1 | ////////////////////////////////////////////////////////////////////////////// |
---|
2 | version="$Id$"; |
---|
3 | category="General purpose"; |
---|
4 | info=" |
---|
5 | LIBRARY: schreyer.lib Helpers for working with the Schreyer induced ordering |
---|
6 | AUTHOR: Oleksandr Motsak <U@D>, where U={motsak}, D={mathematik.uni-kl.de} |
---|
7 | |
---|
8 | PROCEDURES: |
---|
9 | Sres(M,l) Schreyer resolution of module M of maximal length l |
---|
10 | Ssyz(M) Schreyer resolution of module M of length 1 |
---|
11 | Scontinue(l) continue the resolution computation by most l steps |
---|
12 | |
---|
13 | KEYWORDS: syzygy; Schreyer induced ordering; Schreyer free resolution |
---|
14 | NOTE: requires the dynamic module: syzextra |
---|
15 | "; |
---|
16 | |
---|
17 | static proc prepareSyz( module I, list # ) |
---|
18 | { |
---|
19 | int i; |
---|
20 | int k = 0; |
---|
21 | int r = nrows(I); |
---|
22 | int c = ncols(I); |
---|
23 | |
---|
24 | |
---|
25 | if( size(#) > 0 ) |
---|
26 | { |
---|
27 | if( typeof(#[1]) == "int" || typeof(#[1]) == "bigint" ) |
---|
28 | { |
---|
29 | k = #[1]; |
---|
30 | } |
---|
31 | } |
---|
32 | |
---|
33 | if( k < r ) |
---|
34 | { |
---|
35 | "// *** Wrong k: ", k, " < nrows: ", r, " => setting k = r = ", r; |
---|
36 | k = r; |
---|
37 | } |
---|
38 | |
---|
39 | // "k: ", k; "c: ", c; "I: ", I; |
---|
40 | |
---|
41 | for( i = c; i > 0; i-- ) |
---|
42 | { |
---|
43 | I[i] = I[i] + gen(k + i); |
---|
44 | } |
---|
45 | |
---|
46 | // DetailedPrint(I); |
---|
47 | |
---|
48 | return(I); |
---|
49 | } |
---|
50 | |
---|
51 | static proc separateSyzGB( module J, int c ) |
---|
52 | { |
---|
53 | module II, G; vector v; int i; |
---|
54 | |
---|
55 | J = simplify(J, 2); |
---|
56 | |
---|
57 | for( i = ncols(J); i > 0; i-- ) |
---|
58 | { |
---|
59 | v = J[i]; |
---|
60 | if( leadcomp(v) > c ) |
---|
61 | { |
---|
62 | II[i] = v; |
---|
63 | } else |
---|
64 | { |
---|
65 | G[i] = v; // leave only gen(i): i <= c |
---|
66 | } |
---|
67 | } |
---|
68 | |
---|
69 | II = simplify(II, 2); |
---|
70 | G = simplify(G, 2); |
---|
71 | |
---|
72 | return (list(G, II)); |
---|
73 | } |
---|
74 | |
---|
75 | static proc splitSyzGB( module J, int c ) |
---|
76 | { |
---|
77 | module JJ; vector v, vv; int i; |
---|
78 | |
---|
79 | for( i = ncols(J); i > 0; i-- ) |
---|
80 | { |
---|
81 | v = J[i]; |
---|
82 | |
---|
83 | vv = 0; |
---|
84 | |
---|
85 | while( leadcomp(v) <= c ) |
---|
86 | { |
---|
87 | vv = vv + lead(v); |
---|
88 | v = v - lead(v); |
---|
89 | } |
---|
90 | |
---|
91 | J[i] = vv; |
---|
92 | JJ[i] = v; |
---|
93 | } |
---|
94 | |
---|
95 | J = simplify(J, 2); |
---|
96 | JJ = simplify(JJ, 2); |
---|
97 | |
---|
98 | return (list(J, JJ)); |
---|
99 | } |
---|
100 | |
---|
101 | |
---|
102 | static proc Sinit(module M) |
---|
103 | { |
---|
104 | def @save = basering; |
---|
105 | |
---|
106 | int @DEBUG = !system("with", "ndebug"); |
---|
107 | if( @DEBUG ) |
---|
108 | { |
---|
109 | "Sinit::Input"; |
---|
110 | type(M); |
---|
111 | DetailedPrint(M); |
---|
112 | attrib(M); |
---|
113 | } |
---|
114 | |
---|
115 | int @RANK = nrows(M); int @SIZE = ncols(M); |
---|
116 | |
---|
117 | int @IS_A_SB = attrib(M, "isSB"); // ??? only if all weights were zero?! |
---|
118 | |
---|
119 | if( !@IS_A_SB ) |
---|
120 | { |
---|
121 | M = std(M); // this should be faster than computing std in S (later on) |
---|
122 | } |
---|
123 | |
---|
124 | def S = MakeInducedSchreyerOrdering(1); // 1 puts history terms to the back |
---|
125 | // TODO: NOTE: +1 causes trouble to Singular interpreter!!!??? |
---|
126 | setring S; // a new ring with a Schreyer ordering |
---|
127 | |
---|
128 | if( @DEBUG ) |
---|
129 | { |
---|
130 | "Sinit::StartingISRing"; |
---|
131 | basering; |
---|
132 | // DetailedPrint(basering); |
---|
133 | } |
---|
134 | |
---|
135 | // Setup the leading syzygy^{-1} module to zero: |
---|
136 | module Z = 0; Z[@RANK] = 0; attrib(Z, "isHomog", intvec(0)); |
---|
137 | |
---|
138 | module MRES = Z; |
---|
139 | |
---|
140 | list RES; RES[1] = Z; |
---|
141 | |
---|
142 | module F = freemodule(@RANK); |
---|
143 | intvec @V = deg(F[1..@RANK]); |
---|
144 | |
---|
145 | module M = imap(@save, M); |
---|
146 | |
---|
147 | attrib(M, "isHomog", @V); |
---|
148 | attrib(M, "isSB", 1); |
---|
149 | |
---|
150 | |
---|
151 | if( @DEBUG ) |
---|
152 | { |
---|
153 | "Sinit::SB_Input: "; |
---|
154 | type(M); |
---|
155 | attrib(M); |
---|
156 | attrib(M, "isHomog"); |
---|
157 | DetailedPrint(M); |
---|
158 | } |
---|
159 | |
---|
160 | if( @DEBUG ) |
---|
161 | { |
---|
162 | // 0^th syz. property |
---|
163 | if( size(module(transpose( transpose(M) * transpose(MRES) ))) > 0 ) |
---|
164 | { |
---|
165 | transpose( transpose(M) * transpose(MRES) ); |
---|
166 | "transpose( transpose(M) * transpose(MRES) ) != 0!!!"; |
---|
167 | $ |
---|
168 | } |
---|
169 | } |
---|
170 | |
---|
171 | RES[size(RES)+1] = M; // list of all syzygy modules |
---|
172 | MRES = MRES, M; |
---|
173 | |
---|
174 | attrib(MRES, "isHomog", @V); |
---|
175 | |
---|
176 | attrib(S, "InducionLeads", lead(M)); |
---|
177 | attrib(S, "InducionStart", @RANK); |
---|
178 | |
---|
179 | if( @DEBUG ) |
---|
180 | { |
---|
181 | "Sinit::MRES"; |
---|
182 | DetailedPrint(MRES); |
---|
183 | attrib(MRES, "isHomog"); |
---|
184 | attrib(S); |
---|
185 | } |
---|
186 | |
---|
187 | export RES; |
---|
188 | export MRES; |
---|
189 | return (S); |
---|
190 | } |
---|
191 | |
---|
192 | static proc Sstep() |
---|
193 | { |
---|
194 | int @DEBUG = !system("with", "ndebug"); |
---|
195 | |
---|
196 | if( @DEBUG ) |
---|
197 | { |
---|
198 | "Sstep::NextInducedRing"; |
---|
199 | DetailedPrint(basering); |
---|
200 | |
---|
201 | attrib(basering, "InducionLeads"); |
