1 | // $Id: invar.lib,v 1.6 1998-05-05 11:55:30 krueger Exp $ |
---|
2 | /////////////////////////////////////////////////////// |
---|
3 | // invar.lib |
---|
4 | // algorithm for computing the ring of invariants under |
---|
5 | // the action of the additive group (C,+) |
---|
6 | // written by Gerhard Pfister |
---|
7 | ////////////////////////////////////////////////////// |
---|
8 | |
---|
9 | version="$Id: invar.lib,v 1.6 1998-05-05 11:55:30 krueger Exp $"; |
---|
10 | info=" |
---|
11 | LIBRARY: invar.lib PROCEDURES FOR COMPUTING INVARIANTS OF (C,+)-ACTIONS |
---|
12 | |
---|
13 | invariantRing(matrix m,poly p,poly q,int choose) |
---|
14 | // ring of invariants of the action of the additive group |
---|
15 | // defined by the vectorfield corresponding to the matrix m |
---|
16 | // (m[i,1] are the coefficients of d/dx(i)) |
---|
17 | // the polys p and q are assumed to be variables x(i) and x(j) |
---|
18 | // such that m[j,1]=0 and m[i,1]=x(j) |
---|
19 | // if choose=0 the computation stops if generators of the ring |
---|
20 | // of invariants are computed (to be used only if you know that |
---|
21 | // the ring of invariants is finitey generated) |
---|
22 | // if choose<>0 it computes invariants up to degree choose |
---|
23 | |
---|
24 | actionIsProper(matrix m) |
---|
25 | // returns 1 if the action of the additive group defined by the |
---|
26 | // matrix m as above i proper and 0 if not. |
---|
27 | "; |
---|
28 | |
---|
29 | /////////////////////////////////////////////////////////////////////////////// |
---|
30 | |
---|
31 | LIB "inout.lib"; |
---|
32 | LIB "general.lib"; |
---|
33 | |
---|
34 | /////////////////////////////////////////////////////////////////////////////// |
---|
35 | |
---|
36 | |
---|
37 | proc sortier(ideal id) |
---|
38 | { |
---|
39 | if(size(id)==0) |
---|
40 | { |
---|
41 | return(id); |
---|
42 | } |
---|
43 | intvec i=sortvec(id); |
---|
44 | int j; |
---|
45 | ideal m; |
---|
46 | for (j=1;j<=size(i);j++) |
---|
47 | { |
---|
48 | m[j] = id[i[j]]; |
---|
49 | } |
---|
50 | return(m); |
---|
51 | } |
---|
52 | example |
---|
53 | { "EXAMPLE:"; echo = 2; |
---|
54 | ring q=0,(x,y,z,u,v,w),dp; |
---|
55 | ideal i=w,x,z,y,v; |
---|
56 | ideal j=sortier(i); |
---|
57 | j; |
---|
58 | } |
---|
59 | |
---|
60 | |
---|
61 | /////////////////////////////////////////////////////////////////////////////// |
---|
62 | |
---|
63 | |
---|
64 | proc der (matrix m, poly f) |
---|
65 | "USAGE: der(m,f); m matrix, f poly |
---|
66 | RETURN: poly= application of the vectorfield m befined by the matrix m |
---|
67 | (m[i,1] are the coefficients of d/dx(i)) to f |
---|
68 | NOTE: |
---|
69 | EXAMPLE: example der; shows an example |
