1 | /////////////////////////////////////////////////////////////////////////// |
---|
2 | version="version schreyer.lib 4.0.0.0 Jun_2013 "; // $Id$ |
---|
3 | category="General purpose"; |
---|
4 | info=" |
---|
5 | LIBRARY: schreyer.lib Helpers for computing a Schreyer resolution in @code{derham.lib} |
---|
6 | AUTHOR: Oleksandr Motsak <U@D>, where U={motsak}, D={mathematik.uni-kl.de} |
---|
7 | KEYWORDS: Schreyer ordering; Schreyer resolution; syzygy |
---|
8 | OVERVIEW: |
---|
9 | @* The library contains helper procedures for computing a Schreyer resoltion (cf. [SFO]), |
---|
10 | originally meant to be used by @code{derham.lib} (which requires resolutions over the homogenized Weyl algebra). |
---|
11 | The library works both in the commutative and non-commutative setting (cf. [MO]). |
---|
12 | Here, we call a free resolution a Schreyer resolution if each syzygy module is given by a Groebner basis |
---|
13 | with respect to the corresponding Schreyer ordering. |
---|
14 | A Schreyer resolution can be much bigger than a minimal resolution of the same module, but may be easier to construct. |
---|
15 | @* The input for the resolution computations is a set of vectors @code{M} in form of a module over some basering @code{R}. |
---|
16 | The ring @code{R} may be non-commutative, in which case the ring ordering should be global. |
---|
17 | @* These procedures produce/work with partial Schreyer resolutions of @code{(R^rank(M))/M} in form of |
---|
18 | a ring (endowed with a special ring ordering that will be extended in the course of a resolution computation) |
---|
19 | containing a list of modules @code{RES} and a module @code{MRES}: |
---|
20 | @* The list of modules @code{RES} contains the images of maps (also called syzygy modules) substituting the |
---|
21 | computed beginning of a Schreyer resolution, that is, each syzygy module is given by a Groebner basis |
---|
22 | with respect to the corresponding Schreyer ordering. |
---|
23 | @* The list @code{RES} starts with a zero map given by @code{rank(M)} zero generators indicating that the image of |
---|
24 | the first differential map is zero. The second map @code{RES[2]} is given by @code{M}, which indicates that |
---|
25 | the resolution of @code{(R^rank(M))/M} is being computed. |
---|
26 | @* The module @code{MRES} is a direct sum of modules from @code{RES} and thus comprises all computed differentials. |
---|
27 | @* Syzygies are shifted so that @code{gen(i)} is mapped to @code{MRES[i]} under the differential map. |
---|
28 | @* The Schreyer ordering succesively extends the starting module ordering on @code{M} (defined in Singular by the basering @code{R}) |
---|
29 | and is extended to higher syzygies using the following definition: |
---|
30 | @* a < b if and only if (d(a) < d(b)) OR ( (d(a) = d(b) AND (comp(a) < comp(b)) ), |
---|
31 | @* where @code{d(a)} is the image of a under the differential (given by @code{MRES}), |
---|
32 | and @code{comp(a)} is the module component, for any module terms @code{a} and @code{b} from the same higher syzygy module. |
---|
33 | REFERENCES: |
---|
34 | [SFO] Schreyer, F.O.: Die Berechnung von Syzygien mit dem verallgemeinerten Weierstrassschen Divisionssatz, |
---|
35 | Master's thesis, Univ. Hamburg, 1980. |
---|
36 | [MO] Motsak, O.: Non-commutative Computer Algebra with applications: Graded commutative algebra and related |
---|
37 | structures in Singular with applications, Ph.D. thesis, TU Kaiserslautern, 2010 |
---|
38 | |
---|
39 | NOTE: requires the dynamic or built-in module @code{syzextra} |
---|
40 | |
---|
41 | PROCEDURES: |
---|
42 | Sres(M,len) compute Schreyer resolution of module M of maximal length len |
---|
43 | Ssyz(M) compute Schreyer resolution of module M of length 1 |
---|
44 | Scontinue(len) extend currently active resolution by (at most) len syszygies |
---|
45 | "; |
---|
46 | |
---|
47 | static proc prepareSyz( module I, list # ) |
---|
48 | { |
---|
49 | int i; |
---|
50 | int k = 0; |
---|
51 | int r = nrows(I); |
---|
52 | int c = ncols(I); |
---|
53 | |
---|
54 | |
---|
55 | if( size(#) > 0 ) |
---|
56 | { |
---|
57 | if( typeof(#[1]) == "int" || typeof(#[1]) == "bigint" ) |
---|
58 | { |
---|
59 | k = #[1]; |
---|
60 | } |
---|
61 | } |
---|
62 | |
---|
63 | if( k < r ) |
---|
64 | { |
---|
65 | "// *** Wrong k: ", k, " < nrows: ", r, " => setting k = r = ", r; |
---|
66 | k = r; |
---|
67 | } |
---|
68 | |
---|
69 | // "k: ", k; "c: ", c; "I: ", I; |
---|
70 | |
---|
71 | for( i = c; i > 0; i-- ) |
---|
72 | { |
---|
73 | I[i] = I[i] + gen(k + i); |
---|
74 | } |
---|
75 | |
---|
76 | // DetailedPrint(I); |
---|
77 | |
---|
78 | return(I); |
---|
79 | } |
---|
80 | |
---|
81 | static proc separateSyzGB( module J, int c ) |
---|
82 | { |
---|
83 | module II, G; vector v; int i; |
---|
84 | |
---|
85 | J = simplify(J, 2); |
---|
86 | |
---|
87 | for( i = ncols(J); i > 0; i-- ) |
---|
88 | { |
---|
89 | v = J[i]; |
---|
90 | if( leadcomp(v) > c ) |
---|
91 | { |
---|
92 | II[i] = v; |
---|
93 | } else |
---|
94 | { |
---|
95 | G[i] = v; // leave only gen(i): i <= c |
---|
96 | } |
---|
97 | } |
---|
98 | |
---|
99 | II = simplify(II, 2); |
---|
100 | G = simplify(G, 2); |
---|
101 | |
---|
102 | return (list(G, II)); |
---|
103 | } |
---|
104 | |
---|
105 | static proc splitSyzGB( module J, int c ) |
---|
106 | { |
---|
107 | module JJ; vector v, vv; int i; |
---|
108 | |
---|
109 | for( i = ncols(J); i > 0; i-- ) |
---|
110 | { |
---|
111 | v = J[i]; |
---|
112 | |
---|
113 | vv = 0; |
---|
114 | |
---|
115 | while( leadcomp(v) <= c ) |
---|
116 | { |
---|
117 | vv = vv + lead(v); |
---|
118 | v = v - lead(v); |
---|
119 | } |
---|
120 | |
---|
121 | J[i] = vv; |
---|
122 | JJ[i] = v; |
---|
123 | } |
---|
124 | |
---|
125 | J = simplify(J, 2); |
---|
126 | JJ = simplify(JJ, 2); |
---|
127 | |
---|
128 | return (list(J, JJ)); |
---|
129 | } |