---|
202 | attrib(basering, "InducionStart"); |
---|
203 | |
---|
204 | GetInducedData(); |
---|
205 | } |
---|
206 | |
---|
207 | // syzygy step: |
---|
208 | |
---|
209 | /* |
---|
210 | // is initial weights are all zeroes! |
---|
211 | def L = lead(M); |
---|
212 | intvec @V = deg(M[1..ncols(M)]); @W; @V; @W = @V; attrib(L, "isHomog", @W); |
---|
213 | SetInducedReferrence(L, @RANK, 0); |
---|
214 | */ |
---|
215 | |
---|
216 | // def L = lead(MRES); |
---|
217 | // @W = @W, @V; |
---|
218 | // attrib(L, "isHomog", @W); |
---|
219 | |
---|
220 | |
---|
221 | // General setting: |
---|
222 | // SetInducedReferrence(MRES, 0, 0); // limit: 0! |
---|
223 | int @l = size(RES); |
---|
224 | |
---|
225 | module M = RES[@l]; |
---|
226 | |
---|
227 | module L = attrib(basering, "InducionLeads"); |
---|
228 | int limit = attrib(basering, "InducionStart"); |
---|
229 | |
---|
230 | // L; limit; |
---|
231 | |
---|
232 | int @RANK = ncols(MRES) - ncols(M); // nrows(M); // what if M is zero?! |
---|
233 | |
---|
234 | /* |
---|
235 | if( @RANK != nrows(M) ) |
---|
236 | { |
---|
237 | type(MRES); |
---|
238 | @RANK; |
---|
239 | type(M); |
---|
240 | pause(); |
---|
241 | } |
---|
242 | */ |
---|
243 | |
---|
244 | intvec @W = attrib(M, "isHomog"); |
---|
245 | intvec @V = deg(M[1..ncols(M)]); |
---|
246 | @V = @W, @V; |
---|
247 | |
---|
248 | if( @DEBUG ) |
---|
249 | { |
---|
250 | "Sstep::NextInput: "; |
---|
251 | M; |
---|
252 | @V; |
---|
253 | @RANK; |
---|
254 | DetailedPrint(MRES); |
---|
255 | attrib(MRES, "isHomog"); |
---|
256 | } |
---|
257 | |
---|
258 | |
---|
259 | |
---|
260 | SetInducedReferrence(L, limit, 0); |
---|
261 | |
---|
262 | def K = prepareSyz(M, @RANK); |
---|
263 | // K; |
---|
264 | |
---|
265 | // attrib(K, "isHomog", @V); DetailedPrint(K, 1000); |
---|
266 | |
---|
267 | // pause(); |
---|
268 | |
---|
269 | K = idPrepare(K, @RANK); // std(K); // ? |
---|
270 | K = simplify(K, 2); |
---|
271 | |
---|
272 | // K; |
---|
273 | |
---|
274 | module N = separateSyzGB(K, @RANK)[2]; // 1^st syz. module: vectors which start in lower part (comp >= @RANK) |
---|
275 | attrib(N, "isHomog", @V); |
---|
276 | |
---|
277 | // "N_0: "; N; DetailedPrint(N, 10); |
---|
278 | |
---|
279 | N = std(N); // TODO: fix "wrong weights"!!!? |
---|
280 | attrib(N, "isHomog", @V); |
---|
281 | |
---|
282 | // N; |
---|
283 | |
---|
284 | if( @DEBUG ) |
---|
285 | { |
---|
286 | if( size(N) > 0 ) |
---|
287 | { |
---|
288 | // next syz. property |
---|
289 | if( size(module(transpose( transpose(N) * transpose(MRES) ))) > 0 ) |
---|
290 | { |
---|
291 | MRES; |
---|
292 | |
---|
293 | "N: "; N; DetailedPrint(N, 10); |
---|
294 | |
---|
295 | "K:"; K; DetailedPrint(K, 10); |
---|
296 | |
---|
297 | "RANKS: ", @RANK; |
---|
298 | |
---|
299 | "transpose( transpose(N) * transpose(MRES) ) != 0!!!"; |
---|
300 | transpose( transpose(N) * transpose(MRES) ); |
---|
301 | |
---|
302 | "transpose(N) * transpose(MRES): "; |
---|
303 | transpose(N) * transpose(MRES); |
---|
304 | DetailedPrint(module(_), 2); |
---|
305 | $ |
---|
306 | } |
---|
307 | } |
---|
308 | } |
---|
309 | |
---|
310 | RES[@l + 1] = N; // list of all syzygy modules |
---|
311 | |
---|
312 | MRES = MRES, N; |
---|
313 | attrib(MRES, "isHomog", @V); |
---|
314 | |
---|
315 | |
---|
316 | L = L, lead(N); |
---|
317 | attrib(basering, "InducionLeads", L); |
---|
318 | |
---|
319 | if( @DEBUG ) |
---|
320 | { |
---|
321 | "Sstep::NextSyzOutput: "; |
---|
322 | DetailedPrint(N); |
---|
323 | attrib(N, "isHomog"); |
---|
324 | } |
---|
325 | |
---|
326 | } |
---|
327 | |
---|
328 | proc Scontinue(int l) |
---|
329 | "USAGE: Scontinue(l) |
---|
330 | RETURN: nothing, instead it changes RES and MRES variables in the current ring |
---|
331 | PURPOSE: computes further (at most l) syzygies |
---|
332 | NOTE: must be used within a ring returned by Sres or Ssyz. RES and MRES are |
---|
333 | explained in Sres |
---|
334 | EXAMPLE: example Scontinue; shows an example |
---|
335 | " |
---|
336 | { |
---|
337 | def data = GetInducedData(); |
---|
338 | |
---|
339 | if( (!defined(RES)) || (!defined(MRES)) || (typeof(data) != "list") || (size(data) != 2) ) |
---|
340 | { |
---|
341 | ERROR("Sorry, but basering does not seem to be returned by Sres or Ssyz"); |
---|
342 | } |
---|
343 | for (; (l != 0) && (size(RES[size(RES)]) > 0); l-- ) |
---|
344 | { |
---|
345 | Sstep(); |
---|
346 | } |
---|
347 | } |
---|
348 | example |
---|
349 | { "EXAMPLE:"; echo = 2; |
---|
350 | ring r; |
---|
351 | module M = maxideal(1); M; |
---|
352 | def S = Ssyz(M); setring S; S; |
---|
353 | "Only the first syzygy: "; |
---|
354 | RES; MRES; |
---|
355 | "More syzygies: "; |
---|
356 | Scontinue(10); |
---|
357 | RES; MRES; |
---|
358 | } |
---|
359 | |
---|
360 | proc Ssyz(module M) |
---|
361 | "USAGE: Ssyz(M) |
---|
362 | RETURN: ring, containing a list of modules RES and a module MRES |
---|
363 | PURPOSE: computes the first syzygy module of M (wrt some Schreyer ordering) |
---|
364 | NOTE: The output is explained in Sres |
---|
365 | EXAMPLE: example Ssyz; shows an example |
---|
366 | " |
---|
367 | { |
---|
368 | def S = Sinit(M); setring S; |
---|
369 | |
---|
370 | Sstep(); // NOTE: what if M is zero? |
---|
371 | |
---|
372 | return (S); |
---|
373 | } |
---|
374 | example |
---|
375 | { "EXAMPLE:"; echo = 2; |
---|
376 | ring r; |
---|
377 | module M = maxideal(1); M; |
---|
378 | def S = Ssyz(M); setring S; S; |
---|
379 | "Only the first syzygy: "; |
---|
380 | RES; |
---|
381 | MRES; // Note gen(i) |
---|
382 | kill S; |
---|
383 | setring r; kill M; |
---|
384 | |
---|
385 | module M = 0; |
---|
386 | def S = Ssyz(M); setring S; S; |
---|
387 | "Only the first syzygy: "; |
---|
388 | RES; |
---|