---|
70 | " |
---|
71 | { |
---|
72 | matrix mh=matrix(jacob(f))*m; |
---|
73 | return(mh[1,1]); |
---|
74 | } |
---|
75 | example |
---|
76 | { "EXAMPLE:"; echo = 2; |
---|
77 | ring q=0,(x,y,z,u,v,w),dp; |
---|
78 | poly f=2xz-y2; |
---|
79 | matrix m[6][1]; |
---|
80 | m[2,1]=x; |
---|
81 | m[3,1]=y; |
---|
82 | m[5,1]=u; |
---|
83 | m[6,1]=v; |
---|
84 | der(m,f); |
---|
85 | } |
---|
86 | |
---|
87 | /////////////////////////////////////////////////////////////////////////////// |
---|
88 | |
---|
89 | |
---|
90 | proc actionIsProper(matrix m) |
---|
91 | "USAGE: actionIsProper(m); m matrix |
---|
92 | RETURN: int= 1 if is proper, 0 else |
---|
93 | NOTE: |
---|
94 | EXAMPLE: example actionIsProper; shows an example |
---|
95 | " |
---|
96 | { |
---|
97 | int i; |
---|
98 | ideal id=maxideal(1); |
---|
99 | def bsr=basering; |
---|
100 | |
---|
101 | //changes the basering bsr to bsr[@t] |
---|
102 | execute "ring s="+charstr(basering)+",("+varstr(basering)+",@t),dp;"; |
---|
103 | poly inv,delta,tee,j; |
---|
104 | ideal id=imap(bsr,id); |
---|
105 | matrix @m[size(id)+1][1]; |
---|
106 | @m=imap(bsr,m),0; |
---|
107 | |
---|
108 | //computes the exp(@t*m)(var(i)) for all i |
---|
109 | for(i=1;i<=nvars(basering)-1;i++) |
---|
110 | { |
---|
111 | inv=var(i); |
---|
112 | delta=der(@m,inv); |
---|
113 | j=1; |
---|
114 | tee=@t; |
---|
115 | while(delta!=0) |
---|
116 | { |
---|
117 | inv=inv+1/j*delta*tee; |
---|
118 | j=j*(j+1); |
---|
119 | tee=tee*@t; |
---|
120 | delta=der(@m,delta); |
---|
121 | } |
---|
122 | id=id+ideal(inv); |
---|
123 | } |
---|
124 | i=inSubring(@t,id)[1]; |
---|
125 | setring(bsr); |
---|
126 | return(i); |
---|
127 | } |
---|
128 | example |
---|
129 | { "EXAMPLE:"; echo = 2; |
---|
130 | |
---|
131 | ring rf=0,(x(1..7)),dp; |
---|
132 | matrix m[7][1]; |
---|
133 | m[4,1]=x(1)^3; |
---|
134 | m[5,1]=x(2)^3; |
---|
135 | m[6,1]=x(3)^3; |
---|
136 | m[7,1]=(x(1)*x(2)*x(3))^2; |
---|
137 | actionIsProper(m); |
---|
138 | |
---|
139 | ring rd=0,(x(1..5)),dp; |
---|
140 | matrix m[5][1]; |
---|
141 | m[3,1]=x(1); |
---|
142 | m[4,1]=x(2); |
---|
143 | m[5,1]=1+x(1)*x(4)^2 |
---|
144 | actionIsProper(m); |
---|
145 | } |
---|
146 | /////////////////////////////////////////////////////////////////////////////// |
---|
147 | |
---|
148 | |
---|
149 | proc reduction(poly p, ideal dom, list #) |
---|
150 | "USAGE: reduction(p,dom,q); p poly, dom ideal, q (optional) monomial |
---|
151 | RETURN: poly= (p-H(f1,...,fr))/q^a, if Lt(p)=H(Lt(f1),...,Lt(fr)) for |
---|
152 | some polynomial H in r variables over the base field, |
---|
153 | a maximal such that q^a devides p-H(f1,...,fr), |
---|
154 | dom =(f1,...,fr) |