---|
130 | |
---|
131 | |
---|
132 | static proc Sinit(module M) |
---|
133 | { |
---|
134 | def @save = basering; |
---|
135 | |
---|
136 | int @DEBUG = !system("with", "ndebug"); |
---|
137 | if( @DEBUG ) |
---|
138 | { |
---|
139 | "Sinit::Input"; |
---|
140 | type(M); |
---|
141 | DetailedPrint(M); |
---|
142 | attrib(M); |
---|
143 | } |
---|
144 | |
---|
145 | int @RANK = nrows(M); int @SIZE = ncols(M); |
---|
146 | |
---|
147 | int @IS_A_SB = attrib(M, "isSB"); // ??? only if all weights were zero?! |
---|
148 | |
---|
149 | if( !@IS_A_SB ) |
---|
150 | { |
---|
151 | M = std(M); // this should be faster than computing std in S (later on) |
---|
152 | } |
---|
153 | |
---|
154 | def S = MakeInducedSchreyerOrdering(1); // 1 puts history terms to the back |
---|
155 | // TODO: NOTE: +1 causes trouble to Singular interpreter!!!??? |
---|
156 | setring S; // a new ring with a Schreyer ordering |
---|
157 | |
---|
158 | if( @DEBUG ) |
---|
159 | { |
---|
160 | "Sinit::StartingISRing"; |
---|
161 | basering; |
---|
162 | // DetailedPrint(basering); |
---|
163 | } |
---|
164 | |
---|
165 | // Setup the leading syzygy^{-1} module to zero: |
---|
166 | module Z = 0; Z[@RANK] = 0; attrib(Z, "isHomog", intvec(0)); |
---|
167 | |
---|
168 | module MRES = Z; |
---|
169 | |
---|
170 | list RES; RES[1] = Z; |
---|
171 | |
---|
172 | module F = freemodule(@RANK); |
---|
173 | intvec @V = deg(F[1..@RANK]); |
---|
174 | |
---|
175 | module M = imap(@save, M); |
---|
176 | |
---|
177 | attrib(M, "isHomog", @V); |
---|
178 | attrib(M, "isSB", 1); |
---|
179 | |
---|
180 | |
---|
181 | if( @DEBUG ) |
---|
182 | { |
---|
183 | "Sinit::SB_Input: "; |
---|
184 | type(M); |
---|
185 | attrib(M); |
---|
186 | attrib(M, "isHomog"); |
---|
187 | DetailedPrint(M); |
---|
188 | } |
---|
189 | |
---|
190 | if( @DEBUG ) |
---|
191 | { |
---|
192 | // 0^th syz. property |
---|
193 | if( size(module(transpose( transpose(M) * transpose(MRES) ))) > 0 ) |
---|
194 | { |
---|
195 | transpose( transpose(M) * transpose(MRES) ); |
---|
196 | "transpose( transpose(M) * transpose(MRES) ) != 0!!!"; |
---|
197 | $ |
---|
198 | } |
---|
199 | } |
---|
200 | |
---|
201 | RES[size(RES)+1] = M; // list of all syzygy modules |
---|
202 | MRES = MRES, M; |
---|
203 | |
---|
204 | attrib(MRES, "isHomog", @V); |
---|
205 | |
---|
206 | attrib(S, "InducionLeads", lead(M)); |
---|
207 | attrib(S, "InducionStart", @RANK); |
---|
208 | |
---|
209 | if( @DEBUG ) |
---|
210 | { |
---|
211 | "Sinit::MRES"; |
---|
212 | DetailedPrint(MRES); |
---|
213 | attrib(MRES, "isHomog"); |
---|
214 | attrib(S); |
---|
215 | } |
---|
216 | |
---|
217 | export RES; |
---|
218 | export MRES; |
---|
219 | return (S); |
---|
220 | } |
---|
221 | |
---|
222 | static proc Sstep() |
---|
223 | { |
---|
224 | int @DEBUG = !system("with", "ndebug"); |
---|
225 | |
---|
226 | if( @DEBUG ) |
---|
227 | { |
---|
228 | "Sstep::NextInducedRing"; |
---|
229 | DetailedPrint(basering); |
---|
230 | |
---|
231 | attrib(basering, "InducionLeads"); |
---|
232 | attrib(basering, "InducionStart"); |
---|
233 | |
---|
234 | GetInducedData(); |
---|
235 | } |
---|
236 | |
---|
237 | // syzygy step: |
---|
238 | |
---|
239 | /* |
---|
240 | // is initial weights are all zeroes! |
---|
241 | def L = lead(M); |
---|
242 | intvec @V = deg(M[1..ncols(M)]); @W; @V; @W = @V; attrib(L, "isHomog", @W); |
---|
243 | SetInducedReferrence(L, @RANK, 0); |
---|
244 | */ |
---|
245 | |
---|
246 | // def L = lead(MRES); |
---|
247 | // @W = @W, @V; |
---|
248 | // attrib(L, "isHomog", @W); |
---|
249 | |
---|
250 | |
---|
251 | // General setting: |
---|
252 | // SetInducedReferrence(MRES, 0, 0); // limit: 0! |
---|
253 | int @l = size(RES); |
---|
254 | |
---|
255 | module M = RES[@l]; |
---|
256 | |
---|
257 | module L = attrib(basering, "InducionLeads"); |
---|
258 | int limit = attrib(basering, "InducionStart"); |
---|
259 | |
---|
260 | // L; limit; |
---|
261 | |
---|
262 | int @RANK = ncols(MRES) - ncols(M); // nrows(M); // what if M is zero?! |
---|
263 | |
---|
264 | /* |
---|
265 | if( @RANK != nrows(M) ) |
---|
266 | { |
---|
267 | type(MRES); |
---|
268 | @RANK; |
---|
269 | type(M); |
---|
270 | pause(); |
---|
271 | } |
---|
272 | */ |
---|
273 | |
---|
274 | intvec @W = attrib(M, "isHomog"); |
---|
275 | intvec @V = deg(M[1..ncols(M)]); |
---|
276 | @V = @W, @V; |
---|
277 | |
---|
278 | if( @DEBUG ) |
---|
279 | { |
---|
280 | "Sstep::NextInput: "; |
---|
281 | M; |
---|
282 | deg(M[1..ncols(M)]); // no use of @W :(? |
---|
283 | @RANK; |
---|
284 | DetailedPrint(MRES); |
---|
285 | attrib(MRES, "isHomog"); @W; |
---|
286 | deg(MRES[1..ncols(MRES)]); |
---|
287 | } |
---|
288 | |
---|
289 | |
---|
290 | |
---|
291 | SetInducedReferrence(L, limit, 0); |
---|
292 | |
---|
293 | def K = prepareSyz(M, @RANK); |
---|
294 | // K; |
---|
295 | |
---|
296 | // attrib(K, "isHomog", @V); DetailedPrint(K, 1000); |
---|
297 | |
---|
298 | // pause(); |
---|
299 | |
---|
300 | K = idPrepare(K, @RANK); // std(K); // ? |
---|
301 | K = simplify(K, 2); |
---|
302 | |
---|
303 | // K; |
---|
304 | |
---|
305 | module N = separateSyzGB(K, @RANK)[2]; // 1^st syz. module: vectors which start in lower part (comp >= @RANK) |
---|
306 | |
---|
307 | // "N_0: "; N; DetailedPrint(N, 10); |
---|
308 | |
---|
309 | // basering; print(@V); type(N); |
---|
310 | // attrib(N, "isHomog", @V); // TODO: fix "wrong weights"!!!? deg is wrong :((( |
---|
311 | N = std(N); |
---|
312 | attrib(N, "isHomog", @V); |
---|
313 | |
---|
314 | // N; |
---|
315 | |
---|
316 | if( @DEBUG ) |
---|
317 | { |
---|
318 | if( size(N) > 0 ) |
---|
319 | { |
---|
320 | // next syz. property |
---|
321 | if( size(module(transpose( transpose(N) * transpose(MRES) ))) > 0 ) |
---|
322 | { |
---|
323 | MRES; |
---|
324 | |
---|
325 | "N: "; N; DetailedPrint(N, 10); |
---|
326 | |
---|
327 | "K:"; K; DetailedPrint(K, 10); |