389 | MRES; |
---|
390 | } |
---|
391 | |
---|
392 | proc Sres(module M, int l) |
---|
393 | "USAGE: Sres(M, l) |
---|
394 | RETURN: ring, containing a list of modules RES and a module MRES |
---|
395 | PURPOSE: computes (at most l) syzygy modules of M wrt the classical Schreyer |
---|
396 | induced ordering with gen(i) > gen(j) if i > j, provided both gens |
---|
397 | are from the same syzygy level. |
---|
398 | NOTE: RES contains the images of maps subsituting the beginning of the |
---|
399 | Schreyer free resolution of baseRing^r/M, while MRES is a sum of |
---|
400 | these images in a big free sum, containing all the syzygy modules. |
---|
401 | The syzygy modules are shifted so that gen(i) correspons to MRES[i]. |
---|
402 | The leading zero module RES[0] indicates the fact that coker of the |
---|
403 | first map is zero. The number of zeroes inducates the rank of input. |
---|
404 | NOTE: If l == 0 then l is set to be nvars(basering) + 1 |
---|
405 | EXAMPLE: example Sres; shows an example |
---|
406 | " |
---|
407 | { |
---|
408 | def S = Sinit(M); setring S; |
---|
409 | |
---|
410 | if (l == 0) |
---|
411 | { |
---|
412 | l = nvars(basering) + 1; // not really an estimate...?! |
---|
413 | } |
---|
414 | |
---|
415 | Sstep(); l = l - 1; |
---|
416 | |
---|
417 | Scontinue(l); |
---|
418 | |
---|
419 | return (S); |
---|
420 | } |
---|
421 | example |
---|
422 | { "EXAMPLE:"; echo = 2; |
---|
423 | ring r; |
---|
424 | module M = maxideal(1); M; |
---|
425 | def S = Sres(M, 0); setring S; S; |
---|
426 | RES; |
---|
427 | MRES; |
---|
428 | kill S; |
---|
429 | setring r; kill M; |
---|
430 | |
---|
431 | def A = nc_algebra(-1,0); setring A; |
---|
432 | ideal Q = var(1)^2, var(2)^2, var(3)^2; |
---|
433 | qring SCA = twostd(Q); |
---|
434 | basering; |
---|
435 | |
---|
436 | module M = maxideal(1); |
---|
437 | def S = Sres(M, 2); setring S; S; |
---|
438 | RES; |
---|
439 | MRES; |
---|
440 | } |
---|
441 | |
---|
442 | |
---|
443 | |
---|
444 | // ================================================================== // |
---|
445 | |
---|
446 | |
---|
447 | LIB "general.lib"; // for sort |
---|
448 | |
---|
449 | /* static proc Tail(def M) // DONE: in C++ (dyn. module: syzextra)! |
---|
450 | { |
---|
451 | int i = ncols(M); def m; |
---|
452 | while (i > 0) |
---|
453 | { |
---|
454 | m = M[i]; |
---|
455 | m = m - lead(m); // m = tail(m) |
---|
456 | M[i] = m; |
---|
457 | i--; |
---|
458 | } |
---|
459 | return (M); |
---|
460 | }*/ |
---|
461 | |
---|
462 | |
---|
463 | /* static */ proc SSinit(def M) |
---|
464 | { |
---|
465 | if( (typeof(M) != "module") && (typeof(M) != "ideal") ) |
---|
466 | { |
---|
467 | ERROR("Sorry: need an ideal or a module for input"); |
---|
468 | } |
---|
469 | |
---|
470 | // TODO! DONE? |
---|
471 | def @save = basering; |
---|
472 | |
---|
473 | int @DEBUG = !system("with", "ndebug"); |
---|
474 | int @SYZCHECK = 1 || @DEBUG; // TODO: only for now!! |
---|
475 | |
---|
476 | if( @DEBUG ) |
---|
477 | { |
---|
478 | "SSinit::Input"; |
---|
479 | type(M); |
---|
480 | // DetailedPrint(M); |
---|
481 | attrib(M); |
---|
482 | } |
---|
483 | |
---|
484 | int @RANK = nrows(M); int @SIZE = ncols(M); |
---|
485 | |
---|
486 | int @IS_A_SB = attrib(M, "isSB"); // ??? only if all weights were zero?! |
---|
487 | |
---|
488 | if( !@IS_A_SB ) |
---|
489 | { |
---|
490 | def opts = option(get); |
---|
491 | option(redSB); option(redTail); |
---|
492 | M = std(M); |
---|
493 | option(set, opts); |
---|
494 | kill opts; |
---|
495 | } else |
---|
496 | { |
---|
497 | M = simplify(M, 2 + 4 + 32); |
---|
498 | } |
---|
499 | |
---|
500 | def LEAD = lead(M); |
---|
501 | |
---|
502 | // sort wrt neg.deg.rev.lex! |
---|
503 | intvec iv_ds = sort(LEAD, "ds", 1)[2]; // ,1 => reversed! |
---|
504 | |
---|
505 | M = M[iv_ds]; // sort M wrt ds on current leading terms |
---|
506 | LEAD = LEAD[iv_ds]; |
---|
507 | |
---|
508 | def TAIL = Tail(M); |
---|
509 | |
---|
510 | intvec @DEGS = deg(M[1..@SIZE]); // store actuall degrees of input elements |
---|
511 | |
---|
512 | // TODO: what about real modules? weighted ones? |
---|
513 | |
---|
514 | list @l = ringlist(@save); |
---|
515 | |
---|
516 | int @z = 0; ideal @m = maxideal(1); intvec @wdeg = deg(@m[1..ncols(@m)]); |
---|
517 | |
---|
518 | // NOTE: @wdeg will be ignored anyway :( |
---|
519 | @l[3] = list(list("C", @z), list("lp", @wdeg)); |
---|
520 | |
---|
521 | kill @z, @wdeg; // since these vars are ring independent! |
---|
522 | |
---|
523 | def S = ring(@l); // --MakeInducedSchreyerOrdering(1); |
---|
524 | |
---|
525 | module F = freemodule(@RANK); |
---|
526 | intvec @V = deg(F[1..@RANK]); |
---|
527 | |
---|
528 | setring S; // ring with an easy divisibility test ("C, lex") |
---|
529 | |
---|
530 | if( @DEBUG ) |
---|
531 | { |
---|
532 | "SSinit::NewRing(C, lex)"; |
---|
533 | basering; |
---|
534 | // DetailedPrint(basering); |
---|
535 | } |
---|
536 | |
---|
537 | // Setup the leading syzygy^{-1} module to zero: |
---|
538 | module Z = 0; Z[@RANK] = 0; attrib(Z, "isHomog", intvec(0)); |
---|
539 | |
---|
540 | module MRES = Z; |
---|
541 | |
---|
542 | list RES; RES[1] = Z; |
---|
543 | list LRES; LRES[1] = Z; |
---|
544 | list TRES; TRES[1] = Z; |
---|
545 | |
---|
546 | def M = imap(@save, M); |
---|
547 | |
---|
548 | attrib(M, "isHomog", @V); |
---|
549 | attrib(M, "isSB", 1); |
---|
550 | attrib(M, "degrees", @DEGS); |
---|
551 | |
---|
552 | def LEAD = imap(@save, LEAD); |
---|
553 | |
---|
554 | attrib(LEAD, "isHomog", @V); |
---|
555 | attrib(LEAD, "isSB", 1); |
---|
556 | |
---|
557 | def TAIL = imap(@save, TAIL); |
---|
558 | |
---|
559 | if( @DEBUG ) |
---|
560 | { |
---|
561 | "SSinit::(sorted) SB_Input: "; |
---|
562 | type(M); |
---|
563 | attrib(M); |
