1 | version="$Id: equising.lib,v 1.5 2000-12-22 13:44:24 greuel Exp $"; |
---|
2 | category="Singularities"; |
---|
3 | info=" |
---|
4 | LIBRARY: equising.lib Equisingularity Stratum of a Family of Plane Curves |
---|
5 | AUTHOR: Andrea Mindnich, e-mail:mindnich@mathematik.uni-kl.de |
---|
6 | |
---|
7 | PROCEDURES: |
---|
8 | esStratum computes the equisingularity stratum of a deformation |
---|
9 | isEquising tests, whether a given deformation is equisingular |
---|
10 | " |
---|
11 | |
---|
12 | LIB "poly.lib"; |
---|
13 | LIB "elim.lib"; |
---|
14 | LIB "hnoether.lib"; |
---|
15 | |
---|
16 | /////////////////////////////////////////////////////////////////////////////// |
---|
17 | // COMPUTES a weight vector. x and y get weight 1 and all other |
---|
18 | // variables get weight 0. |
---|
19 | static proc xyVector() |
---|
20 | { |
---|
21 | intvec iv ; |
---|
22 | iv[nvars(basering)]=0 ; |
---|
23 | iv[rvar(x)] =1; |
---|
24 | iv[rvar(y)] =1; |
---|
25 | return (iv); |
---|
26 | } |
---|
27 | /////////////////////////////////////////////////////////////////////////////// |
---|
28 | // exchanges the variables x and y in the polynomial p_f |
---|
29 | static proc swapXY(poly f) |
---|
30 | { |
---|
31 | def r_base = basering; |
---|
32 | ideal MI = maxideal(1); |
---|
33 | MI[rvar(x)]=y; |
---|
34 | MI[rvar(y)]=x; |
---|
35 | map phi = r_base, MI; |
---|
36 | f=phi(f); |
---|
37 | return (f); |
---|
38 | } |
---|
39 | /////////////////////////////////////////////////////////////////////////////// |
---|
40 | // ASSUME: p_mon is a monomial |
---|
41 | // and p_g is the product of the variables in p_mon. |
---|
42 | // COMPUTES the coefficient of p_mon in p_h. |
---|
43 | static proc coefficient(poly p_h, poly p_mon, poly p_g) |
---|
44 | { |
---|
45 | matrix coefMatrix = coef(p_h,p_g); |
---|
46 | int nc = ncols(coefMatrix); |
---|
47 | int ii=1; |
---|
48 | poly p_c=0; |
---|
49 | while(ii<=nc) |
---|
50 | { |
---|
51 | if (coefMatrix[1,ii] == p_mon) |
---|
52 | { |
---|
53 | p_c = coefMatrix[2,ii]; |
---|
54 | break; |
---|
55 | } |
---|
56 | ii++; |
---|
57 | } |
---|
58 | return (p_c); |
---|
59 | } |
---|
60 | /////////////////////////////////////////////////////////////////////////////// |
---|
61 | // in p_F the variable p_vari is substituted by the polynomial p_g. |
---|
62 | static proc eSubst(poly p_F, poly p_vari, poly p_g) |
---|
63 | { |
---|
64 | def r_base = basering; |
---|
65 | ideal MI; |
---|
66 | map phi; |
---|
67 | int ii = rvar(p_vari); |
---|
68 | if (ii != 0) |
---|
69 | { |
---|
70 | MI = maxideal(1); |
---|
71 | MI[ii] = p_g; |
---|
72 | phi = r_base, MI; |
---|
73 | p_F = phi(p_F); |
---|
74 | } |
---|
75 | return (p_F); |
---|
76 | } |
---|
77 | /////////////////////////////////////////////////////////////////////////////// |
---|
78 | // All ring variables of p_F which occur in (the set of generators of) the |
---|
79 | // ideal Id, are substituted by 0 |
---|
80 | static proc SimplifyF(poly p_F, ideal Id) |
---|
81 | { |
---|
82 | int i=1; |
---|
83 | int si = size(Id); |
---|
84 | for (i=1; i <= si; i++) |
---|
85 | { |
---|
86 | if (rvar(Id[i])) |
---|
87 | { |
---|
88 | p_F = subst(p_F, Id[i], 0); |
---|
89 | } |
---|
90 | } |
---|
91 | return(p_F); |
---|
92 | } |
---|
93 | |
---|
94 | |
---|
95 | /////////////////////////////////////////////////////////////////////////////// |
---|
96 | |
---|
97 | |
---|
98 | // Checks, if the basering is admissible. |
---|
99 | static proc checkBasering () |
---|
100 | { |
---|
101 | int error = 0; |
---|
102 | if(find(charstr(basering),"real")) |
---|
103 | { |
---|
104 | ERROR ("cannot compute esStratum with 'real' as coefficient field"); |
---|
105 | } |
---|
106 | if (nvars(basering) <= 2) |
---|
107 | { |
---|
108 | ERROR ("there are to less ringvariables to compute esStratum") |
---|
109 | } |
---|
110 | error = checkQIdeal(ideal(basering)); |
---|
111 | return(error); |
---|
112 | } |
---|
113 | /////////////////////////////////////////////////////////////////////////////// |
---|
114 | static proc getInput (list #) |
---|
115 | { |
---|
116 | def r_base = basering; |
---|
117 | |
---|
118 | int maxStep = -1; |
---|
119 | |
---|
120 | if (size(#) >= 1) |
---|
121 | { |
---|
122 | if (typeof(#[1]) == "int") |
---|
123 | { |
---|
124 | maxStep = #[1]; |
---|
125 | } |
---|
126 | |
---|
127 | else |
---|
128 | { |
---|
129 | ERROR("expected esStratum('poly', 'int') "); |
---|
130 | } |
---|
131 | } |
---|
132 | |
---|
133 | return(maxStep); |
---|
134 | } |
---|
135 | ////////////////////////////////////////////////////////////////////////////// |
---|
136 | // RETURNS: 0, if the ideal cond only depends on the deformation parameters |
---|
137 | // 1, otherwise. |
---|
138 | static proc checkQIdeal (ideal cond) |
---|
139 | { |
---|
140 | def r_base = basering; |
---|
141 | |
---|
142 | int i_print = printlevel-voice + 4; |
---|
143 | int i_nvars = nvars(basering); |
---|
144 | |
---|
145 | ideal id_help = subst(cond,var(i_nvars),0,var(i_nvars-1),0) - cond; |
---|
146 | |
---|
147 | // cond depends only on the first i_nvars-2 variables <=> |
---|
148 | // id_help == <0> |
---|
149 | if ( simplify(id_help, 2) != 0) |
---|
150 | { |