---|
328 | |
---|
329 | "RANKS: ", @RANK; |
---|
330 | |
---|
331 | "transpose( transpose(N) * transpose(MRES) ) != 0!!!"; |
---|
332 | transpose( transpose(N) * transpose(MRES) ); |
---|
333 | |
---|
334 | "transpose(N) * transpose(MRES): "; |
---|
335 | transpose(N) * transpose(MRES); |
---|
336 | DetailedPrint(module(_), 2); |
---|
337 | $ |
---|
338 | } |
---|
339 | } |
---|
340 | } |
---|
341 | |
---|
342 | RES[@l + 1] = N; // list of all syzygy modules |
---|
343 | |
---|
344 | MRES = MRES, N; |
---|
345 | attrib(MRES, "isHomog", @V); |
---|
346 | |
---|
347 | |
---|
348 | L = L, lead(N); |
---|
349 | attrib(basering, "InducionLeads", L); |
---|
350 | |
---|
351 | if( @DEBUG ) |
---|
352 | { |
---|
353 | "Sstep::NextSyzOutput: "; |
---|
354 | DetailedPrint(N); |
---|
355 | attrib(N, "isHomog"); |
---|
356 | } |
---|
357 | |
---|
358 | } |
---|
359 | |
---|
360 | proc Scontinue(int l) |
---|
361 | "USAGE: Scontinue(int len) |
---|
362 | RETURN: nothing, instead it changes the currently active resolution |
---|
363 | PURPOSE: extends the currently active resolution by at most len syzygies |
---|
364 | ASSUME: must be used within a ring returned by Sres or Ssyz |
---|
365 | EXAMPLE: example Scontinue; shows an example |
---|
366 | " |
---|
367 | { |
---|
368 | def data = GetInducedData(); |
---|
369 | |
---|
370 | if( (!defined(RES)) || (!defined(MRES)) || (typeof(data) != "list") || (size(data) != 2) ) |
---|
371 | { |
---|
372 | ERROR("Sorry, but basering does not seem to be returned by Sres or Ssyz"); |
---|
373 | } |
---|
374 | for (; (l != 0) && (size(RES[size(RES)]) > 0); l-- ) |
---|
375 | { |
---|
376 | Sstep(); |
---|
377 | } |
---|
378 | } |
---|
379 | example |
---|
380 | { "EXAMPLE:"; echo = 2; |
---|
381 | ring r; |
---|
382 | module M = maxideal(1); M; |
---|
383 | def S = Ssyz(M); setring S; S; |
---|
384 | "Only the first syzygy: "; |
---|
385 | RES; MRES; |
---|
386 | "More syzygies: "; |
---|
387 | Scontinue(10); |
---|
388 | RES; MRES; |
---|
389 | } |
---|
390 | |
---|
391 | proc Ssyz(module M) |
---|
392 | "USAGE: Ssyz(module M) |
---|
393 | RETURN: ring, containing a Schreyer resolution |
---|
394 | PURPOSE: computes a Schreyer resolution of M of length 1 (see the library overview) |
---|
395 | SEE ALSO: Sres |
---|
396 | EXAMPLE: example Ssyz; shows an example |
---|
397 | " |
---|
398 | { |
---|
399 | def S = Sinit(M); setring S; |
---|
400 | |
---|
401 | Sstep(); // NOTE: what if M is zero? |
---|
402 | |
---|
403 | return (S); |
---|
404 | } |
---|
405 | example |
---|
406 | { "EXAMPLE:"; echo = 2; |
---|
407 | ring r; |
---|
408 | module M = maxideal(1); M; |
---|
409 | def S = Ssyz(M); setring S; S; |
---|
410 | "Only the first syzygy: "; |
---|
411 | RES; |
---|
412 | MRES; // Note gen(i) |
---|
413 | kill S; |
---|
414 | setring r; kill M; |
---|
415 | |
---|
416 | module M = 0; |
---|
417 | def S = Ssyz(M); setring S; S; |
---|
418 | "Only the first syzygy: "; |
---|
419 | RES; |
---|
420 | MRES; |
---|
421 | } |
---|
422 | |
---|
423 | proc Sres(module M, int l) |
---|
424 | "USAGE: Sres(module M, int len) |
---|
425 | RETURN: ring, containing a Schreyer resolution |
---|
426 | PURPOSE: computes a Schreyer resolution of M of length at most len (see the library overview) |
---|
427 | NOTE: If given len is zero then nvars(basering) + 1 is used instead. |
---|
428 | SEE ALSO: Ssyz |
---|
429 | EXAMPLE: example Sres; shows an example |
---|
430 | " |
---|
431 | { |
---|
432 | def S = Sinit(M); setring S; |
---|
433 | |
---|
434 | if (l == 0) |
---|
435 | { |
---|
436 | l = nvars(basering) + 1; // not really an estimate...?! |
---|
437 | } |
---|
438 | |
---|
439 | Sstep(); l = l - 1; |
---|
440 | |
---|
441 | Scontinue(l); |
---|
442 | |
---|
443 | return (S); |
---|
444 | } |
---|
445 | example |
---|
446 | { "EXAMPLE:"; echo = 2; |
---|
447 | ring r; |
---|
448 | module M = maxideal(1); M; |
---|
449 | def S = Sres(M, 0); setring S; S; |
---|
450 | RES; |
---|
451 | MRES; |
---|
452 | kill S; |
---|
453 | setring r; kill M; |
---|
454 | |
---|
455 | def A = nc_algebra(-1,0); setring A; |
---|
456 | ideal Q = var(1)^2, var(2)^2, var(3)^2; |
---|
457 | qring SCA = twostd(Q); |
---|
458 | basering; |
---|
459 | |
---|
460 | module M = maxideal(1); |
---|
461 | def S = Sres(M, 2); setring S; S; |
---|
462 | RES; |
---|
463 | MRES; |
---|
464 | } |
---|
465 | |
---|
466 | |
---|
467 | |
---|
468 | // ================================================================== // |
---|
469 | |
---|
470 | |
---|
471 | LIB "general.lib"; // for sort |
---|
472 | |
---|
473 | /* static proc Tail(def M) // DONE: in C++ (dyn. module: syzextra)! |
---|
474 | { |
---|
475 | int i = ncols(M); def m; |
---|
476 | while (i > 0) |
---|
477 | { |
---|
478 | m = M[i]; |
---|
479 | m = m - lead(m); // m = tail(m) |
---|
480 | M[i] = m; |
---|
481 | i--; |
---|
482 | } |
---|
483 | return (M); |
---|
484 | }*/ |
---|
485 | |
---|
486 | /* static */ |
---|
487 | proc MySort(def M) |
---|
488 | { |
---|
489 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
490 | { |
---|
491 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
492 | } else |
---|
493 | { |
---|
494 | int @DEBUG = !system("with", "ndebug"); |
---|
495 | } |
---|
496 | |
---|
497 | if( typeof( attrib(basering, "SYZCHECK") ) == "int" ) |
---|
498 | { |
---|
499 | int @SYZCHECK = attrib(basering, "SYZCHECK"); |
---|
500 | } else |
---|
501 | { |
---|
502 | int @SYZCHECK = @DEBUG; |
---|
503 | } |
---|
504 | |
---|
505 | |
---|
506 | if( @DEBUG ) |
---|
507 | { |
---|
508 | "MySort:: Input: "; M; |
---|
509 | } |
---|
510 | |
---|
511 | def @N = M; Sort_c_ds(@N); |
---|
512 | |
---|
513 | if( @SYZCHECK ) |
---|
514 | { |
---|
515 | def iv = sort(lead(M), "c,ds", 1)[2]; // ,1 => reversed! // TODO: not needed? |
---|
516 | def @M = M; |
---|
517 | @M = M[iv]; |
---|
518 | |
---|
519 | // 0^th syz. property |