---|
564 | attrib(M, "isHomog"); |
---|
565 | // DetailedPrint(M); |
---|
566 | } |
---|
567 | |
---|
568 | if( @SYZCHECK ) |
---|
569 | { |
---|
570 | // 0^th syz. property |
---|
571 | if( size(module(transpose( transpose(M) * transpose(MRES) ))) > 0 ) |
---|
572 | { |
---|
573 | transpose( transpose(M) * transpose(MRES) ); |
---|
574 | "ERROR: transpose( transpose(M) * transpose(MRES) ) != 0!!!"; |
---|
575 | $ |
---|
576 | } |
---|
577 | } |
---|
578 | |
---|
579 | RES[size(RES)+1] = M; // list of all syzygy modules |
---|
580 | LRES[size(LRES)+1] = LEAD; // list of all syzygy modules |
---|
581 | TRES[size(TRES)+1] = TAIL; // list of all syzygy modules |
---|
582 | |
---|
583 | MRES = MRES, M; //? |
---|
584 | |
---|
585 | attrib(MRES, "isHomog", @V); |
---|
586 | |
---|
587 | // attrib(S, "InducionStart", @RANK); |
---|
588 | attrib(S, "LEAD2SYZ", 1); |
---|
589 | attrib(S, "TAILREDSYZ", 0); |
---|
590 | attrib(S, "DEBUG", @DEBUG); |
---|
591 | attrib(S, "SYZCHECK", @SYZCHECK); |
---|
592 | |
---|
593 | if( @DEBUG ) |
---|
594 | { |
---|
595 | "SSinit::MRES"; |
---|
596 | MRES; |
---|
597 | // DetailedPrint(MRES); |
---|
598 | attrib(MRES, "isHomog"); |
---|
599 | attrib(S); |
---|
600 | } |
---|
601 | |
---|
602 | export RES; |
---|
603 | export MRES; |
---|
604 | export LRES; |
---|
605 | export TRES; |
---|
606 | return (S); |
---|
607 | } |
---|
608 | example |
---|
609 | { "EXAMPLE:"; echo = 2; |
---|
610 | ring R = 0, (w, x, y, z), dp; |
---|
611 | |
---|
612 | def M = maxideal(1); |
---|
613 | def S = SSinit(M); setring S; S; |
---|
614 | |
---|
615 | "Only the first initialization: "; |
---|
616 | RES; LRES; TRES; |
---|
617 | MRES; |
---|
618 | |
---|
619 | kill S; setring R; kill M; |
---|
620 | |
---|
621 | ideal M = w^2 - x*z, w*x - y*z, x^2 - w*y, x*y - z^2, y^2 - w*z; |
---|
622 | def S = SSinit(M); setring S; S; |
---|
623 | |
---|
624 | "Only the first initialization: "; |
---|
625 | RES; LRES; TRES; |
---|
626 | MRES; |
---|
627 | |
---|
628 | kill S; setring R; kill M; |
---|
629 | } |
---|
630 | |
---|
631 | |
---|
632 | LIB "poly.lib"; // for lcm |
---|
633 | |
---|
634 | |
---|
635 | |
---|
636 | /// Compute L(Syz(L)) |
---|
637 | proc SSComputeLeadingSyzygyTerms(def L) |
---|
638 | { |
---|
639 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
640 | { |
---|
641 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
642 | } else |
---|
643 | { |
---|
644 | int @DEBUG = !system("with", "ndebug"); |
---|
645 | } |
---|
646 | |
---|
647 | if( @DEBUG ) |
---|
648 | { |
---|
649 | "SSComputeLeadingSyzygyTerms::Input: "; |
---|
650 | L; |
---|
651 | } |
---|
652 | |
---|
653 | int i, j, r; intvec iv_ds; |
---|
654 | int N = ncols(L); |
---|
655 | def a, b; |
---|
656 | poly aa, bb; |
---|
657 | |
---|
658 | bigint c; |
---|
659 | |
---|
660 | ideal M; |
---|
661 | |
---|
662 | module S = 0; |
---|
663 | |
---|
664 | for(i = 1; i <= N; i++) |
---|
665 | { |
---|
666 | a = L[i]; |
---|
667 | // "a: ", a; |
---|
668 | c = leadcomp(a); |
---|
669 | r = int(c); |
---|
670 | |
---|
671 | if(r > 0) |
---|
672 | { |
---|
673 | aa = a[r]; |
---|
674 | } else |
---|
675 | { |
---|
676 | aa = a; |
---|
677 | } |
---|
678 | |
---|
679 | M = 0; |
---|
680 | |
---|
681 | for(j = i-1; j > 0; j--) |
---|
682 | { |
---|
683 | b = L[j]; |
---|
684 | // "b: ", b; |
---|
685 | |
---|
686 | if( leadcomp(b) == c ) |
---|
687 | { |
---|
688 | if(r > 0) |
---|
689 | { |
---|
690 | bb = b[r]; |
---|
691 | } else |
---|
692 | { |
---|
693 | bb = b; |
---|
694 | } |
---|
695 | |
---|
696 | M[j] = (lcm(aa, bb) / aa); |
---|
697 | } |
---|
698 | } |
---|
699 | |
---|
700 | // TODO: add quotient relations here... |
---|
701 | |
---|
702 | M = simplify(M, 1 + 2 + 32); |
---|
703 | |
---|
704 | iv_ds = sort(M, "ds", 1)[2]; // ,1 => reversed! |
---|
705 | |
---|
706 | M = M[iv_ds]; |
---|
707 | |
---|
708 | S = S, M * gen(i); |
---|
709 | } |
---|
710 | |
---|
711 | S = simplify(S, 2); |
---|
712 | |
---|
713 | S = sort(S, "ds", 1)[1]; // ,1 => reversed! // TODO: not needed? |
---|
714 | |
---|
715 | if( @DEBUG ) |
---|
716 | { |
---|
717 | "SSComputeLeadingSyzygyTerms::Output: "; |
---|
718 | S; |
---|
719 | } |
---|
720 | |
---|
721 | attrib(S, "isSB", 1); |
---|
722 | |
---|
723 | return (S); |
---|
724 | } |
---|
725 | |
---|
726 | /// Compute Syz(L), where L is a monomial (leading) module |
---|
727 | proc SSCompute2LeadingSyzygyTerms(def L, int @TAILREDSYZ) |
---|
728 | { |
---|
729 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
730 | { |
---|
731 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
732 | } else |
---|
733 | { |
---|
734 | int @DEBUG = !system("with", "ndebug"); |
---|
735 | } |
---|
736 | |
---|
737 | if( typeof( attrib(basering, "SYZCHECK") ) == "int" ) |
---|
738 | { |
---|
739 | int @SYZCHECK = attrib(basering, "SYZCHECK"); |
---|
740 | } else |
---|
741 | { |
---|
742 | int @SYZCHECK = @DEBUG; |
---|
743 | } |
---|
744 | |
---|
745 | |
---|
746 | if( @DEBUG ) |
---|
747 | { |
---|
748 | "SSCompute2LeadingSyzygyTerms::Input: "; |
---|
749 | L; |
---|
750 | "@TAILREDSYZ: ", @TAILREDSYZ; |
---|
751 | } |
---|
752 | |
---|
753 | int i, j, r; |
---|
754 | int N = ncols(L); |
---|
755 | def a, b; |
---|
756 | |
---|
757 | poly aa, bb, @lcm; |
---|
758 | |
---|
759 | bigint c; |
---|
760 | |
---|
761 | module M; |
---|
762 | |
---|
763 | module S = 0; |
---|
764 | |
---|
765 | for(i = 1; i <= N; i++) |
---|
766 | { |
---|
767 | a = L[i]; |
---|
768 | // "a: ", a; |
---|
769 | c = leadcomp(a); |
---|
770 | r = int(c); |
---|
771 | |
---|
772 | aa = leadmonomial(a); |
---|
773 | |
---|
774 | M = 0; |
---|
775 | |
---|
776 | for(j = i-1; j > 0; j--) |
---|
777 | { |
---|
778 | b = L[j]; |
---|
779 | // "b: ", b; |
---|
780 | |