---|
151 | dbprint(i_print, |
---|
152 | "ideal(basering) must only depend on the deformation parameters"); |
---|
153 | return(1); |
---|
154 | } |
---|
155 | |
---|
156 | return(0); |
---|
157 | } |
---|
158 | |
---|
159 | |
---|
160 | /////////////////////////////////////////////////////////////////////////////// |
---|
161 | // COMPUTES string(minpoly) and substitutes the parameter by newParName |
---|
162 | static proc makeMinPolyString (string newParName) |
---|
163 | { |
---|
164 | int i; |
---|
165 | string parName = parstr(basering); |
---|
166 | int parNameSize = size(parName); |
---|
167 | |
---|
168 | string oldMinPolyStr = string (minpoly); |
---|
169 | int minPolySize = size(oldMinPolyStr); |
---|
170 | |
---|
171 | string newMinPolyStr = ""; |
---|
172 | |
---|
173 | for (i=1;i <= minPolySize; i++) |
---|
174 | { |
---|
175 | if (oldMinPolyStr[i,parNameSize] == parName) |
---|
176 | { |
---|
177 | newMinPolyStr = newMinPolyStr + newParName; |
---|
178 | i = i + parNameSize-1; |
---|
179 | } |
---|
180 | else |
---|
181 | { |
---|
182 | newMinPolyStr = newMinPolyStr + oldMinPolyStr[i]; |
---|
183 | } |
---|
184 | } |
---|
185 | |
---|
186 | return(newMinPolyStr); |
---|
187 | } |
---|
188 | |
---|
189 | |
---|
190 | /////////////////////////////////////////////////////////////////////////////// |
---|
191 | // Defines a new ring without deformation-parameters. |
---|
192 | static proc createHNERing() |
---|
193 | { |
---|
194 | string str; |
---|
195 | string minpolyStr = string(minpoly); |
---|
196 | |
---|
197 | str = " ring HNERing = (" + charstr(basering) + "), (x,y), ls;"; |
---|
198 | execute (str); |
---|
199 | |
---|
200 | str = "minpoly ="+ minpolyStr+";"; |
---|
201 | execute(str); |
---|
202 | |
---|
203 | keepring(HNERing); |
---|
204 | } |
---|
205 | /////////////////////////////////////////////////////////////////////////////// |
---|
206 | // RETURNS: 1, if p_f = 0 or char(basering) divides the order of p_f |
---|
207 | // or p_f is not squarefree. |
---|
208 | // 0, otherwise |
---|
209 | static proc checkPoly (poly p_f) |
---|
210 | { |
---|
211 | int i_print = printlevel - voice + 3; |
---|
212 | int i_ord; |
---|
213 | |
---|
214 | if (p_f == 0) |
---|
215 | { |
---|
216 | dbprint(i_print,"The Input is a 'deformation' of the zero polynomial"); |
---|
217 | return(1); |
---|
218 | } |
---|
219 | |
---|
220 | i_ord = mindeg1(p_f); |
---|
221 | |
---|
222 | if (number(i_ord) == 0) |
---|
223 | { |
---|
224 | dbprint(i_print, |
---|
225 | "The characteristic of the coefficient field |
---|
226 | divides the order of the equation"); |
---|
227 | return(1); |
---|
228 | } |
---|
229 | |
---|
230 | if (squarefree(p_f) != p_f) |
---|
231 | { |
---|
232 | dbprint(i_print, "The curve is reducible"); |
---|
233 | return(1); |
---|
234 | } |
---|
235 | |
---|
236 | return(0); |
---|
237 | } |
---|
238 | /////////////////////////////////////////////////////////////////////////////// |
---|
239 | // COMPUTES the multiplicity sequence of p_f |
---|
240 | static proc calcMultSequence (poly p_f) |
---|
241 | { |
---|
242 | int i_print = printlevel-voice + 3; |
---|
243 | intvec multSeq=0; |
---|
244 | list hneList; |
---|
245 | int xNotTransversal; |
---|
246 | int fIrreducible = 1; |
---|
247 | |
---|
248 | // if basering = (p,a) or (p,a(1..s)), |
---|
249 | // p prime, a algebraic, a(1..s) transcendent use reddevelop |
---|
250 | // otherwise use develop |
---|
251 | if (char(basering) != 0 |
---|
252 | && npars(basering) !=0 |
---|
253 | && charstr(basering) == string(char(basering)) + "," + parstr(basering)) |
---|
254 | { |
---|
255 | hneList = reddevelop(p_f, -1); |
---|
256 | if ( size(hneList)>=2) |
---|
257 | { |
---|
258 | fIrreducible = 0; |
---|
259 | dbprint(i_print, "The curve is reducible"); |
---|
260 | return(multSeq, xNotTransversal, fIrreducible); |
---|
261 | } |
---|
262 | |
---|
263 | hneList = hneList[1]; |
---|
264 | |
---|
265 | xNotTransversal= hneList[3]; |
---|
266 | } |
---|
267 | else |
---|
268 | { |
---|
269 | hneList = develop(p_f, -1); |
---|
270 | |
---|
271 | xNotTransversal= hneList[3]; |
---|
272 | fIrreducible = hneList[5]; |
---|
273 | } |
---|
274 | |
---|
275 | if ( ! fIrreducible) |
---|
276 | { |
---|
277 | dbprint(i_print, "The curve is reducible"); |
---|
278 | return(multSeq, xNotTransversal, fIrreducible); |
---|
279 | } |
---|
280 | |
---|
281 | multSeq = multsequence (hneList); |
---|
282 | return(multSeq, xNotTransversal, fIrreducible); |
---|
283 | } |
---|
284 | /////////////////////////////////////////////////////////////////////////////// |
---|
285 | // ASSUME: The basering is no qring and has at least 3 variables |
---|
286 | // DEFINES: A new basering, "myRing", |
---|
287 | // with new names for the parameters and variables. |
---|
288 | // The new names for the parameters are a(1..k), |
---|
289 | // and t(1..s),x,y for the variables |
---|
290 | // The ring ordering is ordStr. |
---|
291 | // NOTE: This proc uses 'execute'. |
---|
292 | static proc createMyRing(poly p_F, string ordStr ) |
---|
293 | { |
---|
294 | def r_old = basering; |
---|
295 | |
---|
296 | int chara = char(basering); |
---|
297 | string charaStr; |
---|
298 | int i; |
