1 | version="$Id: equising.lib,v 1.17 2006-07-18 15:48:12 Singular Exp $"; |
---|
2 | category="Singularities"; |
---|
3 | info=" |
---|
4 | LIBRARY: equising.lib Equisingularity Stratum of a Family of Plane Curves |
---|
5 | AUTHOR: Christoph Lossen, lossen@mathematik.uni-kl.de |
---|
6 | Andrea Mindnich, mindnich@mathematik.uni-kl.de |
---|
7 | |
---|
8 | MAIN PROCEDURES: |
---|
9 | tau_es(f); codim of mu-const stratum in semi-universal def. base |
---|
10 | esIdeal(f); (Wahl's) equisingularity ideal of f |
---|
11 | esStratum(F[,m,L]); equisingularity stratum of a family F |
---|
12 | isEquising(F[,m,L]); tests if a given deformation is equisingular |
---|
13 | |
---|
14 | AUXILIARY PROCEDURE: |
---|
15 | control_Matrix(M); computes list of blowing-up data |
---|
16 | "; |
---|
17 | |
---|
18 | LIB "hnoether.lib"; |
---|
19 | LIB "poly.lib"; |
---|
20 | LIB "elim.lib"; |
---|
21 | LIB "deform.lib"; |
---|
22 | LIB "sing.lib"; |
---|
23 | |
---|
24 | //////////////////////////////////////////////////////////////////////////////// |
---|
25 | // |
---|
26 | // The following (static) procedures are used by esComputation |
---|
27 | // |
---|
28 | //////////////////////////////////////////////////////////////////////////////// |
---|
29 | // COMPUTES a weight vector. x and y get weight 1 and all other |
---|
30 | // variables get weight 0. |
---|
31 | static proc xyVector() |
---|
32 | { |
---|
33 | intvec iv ; |
---|
34 | iv[nvars(basering)]=0 ; |
---|
35 | iv[rvar(x)] =1; |
---|
36 | iv[rvar(y)] =1; |
---|
37 | return (iv); |
---|
38 | } |
---|
39 | |
---|
40 | //////////////////////////////////////////////////////////////////////////////// |
---|
41 | // exchanges the variables x and y in the polynomial f |
---|
42 | static proc swapXY(poly f) |
---|
43 | { |
---|
44 | def r_base = basering; |
---|
45 | ideal MI = maxideal(1); |
---|
46 | MI[rvar(x)]=y; |
---|
47 | MI[rvar(y)]=x; |
---|
48 | map phi = r_base, MI; |
---|
49 | f=phi(f); |
---|
50 | return (f); |
---|
51 | } |
---|
52 | |
---|
53 | //////////////////////////////////////////////////////////////////////////////// |
---|
54 | // computes m-jet w.r.t. the variables x,y (other variables weighted 0 |
---|
55 | static proc m_Jet(poly F,int m); |
---|
56 | { |
---|
57 | intvec w=xyVector(); |
---|
58 | poly Fd=jet(F,m,w); |
---|
59 | return(Fd); |
---|
60 | } |
---|
61 | |
---|
62 | |
---|
63 | //////////////////////////////////////////////////////////////////////////////// |
---|
64 | // computes the 4 control matrices (input is multsequence(L)) |
---|
65 | proc control_Matrix(list M); |
---|
66 | "USAGE: control_Matrix(L); L list |
---|
67 | ASSUME: L is the output of multsequence(hnexpansion(f)). |
---|
68 | RETURN: list M of 4 intmat's: |
---|
69 | @format |
---|
70 | M[1] contains the multiplicities at the respective infinitely near points |
---|
71 | p[i,j] (i=step of blowup+1, j=branch) -- if branches j=k,...,k+m pass |
---|
72 | through the same p[i,j] then the multiplicity is stored in M[1][k,j], |
---|
73 | while M[1][k+1]=...=M[1][k+m]=0. |
---|
74 | M[2] contains the number of branches meeting at p[i,j] (again, the information |
---|
75 | is stored according to the above rule) |
---|
76 | M[3] contains the information about the splitting of M[1][i,j] with respect to |
---|
77 | different tangents of branches at p[i,j] (information is stored only for |
---|
78 | minimal j>=k corresponding to a new tangent direction). |
---|
79 | The entries are the sum of multiplicities of all branches with the |
---|
80 | respective tangent. |
---|
81 | M[4] contains the maximal sum of higher multiplicities for a branch passing |
---|
82 | through p[i,j] ( = degree Bound for blowing up) |
---|
83 | @end format |
---|
84 | NOTE: the branches are ordered in such a way that only consecutive branches |
---|
85 | can meet at an infinitely near point. @* |
---|
86 | the final rows of the matrices M[1],...,M[3] is (1,1,1,...,1), and |
---|
87 | correspond to infinitely near points such that the strict transforms |
---|
88 | of the branches are smooth and intersect the exceptional divisor |
---|
89 | transversally. |
---|
90 | SEE ALSO: multsequence |
---|
91 | EXAMPLE: example control_Matrix; shows an example |
---|
92 | " |
---|
93 | { |
---|
94 | int i,j,k,dummy; |
---|
95 | |
---|
96 | dummy=0; |
---|
97 | for (j=1;j<=ncols(M[2]);j++) |
---|
98 | { |
---|
99 | dummy=dummy+M[1][nrows(M[1])-1,j]-M[1][nrows(M[1]),j]; |
---|
100 | } |
---|
101 | intmat S[nrows(M[1])+dummy][ncols(M[1])]; |
---|
102 | intmat T[nrows(M[1])+dummy][ncols(M[1])]; |
---|
103 | intmat U[nrows(M[1])+dummy][ncols(M[1])]; |
---|
104 | intmat maxDeg[nrows(M[1])+dummy][ncols(M[1])]; |
---|
105 | |
---|
106 | for (i=1;i<=nrows(M[2]);i++) |
---|
107 | { |
---|
108 | dummy=1; |
---|
109 | for (j=1;j<=ncols(M[2]);j++) |
---|
110 | { |
---|
111 | for (k=dummy;k<dummy+M[2][i,j];k++) |
---|
112 | { |
---|
113 | T[i,dummy]=T[i,dummy]+1; |
---|
114 | S[i,dummy]=S[i,dummy]+M[1][i,k]; |
---|
115 | if (i>1) |
---|
116 | { |
---|
117 | U[i-1,dummy]=U[i-1,dummy]+M[1][i-1,k]; |
---|
118 | } |
---|
119 | } |
---|
120 | dummy=k; |
---|
121 | } |
---|
122 | } |
---|
123 | |
---|
124 | // adding an extra row (in some cases needed to control ES-Stratum |
---|
125 | // computation) |
---|
126 | for (i=nrows(M[1]);i<=nrows(S);i++) |
---|
127 | { |
---|
128 | for (j=1;j<=ncols(M[2]);j++) |
---|
129 | { |
---|
130 | S[i,j]=1; |
---|
131 | T[i,j]=1; |
---|
132 | U[i,j]=1; |
---|
133 | } |
---|
134 | } |
---|
135 | |
---|
136 | // Computing the degree Bounds to be stored in M[4]: |
---|
137 | for (i=1;i<=nrows(S);i++) |
---|
138 | { |
---|
139 | dummy=1; |
---|
140 | for (j=1;j<=ncols(S);j++) |
---|
141 | { |
---|
142 | for (k=dummy;k<dummy+T[i,j];k++) |
---|
143 | { |
---|
144 | maxDeg[i,k]=S[i,dummy]; // multiplicity at i-th blowup |
---|
145 | } |
---|
146 | dummy=k; |
---|
147 | } |
---|
148 | } |
---|
149 | // adding up multiplicities |
---|
150 | for (i=nrows(S);i>=2;i--) |
---|
151 | { |
---|
152 | for (j=1;j<=ncols(S);j++) |
---|
153 | { |
---|
154 | maxDeg[i-1,j]=maxDeg[i-1,j]+maxDeg[i,j]; |
---|
155 | } |
---|
156 | } |
---|
157 | |
---|
158 | list L=S,T,U,maxDeg; |
---|
159 | return(L); |
---|
160 | } |
---|
161 | |
---|
162 | |
---|
163 | //////////////////////////////////////////////////////////////////////////////// |
---|
164 | // matrix of higher tangent directions: |
---|
165 | // returns list: 1) tangent directions |
---|
166 | // 2) swapping information (x <--> y) |
---|
167 | static proc inf_Tangents(list L,int s); // L aus hnexpansion, |
---|
168 | { |
---|
169 | int nv=nvars(basering); |
---|
170 | matrix M; |
---|
171 | matrix B[s][size(L)]; |
---|
172 | intvec V; |
---|
173 | intmat Mult=multsequence(L)[1]; |
---|
174 | |
---|
175 | int i,j,k,counter,e; |
---|
176 | for (k=1;k<=size(L);k++) |
---|
177 | { |
---|
178 | V[k]=L[k][3]; // switch: 0 --> tangent 2nd parameter |
---|
179 | // 1 --> tangent 1st parameter |
---|
180 | e=0; |
---|
181 | M=L[k][1]; |
---|
182 | counter=1; |
---|
183 | B[counter,k]=M[1,1]; |
---|
184 | |
---|
185 | for (i=1;i<=nrows(M);i++) |
---|
186 | { |
---|
187 | for (j=2;j<=ncols(M);j++) |
---|
188 | { |
---|
189 | counter=counter+1; |
---|
190 | if (M[i,j]==var(nv-1)) |
---|
191 | { |
---|
192 | if (i<>nrows(M)) |
---|
193 | { |
---|
194 | B[counter,k]=M[i,j]; |
---|
195 | j=ncols(M)+1; // goto new row of HNmatrix... |
---|
196 | if (counter<>s) |
---|
197 | { |
---|
198 | if (counter+1<=nrows(Mult)) |
---|
199 | { |
---|
200 | e=Mult[counter-1,k]-Mult[counter,k]-Mult[counter+1,k]; |
---|
201 | } |
---|
202 | else |
---|
203 | { |
---|
204 | e=Mult[counter-1,k]-Mult[counter,k]-1; |
---|
205 | } |
---|
206 | } |
---|
207 | } |
---|
208 | else |
---|
209 | { |
---|
210 | B[counter,k]=0; |
---|
211 | j=ncols(M)+1; // goto new row of HNmatrix... |
---|
212 | } |
---|
213 | } |
---|
214 | else |
---|
215 | { |
---|
216 | if (e<=0) |
---|
217 | { |
---|
218 | B[counter,k]=M[i,j]; |
---|
219 | } |
---|
220 | else // point is still proximate to an earlier point |
---|
221 | { |
---|
222 | B[counter,k]=y; // marking proximity (without swap....) |
---|
223 | if (counter+1<=nrows(Mult)) |
---|
224 | { |
---|
225 | e=e-Mult[counter+1,k]; |
---|
226 | } |
---|
227 | else |
---|
228 | { |
---|
229 | e=e-1; |
---|
230 | } |
---|
231 | } |
---|
232 | } |
---|
233 | |
---|
234 | if (counter==s) // given number of points determined |
---|
235 | { |
---|
236 | j=ncols(M)+1; |
---|
237 | i=nrows(M)+1; |
---|
238 | // leave procedure |
---|
239 | } |
---|
240 | } |
---|
241 | } |
---|
242 | } |
---|
243 | L=B,V; |
---|
244 | return(L); |
---|
245 | } |
---|
246 | |
---|
247 | //////////////////////////////////////////////////////////////////////////////// |
---|
248 | // compute "good" upper bound for needed number of help variables |
---|
249 | // |
---|
250 | static proc Determine_no_b(intmat U,matrix B) |
---|
251 | // U is assumed to be 3rd output of control_Matrix |
---|
252 | // B is assumed to be 1st output of inf_Tangents |
---|
253 | { |
---|
254 | int nv=nvars(basering); |
---|
255 | int i,j,counter; |
---|
256 | for (j=1;j<=ncols(U);j++) |
---|
257 | { |
---|
258 | for (i=1;i<=nrows(U);i++) |
---|
259 | { |
---|
260 | if (U[i,j]>1) |
---|
261 | { |
---|
262 | if (B[i,j]<>var(nv-1) and B[i,j]<>var(nv)) |
---|
263 | { |
---|
264 | counter=counter+1; |