---|
520 | if( size(module( matrix(@N) - matrix(@M) )) > 0 ) |
---|
521 | { |
---|
522 | "@M:"; @M; |
---|
523 | "@N:"; @N; |
---|
524 | |
---|
525 | "module( matrix(@N) - matrix(@M) ): "; |
---|
526 | module( matrix(@N) - matrix(@M) ); |
---|
527 | |
---|
528 | "ERROR: wrong sorting in 'MySort': @N != @M!!!"; |
---|
529 | $ |
---|
530 | } |
---|
531 | } |
---|
532 | |
---|
533 | if( @DEBUG ) |
---|
534 | { |
---|
535 | "MySort:: Ouput: "; @N; |
---|
536 | } |
---|
537 | |
---|
538 | return (@N); |
---|
539 | } |
---|
540 | |
---|
541 | |
---|
542 | /* static */ proc SSinit(def M) |
---|
543 | { |
---|
544 | if( (typeof(M) != "module") && (typeof(M) != "ideal") ) |
---|
545 | { |
---|
546 | ERROR("Sorry: need an ideal or a module for input"); |
---|
547 | } |
---|
548 | |
---|
549 | // TODO! DONE? |
---|
550 | def @save = basering; |
---|
551 | |
---|
552 | int @DEBUG = !system("with", "ndebug"); |
---|
553 | int @SYZCHECK = 1 || @DEBUG; // TODO: only for now!! |
---|
554 | |
---|
555 | if( @DEBUG ) |
---|
556 | { |
---|
557 | "SSinit::Input"; |
---|
558 | type(M); |
---|
559 | // DetailedPrint(M); |
---|
560 | attrib(M); |
---|
561 | } |
---|
562 | |
---|
563 | int @RANK = nrows(M); int @SIZE = ncols(M); |
---|
564 | |
---|
565 | int @IS_A_SB = attrib(M, "isSB"); // ??? only if all weights were zero?! |
---|
566 | |
---|
567 | if( !@IS_A_SB ) |
---|
568 | { |
---|
569 | def opts = option(get); |
---|
570 | option(redSB); option(redTail); |
---|
571 | M = std(M); |
---|
572 | option(set, opts); |
---|
573 | kill opts; |
---|
574 | } else |
---|
575 | { |
---|
576 | M = simplify(M, 2 + 4 + 32); |
---|
577 | } |
---|
578 | |
---|
579 | def @N = MySort(M); // TODO: replace with inplace sorting!!! |
---|
580 | def LEAD = lead(@N); |
---|
581 | |
---|
582 | if( @SYZCHECK ) |
---|
583 | { |
---|
584 | def @LEAD = lead(M); |
---|
585 | |
---|
586 | // sort wrt neg.deg.rev.lex! |
---|
587 | intvec iv_ds = sort(@LEAD, "c,ds", 1)[2]; // ,1 => reversed! |
---|
588 | |
---|
589 | M = M[iv_ds]; // sort M wrt ds on current leading terms |
---|
590 | @LEAD = @LEAD[iv_ds]; |
---|
591 | |
---|
592 | // 0^th syz. property |
---|
593 | if( size(module( matrix(@N) - matrix(M) )) > 0 ) |
---|
594 | { |
---|
595 | "M:"; M; |
---|
596 | "@N:"; @N; |
---|
597 | |
---|
598 | "module( matrix(@N) - matrix(M) ): "; |
---|
599 | module( matrix(@N) - matrix(M) ); |
---|
600 | |
---|
601 | "ERROR: wrong sorting (in SSnit): @N != M!!!"; |
---|
602 | $ |
---|
603 | } |
---|
604 | |
---|
605 | // 0^th syz. property |
---|
606 | if( size(module( matrix(@LEAD) - matrix(LEAD) )) > 0 ) |
---|
607 | { |
---|
608 | "LEAD:"; LEAD; |
---|
609 | "@LEAD:"; @LEAD; |
---|
610 | |
---|
611 | "module( matrix(@LEAD) - matrix(LEAD) ): "; |
---|
612 | module( matrix(@LEAD) - matrix(LEAD) ); |
---|
613 | |
---|
614 | "ERROR: wrong sorting (in SSnit): @LEAD != LEAD!!!"; |
---|
615 | $ |
---|
616 | } |
---|
617 | |
---|
618 | } |
---|
619 | |
---|
620 | M = @N; |
---|
621 | |
---|
622 | |
---|
623 | |
---|
624 | |
---|
625 | def TAIL = Tail(M); |
---|
626 | |
---|
627 | intvec @DEGS = deg(M[1..@SIZE]); // store actuall degrees of input elements |
---|
628 | |
---|
629 | // TODO: what about real modules? weighted ones? |
---|
630 | |
---|
631 | list @l = ringlist(@save); |
---|
632 | |
---|
633 | int @z = 0; ideal @m = maxideal(1); intvec @wdeg = deg(@m[1..ncols(@m)]); |
---|
634 | |
---|
635 | // NOTE: @wdeg will be ignored anyway :( |
---|
636 | @l[3] = list(list("C", @z), list("lp", @wdeg)); |
---|
637 | |
---|
638 | kill @z, @wdeg; // since these vars are ring independent! |
---|
639 | |
---|
640 | def S = ring(@l); // --MakeInducedSchreyerOrdering(1); |
---|
641 | |
---|
642 | module F = freemodule(@RANK); |
---|
643 | intvec @V = deg(F[1..@RANK]); |
---|
644 | |
---|
645 | setring S; // ring with an easy divisibility test ("C, lex") |
---|
646 | |
---|
647 | if( @DEBUG ) |
---|
648 | { |
---|
649 | "SSinit::NewRing(C, lex)"; |
---|
650 | basering; |
---|
651 | DetailedPrint(basering); |
---|
652 | } |
---|
653 | |
---|
654 | // Setup the leading syzygy^{-1} module to zero: |
---|
655 | module Z = 0; Z[@RANK] = 0; attrib(Z, "isHomog", intvec(0)); |
---|
656 | |
---|
657 | module MRES = Z; |
---|
658 | |
---|
659 | list RES; RES[1] = Z; |
---|
660 | list LRES; LRES[1] = Z; |
---|
661 | list TRES; TRES[1] = Z; |
---|
662 | |
---|
663 | def M = imap(@save, M); |
---|
664 | |
---|
665 | attrib(M, "isHomog", @V); |
---|
666 | attrib(M, "isSB", 1); |
---|
667 | attrib(M, "degrees", @DEGS); |
---|
668 | |
---|
669 | def LEAD = imap(@save, LEAD); |
---|
670 | |
---|
671 | attrib(LEAD, "isHomog", @V); |
---|
672 | attrib(LEAD, "isSB", 1); |
---|
673 | |
---|
674 | def TAIL = imap(@save, TAIL); |
---|
675 | |
---|
676 | if( @DEBUG ) |
---|
677 | { |
---|
678 | "SSinit::(sorted) SB_Input: "; |
---|
679 | type(M); |
---|
680 | attrib(M); |
---|
681 | attrib(M, "isHomog"); |
---|
682 | // DetailedPrint(M); |
---|
683 | } |
---|
684 | |
---|
685 | if( @SYZCHECK ) |
---|
686 | { |
---|
687 | // 0^th syz. property |
---|
688 | if( size(module(transpose( transpose(M) * transpose(MRES) ))) > 0 ) |
---|
689 | { |
---|
690 | transpose( transpose(M) * transpose(MRES) ); |
---|
691 | "ERROR: transpose( transpose(M) * transpose(MRES) ) != 0!!!"; |
---|
692 | $ |
---|
693 | } |
---|
694 | } |
---|
695 | |
---|
696 | RES [size(RES)+1] = M; // list of all syzygy modules |
---|
697 | LRES[size(LRES)+1] = LEAD; // list of all syzygy modules |
---|
698 | TRES[size(TRES)+1] = TAIL; // list of all syzygy modules |
---|
699 | |
---|
700 | MRES = MRES, M; //? |
---|
701 | |
---|
702 | attrib(MRES, "isHomog", @V); |
---|
703 | |
---|
704 | // attrib(S, "InducionStart", @RANK); |
---|
705 | attrib(S, "LEAD2SYZ", 1); |
---|
706 | attrib(S, "TAILREDSYZ", 1); |
---|
707 | attrib(S, "DEBUG", @DEBUG); |
---|
708 | attrib(S, "SYZCHECK", @SYZCHECK); |
---|
709 | |