---|
299 | string minPolyStr = ""; |
---|
300 | string helpStr; |
---|
301 | int nDefParams = nvars(r_old)-2; |
---|
302 | |
---|
303 | |
---|
304 | ideal qIdeal = ideal(basering); |
---|
305 | |
---|
306 | if (npars(basering) == 0) |
---|
307 | { |
---|
308 | helpStr = "ring myRing =" |
---|
309 | + string(chara)+ ", (t(1..nDefParams), x, y),"+ ordStr +";"; |
---|
310 | execute(helpStr); |
---|
311 | } |
---|
312 | |
---|
313 | else |
---|
314 | { |
---|
315 | charaStr = charstr(basering); |
---|
316 | if (charaStr == string(chara) + "," + parstr(basering)) |
---|
317 | { |
---|
318 | if (minpoly !=0) |
---|
319 | { |
---|
320 | minPolyStr = makeMinPolyString("a"); |
---|
321 | |
---|
322 | helpStr = "ring myRing = |
---|
323 | (" + string(chara) + ",a), |
---|
324 | (t(1..nDefParams), x, y)," + ordStr + ";"; |
---|
325 | execute(helpStr); |
---|
326 | |
---|
327 | helpStr = "minpoly =" + minPolyStr + ";"; |
---|
328 | execute (helpStr); |
---|
329 | } |
---|
330 | else |
---|
331 | { |
---|
332 | helpStr = "ring myRing = |
---|
333 | (" + string(chara) + ",a(1..npars(basering)) ), |
---|
334 | (t(1..nDefParams), x, y)," + ordStr + ";"; |
---|
335 | execute(helpStr); |
---|
336 | } |
---|
337 | } |
---|
338 | else |
---|
339 | { |
---|
340 | i = find (charaStr,","); |
---|
341 | |
---|
342 | helpStr = " ring myRing = |
---|
343 | (" + charaStr[1,i] + "a), |
---|
344 | (t(1..nDefParams), x, y)," + ordStr + ";"; |
---|
345 | |
---|
346 | execute (helpStr); |
---|
347 | } |
---|
348 | } |
---|
349 | |
---|
350 | ideal qIdeal = fetch(r_old, qIdeal); |
---|
351 | |
---|
352 | if(qIdeal != 0) |
---|
353 | { |
---|
354 | def r_base = basering; |
---|
355 | kill myRing; |
---|
356 | qring myRing = std(qIdeal); |
---|
357 | } |
---|
358 | |
---|
359 | poly p_F = fetch(r_old, p_F); |
---|
360 | ideal ES; |
---|
361 | |
---|
362 | keepring(myRing); |
---|
363 | } |
---|
364 | |
---|
365 | /////////////////////////////////////////////////////////////////////////////// |
---|
366 | /////////// procedures to compute the equisingularity stratum ///////////////// |
---|
367 | /////////////////////////////////////////////////////////////////////////////// |
---|
368 | // DEFINES a new basering, myRing, which has one variable |
---|
369 | // more than the old ring. |
---|
370 | // The name for the new variable is "H(nhelpV)". |
---|
371 | // p_F and ES are "imaped" into the new ring. |
---|
372 | static proc extendRing (poly p_F, ideal ES, int nHelpV, ideal HCond) |
---|
373 | { |
---|
374 | def r_old = basering; |
---|
375 | |
---|
376 | string helpStr; |
---|
377 | string minPolyStr = ""; |
---|
378 | |
---|
379 | ideal qIdeal = ideal(basering); |
---|
380 | |
---|
381 | if (minpoly != 0) |
---|
382 | { |
---|
383 | if (charstr(basering) == string(char(basering)) + "," + parstr(basering)) |
---|
384 | { |
---|
385 | minPolyStr = string(minpoly); |
---|
386 | } |
---|
387 | } |
---|
388 | |
---|
389 | string str = |
---|
390 | "ring myRing = (" + charstr(r_old) + "), |
---|
391 | (H(" + string(nHelpV)+ ")," + string(maxideal(1)) + "), |
---|
392 | (dp(" + string(nHelpV) + "),dp);"; |
---|
393 | execute (str); |
---|
394 | |
---|
395 | if (minPolyStr != "") |
---|
396 | { |
---|
397 | helpStr = "minpoly =" + minPolyStr + ";"; |
---|
398 | execute(helpStr); |
---|
399 | } |
---|
400 | |
---|
401 | ideal qIdeal = imap(r_old, qIdeal); |
---|
402 | if(qIdeal != 0) |
---|
403 | { |
---|
404 | def r_base = basering; |
---|
405 | |
---|
406 | kill myRing; |
---|
407 | |
---|
408 | attrib(qIdeal,"isSB",1); |
---|
409 | qring myRing = qIdeal; |
---|
410 | } |
---|
411 | |
---|
412 | |
---|
413 | poly p_F = imap(r_old, p_F); |
---|
414 | ideal ES = imap(r_old, ES); |
---|
415 | ideal HCond = imap(r_old, HCond); |
---|
416 | |
---|
417 | keepring(myRing); |
---|
418 | } |
---|
419 | /////////////////////////////////////////////////////////////////////////////// |
---|
420 | // COMPUTES an ideal equimultCond, such that F_(n-1) mod equimultCond =0, |
---|
421 | // where F_(n-1) is the (nNew-1)-jet of p_F with respect to x,y. |
---|
422 | static proc calcEquimultCond(poly p_F, int nNew) |
---|
423 | { |
---|
424 | ideal equimultCond = 0; |
---|
425 | poly p_FnMinus1; |
---|
426 | matrix coefMatrix; |
---|
427 | int nc; |
---|
428 | int ii = 1; |
---|
429 | |
---|
430 | p_FnMinus1 = jet(p_F, nNew-1, xyVector()); |
---|
431 | |
---|
432 | coefMatrix = coef(p_FnMinus1, xy); |
---|
433 | |
---|
434 | nc = ncols(coefMatrix); |
---|
435 | |
---|
436 | for (ii=1; ii<=nc; ii++) |
---|
437 | { |
---|
438 | equimultCond[ii] = NF(coefMatrix[2,ii],std(0)); |
---|
439 | } |
---|
440 | |
---|
441 | p_F = p_F - p_FnMinus1; |
---|
442 | p_F = SimplifyF(p_F, equimultCond); |
---|
443 | |
---|
444 | return(equimultCond, p_F); |
---|
445 | } |
---|
446 | /////////////////////////////////////////////////////////////////////////////// |
---|
447 | // COMPUTES smallest integer >= nNew/nOld -1 |
---|
448 | static proc calcNZeroSteps (int nOld,int nNew) |
---|
449 | { |
---|
450 | int nZeroSteps; |
---|
451 | |
---|
452 | if (nOld mod nNew == 0) |
---|
453 | { |
---|
454 | nZeroSteps = nOld div nNew -1; |
---|
455 | } |
---|
456 | |
---|
457 | else |
---|