---|
265 | } |
---|
266 | } |
---|
267 | |
---|
268 | } |
---|
269 | } |
---|
270 | counter=counter+ncols(U); |
---|
271 | return(counter); |
---|
272 | } |
---|
273 | |
---|
274 | //////////////////////////////////////////////////////////////////////////////// |
---|
275 | // compute number of infinitely near free points corresponding to non-zero |
---|
276 | // entries in control_Matrix[1] (except first row) |
---|
277 | // |
---|
278 | static proc no_freePoints(intmat Mult,matrix B) |
---|
279 | // Mult is assumed to be 1st output of control_Matrix |
---|
280 | // U is assumed to be 3rd output of control_Matrix |
---|
281 | // B is assumed to be 1st output of inf_Tangents |
---|
282 | { |
---|
283 | int i,j,k,counter; |
---|
284 | for (j=1;j<=ncols(Mult);j++) |
---|
285 | { |
---|
286 | for (i=2;i<=nrows(Mult);i++) |
---|
287 | { |
---|
288 | if (Mult[i,j]>=1) |
---|
289 | { |
---|
290 | if (B[i-1,j]<>x and B[i-1,j]<>y) |
---|
291 | { |
---|
292 | counter=counter+1; |
---|
293 | } |
---|
294 | } |
---|
295 | } |
---|
296 | } |
---|
297 | return(counter); |
---|
298 | } |
---|
299 | |
---|
300 | |
---|
301 | /////////////////////////////////////////////////////////////////////////////// |
---|
302 | // COMPUTES string(minpoly) and substitutes the parameter by newParName |
---|
303 | static proc makeMinPolyString (string newParName) |
---|
304 | { |
---|
305 | int i; |
---|
306 | string parName = parstr(basering); |
---|
307 | int parNameSize = size(parName); |
---|
308 | |
---|
309 | string oldMinPolyStr = string (minpoly); |
---|
310 | int minPolySize = size(oldMinPolyStr); |
---|
311 | |
---|
312 | string newMinPolyStr = ""; |
---|
313 | |
---|
314 | for (i=1;i <= minPolySize; i++) |
---|
315 | { |
---|
316 | if (oldMinPolyStr[i,parNameSize] == parName) |
---|
317 | { |
---|
318 | newMinPolyStr = newMinPolyStr + newParName; |
---|
319 | i = i + parNameSize-1; |
---|
320 | } |
---|
321 | else |
---|
322 | { |
---|
323 | newMinPolyStr = newMinPolyStr + oldMinPolyStr[i]; |
---|
324 | } |
---|
325 | } |
---|
326 | |
---|
327 | return(newMinPolyStr); |
---|
328 | } |
---|
329 | |
---|
330 | |
---|
331 | /////////////////////////////////////////////////////////////////////////////// |
---|
332 | // |
---|
333 | // DEFINES: A new basering, "myRing", |
---|
334 | // with new names for the parameters and variables. |
---|
335 | // The new names for the parameters are a(1..k), |
---|
336 | // and t(1..s),x,y for the variables |
---|
337 | // The ring ordering is ordStr. |
---|
338 | // NOTE: This proc uses 'execute'. |
---|
339 | static proc createMyRing_new(poly p_F, string ordStr, |
---|
340 | string minPolyStr, int no_b) |
---|
341 | { |
---|
342 | def r_old = basering; |
---|
343 | |
---|
344 | int chara = char(basering); |
---|
345 | string charaStr; |
---|
346 | int i; |
---|
347 | string helpStr; |
---|
348 | int nDefParams = nvars(r_old)-2; |
---|
349 | |
---|
350 | ideal qIdeal = ideal(basering); |
---|
351 | |
---|
352 | if ((npars(basering)==0) and (minPolyStr=="")) |
---|
353 | { |
---|
354 | helpStr = "ring myRing1 =" |
---|
355 | + string(chara)+ ", (t(1..nDefParams), x, y),("+ ordStr +");"; |
---|
356 | execute(helpStr); |
---|
357 | } |
---|
358 | else |
---|
359 | { |
---|
360 | charaStr = charstr(basering); |
---|
361 | if (charaStr == string(chara) + "," + parstr(basering) or minPolyStr<>"") |
---|
362 | { |
---|
363 | if (minPolyStr<>"") |
---|
364 | { |
---|
365 | helpStr = "ring myRing1 = |
---|
366 | (" + string(chara) + ",a), |
---|
367 | (t(1..nDefParams), x, y),(" + ordStr + ");"; |
---|
368 | execute(helpStr); |
---|
369 | |
---|
370 | execute (minPolyStr); |
---|
371 | } |
---|
372 | else // no minpoly given |
---|
373 | { |
---|
374 | helpStr = "ring myRing1 = |
---|
375 | (" + string(chara) + ",a(1..npars(basering)) ), |
---|
376 | (t(1..nDefParams), x, y),(" + ordStr + ");"; |
---|
377 | execute(helpStr); |
---|
378 | } |
---|
379 | } |
---|
380 | else |
---|
381 | { |
---|
382 | // ground field is of type (p^k,a).... |
---|
383 | i = find (charaStr,","); |
---|
384 | helpStr = "ring myRing1 = (" + charaStr[1,i] + "a), |
---|
385 | (t(1..nDefParams), x, y),(" + ordStr + ");"; |
---|
386 | execute (helpStr); |
---|
387 | } |
---|
388 | } |
---|
389 | |
---|
390 | ideal mIdeal = maxideal(1); |
---|
391 | ideal qIdeal = fetch(r_old, qIdeal); |
---|
392 | poly p_F = fetch(r_old, p_F); |
---|
393 | export p_F,mIdeal; |
---|
394 | |
---|
395 | // Extension by no_b auxiliary variables |
---|
396 | if (no_b>0) |
---|
397 | { |
---|
398 | if (npars(basering) == 0) |
---|
399 | { |
---|
400 | ordStr = "(dp("+string(no_b)+"),"+ordStr+")"; |
---|
401 | helpStr = "ring myRing =" |
---|
402 | + string(chara)+ ", (b(1..no_b), t(1..nDefParams), x, y)," |
---|
403 | + ordStr +";"; |
---|
404 | execute(helpStr); |
---|
405 | } |
---|
406 | else |
---|
407 | { |
---|
408 | charaStr = charstr(basering); |
---|
409 | if (charaStr == string(chara) + "," + parstr(basering)) |
---|
410 | { |
---|
411 | if (minpoly !=0) |
---|
412 | { |
---|
413 | ordStr = "(dp(" + string(no_b) + ")," + ordStr + ")"; |
---|
414 | minPolyStr = makeMinPolyString("a"); |
---|
415 | helpStr = "ring myRing = |
---|
416 | (" + string(chara) + ",a), |
---|
417 | (b(1..no_b), t(1..nDefParams), x, y)," + ordStr + ";"; |
---|
418 | execute(helpStr); |
---|
419 | |
---|
420 | helpStr = "minpoly =" + minPolyStr + ";"; |
---|
421 | execute (helpStr); |
---|
422 | } |
---|
423 | else // no minpoly given |
---|
424 | { |
---|
425 | ordStr = "(dp(" + string(no_b) + ")," + ordStr + ")"; |
---|
426 | helpStr = "ring myRing = |
---|
427 | (" + string(chara) + ",a(1..npars(basering)) ), |
---|
428 | (b(1..no_b), t(1..nDefParams), x, y)," + ordStr + ";"; |
---|
429 | execute(helpStr); |
---|
430 | } |
---|
431 | } |
---|
432 | else |
---|
433 | { |
---|
434 | i = find (charaStr,","); |
---|
435 | ordStr = "(dp(" + string(no_b) + ")," + ordStr + ")"; |
---|
436 | helpStr = "ring myRing = |
---|
437 | (" + charaStr[1,i] + "a), |
---|
438 | (b(1..no_b), t(1..nDefParams), x, y)," + ordStr + ";"; |
---|
439 | execute (helpStr); |
---|
440 | } |
---|
441 | } |
---|
442 | ideal qIdeal = imap(myRing1, qIdeal); |
---|
443 | |
---|
444 | if(qIdeal != 0) |
---|
445 | { |
---|
446 | def r_base = basering; |
---|
447 | setring r_base; |
---|
448 | kill myRing; |
---|
449 | qring myRing = std(qIdeal); |
---|
450 | } |
---|
451 | |
---|
452 | poly p_F = imap(myRing1, p_F); |
---|
453 | ideal mIdeal = imap(myRing1, mIdeal); |
---|
454 | export p_F,mIdeal; |
---|
455 | kill myRing1; |
---|
456 | } |
---|
457 | else |
---|
458 | { |
---|
459 | if(qIdeal != 0) |
---|
460 | { |
---|
461 | def r_base = basering; |
---|
462 | setring r_base; |
---|
463 | kill myRing1; |
---|
464 | qring myRing = std(qIdeal); |
---|
465 | poly p_F = imap(myRing1, p_F); |
---|
466 | ideal mIdeal = imap(myRing1, mIdeal); |
---|
467 | export p_F,mIdeal; |
---|
468 | } |
---|
469 | else |
---|
470 | { |
---|
471 | def myRing=myRing1; |
---|
472 | } |
---|
473 | kill myRing1; |
---|
474 | } |
---|
475 | |
---|
476 | setring r_old; |
---|
477 | return(myRing); |
---|
478 | } |
---|
479 | |
---|
480 | //////////////////////////////////////////////////////////////////////////////// |
---|
481 | // returns list of coef, leadmonomial |
---|
482 | // |
---|
483 | static proc determine_coef (poly Fm) |
---|
484 | { |
---|
485 | def r_base = basering; // is assumed to be the result of createMyRing |
---|
486 | |
---|
487 | int chara = char(basering); |
---|
488 | string charaStr; |
---|
489 | int i; |
---|
490 | string minPolyStr = ""; |
---|
491 | string helpStr = ""; |
---|
492 | |
---|
493 | if (npars(basering) == 0) |
---|
494 | { |
---|
495 | helpStr = "ring myRing1 =" |
---|
496 | + string(chara)+ ", (y,x),ds;"; |
---|
497 | execute(helpStr); |
---|
498 | } |
---|
499 | else |
---|
500 | { |
---|
501 | charaStr = charstr(basering); |
---|
502 | if (charaStr == string(chara) + "," + parstr(basering)) |
---|
503 | { |
---|
504 | if (minpoly !=0) |
---|
505 | { |
---|
506 | minPolyStr = makeMinPolyString("a"); |
---|
507 | helpStr = "ring myRing1 = (" + string(chara) + ",a), (y,x),ds;"; |
---|
508 | execute(helpStr); |
---|
509 | |
---|
510 | helpStr = "minpoly =" + minPolyStr + ";"; |
---|
511 | execute (helpStr); |
---|
512 | } |
---|
513 | else // no minpoly given |
---|
514 | { |
---|
515 | helpStr = "ring myRing1 = |
---|
516 | (" + string(chara) + ",a(1..npars(basering)) ), (y,x),ds;"; |
---|
517 | execute(helpStr); |
---|
518 | } |
---|
519 | } |
---|
520 | else |
---|
521 | { |
---|
522 | i = find (charaStr,","); |
---|
523 | |
---|
524 | helpStr = " ring myRing1 = (" + charaStr[1,i] + "a), (y,x),ds;"; |
---|
525 | execute (helpStr); |
---|
526 | } |
---|
527 | } |
---|
528 | poly f=imap(r_base,Fm); |
---|
529 | poly g=leadmonom(f); |
---|
530 | setring r_base; |
---|
531 | poly g=imap(myRing1,g); |
---|
532 | kill myRing1; |
---|
533 | def M=coef(Fm,xy); |
---|
534 | |
---|
535 | for (i=1; i<=ncols(M); i++) |
---|
536 | { |
---|
537 | if (M[1,i]==g) |
---|
538 | { |
---|
539 | poly h=M[2,i]; // determine coefficient of leading monomial (in K[t]) |
---|
540 | i=ncols(M)+1; |
---|
541 | } |
---|
542 | } |
---|
543 | return(list(h,g)); |
---|
544 | } |
---|
545 | |
---|
546 | /////////////////////////////////////////////////////////////////////////////// |