---|
155 | NOTE: |
---|
156 | EXAMPLE: example reduction; shows an example |
---|
157 | " |
---|
158 | { |
---|
159 | int i; |
---|
160 | int z=size(dom); |
---|
161 | def bsr=basering; |
---|
162 | |
---|
163 | //arranges the monomial v for elimination |
---|
164 | poly v=var(1); |
---|
165 | for (i=2;i<=nvars(basering);i=i+1) |
---|
166 | { |
---|
167 | v=v*var(i); |
---|
168 | } |
---|
169 | |
---|
170 | //changes the basering bsr to bsr[@(0),...,@(z)] |
---|
171 | execute "ring s="+charstr(basering)+",("+varstr(basering)+",@(0..z)),dp;"; |
---|
172 | |
---|
173 | //costructes the ideal dom=(p-@(0),dom[1]-@(1),...,dom[z]-@(z)) |
---|
174 | ideal dom=imap(bsr,dom); |
---|
175 | for (i=1;i<=z;i++) |
---|
176 | { |
---|
177 | dom[i]=lead(dom[i])-var(nvars(bsr)+i+1); |
---|
178 | } |
---|
179 | dom=lead(imap(bsr,p))-@(0),dom; |
---|
180 | |
---|
181 | //eliminates the variables of the basering bsr |
---|
182 | //i.e. computes dom intersected with K[@(0),...,@(z)] |
---|
183 | ideal kern=eliminate(dom,imap(bsr,v)); |
---|
184 | |
---|
185 | // test wether @(0)-h(@(1),...,@(z)) is in ker for some poly h |
---|
186 | poly h; |
---|
187 | z=size(kern); |
---|
188 | for (i=1;i<=z;i++) |
---|
189 | { |
---|
190 | h=kern[i]/@(0); |
---|
191 | if (deg(h)==0) |
---|
192 | { |
---|
193 | h=(1/h)*kern[i]; |
---|
194 | // defines the map psi : s ---> bsr defined by @(i) ---> p,dom[i] |
---|
195 | setring bsr; |
---|
196 | map psi=s,maxideal(1),p,dom; |
---|
197 | poly re=psi(h); |
---|
198 | |
---|
199 | // devides by the maximal power of #[1] |
---|
200 | if (size(#)>0) |
---|
201 | { |
---|
202 | while ((re!=0) && (re!=#[1]) &&(subst(re,#[1],0)==0)) |
---|
203 | { |
---|
204 | re=re/#[1]; |
---|
205 | } |
---|
206 | } |
---|
207 | |
---|
208 | return(re); |
---|
209 | } |
---|
210 | } |
---|
211 | setring bsr; |
---|
212 | return(p); |
---|
213 | } |
---|
214 | example |
---|
215 | { "EXAMPLE:"; echo = 2; |
---|
216 | ring q=0,(x,y,z,u,v,w),dp; |
---|
217 | poly p=x2yz-x2v; |
---|
218 | ideal dom =x-w,u2w+1,yz-v; |
---|
219 | reduction(p,dom); |
---|
220 | reduction(p,dom,w); |
---|
221 | } |
---|
222 | |
---|
223 | /////////////////////////////////////////////////////////////////////////////// |
---|
224 | |
---|
225 | proc completeReduction(poly p, ideal dom, list #) |
---|
226 | "USAGE: completeReduction(p,dom,q); p poly, dom ideal, |
---|
227 | q (optional) monomial |
---|
228 | RETURN: poly= the polynomial p reduced with dom via the procedure |
---|
229 | reduction as long as possible |
---|
230 | NOTE: |
---|
231 | EXAMPLE: example completeReduction; shows an example |
---|
232 | " |
---|
233 | { |
---|
234 | poly p1=p; |
---|