458 | { |
---|
459 | nZeroSteps = nOld div nNew; |
---|
460 | } |
---|
461 | |
---|
462 | return(nZeroSteps); |
---|
463 | } |
---|
464 | /////////////////////////////////////////////////////////////////////////////// |
---|
465 | // ASSUME: ord_(X,Y)(F)=nNew |
---|
466 | // COMPUTES an ideal I such that (p_F mod I)_nNew = p_c*y^nNew. |
---|
467 | static proc purePowerOfY (poly p_F, int nNew) |
---|
468 | { |
---|
469 | ideal id_help = 0; |
---|
470 | poly p_Fn; |
---|
471 | matrix coefMatrix; |
---|
472 | int nc; |
---|
473 | poly p_c; |
---|
474 | int ii=1; |
---|
475 | |
---|
476 | p_Fn = jet(p_F, nNew, xyVector()); |
---|
477 | |
---|
478 | coefMatrix = coef(p_Fn, xy); |
---|
479 | nc = ncols(coefMatrix); |
---|
480 | |
---|
481 | p_c = coefMatrix[2,nc]; |
---|
482 | |
---|
483 | for (ii=1; ii <= nc-1; ii++) |
---|
484 | { |
---|
485 | id_help[ii] = NF(coefMatrix[2,ii],std(0)); |
---|
486 | } |
---|
487 | |
---|
488 | p_F = p_F - p_Fn + p_c*y^nNew; |
---|
489 | |
---|
490 | p_F = SimplifyF(p_F, id_help); |
---|
491 | |
---|
492 | return(id_help, p_F, p_c); |
---|
493 | } |
---|
494 | /////////////////////////////////////////////////////////////////////////////// |
---|
495 | // ASSUME: ord_(X,Y)(F)=nNew |
---|
496 | // COMPUTES an ideal I such that p_Fn mod I = p_c*(y-p_a*x)^nNew, |
---|
497 | // where p_Fn is the homogeneous part of p_F of order nNew. |
---|
498 | static proc purePowerOfLin (poly p_F, ideal HCond, int nNew, int nHelpV) |
---|
499 | { |
---|
500 | ideal id_help = 0; |
---|
501 | poly p_Fn; |
---|
502 | matrix coefMatrix; |
---|
503 | poly p_c; |
---|
504 | poly p_ca; |
---|
505 | poly p_help; |
---|
506 | poly p_a; |
---|
507 | int ii; |
---|
508 | int jj; |
---|
509 | int bico; |
---|
510 | |
---|
511 | p_Fn = jet(p_F, nNew, xyVector()); |
---|
512 | |
---|
513 | coefMatrix = coeffs(subst(p_Fn,x,1),y); |
---|
514 | |
---|
515 | p_c = coefMatrix[nNew+1,1]; |
---|
516 | p_ca = coefMatrix[nNew,1]/(-nNew); |
---|
517 | |
---|
518 | if (npars(basering)==1 && charstr(basering) != string(char(basering)) + "," + parstr(basering)) |
---|
519 | { |
---|
520 | p_a = H(nHelpV); |
---|
521 | HCond = HCond + ideal(p_ca - p_a*p_c); |
---|
522 | } |
---|
523 | else |
---|
524 | { |
---|
525 | p_help = p_ca/p_c; |
---|
526 | if (p_help * p_c == p_ca) |
---|
527 | { |
---|
528 | p_a = p_help; |
---|
529 | } |
---|
530 | else |
---|
531 | { |
---|
532 | p_a = H(nHelpV); |
---|
533 | HCond = HCond + ideal(p_ca - p_a*p_c); |
---|
534 | } |
---|
535 | } |
---|
536 | |
---|
537 | bico = (nNew*(nNew-1))/2; |
---|
538 | |
---|
539 | for (ii = 2; ii <= nNew ; ii++) |
---|
540 | { |
---|
541 | |
---|
542 | if (coefMatrix[nNew+1-ii,1] == 0) |
---|
543 | { |
---|
544 | if (number(bico) != 0) |
---|
545 | // Then a^i=0 since c is a unit |
---|
546 | { |
---|
547 | id_help = id_help + ideal(NF(p_a^(ii),std(0))); |
---|
548 | for (jj = ii+1; jj <= nNew; jj++) |
---|
549 | // the remaining coefficients (of y^(nnew-k)*x^k with k>i) |
---|
550 | // are also zero. |
---|
551 | { |
---|
552 | id_help = id_help |
---|
553 | + ideal(NF(coefMatrix[nNew+1-jj,1],std(0))); |
---|
554 | } |
---|
555 | break; |
---|
556 | } |
---|
557 | } |
---|
558 | |
---|
559 | else |
---|
560 | { |
---|
561 | id_help = id_help + |
---|
562 | ideal(NF(coefMatrix[nNew+1-ii,1] - bico*p_c*(-p_a)^ii,std(0))); |
---|
563 | } |
---|
564 | |
---|
565 | bico = (bico*(nNew-ii))/(ii+1); |
---|
566 | } |
---|
567 | |
---|
568 | p_F = SimplifyF(p_F, id_help); |
---|
569 | |
---|
570 | return(id_help, HCond, p_F, p_c, p_a); |
---|
571 | } |
---|
572 | /////////////////////////////////////////////////////////////////////////////// |
---|
573 | // eliminates the variables H(1),..,H(nHelpV) from the ideal ES + HCond |
---|
574 | static proc helpVarElim(ideal ES, ideal HCond, int nHelpV) |
---|
575 | { |
---|
576 | ES = ES + HCond; |
---|
577 | ES = std(ES); |
---|
578 | ES = nselect(ES,1,nHelpV); |
---|
579 | |
---|
580 | return(ES); |
---|
581 | } |
---|
582 | /////////////////////////////////////////////////////////////////////////////// |
---|
583 | // ASSUME: ord(F)=nNew and p_c(y-p_a*x)^n is the nNew-jet of F with respect |
---|
584 | // to X,Y |
---|
585 | // COMPUTES F(x,yx+a*x)/x^n |
---|
586 | static proc formalBlowUp(poly p_F, poly p_c, poly p_a, int nNew) |
---|
587 | { |
---|
588 | |
---|
589 | p_F = p_F - jet(p_F, nNew, xyVector()); |
---|
590 | |
---|
591 | if (p_a != 0) |
---|
592 | { |
---|
593 | p_F = eSubst(p_F, y , yx + p_a*x); |
---|
594 | } |
---|
595 | else |
---|
596 | { |
---|
597 | p_F = subst(p_F, y, xy); |
---|
598 | } |
---|
599 | |
---|
600 | p_F = p_F/(x^nNew); |
---|
601 | |
---|
602 | p_F = p_F + p_c * y^nNew; |
---|
603 | |
---|
604 | return (p_F); |
---|
605 | } |
---|
606 | /////////////////////////////////////////////////////////////////////////////// |
---|
607 | // ASSUME: p_F in K[t(1)..t(s),x,y] |
---|
608 | // COMPUTES the minimal ideal ES, such that the deformation p_F mod ES is |
---|
609 | // equisingular. |
---|
610 | // The computation is done up to iteration step 'maxstep'. |
---|
611 | // RETURNS: list l, such that |
---|
612 | // l[1]=1 if some error has occured, |
---|
613 | // l[1]=0 otherwise and then l[2] = es_cond. |