---|
547 | // RETURNS: 1, if p_f = 0 or char(basering) divides the order of p_f |
---|
548 | // or p_f is not squarefree. |
---|
549 | // 0, otherwise |
---|
550 | static proc checkPoly (poly p_f) |
---|
551 | { |
---|
552 | int i_print = printlevel - voice + 3; |
---|
553 | int i_ord; |
---|
554 | |
---|
555 | if (p_f == 0) |
---|
556 | { |
---|
557 | print("Input is a 'deformation' of the zero polynomial!"); |
---|
558 | return(1); |
---|
559 | } |
---|
560 | |
---|
561 | i_ord = mindeg1(p_f); |
---|
562 | |
---|
563 | if (number(i_ord) == 0) |
---|
564 | { |
---|
565 | print("Characteristic of coefficient field " |
---|
566 | +"divides order of zero-fiber !"); |
---|
567 | return(1); |
---|
568 | } |
---|
569 | |
---|
570 | if (squarefree(p_f) != p_f) |
---|
571 | { |
---|
572 | print("Original polynomial (= zero-fiber) is not reduced!"); |
---|
573 | return(1); |
---|
574 | } |
---|
575 | |
---|
576 | return(0); |
---|
577 | } |
---|
578 | |
---|
579 | //////////////////////////////////////////////////////////////////////////////// |
---|
580 | static proc make_ring_small(ideal J) |
---|
581 | // returns varstr for new ring, the map and the number of vars |
---|
582 | { |
---|
583 | attrib(J,"isSB",1); |
---|
584 | int counter=0; |
---|
585 | ideal newmap; |
---|
586 | string newvar=""; |
---|
587 | for (int i=1; i<=nvars(basering); i++) |
---|
588 | { |
---|
589 | if (reduce(var(i),J)<>0) |
---|
590 | { |
---|
591 | newmap[i]=var(i); |
---|
592 | |
---|
593 | if (newvar=="") |
---|
594 | { |
---|
595 | newvar=newvar+string(var(i)); |
---|
596 | counter=counter+1; |
---|
597 | } |
---|
598 | else |
---|
599 | { |
---|
600 | newvar=newvar+","+string(var(i)); |
---|
601 | counter=counter+1; |
---|
602 | } |
---|
603 | } |
---|
604 | else |
---|
605 | { |
---|
606 | newmap[i]=0; |
---|
607 | } |
---|
608 | } |
---|
609 | list L=newvar,newmap,counter; |
---|
610 | attrib(J,"isSB",0); |
---|
611 | return(L); |
---|
612 | } |
---|
613 | |
---|
614 | /////////////////////////////////////////////////////////////////////////////// |
---|
615 | // The following procedure is called by esStratum (typ=0), resp. by |
---|
616 | // isEquising (typ=1) |
---|
617 | /////////////////////////////////////////////////////////////////////////////// |
---|
618 | |
---|
619 | static proc esComputation (int typ, poly p_F, list #) |
---|
620 | { |
---|
621 | intvec ov=option(get); // store options set at beginning |
---|
622 | option(redSB); |
---|
623 | // Initialize variables |
---|
624 | int branch=1; |
---|
625 | int blowup=1; |
---|
626 | int auxVar=1; |
---|
627 | int nVars; |
---|
628 | |
---|
629 | intvec upper_bound, upper_bound_old, fertig, soll; |
---|
630 | list blowup_string; |
---|
631 | int i_print= printlevel-voice+2; |
---|
632 | |
---|
633 | int no_b, number_of_branches, swapped; |
---|
634 | int i,j,k,m, counter, dummy; |
---|
635 | string helpStr = ""; |
---|
636 | string ordStr = ""; |
---|
637 | string MinPolyStr = ""; |
---|
638 | |
---|
639 | if (nvars(basering)<=2) |
---|
640 | { |
---|
641 | print("family is trivial (no deformation parameters)!"); |
---|
642 | if (typ==1) //isEquising |
---|
643 | { |
---|
644 | option(set,ov); |
---|
645 | return(1); |
---|
646 | } |
---|
647 | else |
---|
648 | { |
---|
649 | option(set,ov); |
---|
650 | return(list(ideal(0),0)); |
---|
651 | } |
---|
652 | } |
---|
653 | |
---|
654 | if (size(#)>0) |
---|
655 | { |
---|
656 | if (typeof(#[1])=="int") |
---|
657 | { |
---|
658 | def artin_bd=#[1]; // compute modulo maxideal(artin_bd) |
---|
659 | if (artin_bd <= 1) |
---|
660 | { |
---|
661 | print("Do you really want to compute over Basering/maxideal(" |
---|
662 | +string(artin_bd)+") ?"); |
---|
663 | print("No computation performed !"); |
---|
664 | if (typ==1) //isEquising |
---|
665 | { |
---|
666 | option(set,ov); |
---|
667 | return(1); |
---|
668 | } |
---|
669 | else |
---|
670 | { |
---|
671 | option(set,ov); |
---|
672 | return(list(ideal(0),int(1))); |
---|
673 | } |
---|
674 | } |
---|
675 | if (size(#)>1) |
---|
676 | { |
---|
677 | if (typeof(#[2])=="list") |
---|
678 | { |
---|
679 | def @L=#[2]; // is assumed to be the Hamburger-Noether matrix |
---|
680 | } |
---|
681 | } |
---|
682 | } |
---|
683 | else |
---|
684 | { |
---|
685 | if (typeof(#)=="list") |
---|
686 | { |
---|
687 | def @L=#; // is assumed to be the Hamburger-Noether matrix |
---|
688 | } |
---|
689 | } |
---|
690 | } |
---|
691 | int ring_is_changed; |
---|
692 | def old_ring=basering; |
---|
693 | if(defined(@L)<=0) |
---|
694 | { |
---|
695 | // define a new ring without deformation-parameters and change to it: |
---|
696 | string str; |
---|
697 | string minpolyStr = string(minpoly); |
---|
698 | str = " ring HNERing = (" + charstr(basering) + "), (x,y), ls;"; |
---|
699 | execute (str); |
---|
700 | str = "minpoly ="+ minpolyStr+";"; |
---|
701 | execute(str); |
---|
702 | ring_is_changed=1; |
---|
703 | // Basering changed to HNERing (variables x,y, with ls ordering) |
---|
704 | |
---|
705 | k=nvars(old_ring); |
---|
706 | matrix Map_Phi[1][k]; |
---|
707 | Map_Phi[1,k-1]=x; |
---|
708 | Map_Phi[1,k]=y; |
---|
709 | map phi=old_ring,Map_Phi; |
---|
710 | poly f=phi(p_F); |
---|
711 | |
---|
712 | // Heuristics: if x,y are transversal parameters then computation of HNE |
---|
713 | // can be much faster when exchanging variables...! |
---|
714 | if (2*size(coeffs(f,x))<size(coeffs(f,y))) |
---|
715 | { |
---|
716 | swapped=1; |
---|
717 | f=swapXY(f); |
---|
718 | } |
---|
719 | |
---|
720 | int error=checkPoly(f); |
---|
721 | if (error) |
---|
722 | { |
---|
723 | setring old_ring; |
---|
724 | if (typ==1) //isEquising |
---|
725 | { |
---|
726 | print("Return value (=0) has no meaning!"); |
---|
727 | option(set,ov); |
---|
728 | return(0); |
---|
729 | } |
---|
730 | else |
---|
731 | { |
---|
732 | option(set,ov); |
---|
733 | return(list( ideal(0),error)); |
---|
734 | } |
---|
735 | } |
---|
736 | |
---|
737 | dbprint(i_print,"// "); |
---|
738 | dbprint(i_print,"// Compute HN expansion"); |
---|
739 | dbprint(i_print,"// ---------------------"); |
---|
740 | i=printlevel; |
---|
741 | printlevel=printlevel-5; |
---|
742 | list LLL=hnexpansion(f); |
---|
743 | |
---|
744 | if (size(LLL)==0) { // empty list returned by hnexpansion |
---|
745 | setring old_ring; |
---|
746 | print(i_print,"Unable to compute HN expansion !"); |
---|
747 | if (typ==1) //isEquising |
---|
748 | { |
---|
749 | print("Return value (=0) has no meaning!"); |
---|
750 | option(set,ov); |
---|
751 | return(0); |
---|
752 | } |
---|
753 | else |
---|
754 | { |
---|
755 | option(set,ov); |
---|
756 | return(list(ideal(0),int(1))); |
---|
757 | } |
---|
758 | option(set,ov); |
---|
759 | return(0); |
---|
760 | } |
---|
761 | else |
---|
762 | { |
---|
763 | if (typeof(LLL[1])=="ring") { |
---|
764 | def HNering = LLL[1]; |
---|
765 | setring HNering; |
---|
766 | def @L=stripHNE(hne); |
---|
767 | } |
---|
768 | else { |
---|
769 | def @L=stripHNE(LLL); |
---|
770 | } |
---|
771 | } |
---|
772 | printlevel=i; |
---|
773 | dbprint(i_print,"// finished"); |
---|
774 | dbprint(i_print,"// "); |
---|
775 | } |
---|
776 | def HNEring=basering; |
---|
777 | list M=multsequence(@L); |
---|
778 | M=control_Matrix(M); // this returns the 4 control matrices |
---|
779 | def maxDeg=M[4]; |
---|
780 | |
---|
781 | list L1=inf_Tangents(@L,nrows(M[1])); |
---|
782 | matrix B=L1[1]; |
---|
783 | intvec V=L1[2]; |
---|
784 | kill L1; |
---|
785 | |
---|
786 | // if we have computed the HNE for f after swapping x and y, we have |
---|
787 | // to reinterprete the (swap) matrix V: |
---|
788 | if (swapped==1) |
---|
789 | { |
---|
790 | for (i=1;i<=size(V);i++) { V[i]=V[i]-1; } // turns 0 into -1, 1 into 0 |
---|
791 | } |
---|
792 | |
---|
793 | // Determine maximal number of needed auxiliary parameters (free tangents): |
---|
794 | no_b=Determine_no_b(M[3],B); |
---|
795 | |
---|
796 | // test whether HNexpansion needed field extension.... |
---|
797 | string minPolyStr = ""; |
---|
798 | if (minpoly !=0) |
---|
799 | { |
---|
800 | minPolyStr = makeMinPolyString("a"); |
---|
801 | minPolyStr = "minpoly =" + minPolyStr + ";"; |
---|
802 | } |
---|
803 | |
---|
804 | setring old_ring; |
---|
805 | |
---|
806 | def myRing=createMyRing_new(p_F,"dp",minPolyStr,no_b); |
---|
807 | setring myRing; // comes with mIdeal |
---|
808 | map hole=HNEring,mIdeal; |
---|
809 | // basering has changed to myRing, in particular, the "old" |
---|
810 | // variable names, e.g., A,B,C,z,y are replaced by t(1),t(2),t(3),x,y |
---|
811 | |
---|
812 | ideal bNodes; |
---|
813 | |
---|
814 | // Initialize some variables: |
---|
815 | map phi; |
---|
816 | poly G, F_save; |
---|
817 | poly b_dummy; |
---|
818 | ideal J,Jnew,final_Map; |
---|
819 | number_of_branches=ncols(M[1]); |
---|
820 | for (i=1;i<=number_of_branches;i++) |
---|
821 | { |
---|
822 | poly F(i); |
---|
823 | ideal bl_Map(i); |
---|
824 | } |
---|
825 | upper_bound[number_of_branches]=0; |
---|
826 | upper_bound[1]=number_of_branches; |
---|
827 | upper_bound_old=upper_bound; |
---|
828 | fertig[number_of_branches]=0; |
---|
829 | for (i=1;i<=number_of_branches;i++){ soll[i]=1; } |
---|
830 | |
---|
831 | // Hole: B = matrix of blowup points |
---|
832 | if (ring_is_changed==0) { matrix B=hole(B); } |
---|
833 | else { matrix B=imap(HNEring,B); } |