235 | poly p2=reduction(p,dom,#); |
---|
236 | while (p1!=p2) |
---|
237 | { |
---|
238 | p1=p2; |
---|
239 | p2=reduction(p1,dom,#); |
---|
240 | } |
---|
241 | return(p2); |
---|
242 | } |
---|
243 | example |
---|
244 | { "EXAMPLE:"; echo = 2; |
---|
245 | ring q=0,(x,y,z,u,v,w),dp; |
---|
246 | poly p=x2yz-x2v; |
---|
247 | ideal dom =x-w,u2w+1,yz-v; |
---|
248 | completeReduction(p,dom); |
---|
249 | completeReduction(p,dom,w); |
---|
250 | } |
---|
251 | /////////////////////////////////////////////////////////////////////////////// |
---|
252 | |
---|
253 | proc inSubring(poly p, ideal dom) |
---|
254 | "USAGE: inSubring(p,dom); p poly, dom ideal |
---|
255 | RETURN: list= 1,string(@(0)-h(@(1),...,@(size(dom)))) :if p = h(dom[1],...,dom[size(dom)]) |
---|
256 | 0,string(h(@(0),...,@(size(dom)))) :if there is only a nonlinear relation |
---|
257 | h(p,dom[1],...,dom[size(dom)])=0. |
---|
258 | NOTE: |
---|
259 | EXAMPLE: example inSubring; shows an example |
---|
260 | " |
---|
261 | { |
---|
262 | int z=size(dom); |
---|
263 | int i; |
---|
264 | def gnir=basering; |
---|
265 | list l; |
---|
266 | poly mile=var(1); |
---|
267 | for (i=2;i<=nvars(basering);i++) |
---|
268 | { |
---|
269 | mile=mile*var(i); |
---|
270 | } |
---|
271 | string eli=string(mile); |
---|
272 | // the intersection of ideal nett=(p-@(0),dom[1]-@(1),...) |
---|
273 | // with the ring k[@(0),...,@(n)] is computed, the result is ker |
---|
274 | execute "ring r1="+charstr(basering)+",("+varstr(basering)+",@(0..z)),dp;"; |
---|
275 | ideal nett=imap(gnir,dom); |
---|
276 | poly p; |
---|
277 | for (i=1;i<=z;i++) |
---|
278 | { |
---|
279 | execute "p=@("+string(i)+");"; |
---|
280 | nett[i]=nett[i]-p; |
---|
281 | } |
---|
282 | nett=imap(gnir,p)-@(0),nett; |
---|
283 | execute "ideal ker=eliminate(nett,"+eli+");"; |
---|
284 | // test wether @(0)-h(@(1),...,@(z)) is in ker |
---|
285 | l[1]=0; |
---|
286 | l[2]=""; |
---|
287 | for (i=1;i<=size(ker);i++) |
---|
288 | { |
---|
289 | if (deg(ker[i]/@(0))==0) |
---|
290 | { |
---|
291 | string str=string(ker[i]); |
---|
292 | setring gnir; |
---|
293 | l[1]=1; |
---|
294 | l[2]=str; |
---|
295 | return(l); |
---|
296 | } |
---|
297 | if (deg(ker[i]/@(0))>0) |
---|
298 | { |
---|
299 | l[2]=l[2]+string(ker[i]); |
---|
300 | } |
---|
301 | } |
---|
302 | setring gnir; |
---|
303 | return(l); |
---|
304 | } |
---|
305 | example |
---|
306 | { "EXAMPLE:"; echo = 2; |
---|
307 | ring q=0,(x,y,z,u,v,w),dp; |
---|
308 | poly p=xyzu2w-1yzu2w2+u4w2-1xu2vw+u2vw2+xyz-1yzw+2u2w-1xv+vw+2; |
---|
309 | ideal dom =x-w,u2w+1,yz-v; |
---|
310 | inSubring(p,dom); |
---|
311 | } |
---|
312 | |
---|
313 | /////////////////////////////////////////////////////////////////////////////// |