---|
781 | if( leadcomp(b) == c ) |
---|
782 | { |
---|
783 | bb = leadmonomial(b); |
---|
784 | @lcm = lcm(aa, bb); |
---|
785 | |
---|
786 | M[j] = (@lcm / aa)* gen(i) - (@lcm / bb)* gen(j); |
---|
787 | } |
---|
788 | } |
---|
789 | |
---|
790 | M = simplify(M, 2); |
---|
791 | |
---|
792 | // TODO: add quotient relations here... |
---|
793 | S = S, M; |
---|
794 | } |
---|
795 | |
---|
796 | if( @TAILREDSYZ ) |
---|
797 | { |
---|
798 | // Make sure that 2nd syzygy terms are not reducible by 1st |
---|
799 | def opts = option(get); |
---|
800 | option(redSB); option(redTail); |
---|
801 | S = std(S); // binomial module |
---|
802 | option(set, opts); |
---|
803 | // kill opts; |
---|
804 | } else |
---|
805 | { |
---|
806 | S = simplify(S, 2 + 32); |
---|
807 | } |
---|
808 | |
---|
809 | S = sort(S, "ds", 1)[1]; // ,1 => reversed! |
---|
810 | |
---|
811 | if( @DEBUG ) |
---|
812 | { |
---|
813 | "SSCompute2LeadingSyzygyTerms::Syz(LEAD): "; S; |
---|
814 | } |
---|
815 | |
---|
816 | if( @SYZCHECK ) |
---|
817 | { |
---|
818 | if( size(S) > 0 and size(L) > 0 ) |
---|
819 | { |
---|
820 | if( size(module(transpose( transpose(S) * transpose(L) ))) > 0 ) |
---|
821 | { |
---|
822 | transpose( transpose(S) * transpose(L) ); |
---|
823 | "ERROR: transpose( transpose(S) * transpose(L) ) != 0!!!"; |
---|
824 | $ |
---|
825 | } |
---|
826 | } |
---|
827 | } |
---|
828 | |
---|
829 | module S2 = Tail(S); |
---|
830 | S = lead(S); // (C,lp) on base ring! |
---|
831 | |
---|
832 | if( @DEBUG ) |
---|
833 | { |
---|
834 | "SSCompute2LeadingSyzygyTerms::Output: "; S; S2; |
---|
835 | } |
---|
836 | |
---|
837 | attrib(S, "isSB", 1); |
---|
838 | |
---|
839 | return (S, S2); |
---|
840 | } |
---|
841 | |
---|
842 | // -------------------------------------------------------- // |
---|
843 | |
---|
844 | /// TODO: save shortcut LM(m) * "t" -> ? |
---|
845 | proc SSReduceTerm(poly m, def t, def L, def T, list #) |
---|
846 | { |
---|
847 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
848 | { |
---|
849 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
850 | } else |
---|
851 | { |
---|
852 | int @DEBUG = !system("with", "ndebug"); |
---|
853 | } |
---|
854 | |
---|
855 | if( @DEBUG ) |
---|
856 | { |
---|
857 | "SSReduce::Input: "; |
---|
858 | |
---|
859 | "mult: ", m; |
---|
860 | "term: ", t; |
---|
861 | "L: ", L; |
---|
862 | "T: ", T; |
---|
863 | if( size(#) > 0 ) |
---|
864 | { |
---|
865 | "LSyz: ", #; |
---|
866 | } |
---|
867 | // "attrib(LS, 'isSB')", attrib(LS, "isSB"); |
---|
868 | } |
---|
869 | |
---|
870 | vector s = 0; |
---|
871 | |
---|
872 | if( t == 0 ) |
---|
873 | { |
---|
874 | return (s); |
---|
875 | } |
---|
876 | |
---|
877 | def product = m * t; |
---|
878 | |
---|
879 | bigint c = leadcomp(t); |
---|
880 | int r = int(c); |
---|
881 | |
---|
882 | def a, b, nf, bb; |
---|
883 | |
---|
884 | // looking for an appropriate reducer |
---|
885 | for( int k = ncols(L); k > 0; k-- ) |
---|
886 | { |
---|
887 | a = L[k]; |
---|
888 | // with the same mod. component |
---|
889 | if( leadcomp(a) == c ) |
---|
890 | { |
---|
891 | b = - (leadmonomial(product) / leadmonomial(L[k])); |
---|
892 | |
---|
893 | // which divides the product |
---|
894 | if( b != 0 ) |
---|
895 | { |
---|
896 | // "b: ", b; |
---|
897 | bb = b * gen(k); |
---|
898 | nf = bb; |
---|
899 | |
---|
900 | if( size(#) > 0 ) |
---|
901 | { |
---|
902 | if( typeof(#[1]) == "module" ) |
---|
903 | { |
---|
904 | nf = NF(bb, #[1]); |
---|
905 | // "NF: ", nf; |
---|
906 | } |
---|
907 | } |
---|
908 | |
---|
909 | // while the complement (the fraction) is not reducible by leading syzygies |
---|
910 | if( nf != 0 ) |
---|
911 | { |
---|
912 | /// TODO: save shortcut LM(m) * T[i] -> ? |
---|
913 | |
---|
914 | // choose ANY such reduction... (with the biggest index?) |
---|
915 | s = bb + SSTraverseTail(b, T[k], L, T, #); |
---|
916 | break; |
---|
917 | } |
---|
918 | } |
---|
919 | } |
---|
920 | } |
---|
921 | if( @DEBUG ) |
---|
922 | { |
---|
923 | "SSReduceTerm::Output: ", s; |
---|
924 | } |
---|
925 | return (s); |
---|
926 | } |
---|
927 | |
---|
928 | // TODO: store m * @tail -.-^-.-^-.--> ? |
---|
929 | proc SSTraverseTail(poly m, def @tail, def L, def T, list #) |
---|
930 | { |
---|
931 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
932 | { |
---|
933 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
934 | } else |
---|
935 | { |
---|
936 | int @DEBUG = !system("with", "ndebug"); |
---|
937 | } |
---|
938 | |
---|
939 | if( @DEBUG ) |
---|
940 | { |
---|
941 | "SSTraverse::Input: "; |
---|
942 | |
---|
943 | "mult: ", m; |
---|
944 | "tail: ", @tail; // T[i]; |
---|
945 | |
---|
946 | if( size(#) > 0 ) |
---|
947 | { |
---|
948 | "LSyz: "; #[1]; |
---|
949 | } |
---|
950 | } |
---|
951 | |
---|
952 | vector s = 0; |
---|
953 | |
---|
954 | def @l; |
---|
955 | |
---|
956 | // iterate tail-terms in ANY order! |
---|
957 | while( size(@tail) > 0 ) |
---|
958 | { |
---|
959 | @l = lead(@tail); |
---|
960 | s = s + SSReduceTerm(m, @l, L, T, #); |
---|
961 | @tail = @tail - @l; |
---|
962 | } |
---|
963 | |
---|
964 | if( @DEBUG ) |
---|
965 | { |
---|
966 | "SSTraverseTail::Output: ", s; |
---|
967 | } |
---|
968 | return (s); |
---|
969 | } |
---|
970 | |
---|
971 | // -------------------------------------------------------- // |
---|
972 | |
---|
973 | // module (N, LL, TT) = SSComputeSyzygy(L, T); |
---|
974 | // Compute Syz(L ++ T) = N = LL ++ TT |
---|
975 | proc SSComputeSyzygy(def L, def T) |
---|
976 | { |
---|
977 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
978 | { |
---|
979 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
980 | } else |
---|
981 | { |
---|
982 | int @DEBUG = !system("with", "ndebug"); |
---|
983 | } |
---|