---|
834 | m=M[1][blowup,branch]; // multiplicity at 0 |
---|
835 | |
---|
836 | // now, we start by checking equimultiplicity along trivial section |
---|
837 | poly Fm=m_Jet(p_F,m-1); |
---|
838 | |
---|
839 | matrix coef_Mat = coef(Fm,xy); |
---|
840 | Jnew=coef_Mat[2,1..ncols(coef_Mat)]; |
---|
841 | J=J,Jnew; |
---|
842 | |
---|
843 | if (defined(artin_bd)) // the artin_bd-th power of the maxideal of |
---|
844 | // deformation parameters can be cutted off |
---|
845 | { |
---|
846 | J=jet(J,artin_bd-1); |
---|
847 | } |
---|
848 | |
---|
849 | J=interred(J); |
---|
850 | if (defined(artin_bd)) { J=jet(J,artin_bd-1); } |
---|
851 | |
---|
852 | // J=std(J); |
---|
853 | |
---|
854 | if (typ==1) // isEquising |
---|
855 | { |
---|
856 | if(ideal(nselect(J,1,no_b))<>0) |
---|
857 | { |
---|
858 | setring old_ring; |
---|
859 | option(set,ov); |
---|
860 | return(0); |
---|
861 | } |
---|
862 | } |
---|
863 | |
---|
864 | F(1)=p_F; |
---|
865 | |
---|
866 | // and reduce the remaining terms in F(1): |
---|
867 | bl_Map(1)=maxideal(1); |
---|
868 | |
---|
869 | attrib(J,"isSB",1); |
---|
870 | bl_Map(1)=reduce(bl_Map(1),J); |
---|
871 | attrib(J,"isSB",0); |
---|
872 | |
---|
873 | phi=myRing,bl_Map(1); |
---|
874 | F(1)=phi(F(1)); |
---|
875 | |
---|
876 | // simplify F(1) |
---|
877 | attrib(J,"isSB",1); |
---|
878 | F(1)=reduce(F(1),J); |
---|
879 | attrib(J,"isSB",0); |
---|
880 | |
---|
881 | // now we compute the m-jet: |
---|
882 | Fm=m_Jet(F(1),m); |
---|
883 | |
---|
884 | G=1; |
---|
885 | counter=branch; |
---|
886 | k=upper_bound[branch]; |
---|
887 | |
---|
888 | F_save=F(1); // is truncated differently in the following loop |
---|
889 | |
---|
890 | while(counter<=k) |
---|
891 | { |
---|
892 | F(counter)=m_Jet(F_save,maxDeg[blowup,counter]); |
---|
893 | if (V[counter]==0) // 2nd ring variable is tangent to this branch |
---|
894 | { |
---|
895 | G=G*(y-(b(auxVar)+B[blowup,counter])*x)^(M[3][blowup,counter]); |
---|
896 | } |
---|
897 | else // 1st ring variable is tangent to this branch |
---|
898 | { |
---|
899 | G=G*(x-(b(auxVar)+B[blowup,counter])*y)^(M[3][blowup,counter]); |
---|
900 | F(counter)=swapXY(F(counter)); |
---|
901 | } |
---|
902 | bl_Map(counter)=maxideal(1); |
---|
903 | bl_Map(counter)[nvars(basering)]=xy+(b(auxVar)+B[blowup,counter])*x; |
---|
904 | |
---|
905 | bNodes[counter]=b(auxVar); |
---|
906 | |
---|
907 | auxVar=auxVar+1; |
---|
908 | upper_bound[counter]=counter+M[2][blowup+1,counter]-1; |
---|
909 | counter=counter+M[2][blowup+1,counter]; |
---|
910 | |
---|
911 | } |
---|
912 | |
---|
913 | list LeadDataFm=determine_coef(Fm); |
---|
914 | def LeadDataG=coef(G,xy); |
---|
915 | |
---|
916 | for (i=1; i<=ncols(LeadDataG); i++) |
---|
917 | { |
---|
918 | if (LeadDataG[1,i]==LeadDataFm[2]) |
---|
919 | { |
---|
920 | poly LeadG = LeadDataG[2,i]; // determine the coefficient of G |
---|
921 | i=ncols(LeadDataG)+1; |
---|
922 | } |
---|
923 | } |
---|
924 | |
---|
925 | G=LeadDataFm[1]*G-LeadG*Fm; // leading terms in y should cancel... |
---|
926 | |
---|
927 | coef_Mat = coef(G,xy); |
---|
928 | Jnew=coef_Mat[2,1..ncols(coef_Mat)]; |
---|
929 | |
---|
930 | // simplification of Jnew |
---|
931 | |
---|
932 | if (defined(artin_bd)) // the artin_bd-th power of the maxideal of |
---|
933 | // deformation parameters can be cutted off |
---|
934 | { |
---|
935 | Jnew=jet(Jnew,artin_bd-1); |
---|
936 | } |
---|
937 | Jnew=interred(Jnew); |
---|
938 | if (defined(artin_bd)) { Jnew=jet(Jnew,artin_bd-1); } |
---|
939 | J=J,Jnew; |
---|
940 | |
---|
941 | if (typ==1) // isEquising |
---|
942 | { |
---|
943 | if(ideal(nselect(J,1,no_b))<>0) |
---|
944 | { |
---|
945 | setring old_ring; |
---|
946 | option(set,ov); |
---|
947 | return(0); |
---|
948 | } |
---|
949 | } |
---|
950 | |
---|
951 | while (fertig<>soll and blowup<nrows(M[3])) |
---|
952 | { |
---|
953 | upper_bound_old=upper_bound; |
---|
954 | dbprint(i_print,"// Blowup Step "+string(blowup)+" completed"); |
---|
955 | blowup=blowup+1; |
---|
956 | |
---|
957 | for (branch=1;branch<=number_of_branches;branch=branch+1) |
---|
958 | { |
---|
959 | Jnew=0; |
---|
960 | |
---|
961 | // First we check if the branch still has to be considered: |
---|
962 | if (branch==upper_bound_old[branch] and fertig[branch]<>1) |
---|
963 | { |
---|
964 | if (M[3][blowup-1,branch]==1 and |
---|
965 | ((B[blowup,branch]<>x and B[blowup,branch]<>y) |
---|
966 | or (blowup==nrows(M[3])) )) |
---|
967 | { |
---|
968 | fertig[branch]=1; |
---|
969 | dbprint(i_print,"// 1 branch finished"); |
---|
970 | } |
---|
971 | } |
---|
972 | |
---|
973 | if (branch<=upper_bound_old[branch] and fertig[branch]<>1) |
---|
974 | { |
---|
975 | for (i=branch;i>=1;i--) |
---|
976 | { |
---|
977 | if (M[1][blowup-1,i]<>0) |
---|
978 | { |
---|
979 | m=M[1][blowup-1,i]; // multiplicity before blowup |
---|
980 | i=0; |
---|
981 | } |
---|
982 | } |
---|
983 | |
---|
984 | // we blow up the branch and take the strict transform: |
---|
985 | attrib(J,"isSB",1); |
---|
986 | bl_Map(branch)=reduce(bl_Map(branch),J); |
---|
987 | attrib(J,"isSB",0); |
---|
988 | |
---|
989 | phi=myRing,bl_Map(branch); |
---|
990 | F(branch)=phi(F(branch))/x^m; |
---|
991 | |
---|
992 | // simplify F |
---|
993 | attrib(Jnew,"isSB",1); |
---|
994 | |
---|
995 | F(branch)=reduce(F(branch),Jnew); |
---|
996 | attrib(Jnew,"isSB",0); |
---|
997 | |
---|
998 | m=M[1][blowup,branch]; // multiplicity after blowup |
---|
999 | Fm=m_Jet(F(branch),m); // homogeneous part of lowest degree |
---|
1000 | |
---|
1001 | |
---|
1002 | // we check for Fm=F[k]*...*F[k+s] where |
---|
1003 | // |
---|
1004 | // F[j]=(y-b'(j)*x)^m(j), respectively F[j]=(-b'(j)*y+x)^m(j) |
---|
1005 | // |
---|
1006 | // according to the entries m(j)= M[3][blowup,j] and |
---|
1007 | // b'(j) mod m_A = B[blowup,j] |
---|
1008 | // computed from the HNE of the special fibre of the family: |
---|
1009 | G=1; |
---|
1010 | counter=branch; |
---|
1011 | k=upper_bound[branch]; |
---|
1012 | |
---|
1013 | F_save=F(branch); |
---|
1014 | |
---|
1015 | while(counter<=k) |
---|
1016 | { |
---|
1017 | F(counter)=m_Jet(F_save,maxDeg[blowup,counter]); |
---|
1018 | |
---|
1019 | if (B[blowup,counter]<>x and B[blowup,counter]<>y) |
---|
1020 | { |
---|
1021 | G=G*(y-(b(auxVar)+B[blowup,counter])*x)^(M[3][blowup,counter]); |
---|
1022 | bl_Map(counter)=maxideal(1); |
---|
1023 | bl_Map(counter)[nvars(basering)]= |
---|
1024 | xy+(b(auxVar)+B[blowup,counter])*x; |
---|
1025 | bNodes[counter]=b(auxVar); |
---|
1026 | auxVar=auxVar+1; |
---|
1027 | } |
---|
1028 | else |
---|
1029 | { |
---|
1030 | if (B[blowup,counter]==x) |
---|
1031 | { |
---|
1032 | G=G*x^(M[3][blowup,counter]); // branch has tangent x !! |
---|
1033 | F(counter)=swapXY(F(counter)); // will turn x to y for blow up |
---|
1034 | bl_Map(counter)=maxideal(1); |
---|
1035 | bl_Map(counter)[nvars(basering)]=xy; |
---|
1036 | } |
---|
1037 | else |
---|
1038 | { |
---|
1039 | G=G*y^(M[3][blowup,counter]); // tangent has to be y |
---|
1040 | bl_Map(counter)=maxideal(1); |
---|
1041 | bl_Map(counter)[nvars(basering)]=xy; |
---|
1042 | } |
---|
1043 | bNodes[counter]=0; |
---|
1044 | } |
---|
1045 | upper_bound[counter]=counter+M[2][blowup+1,counter]-1; |
---|
1046 | counter=counter+M[2][blowup+1,counter]; |
---|
1047 | } |
---|
1048 | G=determine_coef(Fm)[1]*G-Fm; // leading terms in y should cancel |
---|
1049 | coef_Mat = coef(G,xy); |
---|
1050 | Jnew=coef_Mat[2,1..ncols(coef_Mat)]; |
---|
1051 | if (defined(artin_bd)) // the artin_bd-th power of the maxideal of |
---|
1052 | // deformation parameters can be cutted off |
---|
1053 | { |
---|
1054 | Jnew=jet(Jnew,artin_bd-1); |
---|
1055 | } |
---|
1056 | |
---|
1057 | // simplification of J |
---|
1058 | Jnew=interred(Jnew); |
---|
1059 | |
---|
1060 | J=J,Jnew; |
---|
1061 | if (typ==1) // isEquising |
---|
1062 | { |
---|
1063 | if (defined(artin_bd)) { J=jet(Jnew,artin_bd-1); } |
---|
1064 | if(ideal(nselect(J,1,no_b))<>0) |
---|
1065 | { |
---|
1066 | setring old_ring; |
---|
1067 | option(set,ov); |
---|
1068 | return(0); |
---|
1069 | } |
---|
1070 | } |
---|
1071 | } |
---|
1072 | } |
---|
1073 | if (number_of_branches>=2) |
---|
1074 | { |
---|
1075 | J=interred(J); |
---|
1076 | if (typ==1) // isEquising |
---|
1077 | { |
---|
1078 | if (defined(artin_bd)) { J=jet(Jnew,artin_bd-1); } |
---|
1079 | if(ideal(nselect(J,1,no_b))<>0) |
---|
1080 | { |
---|
1081 | setring old_ring; |
---|
1082 | option(set,ov); |
---|
1083 | return(0); |
---|
1084 | } |
---|
1085 | } |
---|
1086 | } |
---|
1087 | } |
---|
1088 | |
---|
1089 | // Computation for all equimultiple sections being trivial (I^s(f)) |
---|
1090 | ideal Jtriv=J; |
---|
1091 | for (i=1;i<=no_b; i++) |
---|
1092 | { |
---|
1093 | if (reduce(b(i),std(bNodes))!=0){ |
---|
1094 | Jtriv=subst(Jtriv,b(i),0); |
---|
1095 | } |
---|
1096 | } |
---|
1097 | Jtriv=std(Jtriv); |
---|
1098 | |
---|
1099 | |
---|
1100 | |
---|
1101 | dbprint(i_print,"// "); |
---|
1102 | dbprint(i_print,"// Elimination starts:"); |
---|
1103 | dbprint(i_print,"// -------------------"); |
---|
1104 | |
---|
1105 | poly gg; |
---|
1106 | int b_left=no_b; |
---|
1107 | |
---|
1108 | for (i=1;i<=no_b; i++) |
---|
1109 | { |
---|
1110 | attrib(J,"isSB",1); |
---|
1111 | gg=reduce(b(i),J); |
---|
1112 | if (gg==0) |
---|
1113 | { |
---|