---|
314 | |
---|
315 | proc localInvar(matrix m, poly p, poly q, poly h) |
---|
316 | "USAGE: localInvar(m,p,q,h); m matrix, p,q,h poly |
---|
317 | RETURN: poly= the invariant of the vectorfield m=Sum m[i,1]*d/dx(i) with respect |
---|
318 | to p,q,h, i.e. |
---|
319 | Sum (-1)^v*(1/v!)*m^v(p)*(q/m(q))^v)*m(q)^N, m^N(q)=0, m^(N-1)(q)<>0 |
---|
320 | it is assumed that m(q) and h are invariant |
---|
321 | the sum above is divided by h as much as possible |
---|
322 | NOTE: |
---|
323 | EXAMPLE: example localInvar; shows an example |
---|
324 | " |
---|
325 | { |
---|
326 | if ((der(m,h) !=0) || (der(m,der(m,q)) !=0)) |
---|
327 | { |
---|
328 | "the last variable defines not an invariant function "; |
---|
329 | return(q); |
---|
330 | } |
---|
331 | poly inv=p; |
---|
332 | poly dif= der(m,inv); |
---|
333 | poly a=der(m,q); |
---|
334 | poly sgn=-1; |
---|
335 | poly coeff=sgn*q; |
---|
336 | int k=1; |
---|
337 | if (dif==0) |
---|
338 | { |
---|
339 | return(inv); |
---|
340 | } |
---|
341 | while (dif!=0) |
---|
342 | { |
---|
343 | inv=(a*inv)+(coeff*dif); |
---|
344 | dif=der(m,dif); |
---|
345 | k=k+1; |
---|
346 | coeff=q*coeff*sgn/k; |
---|
347 | } |
---|
348 | while ((inv!=0) && (inv!=h) &&(subst(inv,h,0)==0)) |
---|
349 | { |
---|
350 | inv=inv/h; |
---|
351 | } |
---|
352 | return(inv); |
---|
353 | } |
---|
354 | example |
---|
355 | { "EXAMPLE:"; echo = 2; |
---|
356 | ring q=0,(x,y,z),dp; |
---|
357 | matrix m[3][1]; |
---|
358 | m[2,1]=x; |
---|
359 | m[3,1]=y; |
---|
360 | poly in=localInvar(m,z,y,x); |
---|
361 | in; |
---|
362 | } |
---|
363 | /////////////////////////////////////////////////////////////////////////////// |
---|
364 | |
---|
365 | |
---|
366 | |
---|
367 | proc furtherInvar(matrix m, ideal id, ideal karl, poly q) |
---|
368 | "USAGE: furtherInvar(m,id,karl,q); m matrix, id,karl ideal,q poly |
---|
369 | RETURN: ideal= further invariants of the vectorfield m=Sum m[i,1]*d/dx(i) with respect |
---|
370 | to id,p,q, i.e. |
---|
371 | the ideal id contains invariants of m and we are looking for elements |
---|
372 | in the subring generated by id which are divisible by q |
---|
373 | it is assumed that m(p) and q are invariant |
---|
374 | the elements mentioned above are computed and divided by q |
---|
375 | as much as possible |
---|
376 | the ideal karl contains all invariants computed yet |
---|
377 | NOTE: |
---|
378 | EXAMPLE: example furtherInvar; shows an example |
---|
379 | " |
---|
380 | { |
---|
381 | int i; |
---|
382 | ideal null; |
---|
383 | int z=size(id); |
---|
384 | intvec v; |
---|
385 | def @r=basering; |
---|
386 | ideal su; |
---|
387 | for (i=1;i<=z;i++) |