---|
710 | if( @DEBUG ) |
---|
711 | { |
---|
712 | "SSinit::MRES"; |
---|
713 | MRES; |
---|
714 | // DetailedPrint(MRES); |
---|
715 | attrib(MRES, "isHomog"); |
---|
716 | attrib(S); |
---|
717 | } |
---|
718 | |
---|
719 | export RES; |
---|
720 | export MRES; |
---|
721 | export LRES; |
---|
722 | export TRES; |
---|
723 | return (S); |
---|
724 | } |
---|
725 | example |
---|
726 | { "EXAMPLE:"; echo = 2; |
---|
727 | ring R = 0, (w, x, y, z), dp; |
---|
728 | |
---|
729 | def M = maxideal(1); |
---|
730 | def S = SSinit(M); setring S; S; |
---|
731 | |
---|
732 | "Only the first initialization: "; |
---|
733 | RES; LRES; TRES; |
---|
734 | MRES; |
---|
735 | |
---|
736 | kill S; setring R; kill M; |
---|
737 | |
---|
738 | ideal M = w^2 - x*z, w*x - y*z, x^2 - w*y, x*y - z^2, y^2 - w*z; |
---|
739 | def S = SSinit(M); setring S; S; |
---|
740 | |
---|
741 | "Only the first initialization: "; |
---|
742 | RES; LRES; TRES; |
---|
743 | MRES; |
---|
744 | |
---|
745 | kill S; setring R; kill M; |
---|
746 | } |
---|
747 | |
---|
748 | |
---|
749 | LIB "poly.lib"; // for lcm |
---|
750 | |
---|
751 | |
---|
752 | |
---|
753 | /// Compute L(Syz(L)) |
---|
754 | proc SSComputeLeadingSyzygyTerms(def L) |
---|
755 | { |
---|
756 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
757 | { |
---|
758 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
759 | } else |
---|
760 | { |
---|
761 | int @DEBUG = !system("with", "ndebug"); |
---|
762 | } |
---|
763 | |
---|
764 | if( typeof( attrib(basering, "SYZCHECK") ) == "int" ) |
---|
765 | { |
---|
766 | int @SYZCHECK = attrib(basering, "SYZCHECK"); |
---|
767 | } else |
---|
768 | { |
---|
769 | int @SYZCHECK = @DEBUG; |
---|
770 | } |
---|
771 | |
---|
772 | if( @DEBUG ) |
---|
773 | { |
---|
774 | "SSComputeLeadingSyzygyTerms::Input: "; |
---|
775 | L; |
---|
776 | } |
---|
777 | |
---|
778 | module SS = ComputeLeadingSyzygyTerms(L); |
---|
779 | |
---|
780 | if( @SYZCHECK ) |
---|
781 | { |
---|
782 | int i, j, r; |
---|
783 | int N = ncols(L); |
---|
784 | def a, b; |
---|
785 | poly aa, bb; |
---|
786 | |
---|
787 | bigint c; |
---|
788 | |
---|
789 | ideal M; |
---|
790 | |
---|
791 | module S = 0; |
---|
792 | |
---|
793 | for(i = 1; i <= N; i++) |
---|
794 | { |
---|
795 | a = L[i]; |
---|
796 | c = leadcomp(a); |
---|
797 | r = int(c); |
---|
798 | |
---|
799 | aa = leadmonomial(a); |
---|
800 | |
---|
801 | M = 0; |
---|
802 | |
---|
803 | for(j = i-1; j > 0; j--) |
---|
804 | { |
---|
805 | b = L[j]; |
---|
806 | |
---|
807 | if( leadcomp(b) == c ) |
---|
808 | { |
---|
809 | bb = leadmonomial(b); |
---|
810 | |
---|
811 | M[j] = (lcm(aa, bb) / aa); |
---|
812 | } |
---|
813 | } |
---|
814 | |
---|
815 | // TODO: add quotient relations here... |
---|
816 | |
---|
817 | M = simplify(M, 1 + 2 + 32); |
---|
818 | |
---|
819 | M = MySort(M); |
---|
820 | |
---|
821 | S = S, M * gen(i); |
---|
822 | } |
---|
823 | |
---|
824 | S = MySort(simplify(S, 2)); |
---|
825 | |
---|
826 | if( size(module(matrix(S) - matrix(SS))) > 0 ) |
---|
827 | { |
---|
828 | "ERROR: S != SS "; |
---|
829 | |
---|
830 | "basering: "; |
---|
831 | DetailedPrint(basering); |
---|
832 | |
---|
833 | "S: "; S; |
---|
834 | DetailedPrint(S, 1); |
---|
835 | "SS: "; SS; |
---|
836 | DetailedPrint(SS, 1); |
---|
837 | |
---|
838 | "DIFF: "; |
---|
839 | print(matrix(S) - matrix(SS)); |
---|
840 | DetailedPrint(module(matrix(S) - matrix(SS)), 2); |
---|
841 | $ |
---|
842 | } |
---|
843 | } |
---|
844 | |
---|
845 | |
---|
846 | if( @DEBUG ) |
---|
847 | { |
---|
848 | "SSComputeLeadingSyzygyTerms::Output: "; |
---|
849 | "SS: "; |
---|
850 | SS; |
---|
851 | } |
---|
852 | |
---|
853 | attrib(SS, "isSB", 1); |
---|
854 | |
---|
855 | return (SS); |
---|
856 | } |
---|
857 | |
---|
858 | /// Compute Syz(L), where L is a monomial (leading) module |
---|
859 | proc SSCompute2LeadingSyzygyTerms(def L) |
---|
860 | { |
---|
861 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
862 | { |
---|
863 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
864 | } else |
---|
865 | { |
---|
866 | int @DEBUG = !system("with", "ndebug"); |
---|
867 | } |
---|
868 | |
---|
869 | if( typeof( attrib(basering, "SYZCHECK") ) == "int" ) |
---|
870 | { |
---|
871 | int @SYZCHECK = attrib(basering, "SYZCHECK"); |
---|
872 | } else |
---|
873 | { |
---|
874 | int @SYZCHECK = @DEBUG; |
---|
875 | } |
---|
876 | |
---|
877 | int @TAILREDSYZ = 1; |
---|
878 | if( typeof( attrib(basering, "TAILREDSYZ") ) == "int" ) |
---|
879 | { |
---|
880 | @TAILREDSYZ = attrib(basering, "TAILREDSYZ"); |
---|
881 | } |
---|
882 | |
---|
883 | |
---|
884 | if( @DEBUG ) |
---|
885 | { |
---|
886 | "SSCompute2LeadingSyzygyTerms::Input: "; |
---|
887 | L; |
---|
888 | } |
---|
889 | |
---|
890 | int i, j, r; |
---|
891 | int N = ncols(L); |
---|
892 | def a, b; |
---|
893 | |
---|
894 | poly aa, bb, @lcm; |
---|
895 | |
---|
896 | bigint c; |
---|
897 | |
---|
898 | module M; |
---|
899 | |
---|
900 | module S = 0; |
---|
901 | |
---|
902 | for(i = 1; i <= N; i++) |
---|
903 | { |
---|
904 | a = L[i]; |
---|
905 | // "a: ", a; |
---|
906 | c = leadcomp(a); |
---|
907 | r = int(c); |
---|
908 | |
---|
909 | aa = leadmonomial(a); |
---|
910 | |
---|
911 | M = 0; |
---|
912 | |
---|
913 | for(j = i-1; j > 0; j--) |
---|
914 | { |
---|
915 | b = L[j]; |
---|
916 | // "b: ", b; |
---|
917 | |
---|
918 | if( leadcomp(b) == c ) |
---|
919 | { |
---|
920 | bb = leadmonomial(b); |
---|
921 | @lcm = lcm(aa, bb); |
---|
922 | |
---|
923 | M[j] = (@lcm / aa)* gen(i) - (@lcm / bb)* gen(j); |
---|
924 | } |
---|
925 | } |
---|
926 | |
---|
927 | M = simplify(M, 2); |
---|
928 | |
---|
929 | // TODO: add quotient relations here... |
---|
930 | S = S, M; |
---|
931 | } |
---|
932 | |
---|
933 | if( @TAILREDSYZ ) |
---|
934 | { |
---|
935 | // Make sure that 2nd syzygy terms are not reducible by 1st |
---|