1114 | b_left = b_left-1; // another b(i) has to be 0 |
---|
1115 | } |
---|
1116 | J = subst(J, b(i), gg); |
---|
1117 | attrib(J,"isSB",0); |
---|
1118 | } |
---|
1119 | J=simplify(J,10); |
---|
1120 | if (typ==1) // isEquising |
---|
1121 | { |
---|
1122 | if (defined(artin_bd)) { J=jet(Jnew,artin_bd-1); } |
---|
1123 | if(ideal(nselect(J,1,no_b))<>0) |
---|
1124 | { |
---|
1125 | setring old_ring; |
---|
1126 | option(set,ov); |
---|
1127 | return(0); |
---|
1128 | } |
---|
1129 | } |
---|
1130 | |
---|
1131 | //new CL 11/06: check in which equations b(k) appears and remove those b(k) |
---|
1132 | // which appear in exactly one of the equations (by removing this |
---|
1133 | // equation) |
---|
1134 | dbprint(i_print,"// "); |
---|
1135 | dbprint(i_print,"// Remove superfluous equations:"); |
---|
1136 | dbprint(i_print,"// -----------------------------"); |
---|
1137 | int Z,App_in; |
---|
1138 | ideal J_Tmp; |
---|
1139 | int ncJ=ncols(J); |
---|
1140 | |
---|
1141 | intmat Mdet[ncJ][1]; |
---|
1142 | for (Z=1;Z<=ncJ;Z++){ Mdet[Z,1]=Z; } |
---|
1143 | |
---|
1144 | for (i=1;i<=no_b; i++) |
---|
1145 | { |
---|
1146 | ideal b_appears_in(i); // Eintraege sind spaeter 1 oder 0 |
---|
1147 | intmat b_app_in(i)[1][ncJ]; // Eintraege sind spaeter 1 oder 0 |
---|
1148 | b_appears_in(i)[ncJ]=0; |
---|
1149 | J_Tmp = J-subst(J,b(i),0); |
---|
1150 | for (Z=1; Z<=ncJ; Z++) { |
---|
1151 | if (J_Tmp[Z]<>0) { // b(i) appear in J_Tmp[Z] |
---|
1152 | b_appears_in(i)[Z]=1; |
---|
1153 | b_app_in(i)[1,Z]=1; |
---|
1154 | } |
---|
1155 | } |
---|
1156 | if (size(b_appears_in(i))==1) { //b(i) appears only in one J_Tmp[Z] |
---|
1157 | App_in = (b_app_in(i)*Mdet)[1,1]; // determines Z |
---|
1158 | J[App_in]=0; |
---|
1159 | b_appears_in(i)[App_in]=0; |
---|
1160 | b_app_in(i)[1,App_in]=0; |
---|
1161 | } |
---|
1162 | } |
---|
1163 | |
---|
1164 | for (i=1;i<=no_b; i++) |
---|
1165 | { |
---|
1166 | if (size(b_appears_in(i))==1) { //b(i) appears only in one J_Tmp[Z] |
---|
1167 | App_in = (b_app_in(i)*Mdet)[1,1]; // determines Z |
---|
1168 | J[App_in]=0; |
---|
1169 | b_appears_in(i)[App_in]=0; |
---|
1170 | b_app_in(i)[1,Z]=1; |
---|
1171 | i=0; |
---|
1172 | } |
---|
1173 | } |
---|
1174 | |
---|
1175 | Jtriv = nselect(Jtriv,1,no_b); |
---|
1176 | ideal J_no_b = nselect(J,1,no_b); |
---|
1177 | if (size(J) > size(J_no_b)) |
---|
1178 | { |
---|
1179 | dbprint(i_print,"// std computation started"); |
---|
1180 | // some b(i) didn't appear in linear conditions and have to be eliminated |
---|
1181 | if (defined(artin_bd)) |
---|
1182 | { |
---|
1183 | // first we make the ring smaller (removing variables, which are |
---|
1184 | // forced to 0 by J |
---|
1185 | list LL=make_ring_small(J); |
---|
1186 | ideal Shortmap=LL[2]; |
---|
1187 | minPolyStr = ""; |
---|
1188 | if (minpoly !=0) |
---|
1189 | { |
---|
1190 | minPolyStr = "minpoly = "+string(minpoly); |
---|
1191 | } |
---|
1192 | ordStr = "dp(" + string(b_left) + "),dp"; |
---|
1193 | ideal qId = ideal(basering); |
---|
1194 | |
---|
1195 | helpStr = "ring Shortring = (" |
---|
1196 | + charstr(basering) + "),("+ LL[1] +") , ("+ ordStr +");"; |
---|
1197 | execute(helpStr); |
---|
1198 | execute(minPolyStr); |
---|
1199 | // ring has changed to "Shortring" |
---|
1200 | |
---|
1201 | ideal MM=maxideal(artin_bd); |
---|
1202 | MM=subst(MM,x,0); |
---|
1203 | MM=subst(MM,y,0); |
---|
1204 | MM=simplify(MM,2); |
---|
1205 | dbprint(i_print-1,"// maxideal("+string(artin_bd)+") has " |
---|
1206 | +string(size(MM))+" elements"); |
---|
1207 | dbprint(i_print-1,"//"); |
---|
1208 | |
---|
1209 | // we change to the qring mod m^artin_bd |
---|
1210 | // first, we have to check if we were in a qring when starting |
---|
1211 | ideal qId = imap(myRing, qId); |
---|
1212 | if (qId == 0) |
---|
1213 | { |
---|
1214 | attrib(MM,"isSB",1); |
---|
1215 | qring QQ=MM; |
---|
1216 | } |
---|
1217 | else |
---|
1218 | { |
---|
1219 | qId=qId,MM; |
---|
1220 | qring QQ = std(qId); |
---|
1221 | } |
---|
1222 | |
---|
1223 | ideal Shortmap=imap(myRing,Shortmap); |
---|
1224 | map phiphi=myRing,Shortmap; |
---|
1225 | |
---|
1226 | ideal J=phiphi(J); |
---|
1227 | option(redSB); |
---|
1228 | J=std(J); |
---|
1229 | J=nselect(J,1,no_b); |
---|
1230 | |
---|
1231 | setring myRing; |
---|
1232 | // back to "myRing" |
---|
1233 | |
---|
1234 | J=nselect(J,1,no_b); |
---|
1235 | Jnew=imap(QQ,J); |
---|
1236 | |
---|
1237 | J=J,Jnew; |
---|
1238 | J=interred(J); |
---|
1239 | if (defined(artin_bd)){ J=jet(J,artin_bd-1); } |
---|
1240 | } |
---|
1241 | else |
---|
1242 | { |
---|
1243 | J=std(J); |
---|
1244 | J=nselect(J,1,no_b); |
---|
1245 | if (defined(artin_bd)){ J=jet(J,artin_bd-1); } |
---|
1246 | } |
---|
1247 | } |
---|
1248 | |
---|
1249 | dbprint(i_print,"// finished"); |
---|
1250 | dbprint(i_print,"// "); |
---|
1251 | |
---|
1252 | minPolyStr = "";option(set,ov); |
---|
1253 | if (minpoly !=0) |
---|
1254 | { |
---|
1255 | minPolyStr = "minpoly = "+string(minpoly); |
---|
1256 | } |
---|
1257 | |
---|
1258 | kill HNEring; |
---|
1259 | |
---|
1260 | if (typ==1) // isEquising |
---|
1261 | { |
---|
1262 | if (defined(artin_bd)) { J=jet(Jnew,artin_bd-1); } |
---|
1263 | if(J<>0) |
---|
1264 | { |
---|
1265 | setring old_ring; |
---|
1266 | option(set,ov); |
---|
1267 | return(0); |
---|
1268 | } |
---|
1269 | else |
---|
1270 | { |
---|
1271 | setring old_ring; |
---|
1272 | option(set,ov); |
---|
1273 | return(1); |
---|
1274 | } |
---|
1275 | } |
---|
1276 | |
---|
1277 | setring old_ring; |
---|
1278 | // we are back in the original ring |
---|
1279 | |
---|
1280 | if (npars(myRing)<>0) |
---|
1281 | { |
---|
1282 | ideal qIdeal = ideal(basering); |
---|
1283 | helpStr = "ring ESSring = (" |
---|
1284 | + string(char(basering))+ "," + parstr(myRing) + |
---|
1285 | ") , ("+ varstr(basering)+") , ("+ ordstr(basering) +");"; |
---|
1286 | execute(helpStr); |
---|
1287 | execute(minPolyStr); |
---|
1288 | // basering has changed to ESSring |
---|
1289 | |
---|
1290 | ideal qIdeal = fetch(old_ring, qIdeal); |
---|
1291 | if(qIdeal != 0) |
---|
1292 | { |
---|
1293 | def r_base = basering; |
---|
1294 | kill ESSring; |
---|
1295 | qring ESSring = std(qIdeal); |
---|
1296 | } |
---|
1297 | kill qIdeal; |
---|
1298 | |
---|
1299 | ideal SSS; |
---|
1300 | for (int ii=1;ii<=nvars(basering);ii++) |
---|
1301 | { |
---|
1302 | SSS[ii+no_b]=var(ii); |
---|
1303 | } |
---|
1304 | map phi=myRing,SSS; // b(i) variables are mapped to zero |
---|
1305 | |
---|
1306 | ideal ES=phi(J); |
---|
1307 | ideal ES_all_triv=phi(Jtriv); |
---|
1308 | kill phi; |
---|
1309 | |
---|
1310 | if (defined(p_F)<=0) |
---|
1311 | { |
---|
1312 | poly p_F=fetch(old_ring,p_F); |
---|
1313 | export(p_F); |
---|
1314 | } |
---|
1315 | export(ES); |
---|
1316 | export(ES_all_triv); |
---|
1317 | setring old_ring; |
---|
1318 | dbprint(i_print+2," |
---|
1319 | // 'esStratum' created a list M of a ring and an integer. |
---|
1320 | // To access the ideal defining the equisingularity stratum, type: |
---|
1321 | def ESSring = M[1]; setring ESSring; ES; "); |
---|
1322 | |
---|
1323 | option(set,ov); |
---|
1324 | return(list(ESSring,0)); |
---|
1325 | } |
---|
1326 | else |
---|
1327 | { |
---|
1328 | // no new ring definition necessary |
---|
1329 | ideal SSS; |
---|
1330 | for (int ii=1;ii<=nvars(basering);ii++) |
---|
1331 | { |
---|
1332 | SSS[ii+no_b]=var(ii); |
---|
1333 | } |
---|
1334 | map phi=myRing,SSS; // b(i) variables are mapped to zero |
---|
1335 | |
---|
1336 | ideal ES=phi(J); |
---|
1337 | ideal ES_all_triv=phi(Jtriv); |
---|
1338 | kill phi; |
---|
1339 | |
---|
1340 | setring old_ring; |
---|
1341 | dbprint(i_print,"// output of 'esStratum' is a list consisting of: |
---|
1342 | // _[1][1] = ideal defining the equisingularity stratum |
---|
1343 | // _[1][2] = ideal defining the part of the equisingularity stratum |
---|
1344 | // where all equimultiple sections are trivial |
---|
1345 | // _[2] = 0"); |
---|
1346 | |
---|
1347 | option(set,ov); |
---|
1348 | return(list(list(ES,ES_all_triv),0)); |
---|
1349 | } |
---|
1350 | |
---|
1351 | } |
---|
1352 | |
---|
1353 | //////////////////////////////////////////////////////////////////////////////// |
---|
1354 | |
---|
1355 | proc tau_es (poly f,list #) |
---|
1356 | "USAGE: tau_es(f); f poly |
---|
1357 | ASSUME: f is a reduced bivariate polynomial, the basering has precisely |
---|
1358 | two variables, is local and no qring. |
---|
1359 | RETURN: int, the codimension of the mu-const stratum in the semi-universal |
---|
1360 | deformation base. |
---|
1361 | NOTE: printlevel>=1 displays additional information. |
---|
1362 | When called with any additional parameter, the computation of the |
---|
1363 | Milnor number is avoided (no check for NND). |
---|
1364 | SEE ALSO: esIdeal, tjurina, invariants |
---|
1365 | EXAMPLE: example tau_es; shows an example. |
---|
1366 | " |
---|
1367 | { |
---|
1368 | int i,j,k,s; |
---|
1369 | int slope_x, slope_y, upper; |
---|
1370 | int i_print = printlevel - voice + 3; |
---|
1371 | string MinPolyStr; |
---|
1372 | |
---|
1373 | // some checks first |
---|
1374 | if ( nvars(basering)<>2 ) |
---|
1375 | { |
---|
1376 | print("// basering has not the correct number (two) of variables !"); |
---|
1377 | print("// computation stopped"); |
---|
1378 | return(0); |
---|
1379 | } |
---|