---|
388 | { |
---|
389 | su[i]=subst(id[i],q,0); |
---|
390 | } |
---|
391 | // defines the map phi : r1 ---> @r defined by |
---|
392 | // y(i) ---> id[i](q=0) |
---|
393 | execute "ring r1="+charstr(basering)+",(y(1..z)),dp"; |
---|
394 | setring @r; |
---|
395 | map phi=r1,su; |
---|
396 | setring r1; |
---|
397 | // computes the kernel of phi |
---|
398 | execute "ideal ker=preimage(@r,phi,null)"; |
---|
399 | // defines the map psi : r1 ---> @r defined by y(i) ---> id[i] |
---|
400 | setring @r; |
---|
401 | map psi=r1,id; |
---|
402 | // computes psi(ker(phi)) |
---|
403 | ideal rel=psi(ker); |
---|
404 | // devides by the maximal power of q |
---|
405 | // and tests wether we really obtain invariants |
---|
406 | for (i=1;i<=size(rel);i++) |
---|
407 | { |
---|
408 | while ((rel[i]!=0) && (rel[i]!=q) &&(subst(rel[i],q,0)==0)) |
---|
409 | { |
---|
410 | rel[i]=rel[i]/q; |
---|
411 | if (der(m,rel[i])!=0) |
---|
412 | { |
---|
413 | "error in furtherInvar, function not invariant"; |
---|
414 | rel[i]; |
---|
415 | } |
---|
416 | } |
---|
417 | rel[i]=simplify(rel[i],1); |
---|
418 | } |
---|
419 | // test whether some variables occur linearly |
---|
420 | // and deletes the corresponding invariant function |
---|
421 | setring r1; |
---|
422 | int j; |
---|
423 | for (i=1;i<=size(ker);i=i+1) |
---|
424 | { |
---|
425 | for (j=1;j<=z;j++) |
---|
426 | { |
---|
427 | if (deg(ker[i]/y(j))==0) |
---|
428 | { |
---|
429 | setring @r; |
---|
430 | rel[i]= completeReduction(rel[i],karl,q); |
---|
431 | if(rel[i]!=0) |
---|
432 | { |
---|
433 | karl[j+1]=rel[i]; |
---|
434 | rel[i]=0; |
---|
435 | } |
---|
436 | setring r1; |
---|
437 | } |
---|
438 | } |
---|
439 | |
---|
440 | } |
---|
441 | setring @r; |
---|
442 | list l=rel+null,karl; |
---|
443 | return(l); |
---|
444 | } |
---|
445 | example |
---|
446 | { "EXAMPLE:"; echo = 2; |
---|
447 | ring r=0,(x,y,z,u),dp; |
---|
448 | matrix m[4][1]; |
---|
449 | m[2,1]=x; |
---|
450 | m[3,1]=y; |
---|
451 | m[4,1]=z; |
---|
452 | ideal id=localInvar(m,z,y,x),localInvar(m,u,y,x); |
---|
453 | ideal karl=id,x; |
---|
454 | list in=furtherInvar(m,id,karl,x); |
---|
455 | in; |
---|
456 | } |
---|
457 | /////////////////////////////////////////////////////////////////////////////// |
---|
458 | |
---|
459 | |
---|
460 | |
---|
461 | proc invariantRing(matrix m, poly p, poly q,list #) |
---|
462 | "USAGE: invariantRing(m,p,q); m matrix, p,q poly |
---|
463 | RETURN: ideal= the invariants of the vectorfield m=Sum m[i,1]*d/dx(i) |
---|
464 | p,q variables with m(p)=q invariant |
---|
465 | NOTE: |
---|
466 | EXAMPLE: example furtherInvar; shows an example |
---|
467 | " |
---|
468 | { |
---|
469 | ideal j; |