936 | def opts = option(get); |
---|
937 | option(redSB); option(redTail); |
---|
938 | S = std(S); // binomial module |
---|
939 | option(set, opts); |
---|
940 | // kill opts; |
---|
941 | } else |
---|
942 | { |
---|
943 | S = simplify(S, 2 + 32); |
---|
944 | } |
---|
945 | |
---|
946 | S = MySort(S); |
---|
947 | |
---|
948 | if( @DEBUG ) |
---|
949 | { |
---|
950 | "SSCompute2LeadingSyzygyTerms::Syz(LEAD): "; S; |
---|
951 | } |
---|
952 | |
---|
953 | if( @SYZCHECK ) |
---|
954 | { |
---|
955 | if( size(S) > 0 and size(L) > 0 ) |
---|
956 | { |
---|
957 | if( size(module(transpose( transpose(S) * transpose(L) ))) > 0 ) |
---|
958 | { |
---|
959 | transpose( transpose(S) * transpose(L) ); |
---|
960 | "ERROR: transpose( transpose(S) * transpose(L) ) != 0!!!"; |
---|
961 | $ |
---|
962 | } |
---|
963 | } |
---|
964 | |
---|
965 | module SS = Compute2LeadingSyzygyTerms(L); |
---|
966 | |
---|
967 | "S: "; DetailedPrint(S); |
---|
968 | "SS: "; DetailedPrint(SS); |
---|
969 | |
---|
970 | if( size(module(matrix(S) - matrix(SS))) > 0 ) |
---|
971 | { |
---|
972 | "ERROR: S != SS "; |
---|
973 | |
---|
974 | "basering: "; |
---|
975 | DetailedPrint(basering); |
---|
976 | |
---|
977 | "S: "; S; |
---|
978 | DetailedPrint(S, 2); |
---|
979 | "SS: "; SS; |
---|
980 | DetailedPrint(SS, 2); |
---|
981 | |
---|
982 | "DIFF: "; |
---|
983 | print(matrix(S) - matrix(SS)); |
---|
984 | DetailedPrint(module(matrix(S) - matrix(SS)), 4); |
---|
985 | $ |
---|
986 | } |
---|
987 | |
---|
988 | } |
---|
989 | |
---|
990 | module S2 = Tail(S); |
---|
991 | S = lead(S); // (C,lp) on base ring! |
---|
992 | |
---|
993 | if( @DEBUG ) |
---|
994 | { |
---|
995 | "SSCompute2LeadingSyzygyTerms::Output: "; S; S2; |
---|
996 | } |
---|
997 | |
---|
998 | attrib(S, "isSB", 1); |
---|
999 | |
---|
1000 | return (S, S2); |
---|
1001 | } |
---|
1002 | |
---|
1003 | // -------------------------------------------------------- // |
---|
1004 | |
---|
1005 | /// TODO: save shortcut LM(m) * "t" -> ? |
---|
1006 | proc SSReduceTerm(poly m, def t, def L, def T, list #) |
---|
1007 | { |
---|
1008 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
1009 | { |
---|
1010 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
1011 | } else |
---|
1012 | { |
---|
1013 | int @DEBUG = !system("with", "ndebug"); |
---|
1014 | } |
---|
1015 | |
---|
1016 | if( @DEBUG ) |
---|
1017 | { |
---|
1018 | "SSReduce::Input: "; |
---|
1019 | |
---|
1020 | "mult: ", m; |
---|
1021 | "term: ", t; |
---|
1022 | "L: ", L; |
---|
1023 | "T: ", T; |
---|
1024 | if( size(#) > 0 ) |
---|
1025 | { |
---|
1026 | "LSyz: ", #; |
---|
1027 | } |
---|
1028 | // "attrib(LS, 'isSB')", attrib(LS, "isSB"); |
---|
1029 | } |
---|
1030 | |
---|
1031 | vector s = 0; |
---|
1032 | |
---|
1033 | if( t == 0 ) |
---|
1034 | { |
---|
1035 | return (s); |
---|
1036 | } |
---|
1037 | |
---|
1038 | def product = m * t; |
---|
1039 | |
---|
1040 | bigint c = leadcomp(t); |
---|
1041 | int r = int(c); |
---|
1042 | |
---|
1043 | def a, b, nf, bb; |
---|
1044 | |
---|
1045 | // looking for an appropriate reducer |
---|
1046 | for( int k = ncols(L); k > 0; k-- ) |
---|
1047 | { |
---|
1048 | a = L[k]; |
---|
1049 | // with the same mod. component |
---|
1050 | if( leadcomp(a) == c ) |
---|
1051 | { |
---|
1052 | b = - (leadmonomial(product) / leadmonomial(L[k])); |
---|
1053 | |
---|
1054 | // which divides the product |
---|
1055 | if( b != 0 ) |
---|
1056 | { |
---|
1057 | // "b: ", b; |
---|
1058 | bb = b * gen(k); |
---|
1059 | nf = bb; |
---|
1060 | |
---|
1061 | if( size(#) > 0 ) |
---|
1062 | { |
---|
1063 | if( typeof(#[1]) == "module" ) |
---|
1064 | { |
---|
1065 | nf = NF(bb, #[1]); |
---|
1066 | // "NF: ", nf; |
---|
1067 | } |
---|
1068 | } |
---|
1069 | |
---|
1070 | // while the complement (the fraction) is not reducible by leading syzygies |
---|
1071 | if( nf != 0 ) |
---|
1072 | { |
---|
1073 | /// TODO: save shortcut LM(m) * T[i] -> ? |
---|
1074 | |
---|
1075 | // choose ANY such reduction... (with the biggest index?) |
---|
1076 | s = bb + SSTraverseTail(b, T[k], L, T, #); |
---|
1077 | break; |
---|
1078 | } |
---|
1079 | } |
---|
1080 | } |
---|
1081 | } |
---|
1082 | if( @DEBUG ) |
---|
1083 | { |
---|
1084 | "SSReduceTerm::Output: ", s; |
---|
1085 | } |
---|
1086 | return (s); |
---|
1087 | } |
---|
1088 | |
---|
1089 | // TODO: store m * @tail -.-^-.-^-.--> ? |
---|
1090 | proc SSTraverseTail(poly m, def @tail, def L, def T, list #) |
---|
1091 | { |
---|
1092 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
1093 | { |
---|
1094 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
1095 | } else |
---|
1096 | { |
---|
1097 | int @DEBUG = !system("with", "ndebug"); |
---|
1098 | } |
---|
1099 | |
---|
1100 | if( @DEBUG ) |
---|
1101 | { |
---|
1102 | "SSTraverse::Input: "; |
---|
1103 | |
---|
1104 | "mult: ", m; |
---|
1105 | "tail: ", @tail; // T[i]; |
---|
1106 | |
---|
1107 | if( size(#) > 0 ) |
---|
1108 | { |
---|
1109 | "LSyz: "; #[1]; |
---|
1110 | } |
---|
1111 | } |
---|
1112 | |
---|
1113 | vector s = 0; |
---|
1114 | |
---|
1115 | def @l; |
---|
1116 | |
---|
1117 | // iterate tail-terms in ANY order! |
---|
1118 | while( size(@tail) > 0 ) |
---|
1119 | { |
---|
1120 | @l = lead(@tail); |
---|
1121 | s = s + SSReduceTerm(m, @l, L, T, #); |
---|
1122 | @tail = @tail - @l; |
---|
1123 | } |
---|
1124 | |
---|
1125 | if( @DEBUG ) |
---|
1126 | { |
---|
1127 | "SSTraverseTail::Output: ", s; |
---|
1128 | } |
---|
1129 | return (s); |
---|
1130 | } |
---|
1131 | |
---|
1132 | // -------------------------------------------------------- // |
---|
1133 | |
---|
1134 | // module (N, LL, TT) = SSComputeSyzygy(L, T); |
---|
1135 | // Compute Syz(L ++ T) = N = LL ++ TT |
---|
1136 | proc SSComputeSyzygy(def L, def T) |
---|
1137 | { |
---|
1138 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
1139 | { |
---|