1380 | if ( mult(std(1+var(1)+var(2))) <> 0) |
---|
1381 | { |
---|
1382 | print("// basering is not local !"); |
---|
1383 | print("// computation stopped"); |
---|
1384 | return(0); |
---|
1385 | } |
---|
1386 | |
---|
1387 | if (mult(std(f))<=1) |
---|
1388 | { |
---|
1389 | // f is rigid |
---|
1390 | return(0); |
---|
1391 | } |
---|
1392 | |
---|
1393 | if ( deg(squarefree(f))!=deg(f) ) |
---|
1394 | { |
---|
1395 | print("// input polynomial was not reduced"); |
---|
1396 | print("// try squarefree(f); first"); |
---|
1397 | return(0); |
---|
1398 | } |
---|
1399 | |
---|
1400 | def old_ring=basering; |
---|
1401 | execute("ring @myRing=("+charstr(basering)+"),("+varstr(basering)+"),ds;"); |
---|
1402 | poly f=imap(old_ring,f); |
---|
1403 | |
---|
1404 | ideal Jacobi_Id = jacob(f); |
---|
1405 | |
---|
1406 | // check for A_k singularity |
---|
1407 | // ---------------------------------------- |
---|
1408 | if (mult(std(f))==2) |
---|
1409 | { |
---|
1410 | dbprint(i_print-1,"// "); |
---|
1411 | dbprint(i_print-1,"// polynomial defined A_k singularity"); |
---|
1412 | dbprint(i_print-1,"// "); |
---|
1413 | return( vdim(std(Jacobi_Id)) ); |
---|
1414 | } |
---|
1415 | |
---|
1416 | // check for D_k singularity |
---|
1417 | // ---------------------------------------- |
---|
1418 | if (mult(std(f))==3 and size(factorize(jet(f,3))[1])>=3) |
---|
1419 | { |
---|
1420 | dbprint(i_print,"// "); |
---|
1421 | dbprint(i_print,"// polynomial defined D_k singularity"); |
---|
1422 | dbprint(i_print,"// "); |
---|
1423 | ideal ES_Id = f, jacob(f); |
---|
1424 | return( vdim(std(Jacobi_Id))); |
---|
1425 | } |
---|
1426 | |
---|
1427 | |
---|
1428 | if (size(#)==0) |
---|
1429 | { |
---|
1430 | // check if Newton polygon non-degenerate |
---|
1431 | // ---------------------------------------- |
---|
1432 | Jacobi_Id=std(Jacobi_Id); |
---|
1433 | int mu = vdim(Jacobi_Id); |
---|
1434 | poly f_tilde=f+var(1)^mu+var(2)^mu; //to obtain convenient Newton-polygon |
---|
1435 | |
---|
1436 | list NP=newtonpoly(f_tilde); |
---|
1437 | dbprint(i_print-1,"// Newton polygon:"); |
---|
1438 | dbprint(i_print-1,NP); |
---|
1439 | dbprint(i_print-1,""); |
---|
1440 | |
---|
1441 | if(is_NND(f,mu,NP)) // f is Newton non-degenerate |
---|
1442 | { |
---|
1443 | upper=NP[1][2]; |
---|
1444 | ideal ES_Id= x^k*y^upper; |
---|
1445 | dbprint(i_print-1,"polynomial is Newton non-degenerate"); |
---|
1446 | dbprint(i_print-1,""); |
---|
1447 | k=0; |
---|
1448 | for (i=1;i<=size(NP)-1;i++) |
---|
1449 | { |
---|
1450 | slope_x=NP[i+1][1]-NP[i][1]; |
---|
1451 | slope_y=NP[i][2]-NP[i+1][2]; |
---|
1452 | for (k=NP[i][1]+1; k<=NP[i+1][1]; k++) |
---|
1453 | { |
---|
1454 | while ( slope_x*upper + slope_y*k >= |
---|
1455 | slope_x*NP[i][2] + slope_y*NP[i][1]) |
---|
1456 | { |
---|
1457 | upper=upper-1; |
---|
1458 | } |
---|
1459 | upper=upper+1; |
---|
1460 | ES_Id=ES_Id, x^k*y^upper; |
---|
1461 | } |
---|
1462 | } |
---|
1463 | ES_Id=std(ES_Id); |
---|
1464 | dbprint(i_print-2,"ideal of monomials above Newton bd. is generated by:"); |
---|
1465 | dbprint(i_print-2,ES_Id); |
---|
1466 | ideal ESfix_Id = ES_Id, f, maxideal(1)*jacob(f); |
---|
1467 | ES_Id = ES_Id, Jacobi_Id; |
---|
1468 | ES_Id = std(ES_Id); |
---|
1469 | dbprint(i_print-1,"// "); |
---|
1470 | dbprint(i_print-1,"// Equisingularity ideal is computed!"); |
---|
1471 | dbprint(i_print-1,""); |
---|
1472 | return(vdim(ES_Id)); |
---|
1473 | } |
---|
1474 | else |
---|
1475 | { |
---|
1476 | dbprint(i_print-1,"polynomial is Newton degenerate !"); |
---|
1477 | dbprint(i_print-1,""); |
---|
1478 | } |
---|
1479 | } |
---|
1480 | |
---|
1481 | // for Newton degenerate polynomials, we compute the HN expansion, and |
---|
1482 | // count the number of free points ..... |
---|
1483 | |
---|
1484 | dbprint(i_print-1,"// "); |
---|
1485 | dbprint(i_print-1,"// Compute HN expansion"); |
---|
1486 | dbprint(i_print-1,"// ---------------------"); |
---|
1487 | i=printlevel; |
---|
1488 | printlevel=printlevel-5; |
---|
1489 | if (2*size(coeffs(f,x))<size(coeffs(f,y))) |
---|
1490 | { |
---|
1491 | f=swapXY(f); |
---|
1492 | } |
---|
1493 | list LLL=hnexpansion(f); |
---|
1494 | if (size(LLL)==0) { // empty list returned by hnexpansion |
---|
1495 | setring old_ring; |
---|
1496 | ERROR("Unable to compute HN expansion !"); |
---|
1497 | } |
---|
1498 | else |
---|
1499 | { |
---|
1500 | if (typeof(LLL[1])=="ring") { |
---|
1501 | def HNering = LLL[1]; |
---|
1502 | setring HNering; |
---|
1503 | def @L=hne; |
---|
1504 | } |
---|
1505 | else { |
---|
1506 | def @L=LLL; |
---|
1507 | } |
---|
1508 | } |
---|
1509 | def HNEring=basering; |
---|
1510 | |
---|
1511 | printlevel=i; |
---|
1512 | dbprint(i_print-1,"// finished"); |
---|
1513 | dbprint(i_print-1,"// "); |
---|
1514 | |
---|
1515 | list M=multsequence(@L); |
---|
1516 | M=control_Matrix(M); // this returns the 4 control matrices |
---|
1517 | intmat Mult=M[1]; |
---|
1518 | |
---|
1519 | list L1=inf_Tangents(@L,nrows(M[1])); |
---|
1520 | matrix B=L1[1]; |
---|
1521 | |
---|
1522 | // determine sum_i m_i(m_i+1)/2 (over inf. near points) |
---|
1523 | int conditions=0; |
---|
1524 | for (i=1;i<=nrows(Mult);i++) |
---|
1525 | { |
---|
1526 | for (j=1;j<=ncols(Mult);j++) |
---|
1527 | { |
---|
1528 | conditions=conditions+(Mult[i,j]*(Mult[i,j]+1)/2); |
---|
1529 | } |
---|
1530 | } |
---|
1531 | int freePts=no_freePoints(M[1],B); |
---|
1532 | int taues=conditions-freePts-2; |
---|
1533 | |
---|
1534 | setring old_ring; |
---|
1535 | return(taues); |
---|
1536 | } |
---|
1537 | example |
---|
1538 | { |
---|
1539 | "EXAMPLE:"; echo=2; |
---|
1540 | ring r=32003,(x,y),ds; |
---|
1541 | poly f=(x4-y4)^2-x10; |
---|
1542 | tau_es(f); |
---|
1543 | } |
---|
1544 | |
---|
1545 | |
---|
1546 | //////////////////////////////////////////////////////////////////////////////// |
---|
1547 | |
---|
1548 | proc esIdeal (poly f,list #) |
---|
1549 | "USAGE: esIdeal(f[,any]]); f poly |
---|
1550 | ASSUME: f is a reduced bivariate polynomial, the basering has precisely |
---|
1551 | two variables, is local and no qring, and the characteristic of |
---|
1552 | the ground field does not divide mult(f). |
---|
1553 | RETURN: if called with only one parameter: list of two ideals, |
---|
1554 | @format |
---|
1555 | _[1]: equisingularity ideal of f (in sense of Wahl), |
---|
1556 | _[2]: ideal of equisingularity with fixed position of the |
---|
1557 | singularity; |
---|
1558 | @end format |
---|
1559 | if called with more than one parameter: list of three ideals, |
---|
1560 | @format |
---|
1561 | _[1]: equisingularity ideal of f (in sense of Wahl) |
---|
1562 | _[2]: ideal of equisingularity with fixed position of the |
---|
1563 | singularity; |
---|
1564 | _[3]: ideal of all g such that the deformation defined by f+eg |
---|
1565 | (e^2=0) is isomorphic to an equisingular deformation |
---|
1566 | of V(f) with all equimultiple sections being trivial. |
---|
1567 | @end format |
---|
1568 | NOTE: if some of the above condition is not satisfied then return |
---|
1569 | value is list(0,0). |
---|
1570 | SEE ALSO: tau_es, esStratum |
---|
1571 | KEYWORDS: equisingularity ideal |
---|
1572 | EXAMPLE: example esIdeal; shows examples. |
---|
1573 | " |
---|
1574 | { |
---|
1575 | |
---|
1576 | int typ; |
---|
1577 | if (size(#)>0) { typ=1; } // I^s is also computed |
---|
1578 | int i,k,s; |
---|
1579 | int slope_x, slope_y, upper; |
---|
1580 | int i_print = printlevel - voice + 3; |
---|
1581 | string MinPolyStr; |
---|
1582 | |
---|
1583 | // some checks first |
---|
1584 | if ( nvars(basering)<>2 ) |
---|
1585 | { |
---|
1586 | print("// basering has not the correct number (two) of variables !"); |
---|
1587 | print("// computation stopped"); |
---|
1588 | return(list(0,0)); |
---|
1589 | } |
---|
1590 | if ( mult(std(1+var(1)+var(2))) <> 0) |
---|
1591 | { |
---|
1592 | print("// basering is not local !"); |
---|
1593 | print("// computation stopped"); |
---|
1594 | return(list(0,0)); |
---|
1595 | } |
---|
1596 | |
---|
1597 | if (mult(std(f))<=1) |
---|
1598 | { |
---|
1599 | // f is rigid |
---|
1600 | if (typ==0) |
---|
1601 | { |
---|
1602 | return(list(ideal(1),ideal(1))); |
---|
1603 | } |
---|
1604 | else |
---|
1605 | { |
---|
1606 | return(list(ideal(1),ideal(1),ideal(1))); |
---|
1607 | } |
---|
1608 | } |
---|
1609 | |
---|
1610 | if ( deg(squarefree(f))!=deg(f) ) |
---|
1611 | { |
---|
1612 | print("// input polynomial was not squarefree"); |
---|
1613 | print("// try squarefree(f); first"); |
---|
1614 | return(list(0,0)); |
---|
1615 | } |
---|
1616 | |
---|
1617 | if (char(basering)<>0) |
---|
1618 | { |
---|
1619 | if (mult(std(f)) mod char(basering)==0) |
---|
1620 | { |
---|
1621 | print("// characteristic of ground field divides " |
---|
1622 | + "multiplicity of polynomial !"); |
---|
1623 | print("// computation stopped"); |
---|
1624 | return(list(0,0)); |
---|
1625 | } |
---|
1626 | } |
---|
1627 | |
---|
1628 | // check for A_k singularity |
---|
1629 | // ---------------------------------------- |
---|
1630 | if (mult(std(f))==2) |
---|
1631 | { |
---|
1632 | dbprint(i_print,"// "); |
---|
1633 | dbprint(i_print,"// polynomial defined A_k singularity"); |
---|
1634 | dbprint(i_print,"// "); |
---|
1635 | ideal ES_Id = f, jacob(f); |
---|
1636 | ES_Id = interred(ES_Id); |
---|