---|
470 | int i,it; |
---|
471 | int bou=-1; |
---|
472 | if(size(#)>0) |
---|
473 | { |
---|
474 | bou=#[1]; |
---|
475 | } |
---|
476 | int z; |
---|
477 | ideal karl; |
---|
478 | ideal k1=1; |
---|
479 | list k2; |
---|
480 | // computation of local invariants |
---|
481 | for (i=1;i<=nvars(basering);i++) |
---|
482 | { |
---|
483 | karl=karl+localInvar(m,var(i),p,q); |
---|
484 | } |
---|
485 | if(bou==0) |
---|
486 | { |
---|
487 | " "; |
---|
488 | "the local invariants:"; |
---|
489 | " "; |
---|
490 | karl; |
---|
491 | // pause; |
---|
492 | " "; |
---|
493 | } |
---|
494 | // computation of further invariants |
---|
495 | it=0; |
---|
496 | while (size(k1)!=0) |
---|
497 | { |
---|
498 | // test if the new invariants are already in the ring generated |
---|
499 | // by the invariants we constructed already |
---|
500 | it++; |
---|
501 | karl=sortier(karl); |
---|
502 | j=q; |
---|
503 | for (i=1;i<=size(karl);i++) |
---|
504 | { |
---|
505 | j=j+ simplify(completeReduction(karl[i],j,q),1); |
---|
506 | } |
---|
507 | karl=j; |
---|
508 | j[1]=0; |
---|
509 | j=simplify(j,2); |
---|
510 | k2=furtherInvar(m,j,karl,q); |
---|
511 | k1=k2[1]; |
---|
512 | karl=k2[2]; |
---|
513 | k1=sortier(k1); |
---|
514 | z=size(k1); |
---|
515 | for (i=1;i<=z;i++) |
---|
516 | { |
---|
517 | k1[i]= completeReduction(k1[i],karl,q); |
---|
518 | if (k1[i]!=0) |
---|
519 | { |
---|
520 | karl=karl+simplify(k1[i],1); |
---|
521 | } |
---|
522 | } |
---|
523 | if(bou==0) |
---|
524 | { |
---|
525 | " "; |
---|
526 | "the invariants after the iteration"; |
---|
527 | it; |
---|
528 | " "; |
---|
529 | karl; |
---|
530 | // pause; |
---|
531 | " "; |
---|
532 | } |
---|
533 | if((bou>0) && (size(k1)>0)) |
---|
534 | { |
---|
535 | if(deg(k1[size(k1)])>bou) |
---|
536 | { |
---|
537 | return(karl); |
---|
538 | } |
---|
539 | } |
---|
540 | } |
---|
541 | return(karl); |
---|
542 | } |
---|
543 | example |
---|
544 | { "EXAMPLE:"; echo = 2; |
---|
545 | |
---|
546 | //Winkelmann: free action but Spec k[x(1),...,x(5)]---> Spec In- |
---|
547 | //variantring is not surjective |
---|
548 | |
---|
549 | ring rw=0,(x(1..5)),dp; |
---|
550 | matrix m[5][1]; |
---|
551 | m[3,1]=x(1); |
---|
552 | m[4,1]=x(2); |
---|
553 | m[5,1]=1+x(1)*x(4)+x(2)*x(3); |
---|
554 | ideal in=invariantRing(m,x(3),x(1),0); |
---|
555 | in; |
---|
556 | |
---|
557 | //Deveney/Finston: The ring of invariants is not finitely generated |
---|
558 | |
---|
559 | ring rf=0,(x(1..7)),dp; |
---|
560 | matrix m[7][1]; |
---|
561 | m[4,1]=x(1)^3; |
---|
562 | m[5,1]=x(2)^3; |
---|
563 | m[6,1]=x(3)^3; |
---|
564 | m[7,1]=(x(1)*x(2)*x(3))^2; |
---|
565 | ideal in=invariantRing(m,x(4),x(1),6); |