1140 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
1141 | } else |
---|
1142 | { |
---|
1143 | int @DEBUG = !system("with", "ndebug"); |
---|
1144 | } |
---|
1145 | |
---|
1146 | if( @DEBUG ) |
---|
1147 | { |
---|
1148 | "SSComputeSyzygy::Input"; |
---|
1149 | "basering: ", basering; attrib(basering); |
---|
1150 | // DetailedPrint(basering); |
---|
1151 | |
---|
1152 | // "iCompShift: ", iCompShift; |
---|
1153 | |
---|
1154 | "L: "; L; |
---|
1155 | "T: "; T; |
---|
1156 | } |
---|
1157 | |
---|
1158 | def a; bigint c; int r, k; poly aa; |
---|
1159 | |
---|
1160 | int @LEAD2SYZ = 0; |
---|
1161 | if( typeof( attrib(basering, "LEAD2SYZ") ) == "int" ) |
---|
1162 | { |
---|
1163 | @LEAD2SYZ = attrib(basering, "LEAD2SYZ"); |
---|
1164 | } |
---|
1165 | |
---|
1166 | int @TAILREDSYZ = 1; |
---|
1167 | if( typeof( attrib(basering, "TAILREDSYZ") ) == "int" ) |
---|
1168 | { |
---|
1169 | @TAILREDSYZ = attrib(basering, "TAILREDSYZ"); |
---|
1170 | // @TAILREDSYZ; |
---|
1171 | } |
---|
1172 | |
---|
1173 | /// Get the critical leading syzygy terms |
---|
1174 | if( @LEAD2SYZ ) // & 2nd syz. term |
---|
1175 | { |
---|
1176 | def a2; int r2; poly aa2; |
---|
1177 | module LL, LL2; |
---|
1178 | (LL, LL2) = SSCompute2LeadingSyzygyTerms(L); // ++ |
---|
1179 | } else |
---|
1180 | { |
---|
1181 | module LL = SSComputeLeadingSyzygyTerms(L); |
---|
1182 | } |
---|
1183 | |
---|
1184 | module TT, SYZ; |
---|
1185 | |
---|
1186 | if( size(LL) > 0 ) |
---|
1187 | { |
---|
1188 | list LS; |
---|
1189 | |
---|
1190 | if( @TAILREDSYZ ) |
---|
1191 | { |
---|
1192 | LS = list(LL); |
---|
1193 | } |
---|
1194 | |
---|
1195 | vector @tail; |
---|
1196 | |
---|
1197 | for(k = ncols(LL); k > 0; k-- ) |
---|
1198 | { |
---|
1199 | // leading syz. term: |
---|
1200 | a = LL[k]; c = leadcomp(a); r = int(c); aa = leadmonomial(a); |
---|
1201 | // "A: ", a, " --->>>> ", aa, " **** [", r, "]: "; |
---|
1202 | |
---|
1203 | /// TODO: save shortcut (aa) * T[r] -> ? |
---|
1204 | @tail = SSTraverseTail(aa, T[r], L, T, LS); |
---|
1205 | |
---|
1206 | // get the 2nd syzygy term... |
---|
1207 | |
---|
1208 | if( @LEAD2SYZ ) // with the 2nd syz. term: |
---|
1209 | { |
---|
1210 | a2 = LL2[k]; c = leadcomp(a2); r2 = int(c); aa2 = leadmonomial(a2); |
---|
1211 | @tail = @tail + |
---|
1212 | /// TODO: save shortcut (aa2) * T[r2] -> ? |
---|
1213 | a2 + SSTraverseTail(aa2, T[r2], L, T, LS); |
---|
1214 | } else |
---|
1215 | { |
---|
1216 | @tail = @tail + SSReduceTerm(aa, L[r], L, T, LS); |
---|
1217 | } |
---|
1218 | |
---|
1219 | TT[k] = @tail; |
---|
1220 | SYZ[k] = a + @tail; |
---|
1221 | } |
---|
1222 | } |
---|
1223 | |
---|
1224 | /* |
---|
1225 | def opts = option(get); option(redSB); option(redTail); |
---|
1226 | module SYZ = std(syz(M)); |
---|
1227 | option(set, opts); kill opts; |
---|
1228 | |
---|
1229 | module LL = lead(SYZ); // TODO: WRONG ORDERING!!!!!!!! |
---|
1230 | module TT = Tail(SYZ); |
---|
1231 | */ |
---|
1232 | |
---|
1233 | if( @DEBUG ) |
---|
1234 | { |
---|
1235 | "SSComputeSyzygy::Output"; |
---|
1236 | |
---|
1237 | "SYZ: "; SYZ; |
---|
1238 | "LL: "; LL; |
---|
1239 | "TT: "; TT; |
---|
1240 | } |
---|
1241 | |
---|
1242 | return (SYZ, LL, TT); |
---|
1243 | } |
---|
1244 | |
---|
1245 | // resolution/syzygy step: |
---|
1246 | static proc SSstep() |
---|
1247 | { |
---|
1248 | if( typeof( attrib(basering, "DEBUG") ) == "int" ) |
---|
1249 | { |
---|
1250 | int @DEBUG = attrib(basering, "DEBUG"); |
---|
1251 | } else |
---|
1252 | { |
---|
1253 | int @DEBUG = !system("with", "ndebug"); |
---|
1254 | } |
---|
1255 | |
---|
1256 | |
---|
1257 | if( typeof( attrib(basering, "SYZCHECK") ) == "int" ) |
---|
1258 | { |
---|
1259 | int @SYZCHECK = attrib(basering, "SYZCHECK"); |
---|
1260 | } else |
---|
1261 | { |
---|
1262 | int @SYZCHECK = @DEBUG; |
---|
1263 | } |
---|
1264 | |
---|
1265 | if( @DEBUG ) |
---|
1266 | { |
---|
1267 | "SSstep::NextInducedRing"; |
---|
1268 | "basering: ", basering; attrib(basering); |
---|
1269 | } |
---|
1270 | |
---|
1271 | /* |
---|
1272 | // is initial weights are all zeroes! |
---|
1273 | def L = lead(M); |
---|
1274 | intvec @V = deg(M[1..ncols(M)]); @W; @V; @W = @V; attrib(L, "isHomog", @W); |
---|
1275 | SetInducedReferrence(L, @RANK, 0); |
---|
1276 | */ |
---|
1277 | |
---|
1278 | // def L = lead(MRES); |
---|
1279 | // @W = @W, @V; |
---|
1280 | // attrib(L, "isHomog", @W); |
---|
1281 | |
---|
1282 | |
---|
1283 | // General setting: |
---|
1284 | // SetInducedReferrence(MRES, 0, 0); // limit: 0! |
---|
1285 | int @l = size(RES); |
---|
1286 | |
---|
1287 | def M = RES[@l]; |
---|
1288 | |
---|
1289 | def L = LRES[@l]; |
---|
1290 | def T = TRES[@l]; |
---|
1291 | |
---|
1292 | |
---|
1293 | //// TODO: wrong !!!!! |
---|
1294 | int @RANK = ncols(MRES) - ncols(M); // nrows(M); // what if M is zero?! |
---|
1295 | |
---|
1296 | |
---|
1297 | |
---|
1298 | /* |
---|
1299 | if( @RANK != nrows(M) ) |
---|
1300 | { |
---|
1301 | type(MRES); |
---|
1302 | @RANK; |
---|
1303 | type(M); |
---|
1304 | pause(); |
---|
1305 | } |
---|
1306 | */ |
---|
1307 | |
---|
1308 | intvec @W = attrib(M, "isHomog"); intvec @V = attrib(M, "degrees"); @V = @W, @V; |
---|
1309 | |
---|
1310 | if( @DEBUG ) |
---|
1311 | { |
---|
1312 | "Sstep::NextInput: "; |
---|
1313 | M; |
---|
1314 | L; |
---|
1315 | @V; |
---|
1316 | @RANK; |
---|
1317 | // DetailedPrint(MRES); |
---|
1318 | attrib(MRES, "isHomog"); |
---|
1319 | } |
---|
1320 | |
---|
1321 | |
---|
1322 | // TODO: N = SYZ( M )!!! |
---|
1323 | module N, LL, TT; |
---|
1324 | (N, LL, TT) = SSComputeSyzygy(/*M, */L, T/*, @RANK*/); |
---|
1325 | |
---|
1326 | // shift syz.comp by @RANK: |
---|
1327 | module Z; |
---|
1328 | Z = 0; Z[@RANK] = 0; Z = Z, transpose(LL); LL = transpose(Z); |
---|
1329 | Z = 0; Z[@RANK] = 0; Z = Z, transpose(TT); TT = transpose(Z); |
---|
1330 | Z = 0; Z[@RANK] = 0; Z = Z, transpose(N); N = transpose(Z); |