---|
614 | static proc calcEsCond(poly p_F, intvec multSeq, int maxStep) |
---|
615 | { |
---|
616 | def r_old = basering; |
---|
617 | |
---|
618 | ideal ES = 0; |
---|
619 | |
---|
620 | int ii; |
---|
621 | int step = 1; |
---|
622 | int nNew = multSeq[step]; |
---|
623 | int nOld = nNew; |
---|
624 | int nZeroSteps; |
---|
625 | int nHelpV = 1; |
---|
626 | ideal HCond = 0; |
---|
627 | int maxDeg = 0; |
---|
628 | int z = printlevel - voice + 3; |
---|
629 | string str; |
---|
630 | |
---|
631 | extendRing(p_F, ES, nHelpV, HCond); |
---|
632 | |
---|
633 | poly p_c; |
---|
634 | poly p_a; |
---|
635 | ideal I; |
---|
636 | |
---|
637 | for (ii = 1; ii <= size(multSeq); ii++) |
---|
638 | { |
---|
639 | maxDeg = maxDeg + multSeq[ii]; |
---|
640 | } |
---|
641 | |
---|
642 | while (step <= maxStep) |
---|
643 | { |
---|
644 | |
---|
645 | nOld = nNew; |
---|
646 | nNew = multSeq[step]; |
---|
647 | |
---|
648 | p_F = jet(p_F, maxDeg, xyVector()); |
---|
649 | maxDeg = maxDeg - nNew; |
---|
650 | |
---|
651 | if (nNew<nOld) |
---|
652 | //start a new line in the HNE of F |
---|
653 | // _ _ |
---|
654 | // for the next | nold/nnew -1 | iteration steps the coefficient 'a' |
---|
655 | // in the leading form Fn = c(y-ax)^n should be zero. |
---|
656 | { |
---|
657 | p_F = swapXY(p_F); |
---|
658 | nZeroSteps = calcNZeroSteps (nOld, nNew); |
---|
659 | } |
---|
660 | |
---|
661 | I, p_F = calcEquimultCond (p_F, nNew); |
---|
662 | ES = ES + I; |
---|
663 | |
---|
664 | if (z>1) |
---|
665 | { |
---|
666 | "// conditions for equimultiplicity in step:", step; |
---|
667 | I; |
---|
668 | if (nHelpV >1) |
---|
669 | { |
---|
670 | str = string(nHelpV); |
---|
671 | "// conditions for help variables H(1),..,H("+str+"):"; |
---|
672 | HCond; |
---|
673 | } |
---|
674 | pause("press <return> to continue"); |
---|
675 | } |
---|
676 | |
---|
677 | if (nZeroSteps > 0) |
---|
678 | { |
---|
679 | nZeroSteps--; |
---|
680 | |
---|
681 | // compute conditions, s.th. Fn = c*y^nnew ? |
---|
682 | I, p_F, p_c = purePowerOfY (p_F, nNew); |
---|
683 | ES = ES + I; |
---|
684 | |
---|
685 | if (z>1) |
---|
686 | { |
---|
687 | "// conditions for pure power in step:", step; |
---|
688 | I; |
---|
689 | if (nHelpV > 1) |
---|
690 | { |
---|
691 | str = string(nHelpV); |
---|
692 | "// conditions for help variables H(1),..,H("+str+"):"; |
---|
693 | HCond; |
---|
694 | } |
---|
695 | pause("press <return> to continue"); |
---|
696 | } |
---|
697 | p_a =0; |
---|
698 | } |
---|
699 | |
---|
700 | else |
---|
701 | { |
---|
702 | I, HCond, p_F, p_c, p_a = purePowerOfLin (p_F, HCond, nNew, nHelpV); |
---|
703 | |
---|
704 | ES = ES + I; |
---|
705 | |
---|
706 | if (z>1) |
---|
707 | { |
---|
708 | "// conditions for pure power in step:", step; |
---|
709 | I; |
---|
710 | str = string(nHelpV); |
---|
711 | "// conditions for help variables H(1),..,H("+str+"):"; |
---|
712 | HCond; |
---|
713 | pause("press <return> to continue"); |
---|
714 | } |
---|
715 | } |
---|
716 | |
---|
717 | p_F = formalBlowUp (p_F, p_c, p_a, nNew); |
---|
718 | |
---|
719 | if (p_a == H(nHelpV)) |
---|
720 | { |
---|
721 | nHelpV++; |
---|
722 | |
---|
723 | def r_base = basering; |
---|
724 | kill myRing; |
---|
725 | |
---|
726 | extendRing(p_F, ES, nHelpV, HCond); |
---|
727 | |
---|
728 | kill r_base; |
---|
729 | |
---|
730 | poly p_a; |
---|
731 | poly p_c; |
---|
732 | ideal I; |
---|
733 | } |
---|
734 | step++; |
---|
735 | } |
---|
736 | if (nHelpV > 1) |
---|
737 | { |
---|
738 | ES = helpVarElim(ES, HCond, nHelpV); |
---|
739 | } |
---|
740 | |
---|
741 | if (nameof(basering)=="myRing") |
---|
742 | { |
---|
743 | setring r_old; |
---|
744 | ES = imap(myRing, ES); |
---|
745 | } |
---|
746 | |
---|
747 | return(ES); |
---|
748 | } |
---|
749 | |
---|
750 | /////////////////////////////////////////////////////////////////////////////// |
---|
751 | // main procedure |
---|
752 | /////////////////////////////////////////////////////////////////////////////// |
---|
753 | |
---|
754 | proc esStratum (poly p_F, list #) |
---|
755 | "USAGE: esStratum(F[,m]); F a polynomial, m an integer |
---|
756 | ASSUME: F defines a deformation of an irreducible bivariate polynomial f |
---|
757 | and that char(basering) does not divide mult(f). |
---|
758 | If nv is the number of variables of the basering, then the first nv-2 |
---|
759 | ringvariables are the deformation parameters. |
---|
760 | If the basering is a qring, ideal(basering) must only depend |
---|
761 | on the deformation parameters. |
---|
762 | RETURN: A list l of an ideal and an integer, where |
---|
763 | l[1] is the ideal in the deformation parameters, defining the |
---|
764 | equisingularity stratum of F, |
---|
765 | l[2] = 1 if some error has occured, l[2] = 0 otherwise. |
---|
766 | If m is given, the computation stops after m steps of the iteration. |
---|
767 | NOTE: printlevel > 0 displays comments and pauses after intermediate |
---|
768 | computations (default: printlevel=0) |
---|
769 | This proc uses 'execute' or calls a procedure using 'execute'. |
---|
770 | EXAMPLE: example esStratum; shows an example |
---|
771 | " |
---|
772 | { |
---|
773 | def r_user = basering; |
---|