---|
566 | in; |
---|
567 | |
---|
568 | |
---|
569 | //Deveney/Finston:Proper Ga-action which is not locally trivial |
---|
570 | //r[x(1),...,x(5)] is not flat over the ring of invariants |
---|
571 | |
---|
572 | ring rd=0,(x(1..5)),dp; |
---|
573 | matrix m[5][1]; |
---|
574 | m[3,1]=x(1); |
---|
575 | m[4,1]=x(2); |
---|
576 | m[5,1]=1+x(1)*x(4)^2; |
---|
577 | ideal in=invariantRing(m,x(3),x(1)); |
---|
578 | in; |
---|
579 | |
---|
580 | actionIsProper(m); |
---|
581 | |
---|
582 | //computes the relations between the invariants |
---|
583 | int z=size(in); |
---|
584 | ideal null; |
---|
585 | ring r1=0,(y(1..z)),dp; |
---|
586 | setring rd; |
---|
587 | map phi=r1,in; |
---|
588 | setring r1; |
---|
589 | ideal ker=preimage(rd,phi,null); |
---|
590 | ker; |
---|
591 | |
---|
592 | //the discriminant |
---|
593 | |
---|
594 | ring r=0,(x(1..2),y(1..2),z,t),dp; |
---|
595 | poly p=z+(1+x(1)*y(2)^2)*t+x(1)*y(1)*y(2)*t^2+(1/3)*x(1)*y(1)^2*t^3; |
---|
596 | |
---|
597 | matrix m[5][5]; |
---|
598 | m[1,1]=z; |
---|
599 | m[1,2]=x(1)*y(2)^2+1; |
---|
600 | m[1,3]=x(1)*y(1)*y(2); |
---|
601 | m[1,4]=1/3*x(1)*y(1)^2; |
---|
602 | m[1,5]=0; |
---|
603 | m[2,1]=0; |
---|
604 | m[2,2]=z; |
---|
605 | m[2,3]=x(1)*y(2)^2+1; |
---|
606 | m[2,4]=x(1)*y(1)*y(2); |
---|
607 | m[2,5]=1/3*x(1)*y(1)^2; |
---|
608 | m[3,1]=x(1)*y(2)^2+1; |
---|
609 | m[3,2]=2*x(1)*y(1)*y(2); |
---|
610 | m[3,3]=x(1)*y(1)^2; |
---|
611 | m[3,4]=0; |
---|
612 | m[3,5]=0; |
---|
613 | m[4,1]=0; |
---|
614 | m[4,2]=x(1)*y(2)^2+1; |
---|
615 | m[4,3]=2*x(1)*y(1)*y(2); |
---|
616 | m[4,4]=x(1)*y(1)^2; |
---|
617 | m[4,5]=0; |
---|
618 | m[5,1]=0; |
---|
619 | m[5,2]=0; |
---|
620 | m[5,3]=x(1)*y(2)^2+1; |
---|
621 | m[5,4]=2*x(1)*y(1)*y(2); |
---|
622 | m[5,5]=x(1)*y(1)^2; |
---|
623 | |
---|
624 | poly disc=9*det(m)/(x(1)^2*y(1)^4); |
---|
625 | |
---|
626 | LIB "invar.lib"; |
---|
627 | matrix n[6][1]; |
---|
628 | n[2,1]=x(1); |
---|
629 | n[4,1]=y(1); |
---|
630 | n[5,1]=1+x(1)*y(2)^2; |
---|
631 | |
---|
632 | der(n,disc); |
---|
633 | |
---|
634 | //x(1)^3*y(2)^6-6*x(1)^2*y(1)*y(2)^3*z+6*x(1)^2*y(2)^4+9*x(1)*y(1)^2*z^2-18*x(1)*y(1)*y(2)*z+9*x(1)*y(2)^2+4 |
---|
635 | |
---|
636 | |
---|
637 | //constructive approach to Weizenbcks theorem |
---|
638 | |
---|
639 | int n=5; |
---|
640 | |
---|
641 | ring w=0,(x(1..n)),wp(1..n); |
---|
642 | |
---|
643 | // definition of the vectorfield m=sum m[i]*d/dx(i) |
---|
644 | matrix m[n][1]; |
---|
645 | int i; |
---|
646 | for (i=1;i<=n-1;i=i+1) |
---|
647 | { |
---|
648 | m[i+1,1]=x(i); |
---|
649 | } |
---|
650 | ideal in=invariantRing(m,x(2),x(1),0); |
---|
651 | in; |
---|
652 | |
---|
653 | |
---|
654 | |
---|
655 | } |
---|