1637 | ideal ESfix_Id = f, maxideal(1)*jacob(f); |
---|
1638 | ESfix_Id= interred(ESfix_Id); |
---|
1639 | if (typ==0) // only for computation of I^es and I^es_fix |
---|
1640 | { |
---|
1641 | return( list(ES_Id,ESfix_Id) ); |
---|
1642 | } |
---|
1643 | else |
---|
1644 | { |
---|
1645 | return( list(ES_Id,ESfix_Id,ES_Id) ); |
---|
1646 | } |
---|
1647 | } |
---|
1648 | |
---|
1649 | // check for D_k singularity |
---|
1650 | // ---------------------------------------- |
---|
1651 | if (mult(std(f))==3 and size(factorize(jet(f,3))[1])>=3) |
---|
1652 | { |
---|
1653 | dbprint(i_print,"// "); |
---|
1654 | dbprint(i_print,"// polynomial defined D_k singularity"); |
---|
1655 | dbprint(i_print,"// "); |
---|
1656 | ideal ES_Id = f, jacob(f); |
---|
1657 | ES_Id = interred(ES_Id); |
---|
1658 | ideal ESfix_Id = f, maxideal(1)*jacob(f); |
---|
1659 | ESfix_Id= interred(ESfix_Id); |
---|
1660 | if (typ==0) // only for computation of I^es and I^es_fix |
---|
1661 | { |
---|
1662 | return( list(ES_Id,ESfix_Id) ); |
---|
1663 | } |
---|
1664 | else |
---|
1665 | { |
---|
1666 | return( list(ES_Id,ESfix_Id,ES_Id) ); |
---|
1667 | } |
---|
1668 | } |
---|
1669 | |
---|
1670 | // check if Newton polygon non-degenerate |
---|
1671 | // ---------------------------------------- |
---|
1672 | int mu = milnor(f); |
---|
1673 | poly f_tilde=f+var(1)^mu+var(2)^mu; //to obtain a convenient Newton-polygon |
---|
1674 | |
---|
1675 | list NP=newtonpoly(f_tilde); |
---|
1676 | dbprint(i_print-1,"// Newton polygon:"); |
---|
1677 | dbprint(i_print-1,NP); |
---|
1678 | dbprint(i_print-1,""); |
---|
1679 | |
---|
1680 | if(is_NND(f,mu,NP)) // f is Newton non-degenerate |
---|
1681 | { |
---|
1682 | upper=NP[1][2]; |
---|
1683 | ideal ES_Id= x^k*y^upper; |
---|
1684 | dbprint(i_print,"polynomial is Newton non-degenerate"); |
---|
1685 | dbprint(i_print,""); |
---|
1686 | k=0; |
---|
1687 | for (i=1;i<=size(NP)-1;i++) |
---|
1688 | { |
---|
1689 | slope_x=NP[i+1][1]-NP[i][1]; |
---|
1690 | slope_y=NP[i][2]-NP[i+1][2]; |
---|
1691 | for (k=NP[i][1]+1; k<=NP[i+1][1]; k++) |
---|
1692 | { |
---|
1693 | while ( slope_x*upper + slope_y*k >= |
---|
1694 | slope_x*NP[i][2] + slope_y*NP[i][1]) |
---|
1695 | { |
---|
1696 | upper=upper-1; |
---|
1697 | } |
---|
1698 | upper=upper+1; |
---|
1699 | ES_Id=ES_Id, x^k*y^upper; |
---|
1700 | } |
---|
1701 | } |
---|
1702 | ES_Id=std(ES_Id); |
---|
1703 | dbprint(i_print-1,"ideal of monomials above Newton bd. is generated by:"); |
---|
1704 | dbprint(i_print-1,ES_Id); |
---|
1705 | ideal ESfix_Id = ES_Id, f, maxideal(1)*jacob(f); |
---|
1706 | ES_Id = ES_Id, f, jacob(f); |
---|
1707 | dbprint(i_print,"// "); |
---|
1708 | dbprint(i_print,"// equisingularity ideal is computed!"); |
---|
1709 | if (typ==0) |
---|
1710 | { |
---|
1711 | return(list(ES_Id,ESfix_Id)); |
---|
1712 | } |
---|
1713 | else |
---|
1714 | { |
---|
1715 | return(list(ES_Id,ESfix_Id,ES_Id)); |
---|
1716 | } |
---|
1717 | } |
---|
1718 | else |
---|
1719 | { |
---|
1720 | dbprint(i_print,"polynomial is Newton degenerate !"); |
---|
1721 | dbprint(i_print,""); |
---|
1722 | } |
---|
1723 | |
---|
1724 | def old_ring=basering; |
---|
1725 | |
---|
1726 | dbprint(i_print,"// "); |
---|
1727 | dbprint(i_print,"// versal deformation with triv. section"); |
---|
1728 | dbprint(i_print,"// ====================================="); |
---|
1729 | dbprint(i_print,"// "); |
---|
1730 | |
---|
1731 | ideal JJ=maxideal(1)*jacob(f); |
---|
1732 | ideal kbase_versal=kbase(std(JJ)); |
---|
1733 | s=size(kbase_versal); |
---|
1734 | string ring_versal="ring @Px = ("+charstr(basering)+"),(t(1.."+string(s)+")," |
---|
1735 | +varstr(basering)+"),(ds("+string(s)+")," |
---|
1736 | +ordstr(basering)+");"; |
---|
1737 | MinPolyStr = string(minpoly); |
---|
1738 | |
---|
1739 | execute(ring_versal); |
---|
1740 | if (MinPolyStr<>"0") |
---|
1741 | { |
---|
1742 | MinPolyStr = "minpoly="+MinPolyStr; |
---|
1743 | execute(MinPolyStr); |
---|
1744 | } |
---|
1745 | // basering has changed to @Px |
---|
1746 | |
---|
1747 | poly F=imap(old_ring,f); |
---|
1748 | ideal kbase_versal=imap(old_ring,kbase_versal); |
---|
1749 | for (i=1; i<=s; i++) |
---|
1750 | { |
---|
1751 | F=F+var(i)*kbase_versal[i]; |
---|
1752 | } |
---|
1753 | dbprint(i_print-1,F); |
---|
1754 | dbprint(i_print-1,""); |
---|
1755 | |
---|
1756 | |
---|
1757 | ideal ES_Id,ES_Id_all_triv; |
---|
1758 | poly Ftriv=F; |
---|
1759 | |
---|
1760 | dbprint(i_print,"// "); |
---|
1761 | dbprint(i_print,"// Compute equisingularity Stratum over Spec(C[t]/t^2)"); |
---|
1762 | dbprint(i_print,"// ==================================================="); |
---|
1763 | dbprint(i_print,"// "); |
---|
1764 | list M=esStratum(F,2); |
---|
1765 | dbprint(i_print,"// finished"); |
---|
1766 | dbprint(i_print,"// "); |
---|
1767 | |
---|
1768 | if (M[2]==1) // error occured during esStratum computation |
---|
1769 | { |
---|
1770 | print("Some error has occured during the computation"); |
---|
1771 | return(list(0,0)); |
---|
1772 | } |
---|
1773 | |
---|
1774 | if ( typeof(M[1])=="list" ) |
---|
1775 | { |
---|
1776 | int defpars = nvars(basering)-2; |
---|
1777 | poly Fred,Ftrivred; |
---|
1778 | poly g; |
---|
1779 | F=reduce(F,std(M[1][1])); |
---|
1780 | Ftriv=reduce(Ftriv,std(M[1][2])); |
---|
1781 | |
---|
1782 | for (i=1; i<=defpars; i++) |
---|
1783 | { |
---|
1784 | Fred=reduce(F,std(var(i))); |
---|
1785 | Ftrivred=reduce(Ftriv,std(var(i))); |
---|
1786 | |
---|
1787 | g=subst(F-Fred,var(i),1); |
---|
1788 | ES_Id=ES_Id, g; |
---|
1789 | F=Fred; |
---|
1790 | |
---|
1791 | g=subst(Ftriv-Ftrivred,var(i),1); |
---|
1792 | ES_Id_all_triv=ES_Id_all_triv, g; |
---|
1793 | Ftriv=Ftrivred; |
---|
1794 | } |
---|
1795 | |
---|
1796 | setring old_ring; |
---|
1797 | // back to original ring |
---|
1798 | |
---|
1799 | ideal ES_Id = imap(@Px,ES_Id); |
---|
1800 | ES_Id = interred(ES_Id); |
---|
1801 | |
---|
1802 | ideal ES_Id_all_triv = imap(@Px,ES_Id_all_triv); |
---|
1803 | ES_Id_all_triv = interred(ES_Id_all_triv); |
---|
1804 | |
---|
1805 | ideal ESfix_Id = ES_Id, f, maxideal(1)*jacob(f); |
---|
1806 | ES_Id = ES_Id, f, jacob(f); |
---|
1807 | ES_Id_all_triv = ES_Id_all_triv, f, jacob(f); |
---|
1808 | |
---|
1809 | if (typ==0) |
---|
1810 | { |
---|
1811 | return(list(ES_Id,ESfix_Id)); |
---|
1812 | } |
---|
1813 | else |
---|
1814 | { |
---|
1815 | return(list(ES_Id,ESfix_Id,ES_Id_all_triv)); |
---|
1816 | } |
---|
1817 | } |
---|
1818 | else |
---|
1819 | { |
---|
1820 | def AuxRing=M[1]; |
---|
1821 | |
---|
1822 | dbprint(i_print,"// "); |
---|
1823 | dbprint(i_print,"// change ring to ESSring"); |
---|
1824 | |
---|
1825 | setring AuxRing; // contains p_F, ES |
---|
1826 | |
---|
1827 | int defpars = nvars(basering)-2; |
---|
1828 | poly Fred,Fredtriv; |
---|
1829 | poly g; |
---|
1830 | ideal ES_Id,ES_Id_all_triv; |
---|
1831 | |
---|
1832 | poly p_Ftriv=p_F; |
---|
1833 | |
---|
1834 | p_F=reduce(p_F,std(ES)); |
---|
1835 | p_Ftriv=reduce(p_Ftriv,std(ES_all_triv)); |
---|
1836 | for (i=1; i<=defpars; i++) |
---|
1837 | { |
---|
1838 | Fred=reduce(p_F,std(var(i))); |
---|
1839 | Fredtriv=reduce(p_Ftriv,std(var(i))); |
---|
1840 | |
---|
1841 | g=subst(p_F-Fred,var(i),1); |
---|
1842 | ES_Id=ES_Id, g; |
---|
1843 | p_F=Fred; |
---|
1844 | |
---|
1845 | g=subst(p_Ftriv-Fredtriv,var(i),1); |
---|
1846 | ES_Id_all_triv=ES_Id_all_triv, g; |
---|
1847 | p_Ftriv=Fredtriv; |
---|
1848 | |
---|
1849 | } |
---|
1850 | |
---|
1851 | dbprint(i_print,"// "); |
---|
1852 | dbprint(i_print,"// back to the original ring"); |
---|
1853 | |
---|
1854 | setring old_ring; |
---|
1855 | // back to original ring |
---|
1856 | |
---|
1857 | ideal ES_Id = imap(AuxRing,ES_Id); |
---|
1858 | ES_Id = interred(ES_Id); |
---|
1859 | |
---|
1860 | ideal ES_Id_all_triv = imap(AuxRing,ES_Id_all_triv); |
---|
1861 | ES_Id_all_triv = interred(ES_Id_all_triv); |
---|
1862 | |
---|
1863 | kill @Px; |
---|
1864 | kill AuxRing; |
---|
1865 | |
---|
1866 | ideal ESfix_Id = ES_Id, f, maxideal(1)*jacob(f); |
---|
1867 | ES_Id = ES_Id, f, jacob(f); |
---|
1868 | ES_Id_all_triv = ES_Id_all_triv, f, jacob(f); |
---|
1869 | dbprint(i_print,"// "); |
---|
1870 | dbprint(i_print,"// equisingularity ideal is computed!"); |
---|
1871 | if (typ==0) |
---|
1872 | { |
---|
1873 | return(list(ES_Id,ESfix_Id)); |
---|
1874 | } |
---|
1875 | else |
---|
1876 | { |
---|
1877 | return(list(ES_Id,ESfix_Id,ES_Id_all_triv)); |
---|
1878 | } |
---|
1879 | } |
---|
1880 | } |
---|
1881 | example |
---|
1882 | { |
---|
1883 | "EXAMPLE:"; echo=2; |
---|
1884 | ring r=0,(x,y),ds; |
---|
1885 | poly f=x7+y7+(x-y)^2*x2y2; |
---|
1886 | list K=esIdeal(f); |
---|
1887 | option(redSB); |
---|
1888 | // Wahl's equisingularity ideal: |
---|
1889 | std(K[1]); |
---|
1890 | |
---|
1891 | ring rr=0,(x,y),ds; |
---|
1892 | poly f=x4+4x3y+6x2y2+4xy3+y4+2x2y15+4xy16+2y17+xy23+y24+y30+y31; |
---|
1893 | list K=esIdeal(f); |
---|
1894 | vdim(std(K[1])); |
---|
1895 | // the latter should be equal to: |
---|
1896 | tau_es(f); |
---|
1897 | } |
---|
1898 | |
---|
1899 | /////////////////////////////////////////////////////////////////////////////// |
---|
1900 | |
---|
1901 | proc esStratum (poly p_F, list #) |
---|
1902 | "USAGE: esStratum(F[,m,L]); F poly, m int, L list |
---|
1903 | ASSUME: F defines a deformation of a reduced bivariate polynomial f |
---|