774 | |
---|
775 | int ii = 1; |
---|
776 | int i_nvars = nvars(basering); |
---|
777 | int error = 0; |
---|
778 | int xNotTransversal; |
---|
779 | int fIrreducible; |
---|
780 | int maxStep; |
---|
781 | int userMaxStep; |
---|
782 | ideal cond; |
---|
783 | intvec multSeq; |
---|
784 | ideal ES = 0; |
---|
785 | |
---|
786 | error = checkBasering(); |
---|
787 | if (error) |
---|
788 | { |
---|
789 | return(list(ES,error)); |
---|
790 | } |
---|
791 | |
---|
792 | userMaxStep = getInput(#); |
---|
793 | |
---|
794 | // define a new basering "myRing" with new names for parameters |
---|
795 | // and variables. |
---|
796 | // The new names are 'a(1)', ..., 'a(npars)' for the parameters |
---|
797 | // and 't(1)', ..., 't(nvars-2)', 'x', 'y' for the variables. |
---|
798 | createMyRing(p_F, "dp"); |
---|
799 | |
---|
800 | // define a ring without deformation parameters, to compute the HNE |
---|
801 | // of F mod <t_1,..,t_s> |
---|
802 | createHNERing(); |
---|
803 | |
---|
804 | poly p_f = imap(myRing,p_F); |
---|
805 | |
---|
806 | error = checkPoly(p_f); |
---|
807 | if (error) |
---|
808 | { |
---|
809 | setring r_user; |
---|
810 | return(list( ideal(0),error)); |
---|
811 | } |
---|
812 | |
---|
813 | // compute the multiplicitysequence of p_f. |
---|
814 | multSeq, xNotTransversal, fIrreducible = calcMultSequence(p_f); |
---|
815 | |
---|
816 | if ( ! fIrreducible) |
---|
817 | { |
---|
818 | setring r_user; |
---|
819 | return(list(ideal(0),1)); |
---|
820 | } |
---|
821 | |
---|
822 | setring myRing; |
---|
823 | |
---|
824 | if (xNotTransversal) |
---|
825 | { |
---|
826 | p_F = swapXY(p_F); |
---|
827 | } |
---|
828 | |
---|
829 | if (userMaxStep != -1 && userMaxStep < size(multSeq)-1) |
---|
830 | { |
---|
831 | maxStep = userMaxStep; |
---|
832 | } |
---|
833 | else |
---|
834 | { |
---|
835 | maxStep = size(multSeq)-1; |
---|
836 | } |
---|
837 | |
---|
838 | ES = calcEsCond(p_F, multSeq, maxStep); |
---|
839 | |
---|
840 | setring r_user; |
---|
841 | ES = fetch(myRing, ES); |
---|
842 | |
---|
843 | return(list(ES, error)); |
---|
844 | } |
---|
845 | |
---|
846 | example |
---|
847 | { |
---|
848 | "EXAMPLE:"; echo=2; |
---|
849 | ring r = 11,(a,b,c,d,e,f,g,x,y),ds; |
---|
850 | poly F = |
---|
851 | xa+yb+x2+2xy+y2c+y^2+y3d+y4e+y5f+y6g+x^7; |
---|
852 | esStratum(F); |
---|
853 | esStratum(F,2); |
---|
854 | ideal I = f-fa,e+b; |
---|
855 | qring q = std(I); |
---|
856 | poly F = imap(r,F); |
---|
857 | esStratum(F); |
---|
858 | } |
---|
859 | |
---|
860 | /////////////////////////////////////////////////////////////////////////////// |
---|
861 | // procedures for equisingularity test |
---|
862 | /////////////////////////////////////////////////////////////////////////////// |
---|
863 | |
---|
864 | // DEFINES a new basering, myRing, which has one variable |
---|
865 | // more than the old ring. |
---|
866 | // The name for the new variable is "H(nhelpV)". |
---|
867 | static proc T_extendRing(poly p_F, int nHelpV, ideal HCond) |
---|
868 | { |
---|
869 | def r_old = basering; |
---|
870 | |
---|
871 | ideal qIdeal = ideal(basering); |
---|
872 | |
---|
873 | string helpStr; |
---|
874 | string minPolyStr = ""; |
---|
875 | |
---|
876 | if(minpoly != 0) |
---|
877 | { |
---|
878 | if (charstr(basering) == string(char(basering)) + "," + parstr(basering)) |
---|
879 | { |
---|
880 | minPolyStr = string(minpoly); |
---|
881 | } |
---|
882 | } |
---|
883 | |
---|
884 | string str = "ring myRing = |
---|
885 | (" + charstr(r_old) + "), |
---|
886 | (H(" + string( nHelpV)+ ")," + string(maxideal(1)) + "), |
---|
887 | (dp(" + string( nHelpV) + "), ds);"; |
---|
888 | execute (str); |
---|
889 | |
---|
890 | if (minPolyStr != "") |
---|
891 | { |
---|
892 | helpStr = "minpoly =" + minPolyStr + ";"; |
---|
893 | execute(helpStr); |
---|
894 | } |
---|
895 | |
---|
896 | ideal qIdeal = imap(r_old, qIdeal); |
---|
897 | if(qIdeal != 0) |
---|
898 | { |
---|
899 | def r_base = basering; |
---|
900 | kill myRing; |
---|
901 | qring myRing = std(qIdeal); |
---|
902 | } |
---|
903 | |
---|
904 | poly p_F =imap(r_old, p_F); |
---|
905 | ideal HCond = imap(r_old, HCond); |
---|
906 | |
---|
907 | keepring(myRing); |
---|
908 | } |
---|
909 | /////////////////////////////////////////////////////////////////////////////// |
---|
910 | // tests, if ord p_F = nNew. |
---|
911 | static proc equimultTest (poly p_F, int nHelpV, int nNew, ideal HCond) |
---|
912 | { |
---|
913 | poly p_FnMinus1; |
---|
914 | ideal id_help; |
---|
915 | matrix coefMatrix; |
---|
916 | int i; |
---|
917 | int nc; |
---|
918 | |
---|
919 | p_FnMinus1 = jet(p_F, nNew-1, xyVector()); |
---|
920 | |
---|
921 | coefMatrix = coef(p_FnMinus1, xy); |
---|
922 | nc = ncols(coefMatrix); |
---|
923 | |
---|
924 | for (i=1; i<=nc; i++) |
---|
925 | { |
---|
926 | id_help[i] = coefMatrix[2,i]; |
---|
927 | } |
---|
928 | |
---|
929 | id_help = T_helpVarElim(id_help, HCond, nHelpV); |
---|
930 | |
---|
931 | if (reduce(id_help, std(0)) !=0 ) |
---|
932 | { |
---|
933 | return(0, p_F); |
---|
934 | } |
---|
935 | |
---|
936 | p_F = p_F - p_FnMinus1; |
---|
937 | |
---|
938 | return(1, p_F); |
---|
939 | } |
---|