984 | |
---|
985 | if( @DEBUG ) |
---|
986 | { |
---|
987 | "SSComputeSyzygy::Input"; |
---|
988 | "basering: ", basering; attrib(basering); |
---|
989 | // DetailedPrint(basering); |
---|
990 | |
---|
991 | // "iCompShift: ", iCompShift; |
---|
992 | |
---|
993 | "L: "; L; |
---|
994 | "T: "; T; |
---|
995 | } |
---|
996 | |
---|
997 | def a; bigint c; int r, k; poly aa; |
---|
998 | |
---|
999 | int @LEAD2SYZ = 0; |
---|
1000 | if( typeof( attrib(basering, "LEAD2SYZ") ) == "int" ) |
---|
1001 | { |
---|
1002 | @LEAD2SYZ = attrib(basering, "LEAD2SYZ"); |
---|
1003 | } |
---|
1004 | |
---|
1005 | int @TAILREDSYZ = 1; |
---|
1006 | if( typeof( attrib(basering, "TAILREDSYZ") ) == "int" ) |
---|
1007 | { |
---|
1008 | @TAILREDSYZ = attrib(basering, "TAILREDSYZ"); |
---|
1009 | // @TAILREDSYZ; |
---|
1010 | } |
---|
1011 | |
---|
1012 | /// Get the critical leading syzygy terms |
---|
1013 | if( @LEAD2SYZ ) // & 2nd syz. term |
---|
1014 | { |
---|
1015 | def a2; int r2; poly aa2; |
---|
1016 | module LL, LL2; |
---|
1017 | (LL, LL2) = SSCompute2LeadingSyzygyTerms(L, @TAILREDSYZ); // ++ |
---|
1018 | } else |
---|
1019 | { |
---|
1020 | module LL = SSComputeLeadingSyzygyTerms(L); |
---|
1021 | } |
---|
1022 | |
---|
1023 | module TT, SYZ; |
---|
1024 | |
---|
1025 | if( size(LL) > 0 ) |
---|
1026 | { |
---|
1027 | list LS; |
---|
1028 | |
---|
1029 | if( @TAILREDSYZ ) |
---|
1030 | { |
---|
1031 | LS = list(LL); |
---|
1032 | } |
---|
1033 | |
---|
1034 | vector @tail; |
---|
1035 | |
---|
1036 | for(k = ncols(LL); k > 0; k-- ) |
---|
1037 | { |
---|
1038 | // leading syz. term: |
---|
1039 | a = LL[k]; c = leadcomp(a); r = int(c); aa = leadmonomial(a); |
---|
1040 | // "A: ", a, " --->>>> ", aa, " **** [", r, "]: "; |
---|
1041 | |
---|
1042 | /// TODO: save shortcut (aa) * T[r] -> ? |
---|
1043 | @tail = SSTraverseTail(aa, T[r], L, T, LS); |
---|
1044 | |
---|
1045 | // get the 2nd syzygy term... |
---|
1046 | |
---|
1047 | if( @LEAD2SYZ ) // with the 2nd syz. term: |
---|
1048 | { |
---|
1049 | a2 = LL2[k]; c = leadcomp(a2); r2 = int(c); aa2 = leadmonomial(a2); |
---|
1050 | @tail = @tail + |
---|
1051 | /// TODO: save shortcut (aa2) * T[r2] -> ? |
---|
1052 | a2 + SSTraverseTail(aa2, T[r2], L, T, LS); |
---|
1053 | } else |
---|
1054 | { |
---|
1055 | @tail = @tail + SSReduceTerm(aa, L[r], L, T, LS); |
---|
1056 | } |
---|
1057 | |
---|
1058 | |
---|
1059 | TT[k] = @tail; |
---|
1060 | SYZ[k] = a + @tail; |
---|
1061 | } |
---|
1062 | } |
---|
1063 | |
---|
1064 | /* |
---|
1065 | def opts = option(get); option(redSB); option(redTail); |
---|
1066 | module SYZ = std(syz(M)); |
---|
1067 | option(set, opts); kill opts; |
---|
1068 | |
---|
1069 | module LL = lead(SYZ); // TODO: WRONG ORDERING!!!!!!!! |
---|
1070 | module TT = Tail(SYZ); |
---|
1071 | */ |
---|
1072 | |
---|
1073 | if( @DEBUG ) |
---|
1074 | { |
---|
1075 | "SSComputeSyzygy::Output"; |
---|
1076 | |
---|
1077 | "SYZ: "; SYZ; |
---|
1078 | "LL: "; LL; |
---|
1079 | "TT: "; TT; |
---|
1080 | } |
---|
1081 | |
---|
1082 | return (SYZ, LL, TT); |
---|
1083 | } |
---|
1084 | |
---|
1085 | // resolution/syzygy step: |
---|
1086 | static proc SSstep() |
---|
1087 | { |
---|
1088 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
1089 | { |
---|
1090 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
1091 | } else |
---|
1092 | { |
---|
1093 | int @DEBUG = !system("with", "ndebug"); |
---|
1094 | } |
---|
1095 | |
---|
1096 | |
---|
1097 | if( typeof( attrib(basering, "SYZCHECK") ) == "int" ) |
---|
1098 | { |
---|
1099 | int @SYZCHECK = attrib(basering, "SYZCHECK"); |
---|
1100 | } else |
---|
1101 | { |
---|
1102 | int @SYZCHECK = @DEBUG; |
---|
1103 | } |
---|
1104 | |
---|
1105 | if( @DEBUG ) |
---|
1106 | { |
---|
1107 | "SSstep::NextInducedRing"; |
---|
1108 | "basering: ", basering; attrib(basering); |
---|
1109 | } |
---|
1110 | |
---|
1111 | /* |
---|
1112 | // is initial weights are all zeroes! |
---|
1113 | def L = lead(M); |
---|
1114 | intvec @V = deg(M[1..ncols(M)]); @W; @V; @W = @V; attrib(L, "isHomog", @W); |
---|
1115 | SetInducedReferrence(L, @RANK, 0); |
---|
1116 | */ |
---|
1117 | |
---|
1118 | // def L = lead(MRES); |
---|
1119 | // @W = @W, @V; |
---|
1120 | // attrib(L, "isHomog", @W); |
---|
1121 | |
---|
1122 | |
---|
1123 | // General setting: |
---|
1124 | // SetInducedReferrence(MRES, 0, 0); // limit: 0! |
---|
1125 | int @l = size(RES); |
---|
1126 | |
---|
1127 | def M = RES[@l]; |
---|
1128 | |
---|
1129 | def L = LRES[@l]; |
---|
1130 | def T = TRES[@l]; |
---|
1131 | |
---|
1132 | |
---|
1133 | //// TODO: wrong !!!!! |
---|
1134 | int @RANK = ncols(MRES) - ncols(M); // nrows(M); // what if M is zero?! |
---|
1135 | |
---|
1136 | |
---|
1137 | |
---|
1138 | /* |
---|
1139 | if( @RANK != nrows(M) ) |
---|
1140 | { |
---|
1141 | type(MRES); |
---|
1142 | @RANK; |
---|
1143 | type(M); |
---|
1144 | pause(); |
---|
1145 | } |
---|
1146 | */ |
---|
1147 | |
---|
1148 | intvec @W = attrib(M, "isHomog"); intvec @V = attrib(M, "degrees"); @V = @W, @V; |
---|
1149 | |
---|
1150 | if( @DEBUG ) |
---|
1151 | { |
---|
1152 | "Sstep::NextInput: "; |
---|
1153 | M; |
---|
1154 | L; |
---|
1155 | @V; |
---|
1156 | @RANK; |
---|
1157 | // DetailedPrint(MRES); |
---|
1158 | attrib(MRES, "isHomog"); |
---|
1159 | } |
---|
1160 | |
---|
1161 | |
---|
1162 | // TODO: N = SYZ( M )!!! |
---|
1163 | module N, LL, TT; |
---|
1164 | (N, LL, TT) = SSComputeSyzygy(/*M, */L, T/*, @RANK*/); |
---|
1165 | |
---|
1166 | // shift syz.comp by @RANK: |
---|
1167 | module Z; |
---|
1168 | Z = 0; Z[@RANK] = 0; Z = Z, transpose(LL); LL = transpose(Z); |
---|
1169 | Z = 0; Z[@RANK] = 0; Z = Z, transpose(TT); TT = transpose(Z); |
---|
1170 | Z = 0; Z[@RANK] = 0; Z = Z, transpose(N); N = transpose(Z); |
