrix:2(1) charpoly:173(60) factor:15(6)) |
---|
5595 | // primdecGTZ-time:14 (5) |
---|
5596 | //char 181:zerodec-time:(87) (matrix:(1) charpoly:(58) factor:(25)) |
---|
5597 | // primdecGTZ-time:(2) |
---|
5598 | //in char181 stimmen Ergebnisse von zerodec und primdecGTZ ueberein (gecheckt) |
---|
5599 | |
---|
5600 | //ring rs= 0,(a,b,c,d,e,f,g),dp; |
---|
5601 | ring rs= 32003,(a,b,c,d,e,f,g),dp; |
---|
5602 | poly f1= 2gb + 2fc + 2ed + a2 + a; |
---|
5603 | poly f2= 2gc + 2fd + e2 + 2ba + b; |
---|
5604 | poly f3= 2gd + 2fe + 2ca + c + b2; |
---|
5605 | poly f4= 2ge + f2 + 2da + d + 2cb; |
---|
5606 | poly f5= 2fg + 2ea + e + 2db + c2; |
---|
5607 | poly f6= g2 + 2fa + f + 2eb + 2dc; |
---|
5608 | poly f7= 2ga + g + 2fb + 2ec + d2; |
---|
5609 | ideal gls= f1,f2,f3,f4,f5,f6,f7; |
---|
5610 | |
---|
5611 | ~/Singular/Singular/Singular -r123456789 -v |
---|
5612 | LIB"./primdec.lib"; |
---|
5613 | timer=1; |
---|
5614 | int time=timer; |
---|
5615 | printlevel =1; |
---|
5616 | option(prot,mem); |
---|
5617 | time=timer; list pr1=zerodec(gls); timer-time; |
---|
5618 | |
---|
5619 | time=timer; list pr =primdecGTZ(gls); timer-time; |
---|
5620 | time=timer; list pr =primdecSY(gls); timer-time; |
---|
5621 | time=timer; ideal ra =radical(gls); timer-time;size(pr); |
---|
5622 | LIB"all.lib"; |
---|
5623 | |
---|
5624 | ring R=0,(a,b,c,d,e,f),dp; |
---|
5625 | ideal I=cyclic(6); |
---|
5626 | minAssGTZ(I); |
---|
5627 | |
---|
5628 | |
---|
5629 | ring S=(2,a,b),(x,y),lp; |
---|
5630 | ideal I=x8-b,y4+a; |
---|
5631 | minAssGTZ(I); |
---|
5632 | |
---|
5633 | ring S1=2,(x,y,a,b),lp; |
---|
5634 | ideal I=x8-b,y4+a; |
---|
5635 | minAssGTZ(I); |
---|
5636 | |
---|
5637 | |
---|
5638 | ring S2=(2,z),(x,y),dp; |
---|
5639 | minpoly=z2+z+1; |
---|
5640 | ideal I=y3+y+1,x4+x+1; |
---|
5641 | primdecGTZ(I); |
---|
5642 | minAssGTZ(I); |
---|
5643 | |
---|
5644 | ring S3=2,(x,y,z),dp; |
---|
5645 | ideal I=y3+y+1,x4+x+1,z2+z+1; |
---|
5646 | primdecGTZ(I); |
---|
5647 | minAssGTZ(I); |
---|
5648 | |
---|
5649 | |
---|
5650 | ring R1=2,(x,y,z),lp; |
---|
5651 | ideal I=y6+y5+y3+y2+1,x4+x+1,z2+z+1; |
---|
5652 | primdecGTZ(I); |
---|
5653 | minAssGTZ(I); |
---|
5654 | |
---|
5655 | |
---|
5656 | ring R2=(2,z),(x,y),lp; |
---|
5657 | minpoly=z3+z+1; |
---|
5658 | ideal I=y2+y+(z2+z+1),x4+x+1; |
---|
5659 | primdecGTZ(I); |
---|
5660 | minAssGTZ(I); |
---|
5661 | |
---|
5662 | */ |
---|