940 | /////////////////////////////////////////////////////////////////////////////// |
---|
941 | // ASSUME: ord(p_F)=nNew |
---|
942 | // tests, if p_F = p_c*y^nNew for some p_c. |
---|
943 | static proc pPOfYTest (poly p_F, int nHelpV, int nNew, ideal HCond) |
---|
944 | { |
---|
945 | poly p_Fn; |
---|
946 | poly p_c; |
---|
947 | ideal id_help; |
---|
948 | int nc; |
---|
949 | int i=1; |
---|
950 | matrix coefMatrix; |
---|
951 | |
---|
952 | p_Fn = jet(p_F, nNew, xyVector()); |
---|
953 | |
---|
954 | coefMatrix = coef(p_Fn, xy); |
---|
955 | nc = ncols(coefMatrix); |
---|
956 | |
---|
957 | p_c = coefMatrix[2,1]; |
---|
958 | |
---|
959 | for (i = 2; i <= nc; i++) |
---|
960 | { |
---|
961 | id_help[i] = coefMatrix[2,i]; |
---|
962 | } |
---|
963 | |
---|
964 | id_help = T_helpVarElim(id_help, HCond, nHelpV); |
---|
965 | |
---|
966 | if (reduce(id_help, std(0)) !=0 ) |
---|
967 | { |
---|
968 | return(0, p_c); |
---|
969 | } |
---|
970 | |
---|
971 | return(1, p_c); |
---|
972 | } |
---|
973 | /////////////////////////////////////////////////////////////////////////////// |
---|
974 | // ASSUME: ord(p_F)=nNew |
---|
975 | // tests, if p_F = p_c*(y - p_a*x)^nNew for some p_a, p_c. |
---|
976 | static proc pPOfLinTest(poly p_F, int nNew, int nHelpV, ideal HCond) |
---|
977 | { |
---|
978 | poly p_Fn; |
---|
979 | poly p_c; |
---|
980 | poly p_ca; |
---|
981 | poly p_help; |
---|
982 | poly p_a; |
---|
983 | ideal id_help; |
---|
984 | |
---|
985 | p_Fn = jet(p_F, nNew, xyVector()); |
---|
986 | |
---|
987 | p_c = coefficient(p_Fn,y^nNew,y); |
---|
988 | p_ca = coefficient(p_Fn,y^(nNew-1)*x,xy)/-nNew; |
---|
989 | |
---|
990 | if (npars(basering)==1 |
---|
991 | && charstr(basering) != string(char(basering)) + "," + parstr(basering)) |
---|
992 | { |
---|
993 | p_a = H(nHelpV); |
---|
994 | HCond = HCond + ideal(p_ca - p_a*p_c); |
---|
995 | } |
---|
996 | else |
---|
997 | { |
---|
998 | p_help = p_ca/p_c; |
---|
999 | if (p_help * p_c == p_ca) |
---|
1000 | { |
---|
1001 | p_a = p_help; |
---|
1002 | } |
---|
1003 | else |
---|
1004 | { |
---|
1005 | p_a = H(nHelpV); |
---|
1006 | HCond = HCond + ideal(p_ca - p_a*p_c); |
---|
1007 | } |
---|
1008 | } |
---|
1009 | |
---|
1010 | id_help = ideal(p_Fn - p_c *(y - p_a *x)^nNew); |
---|
1011 | id_help = T_helpVarElim(id_help, HCond, nHelpV); |
---|
1012 | |
---|
1013 | if (reduce(id_help, std(0)) != 0 ) |
---|
1014 | { |
---|
1015 | return(0, p_F, p_c, p_a, HCond); |
---|
1016 | } |
---|
1017 | |
---|
1018 | return(1, p_F, p_c, p_a, HCond); |
---|
1019 | } |
---|
1020 | ////////////////////////////////////////////////////////////////////////////// |
---|
1021 | // eliminates the variables H(1),..,H(nHelpV) from the ideal ES + HCond |
---|
1022 | static proc T_helpVarElim(ideal ES, ideal HCond, int nHelpV) |
---|
1023 | { |
---|
1024 | |
---|
1025 | def r_old = basering; |
---|
1026 | |
---|
1027 | ideal qIdeal = ideal(basering); |
---|
1028 | |
---|
1029 | string helpStr; |
---|
1030 | string minPolyStr = ""; |
---|
1031 | |
---|
1032 | if(minpoly != 0) |
---|
1033 | { |
---|
1034 | if (charstr(basering) == string(char(basering)) + "," + parstr(basering)) |
---|
1035 | { |
---|
1036 | minPolyStr = string(minpoly); |
---|
1037 | } |
---|
1038 | } |
---|
1039 | |
---|
1040 | string str = "ring myRing = |
---|
1041 | (" + charstr(r_old) + "),(" + string(maxideal(1)) + "), |
---|
1042 | (dp(" + string( nHelpV) + "), dp);"; |
---|
1043 | execute (str); |
---|
1044 | |
---|
1045 | if (minPolyStr != "") |
---|
1046 | { |
---|
1047 | helpStr = "minpoly =" + minPolyStr + ";"; |
---|
1048 | execute(helpStr); |
---|
1049 | } |
---|
1050 | |
---|
1051 | ideal qIdeal = imap(r_old, qIdeal); |
---|
1052 | if(qIdeal != 0) |
---|
1053 | { |
---|
1054 | def r_base = basering; |
---|
1055 | kill myRing; |
---|
1056 | qring myRing = std(qIdeal); |
---|
1057 | } |
---|
1058 | |
---|
1059 | ideal ES = imap(r_old, ES); |
---|
1060 | ideal HCond = imap(r_old, HCond); |
---|
1061 | |
---|
1062 | ES = ES + HCond; |
---|
1063 | ES = std(ES); |
---|
1064 | ES = nselect(ES,1,nHelpV); |
---|
1065 | |
---|
1066 | setring r_old; |
---|
1067 | ES = imap (myRing,ES); |
---|
1068 | |
---|
1069 | return(ES); |
---|
1070 | } |
---|
1071 | /////////////////////////////////////////////////////////////////////////////// |
---|
1072 | // ASSUME: F in K[t(1)..t(s),x,y], |
---|
1073 | // the ringordering is ds |
---|
1074 | // RETURNS: list l, such that |
---|
1075 | // l[1]=1 if some error has occured, |
---|
1076 | // l[1]=0 otherwise and then |
---|
1077 | // l[2] = 1, if the deformation is equisingular and |
---|
1078 | // l[2] = 0 otherwise. |
---|
1079 | static proc equisingTest (poly p_F, intvec multSeq, int maxStep) |
---|
1080 | { |
---|
1081 | def r_old = basering; |
---|
1082 | |
---|
1083 | ideal id_Es = 0; |
---|
1084 | |
---|
1085 | int isES = 1; |
---|
1086 | int step = 1; |
---|
1087 | int nNew = multSeq[step]; |
---|
1088 | int nOld = nNew; |
---|
1089 | int zeroSteps; |
---|
1090 | ideal HCond = 0; |
---|
1091 | int nHelpV = 1; |
---|
1092 | |
---|
1093 | T_extendRing (p_F, nHelpV, HCond); |
---|
1094 | |
---|
1095 | poly p_c; |
---|
1096 | poly p_a; |
---|
1097 | |
---|
1098 | while (step <= maxStep) |
---|