---|
1171 | |
---|
1172 | |
---|
1173 | if( @SYZCHECK ) |
---|
1174 | { |
---|
1175 | if( size(N) > 0 ) |
---|
1176 | { |
---|
1177 | // next syz. property |
---|
1178 | if( size(module(transpose( transpose(N) * transpose(MRES) ))) > 0 ) |
---|
1179 | { |
---|
1180 | "MRES", MRES; |
---|
1181 | |
---|
1182 | "N: "; N; // DetailedPrint(N, 2); |
---|
1183 | |
---|
1184 | "LL:"; LL; // DetailedPrint(LL, 1); |
---|
1185 | "TT:"; TT; // DetailedPrint(TT, 10); |
---|
1186 | |
---|
1187 | "RANKS: ", @RANK; |
---|
1188 | |
---|
1189 | "transpose( transpose(N) * transpose(MRES) ) != 0!!!"; |
---|
1190 | transpose( transpose(N) * transpose(MRES) ); |
---|
1191 | |
---|
1192 | "transpose(N) * transpose(MRES): "; |
---|
1193 | transpose(N) * transpose(MRES); |
---|
1194 | // DetailedPrint(module(_), 2); |
---|
1195 | $ |
---|
1196 | } |
---|
1197 | } |
---|
1198 | } |
---|
1199 | |
---|
1200 | attrib(N, "isHomog", @V); |
---|
1201 | |
---|
1202 | // TODO: correct the following: |
---|
1203 | intvec @DEGS = deg(N[1..ncols(N)]); // no mod. comp. weights :( |
---|
1204 | |
---|
1205 | |
---|
1206 | attrib(N, "degrees", @DEGS); |
---|
1207 | |
---|
1208 | RES[@l + 1] = N; // list of all syzygy modules |
---|
1209 | LRES[@l + 1] = LL; // list of all syzygy modules |
---|
1210 | TRES[@l + 1] = TT; // list of all syzygy modules |
---|
1211 | |
---|
1212 | MRES = MRES, N; |
---|
1213 | |
---|
1214 | attrib(MRES, "isHomog", @V); |
---|
1215 | |
---|
1216 | // L = L, lead(N); attrib(basering, "InducionLeads", L); |
---|
1217 | |
---|
1218 | if( @DEBUG ) |
---|
1219 | { |
---|
1220 | "SSstep::NextSyzOutput: "; |
---|
1221 | N; |
---|
1222 | // DetailedPrint(N); |
---|
1223 | attrib(N); |
---|
1224 | } |
---|
1225 | |
---|
1226 | } |
---|
1227 | |
---|
1228 | proc SScontinue(int l) |
---|
1229 | "USAGE: SScontinue(l) |
---|
1230 | RETURN: nothing, instead it changes RES and MRES variables in the current ring |
---|
1231 | PURPOSE: computes further (at most l) syzygies |
---|
1232 | NOTE: must be used within a ring returned by Sres or Ssyz. RES and MRES are |
---|
1233 | explained in Sres |
---|
1234 | EXAMPLE: example Scontinue; shows an example |
---|
1235 | " |
---|
1236 | { |
---|
1237 | |
---|
1238 | /// TODO! |
---|
1239 | // def data = GetInducedData(); |
---|
1240 | |
---|
1241 | if( (!defined(RES)) || (!defined(MRES)) ) /* || (typeof(data) != "list") || (size(data) != 2) */ |
---|
1242 | { |
---|
1243 | ERROR("Sorry, but basering does not seem to be returned by Sres or Ssyz"); |
---|
1244 | } |
---|
1245 | for (; (l != 0) && (size(RES[size(RES)]) > 0); l-- ) |
---|
1246 | { |
---|
1247 | SSstep(); |
---|
1248 | } |
---|
1249 | } |
---|
1250 | example |
---|
1251 | { "EXAMPLE:"; echo = 2; |
---|
1252 | ring r; |
---|
1253 | module M = maxideal(1); M; |
---|
1254 | def S = SSsyz(M); setring S; S; |
---|
1255 | "Only the first syzygy: "; |
---|
1256 | RES; MRES; |
---|
1257 | "More syzygies: "; |
---|
1258 | SScontinue(10); |
---|
1259 | RES; MRES; |
---|
1260 | } |
---|
1261 | |
---|
1262 | proc SSsyz(def M) |
---|
1263 | "USAGE: SSsyz(M) |
---|
1264 | RETURN: ring, containing a list of modules RES and a module MRES |
---|
1265 | PURPOSE: computes the first syzygy module of M (wrt some Schreyer ordering)? |
---|
1266 | NOTE: The output is explained in Sres |
---|
1267 | EXAMPLE: example Ssyz; shows an example |
---|
1268 | " |
---|
1269 | { |
---|
1270 | if( (typeof(M) != "module") && (typeof(M) != "ideal") ) |
---|
1271 | { |
---|
1272 | ERROR("Sorry: need an ideal or a module for input"); |
---|
1273 | } |
---|
1274 | |
---|
1275 | def SS = SSinit(M); setring SS; |
---|
1276 | |
---|
1277 | SSstep(); // NOTE: what if M is zero? |
---|
1278 | |
---|
1279 | return (SS); |
---|
1280 | } |
---|
1281 | example |
---|
1282 | { "EXAMPLE:"; echo = 2; |
---|
1283 | ring r; |
---|
1284 | |
---|
1285 | /* ideal M = 0; |
---|
1286 | def S = SSsyz(M); setring S; S; |
---|
1287 | "Only the first syzygy: "; |
---|
1288 | RES; LRES; TRES; |
---|
1289 | MRES; |
---|
1290 | |
---|
1291 | kill S; setring r; kill M; |
---|
1292 | */ |
---|
1293 | |
---|
1294 | module M = maxideal(1); M; |
---|
1295 | def S = SSres(M, 0); setring S; S; |
---|
1296 | MRES; |
---|
1297 | RES; |
---|
1298 | ""; |
---|
1299 | LRES; |
---|
1300 | ""; |
---|
1301 | TRES; |
---|
1302 | |
---|
1303 | kill S; setring r; kill M; |
---|
1304 | |
---|
1305 | kill r; |
---|
1306 | |
---|
1307 | ring R = 0, (w, x, y, z), dp; |
---|
1308 | ideal M = w^2 - x*z, w*x - y*z, x^2 - w*y, x*y - z^2, y^2 - w*z; |
---|
1309 | |
---|
1310 | def S = SSres(M, 0); setring S; S; |
---|
1311 | MRES; |
---|
1312 | RES; |
---|
1313 | ""; |
---|
1314 | LRES; |
---|
1315 | ""; |
---|
1316 | TRES; |
---|
1317 | } |
---|
1318 | |
---|
1319 | proc SSres(def M, int l) |
---|
1320 | "USAGE: SSres(I, l) |
---|
1321 | RETURN: ring, containing a list of modules RES and a module MRES |
---|
1322 | PURPOSE: computes (at most l) syzygy modules of M wrt the classical Schreyer |
---|
1323 | induced ordering with gen(i) > gen(j) if i > j, provided both gens |
---|
1324 | are from the same syzygy level.??? |
---|
1325 | NOTE: RES contains the images of maps subsituting the beginning of the |
---|
1326 | Schreyer free resolution of baseRing^r/M, while MRES is a sum of |
---|
1327 | these images in a big free sum, containing all the syzygy modules. |
---|
1328 | The syzygy modules are shifted so that gen(i) correspons to MRES[i]. |
---|
1329 | The leading zero module RES[0] indicates the fact that coker of the |
---|
1330 | first map is zero. The number of zeroes inducates the rank of input. |
---|
1331 | NOTE: If l == 0 then l is set to be nvars(basering) + 1 |
---|
1332 | EXAMPLE: example SSres; shows an example |
---|
1333 | " |
---|
1334 | { |