1904 | and the characteristic of the basering does not divide mult(f). @* |
---|
1905 | If nv is the number of variables of the basering, then the first |
---|
1906 | nv-2 variables are the deformation parameters. @* |
---|
1907 | If the basering is a qring, ideal(basering) must only depend |
---|
1908 | on the deformation parameters. |
---|
1909 | COMPUTE: equations for the stratum of equisingular deformations with |
---|
1910 | fixed (trivial) section. |
---|
1911 | RETURN: list l: either consisting of a list and an integer, where |
---|
1912 | @format |
---|
1913 | l[1][1]=ideal defining the equisingularity stratum |
---|
1914 | l[1][2]=ideal defining the part of the equisingularity stratum where all |
---|
1915 | equimultiple sections through the non-nodes of the reduced total |
---|
1916 | transform are trivial sections |
---|
1917 | l[2]=1 if some error has occured, l[2]=0 otherwise; |
---|
1918 | @end format |
---|
1919 | or consisting of a ring and an integer, where |
---|
1920 | @format |
---|
1921 | l[1]=ESSring is a ring extension of basering containing the ideal ES |
---|
1922 | (describing the ES-stratum), the ideal ES_all_triv (describing the |
---|
1923 | part with trival equimultiple sections) and the poly p_F=F, |
---|
1924 | l[2]=1 if some error has occured, l[2]=0 otherwise. |
---|
1925 | @end format |
---|
1926 | NOTE: L is supposed to be the output of hnexpansion (with the given ordering |
---|
1927 | of the variables appearing in f). @* |
---|
1928 | If m is given, the ES Stratum over A/maxideal(m) is computed. @* |
---|
1929 | This procedure uses @code{execute} or calls a procedure using |
---|
1930 | @code{execute}. |
---|
1931 | printlevel>=2 displays additional information. |
---|
1932 | SEE ALSO: esIdeal, isEquising |
---|
1933 | KEYWORDS: equisingularity stratum |
---|
1934 | EXAMPLE: example esStratum; shows examples. |
---|
1935 | " |
---|
1936 | { |
---|
1937 | list l=esComputation (0,p_F,#); |
---|
1938 | return(l); |
---|
1939 | } |
---|
1940 | example |
---|
1941 | { |
---|
1942 | "EXAMPLE:"; echo=2; |
---|
1943 | int p=printlevel; |
---|
1944 | printlevel=1; |
---|
1945 | ring r = 0,(a,b,c,d,e,f,g,x,y),ds; |
---|
1946 | poly F = (x2+2xy+y2+x5)+ax+by+cx2+dxy+ey2+fx3+gx4; |
---|
1947 | list M = esStratum(F); |
---|
1948 | M[1][1]; |
---|
1949 | |
---|
1950 | printlevel=3; // displays additional information |
---|
1951 | esStratum(F,2) ; // ES-stratum over Q[a,b,c,d,e,f,g] / <a,b,c,d,e,f,g>^2 |
---|
1952 | |
---|
1953 | ideal I = f-fa,e+b; |
---|
1954 | qring q = std(I); |
---|
1955 | poly F = imap(r,F); |
---|
1956 | esStratum(F); |
---|
1957 | printlevel=p; |
---|
1958 | } |
---|
1959 | |
---|
1960 | /////////////////////////////////////////////////////////////////////////////// |
---|
1961 | |
---|
1962 | proc isEquising (poly p_F, list #) |
---|
1963 | "USAGE: isEquising(F[,m,L]); F poly, m int, L list |
---|
1964 | ASSUME: F defines a deformation of a reduced bivariate polynomial f |
---|
1965 | and the characteristic of the basering does not divide mult(f). @* |
---|
1966 | If nv is the number of variables of the basering, then the first |
---|
1967 | nv-2 variables are the deformation parameters. @* |
---|
1968 | If the basering is a qring, ideal(basering) must only depend |
---|
1969 | on the deformation parameters. |
---|
1970 | COMPUTE: tests if the given family is equisingular along the trivial |
---|
1971 | section. |
---|
1972 | RETURN: int: 1 if the family is equisingular, 0 otherwise. |
---|
1973 | NOTE: L is supposed to be the output of hnexpansion (with the given ordering |
---|
1974 | of the variables appearing in f). @* |
---|
1975 | If m is given, the family is considered over A/maxideal(m). @* |
---|
1976 | This procedure uses @code{execute} or calls a procedure using |
---|
1977 | @code{execute}. |
---|
1978 | printlevel>=2 displays additional information. |
---|
1979 | EXAMPLE: example isEquising; shows examples. |
---|
1980 | " |
---|
1981 | { |
---|
1982 | int check=esComputation (1,p_F,#); |
---|
1983 | return(check); |
---|
1984 | } |
---|
1985 | example |
---|
1986 | { |
---|
1987 | "EXAMPLE:"; echo=2; |
---|
1988 | ring r = 0,(a,b,x,y),ds; |
---|
1989 | poly F = (x2+2xy+y2+x5)+ay3+bx5; |
---|
1990 | isEquising(F); |
---|
1991 | ideal I = ideal(a); |
---|
1992 | qring q = std(I); |
---|
1993 | poly F = imap(r,F); |
---|
1994 | isEquising(F); |
---|
1995 | |
---|
1996 | ring rr=0,(A,B,C,x,y),ls; |
---|
1997 | poly f=x7+y7+(x-y)^2*x2y2; |
---|
1998 | poly F=f+A*y*diff(f,x)+B*x*diff(f,x); |
---|
1999 | isEquising(F); |
---|
2000 | isEquising(F,2); // computation over Q[a,b] / <a,b>^2 |
---|
2001 | } |
---|
2002 | |
---|
2003 | |
---|
2004 | |
---|
2005 | /* Examples: |
---|
2006 | |
---|
2007 | LIB "equising.lib"; |
---|
2008 | ring r = 0,(x,y),ds; |
---|
2009 | poly p1 = y^2+x^3; |
---|
2010 | poly p2 = p1^2+x5y; |
---|
2011 | poly p3 = p2^2+x^10*p1; |
---|
2012 | poly p=p3^2+x^20*p2; |
---|
2013 | p; |
---|
2014 | list L=versal(p); |
---|
2015 | def Px=L[1]; |
---|
2016 | setring Px; |
---|
2017 | poly F=Fs[1,1]; |
---|
2018 | int t=timer; |
---|
2019 | list M=esStratum(F); |
---|
2020 | timer-t; //-> 3 |
---|
2021 | |
---|
2022 | LIB "equising.lib"; |
---|
2023 | option(prot); |
---|
2024 | printlevel=2; |
---|
2025 | ring r=0,(x,y),ds; |
---|
2026 | poly f=(x-yx+y2)^2-(y+x)^31; |
---|
2027 | list L=versal(f); |
---|
2028 | def Px=L[1]; |
---|
2029 | setring Px; |
---|
2030 | poly F=Fs[1,1]; |
---|
2031 | int t=timer; |
---|
2032 | list M=esStratum(F); |
---|
2033 | timer-t; //-> 233 |
---|
2034 | |
---|
2035 | |
---|
2036 | LIB "equising.lib"; |
---|
2037 | printlevel=2; |
---|
2038 | option(prot); |
---|
2039 | timer=1; |
---|
2040 | ring r=0,(x,y),ls; |
---|
2041 | poly f=(x4-y4)^2-x10; |
---|
2042 | list L=versal(f); |
---|
2043 | def Px=L[1]; |
---|
2044 | setring Px; |
---|
2045 | poly F=Fs[1,1]; |
---|
2046 | int t=timer; |
---|
2047 | list M=esStratum(F,3); |
---|
2048 | timer-t; //-> 8 |
---|
2049 | |
---|
2050 | LIB "equising.lib"; |
---|
2051 | printlevel=2; |
---|
2052 | timer=1; |
---|
2053 | ring rr=0,(x,y),ls; |
---|
2054 | poly f=x7+y7+(x-y)^2*x2y2; |
---|
2055 | list K=esIdeal(f); |
---|
2056 | // tau_es |
---|
2057 | vdim(std(K[1])); //-> 22 |
---|
2058 | // tau_es_fix |
---|
2059 | vdim(std(K[2])); //-> 24 |
---|
2060 | |
---|
2061 | |
---|
2062 | LIB "equising.lib"; |
---|
2063 | printlevel=2; |
---|
2064 | timer=1; |
---|
2065 | ring rr=0,(x,y),ls; |
---|
2066 | poly f=x7+y7+(x-y)^2*x2y2+x2y4; // Newton non-deg. |
---|
2067 | list K=esIdeal(f); |
---|
2068 | // tau_es |
---|
2069 | vdim(std(K[1])); //-> 21 |
---|
2070 | // tau_es_fix |
---|
2071 | vdim(std(K[2])); //-> 23 |
---|
2072 | |
---|
2073 | LIB "equising.lib"; |
---|
2074 | ring r=0,(w,v),ds; |
---|
2075 | poly f=w2-v199; |
---|
2076 | list L=hnexpansion(f); |
---|
2077 | list LL=versal(f); |
---|
2078 | def Px=LL[1]; |
---|
2079 | setring Px; |
---|
2080 | list L=imap(r,L); |
---|
2081 | poly F=Fs[1,1]; |
---|
2082 | list M=esStratum(F,2,L); |
---|
2083 | |
---|
2084 | LIB "equising.lib"; |
---|
2085 | printlevel=2; |
---|
2086 | timer=1; |
---|
2087 | ring rr=0,(A,B,C,x,y),ls; |
---|
2088 | poly f=x7+y7+(x-y)^2*x2y2; |
---|
2089 | poly F=f+A*y*diff(f,x)+B*x*diff(f,x)+C*diff(f,y); |
---|
2090 | list M=esStratum(F,6); |
---|
2091 | std(M[1][1]); // standard basis of equisingularity ideal |
---|
2092 | |
---|
2093 | LIB "equising.lib"; |
---|
2094 | printlevel=2; |
---|
2095 | timer=1; |
---|
2096 | ring rr=0,(x,y),ls; |
---|
2097 | poly f=x20+y7+(x-y)^2*x2y2+x2y4; // Newton non-degenerate |
---|
2098 | list K=esIdeal(f); |
---|
2099 | K; |
---|
2100 | |
---|
2101 | ring rr=0,(x,y),ls; |
---|
2102 | poly f=x6y-3x4y4-x4y5+3x2y7-x4y6+2x2y8-y10+2x2y9-y11+x2y10-y12-y13; |
---|
2103 | list K=esIdeal(f); |
---|
2104 | list L=versal(f); |
---|
2105 | def Px=L[1]; |
---|
2106 | setring Px; |
---|
2107 | poly F=Fs[1,1]; |
---|
2108 | list M=esStratum(F,2); |
---|
2109 | |
---|
2110 | LIB "equising.lib"; |
---|
2111 | ring R=0,(A,B,C,D,x,y),ds; |
---|
2112 | poly f=x6y-3x4y4-x4y5+3x2y7-x4y6+2x2y8-y10+2x2y9-y11+x2y10-y12-y13; |
---|
2113 | poly F=f+Ax9+Bx7y2+Cx9y+Dx8y2; |
---|
2114 | list M=esStratum(F,2); |
---|
2115 | |
---|
2116 | |
---|
2117 | LIB "equising.lib"; |
---|
2118 | printlevel=2; |
---|
2119 | ring rr=0,(x,y),ls; |
---|
2120 | poly f=x6y-3x4y4-x4y5+3x2y7-x4y6+2x2y8-y10+2x2y9-y11+x2y10-y12-y13; |
---|
2121 | list K=esIdeal(f); |
---|
2122 | vdim(std(K[1])); //-> 51 |
---|
2123 | tau_es(f); //-> 51 |
---|
2124 | |
---|
2125 | printlevel=3; |
---|
2126 | f=f*(y-x2)*(y2-x3)*(x-y5); |
---|
2127 | int t=timer; |
---|
2128 | list L=esIdeal(f); |
---|
2129 | vdim(std(L[1])); //-> 99 |
---|
2130 | timer-t; //-> 42 |
---|
2131 | t=timer; |
---|
2132 | tau_es(f); //-> 99 |
---|
2133 | timer-t; //-> 23 |
---|
2134 | |
---|
2135 | |
---|
2136 | LIB "equising.lib"; |
---|
2137 | printlevel=3; |
---|
2138 | ring rr=0,(x,y),ds; |
---|
2139 | poly f=x4+4x3y+6x2y2+4xy3+y4+2x2y15+4xy16+2y17+xy23+y24+y30+y31; |
---|
2140 | list K=esIdeal(f); |
---|
2141 | vdim(std(K[1])); //-> 68 |
---|
2142 | tau_es(f); //-> 68 |
---|
2143 | |
---|
2144 | list L=versal(f); |
---|
2145 | def Px=L[1]; |
---|
2146 | setring Px; |
---|
2147 | poly F=Fs[1,1]; |
---|
2148 | list M=esStratum(F); |
---|
2149 | timer-t; //-> 0 |
---|
2150 | |
---|
2151 | |
---|
2152 | |
---|
2153 | */ |
---|