1099 | { |
---|
1100 | nOld = nNew; |
---|
1101 | nNew = multSeq[step]; |
---|
1102 | |
---|
1103 | if (nNew < nOld) |
---|
1104 | //start a new line in the HNE of F |
---|
1105 | // _ _ |
---|
1106 | // for the next | nold/nnew -1 | iteration steps the coefficient 'a' |
---|
1107 | // in the leading form Fn = c(y-ax) should be zero |
---|
1108 | { |
---|
1109 | p_F = swapXY(p_F); |
---|
1110 | zeroSteps = calcNZeroSteps (nOld, nNew); |
---|
1111 | } |
---|
1112 | |
---|
1113 | isES, p_F = equimultTest (p_F, nHelpV, nNew, HCond); |
---|
1114 | |
---|
1115 | if (! isES) |
---|
1116 | { |
---|
1117 | return(0); |
---|
1118 | } |
---|
1119 | |
---|
1120 | if (zeroSteps > 0) |
---|
1121 | { |
---|
1122 | zeroSteps--; |
---|
1123 | |
---|
1124 | isES, p_c = pPOfYTest (p_F, nHelpV, nNew, HCond); |
---|
1125 | p_a = 0; |
---|
1126 | } |
---|
1127 | else |
---|
1128 | { |
---|
1129 | isES, p_F, p_c, p_a, HCond = pPOfLinTest (p_F, nNew, nHelpV, HCond); |
---|
1130 | } |
---|
1131 | |
---|
1132 | if (! isES) |
---|
1133 | { |
---|
1134 | return(0); |
---|
1135 | } |
---|
1136 | |
---|
1137 | p_F = formalBlowUp (p_F, p_c, p_a, nNew); |
---|
1138 | |
---|
1139 | if (p_a == H(nHelpV)) |
---|
1140 | { |
---|
1141 | nHelpV++; |
---|
1142 | |
---|
1143 | def r_base = basering; |
---|
1144 | kill myRing; |
---|
1145 | |
---|
1146 | T_extendRing(p_F, nHelpV, HCond); |
---|
1147 | |
---|
1148 | kill r_base; |
---|
1149 | |
---|
1150 | poly p_a; |
---|
1151 | poly p_c; |
---|
1152 | } |
---|
1153 | |
---|
1154 | step++; |
---|
1155 | } |
---|
1156 | |
---|
1157 | return(1); |
---|
1158 | } |
---|
1159 | /////////////////////////////////////////////////////////////////////////////// |
---|
1160 | |
---|
1161 | proc isEquising (poly p_F, list #) |
---|
1162 | "USAGE: esStratum(F[,m]); F a polynomial, m an integer |
---|
1163 | ASSUME: F defines a deformation of an irreducible bivariate polynomial f |
---|
1164 | and that char(basering) does not divide mult(f). |
---|
1165 | If nv is the number of variables of the basering, then the first nv-2 |
---|
1166 | ringvariables are the deformation parameters. |
---|
1167 | If the basering is a qring, ideal(basering) must only depend |
---|
1168 | on the deformation parameters. |
---|
1169 | RETURN: A list l of two integers, where |
---|
1170 | l[1] = 1 if F is an equisingular deformation,l[1] = 0 otherwise. |
---|
1171 | l[2] = 1 if some error has occured, l[2] = 0 otherwise. |
---|
1172 | If m is given, the computation stops after m steps of the iteration. |
---|
1173 | NOTE: This proc uses 'execute' or calls a procedure using 'execute'. |
---|
1174 | EXAMPLE: example isEquising; shows an example |
---|
1175 | " |
---|
1176 | { |
---|
1177 | def r_user = basering; |
---|
1178 | |
---|
1179 | int ii = 1; |
---|
1180 | int i_nvars = nvars(basering); |
---|
1181 | int error = 0; |
---|
1182 | int maxStep; |
---|
1183 | int userMaxStep; |
---|
1184 | int xNotTransversal = 0; |
---|
1185 | int fIrreducible = 1; |
---|
1186 | intvec multSeq; |
---|
1187 | ideal isES = 1; |
---|
1188 | |
---|
1189 | error = checkBasering(); |
---|
1190 | if (error) |
---|
1191 | { |
---|
1192 | return(0,1); |
---|
1193 | } |
---|
1194 | |
---|
1195 | userMaxStep = getInput(#); |
---|
1196 | |
---|
1197 | // define a new basering "myRing" with new names for parameters |
---|
1198 | // and variables. |
---|
1199 | // The new names are 'a(1)', ..., 'a(npars)' for the parameters |
---|
1200 | // and 't(1)', ..., 't(nvars-2)', 'x', 'y' for the variables. |
---|
1201 | createMyRing(p_F, "ds"); |
---|
1202 | |
---|
1203 | createHNERing(); |
---|
1204 | |
---|
1205 | poly p_f = imap(myRing,p_F); |
---|
1206 | |
---|
1207 | error = checkPoly(p_f); |
---|
1208 | if (error) |
---|
1209 | { |
---|
1210 | return(0,1); |
---|
1211 | } |
---|
1212 | |
---|
1213 | // compute the multiplicity sequence of p_f. |
---|
1214 | multSeq, xNotTransversal, fIrreducible = calcMultSequence(p_f); |
---|
1215 | |
---|
1216 | if ( ! fIrreducible) |
---|
1217 | { |
---|
1218 | return(list(0,1)); |
---|
1219 | } |
---|
1220 | |
---|
1221 | setring myRing; |
---|
1222 | |
---|
1223 | if (xNotTransversal) |
---|
1224 | { |
---|
1225 | p_F = swapXY(p_F); |
---|
1226 | } |
---|
1227 | |
---|
1228 | if (userMaxStep != -1 && userMaxStep < size(multSeq)-1) |
---|
1229 | { |
---|
1230 | maxStep = userMaxStep; |
---|
1231 | } |
---|
1232 | else |
---|
1233 | { |
---|
1234 | maxStep = size(multSeq)-1; |
---|
1235 | } |
---|
1236 | |
---|
1237 | int isES = equisingTest(p_F, multSeq, maxStep); |
---|
1238 | |
---|
1239 | return(list(isES, error)); |
---|
1240 | } |
---|
1241 | |
---|
1242 | example |
---|
1243 | { |
---|
1244 | "EXAMPLE:"; echo=2; |
---|
1245 | ring r = 11,(T,x,y),ds; |
---|
1246 | poly F = (x+y)^2+y^3*T+x^7; |
---|
1247 | isEquising(F); |
---|
1248 | isEquising(F,1); |
---|
1249 | isEquising(F,2); |
---|
1250 | ideal I = ideal(T); |
---|
1251 | qring q = std(I); |
---|
1252 | poly F = imap(r,F); |
---|
1253 | isEquising(F,2); |
---|
1254 | } |
---|
1255 | /////////////////////////////////////////////////////////////////////////////// |
---|
1256 | /* |
---|
1257 | Weiter Beispiele aus Dipl. von A. Mindnich einfuegen |
---|
1258 | */ |
---|