---|
1335 | if( (typeof(M) != "module") && (typeof(M) != "ideal") ) |
---|
1336 | { |
---|
1337 | ERROR("Sorry: need an ideal or a module for input"); |
---|
1338 | } |
---|
1339 | |
---|
1340 | def SS = SSinit(M); setring SS; |
---|
1341 | |
---|
1342 | if (l == 0) |
---|
1343 | { |
---|
1344 | l = nvars(basering) + 1; // not really an estimate...?! |
---|
1345 | } |
---|
1346 | |
---|
1347 | SSstep(); l = l - 1; |
---|
1348 | |
---|
1349 | SScontinue(l); |
---|
1350 | |
---|
1351 | return (SS); |
---|
1352 | } |
---|
1353 | example |
---|
1354 | { "EXAMPLE:"; echo = 2; |
---|
1355 | ring r; |
---|
1356 | module M = maxideal(1); M; |
---|
1357 | def S = SSres(M, 0); setring S; S; |
---|
1358 | RES; |
---|
1359 | MRES; |
---|
1360 | kill S; |
---|
1361 | setring r; kill M; |
---|
1362 | |
---|
1363 | def A = nc_algebra(-1,0); setring A; |
---|
1364 | ideal Q = var(1)^2, var(2)^2, var(3)^2; |
---|
1365 | qring SCA = twostd(Q); |
---|
1366 | basering; |
---|
1367 | |
---|
1368 | module M = maxideal(1); |
---|
1369 | def S = SSres(M, 2); setring S; S; |
---|
1370 | RES; |
---|
1371 | MRES; |
---|
1372 | } |
---|
1373 | |
---|
1374 | |
---|
1375 | |
---|
1376 | static proc loadme() |
---|
1377 | { |
---|
1378 | int @DEBUG = !system("with", "ndebug"); |
---|
1379 | |
---|
1380 | if( @DEBUG ) |
---|
1381 | { |
---|
1382 | |
---|
1383 | "ndebug?: ", system("with", "ndebug"); |
---|
1384 | "om_ndebug?: ", system("with", "om_ndebug"); |
---|
1385 | |
---|
1386 | listvar(Top); |
---|
1387 | listvar(Schreyer); |
---|
1388 | } |
---|
1389 | // listvar(Syzextra); listvar(Syzextra_g); |
---|
1390 | |
---|
1391 | if( !defined(DetailedPrint) ) |
---|
1392 | { |
---|
1393 | if( !@DEBUG ) |
---|
1394 | { |
---|
1395 | |
---|
1396 | if( @DEBUG ) |
---|
1397 | { |
---|
1398 | "Loading the Release version!"; |
---|
1399 | } |
---|
1400 | load("syzextra.so"); |
---|
1401 | |
---|
1402 | if( @DEBUG ) |
---|
1403 | { |
---|
1404 | listvar(Syzextra); |
---|
1405 | } |
---|
1406 | |
---|
1407 | exportto(Top, Syzextra::ClearContent); |
---|
1408 | exportto(Top, Syzextra::ClearDenominators); |
---|
1409 | |
---|
1410 | // export Syzextra; |
---|
1411 | |
---|
1412 | // exportto(Schreyer, Syzextra::noop); |
---|
1413 | exportto(Schreyer, Syzextra::DetailedPrint); |
---|
1414 | exportto(Schreyer, Syzextra::leadmonomial); |
---|
1415 | exportto(Schreyer, Syzextra::leadcomp); |
---|
1416 | // exportto(Schreyer, Syzextra::leadrawexp); |
---|
1417 | // exportto(Schreyer, Syzextra::ISUpdateComponents); |
---|
1418 | exportto(Schreyer, Syzextra::SetInducedReferrence); |
---|
1419 | exportto(Schreyer, Syzextra::GetInducedData); |
---|
1420 | // exportto(Schreyer, Syzextra::GetAMData); |
---|
1421 | // exportto(Schreyer, Syzextra::SetSyzComp); |
---|
1422 | exportto(Schreyer, Syzextra::MakeInducedSchreyerOrdering); |
---|
1423 | // exportto(Schreyer, Syzextra::MakeSyzCompOrdering); |
---|
1424 | exportto(Schreyer, Syzextra::idPrepare); |
---|
1425 | // exportto(Schreyer, Syzextra::reduce_syz); |
---|
1426 | // exportto(Schreyer, Syzextra::p_Content); |
---|
1427 | |
---|
1428 | exportto(Schreyer, Syzextra::Tail); |
---|
1429 | } |
---|
1430 | else |
---|
1431 | { |
---|
1432 | if( @DEBUG ) |
---|
1433 | { |
---|
1434 | "Loading the Debug version!"; |
---|
1435 | } |
---|
1436 | |
---|
1437 | load("syzextra_g.so"); |
---|
1438 | |
---|
1439 | if( @DEBUG ) |
---|
1440 | { |
---|
1441 | listvar(Syzextra_g); |
---|
1442 | } |
---|
1443 | |
---|
1444 | exportto(Top, Syzextra_g::ClearContent); |
---|
1445 | exportto(Top, Syzextra_g::ClearDenominators); |
---|
1446 | |
---|
1447 | // export Syzextra_g; |
---|
1448 | // exportto(Schreyer, Syzextra_g::noop); |
---|
1449 | exportto(Schreyer, Syzextra_g::DetailedPrint); |
---|
1450 | exportto(Schreyer, Syzextra_g::leadmonomial); |
---|
1451 | exportto(Schreyer, Syzextra_g::leadcomp); |
---|
1452 | // exportto(Schreyer, Syzextra_g::leadrawexp); |
---|
1453 | // exportto(Schreyer, Syzextra_g::ISUpdateComponents); |
---|
1454 | exportto(Schreyer, Syzextra_g::SetInducedReferrence); |
---|
1455 | exportto(Schreyer, Syzextra_g::GetInducedData); |
---|
1456 | // exportto(Schreyer, Syzextra_g::GetAMData); |
---|
1457 | // exportto(Schreyer, Syzextra_g::SetSyzComp); |
---|
1458 | exportto(Schreyer, Syzextra_g::MakeInducedSchreyerOrdering); |
---|
1459 | // exportto(Schreyer, Syzextra_g::MakeSyzCompOrdering); |
---|
1460 | exportto(Schreyer, Syzextra_g::idPrepare); |
---|
1461 | // exportto(Schreyer, Syzextra_g::reduce_syz); |
---|
1462 | // exportto(Schreyer, Syzextra_g::p_Content); |
---|
1463 | |
---|
1464 | exportto(Schreyer, Syzextra_g::Tail); |
---|
1465 | |
---|
1466 | } |
---|
1467 | |
---|
1468 | exportto(Top, DetailedPrint); |
---|
1469 | exportto(Top, GetInducedData); |
---|
1470 | |
---|
1471 | if( @DEBUG ) |
---|
1472 | { |
---|
1473 | listvar(Top); |
---|
1474 | listvar(Schreyer); |
---|
1475 | } |
---|
1476 | } |
---|
1477 | |
---|
1478 | if( !defined(GetInducedData) ) |
---|
1479 | { |
---|
1480 | ERROR("Sorry but we are missing the dynamic module (syzextra(_g)?.so)..."); |
---|
1481 | } |
---|
1482 | |
---|
1483 | } |
---|
1484 | |
---|
1485 | static proc mod_init() |
---|
1486 | { |
---|
1487 | loadme(); |
---|
1488 | } |
---|
1489 | |
---|
1490 | |
---|
1491 | proc testallSexamples() |
---|
1492 | { |
---|
1493 | example Ssyz; |
---|
1494 | example Scontinue; |
---|
1495 | example Sres; |
---|
1496 | } |
---|
1497 | |
---|
1498 | proc testallSSexamples() |
---|
1499 | { |
---|
1500 | example SSsyz; |
---|
1501 | example SScontinue; |
---|
1502 | example SSres; |
---|
1503 | } |
---|
1504 | |
---|
1505 | example |
---|
1506 | { "EXAMPLE:"; echo = 2; |
---|
1507 | testallSexamples(); |
---|
1508 | testallSSexamples(); |
---|
1509 | } |
---|