---|
1331 | |
---|
1332 | |
---|
1333 | if( @SYZCHECK ) |
---|
1334 | { |
---|
1335 | if( size(N) > 0 ) |
---|
1336 | { |
---|
1337 | // next syz. property |
---|
1338 | if( size(module(transpose( transpose(N) * transpose(MRES) ))) > 0 ) |
---|
1339 | { |
---|
1340 | "MRES", MRES; |
---|
1341 | |
---|
1342 | "N: "; N; // DetailedPrint(N, 2); |
---|
1343 | |
---|
1344 | "LL:"; LL; // DetailedPrint(LL, 1); |
---|
1345 | "TT:"; TT; // DetailedPrint(TT, 10); |
---|
1346 | |
---|
1347 | "RANKS: ", @RANK; |
---|
1348 | |
---|
1349 | "transpose( transpose(N) * transpose(MRES) ) != 0!!!"; |
---|
1350 | transpose( transpose(N) * transpose(MRES) ); |
---|
1351 | |
---|
1352 | "transpose(N) * transpose(MRES): "; |
---|
1353 | transpose(N) * transpose(MRES); |
---|
1354 | // DetailedPrint(module(_), 2); |
---|
1355 | $ |
---|
1356 | } |
---|
1357 | } |
---|
1358 | } |
---|
1359 | |
---|
1360 | attrib(N, "isHomog", @V); |
---|
1361 | |
---|
1362 | // TODO: correct the following: |
---|
1363 | intvec @DEGS = deg(N[1..ncols(N)]); // no mod. comp. weights :( |
---|
1364 | |
---|
1365 | |
---|
1366 | attrib(N, "degrees", @DEGS); |
---|
1367 | |
---|
1368 | RES[@l + 1] = N; // list of all syzygy modules |
---|
1369 | LRES[@l + 1] = LL; // list of all syzygy modules |
---|
1370 | TRES[@l + 1] = TT; // list of all syzygy modules |
---|
1371 | |
---|
1372 | MRES = MRES, N; |
---|
1373 | |
---|
1374 | attrib(MRES, "isHomog", @V); |
---|
1375 | |
---|
1376 | // L = L, lead(N); attrib(basering, "InducionLeads", L); |
---|
1377 | |
---|
1378 | if( @DEBUG ) |
---|
1379 | { |
---|
1380 | "SSstep::NextSyzOutput: "; |
---|
1381 | N; |
---|
1382 | // DetailedPrint(N); |
---|
1383 | attrib(N); |
---|
1384 | } |
---|
1385 | |
---|
1386 | } |
---|
1387 | |
---|
1388 | proc SScontinue(int l) |
---|
1389 | "USAGE: SScontinue(l) |
---|
1390 | RETURN: nothing, instead it changes RES and MRES variables in the current ring |
---|
1391 | PURPOSE: computes further (at most l) syzygies |
---|
1392 | NOTE: must be used within a ring returned by Sres or Ssyz. RES and MRES are |
---|
1393 | explained in Sres |
---|
1394 | EXAMPLE: example Scontinue; shows an example |
---|
1395 | " |
---|
1396 | { |
---|
1397 | |
---|
1398 | /// TODO! |
---|
1399 | // def data = GetInducedData(); |
---|
1400 | |
---|
1401 | if( (!defined(RES)) || (!defined(MRES)) ) /* || (typeof(data) != "list") || (size(data) != 2) */ |
---|
1402 | { |
---|
1403 | ERROR("Sorry, but basering does not seem to be returned by Sres or Ssyz"); |
---|
1404 | } |
---|
1405 | for (; (l != 0) && (size(RES[size(RES)]) > 0); l-- ) |
---|
1406 | { |
---|
1407 | SSstep(); |
---|
1408 | } |
---|
1409 | } |
---|
1410 | example |
---|
1411 | { "EXAMPLE:"; echo = 2; |
---|
1412 | ring r; |
---|
1413 | module M = maxideal(1); M; |
---|
1414 | def S = SSsyz(M); setring S; S; |
---|
1415 | "Only the first syzygy: "; |
---|
1416 | RES; MRES; |
---|
1417 | "More syzygies: "; |
---|
1418 | SScontinue(10); |
---|
1419 | RES; MRES; |
---|
1420 | } |
---|
1421 | |
---|
1422 | proc SSsyz(def M) |
---|
1423 | "USAGE: SSsyz(M) |
---|
1424 | RETURN: ring, containing a list of modules RES and a module MRES |
---|
1425 | PURPOSE: computes the first syzygy module of M (wrt some Schreyer ordering)? |
---|
1426 | NOTE: The output is explained in Sres |
---|
1427 | EXAMPLE: example Ssyz; shows an example |
---|
1428 | " |
---|
1429 | { |
---|
1430 | if( (typeof(M) != "module") && (typeof(M) != "ideal") ) |
---|
1431 | { |
---|
1432 | ERROR("Sorry: need an ideal or a module for input"); |
---|
1433 | } |
---|
1434 | |
---|
1435 | def SS = SSinit(M); setring SS; |
---|
1436 | |
---|
1437 | SSstep(); // NOTE: what if M is zero? |
---|
1438 | |
---|
1439 | return (SS); |
---|
1440 | } |
---|
1441 | example |
---|
1442 | { "EXAMPLE:"; echo = 2; |
---|
1443 | ring r; |
---|
1444 | |
---|
1445 | /* ideal M = 0; |
---|
1446 | def S = SSsyz(M); setring S; S; |
---|
1447 | "Only the first syzygy: "; |
---|
1448 | RES; LRES; TRES; |
---|
1449 | MRES; |
---|
1450 | |
---|
1451 | kill S; setring r; kill M; |
---|
1452 | */ |
---|
1453 | |
---|
1454 | ideal M = maxideal(1); M; |
---|
1455 | def S = SSres(M, 0); setring S; S; |
---|
1456 | MRES; |
---|
1457 | RES; |
---|
1458 | ""; |
---|
1459 | LRES; |
---|
1460 | ""; |
---|
1461 | TRES; |
---|
1462 | |
---|
1463 | kill S; setring r; kill M; |
---|
1464 | |
---|
1465 | kill r; |
---|
1466 | |
---|
1467 | ring R = 0, (w, x, y, z), dp; |
---|
1468 | ideal M = w^2 - x*z, w*x - y*z, x^2 - w*y, x*y - z^2, y^2 - w*z; |
---|
1469 | |
---|
1470 | def S = SSres(M, 0); setring S; S; |
---|
1471 | MRES; |
---|
1472 | RES; |
---|
1473 | ""; |
---|
1474 | LRES; |
---|
1475 | ""; |
---|
1476 | TRES; |
---|
1477 | } |
---|
1478 | |
---|
1479 | proc SSres(def M, int l) |
---|
1480 | "USAGE: SSres(I, l) |
---|
1481 | RETURN: ring, containing a list of modules RES and a module MRES |
---|
1482 | PURPOSE: computes (at most l) syzygy modules of M wrt the classical Schreyer |
---|
1483 | induced ordering with gen(i) > gen(j) if i > j, provided both gens |
---|
1484 | are from the same syzygy level.??? |
---|
1485 | NOTE: RES contains the images of maps subsituting the beginning of the |
---|
1486 | Schreyer free resolution of baseRing^r/M, while MRES is a sum of |
---|
1487 | these images in a big free sum, containing all the syzygy modules. |
---|
1488 | The syzygy modules are shifted so that gen(i) correspons to MRES[i]. |
---|
1489 | The leading zero module RES[0] indicates the fact that coker of the |
---|
1490 | first map is zero. The number of zeroes inducates the rank of input. |
---|
1491 | NOTE: If l == 0 then l is set to be nvars(basering) + 1 |
---|
1492 | EXAMPLE: example SSres; shows an example |
---|
1493 | " |
---|
1494 | { |
---|
1495 | if( (typeof(M) != "module") && (typeof(M) != "ideal") ) |
---|
1496 | { |
---|
1497 | ERROR("Sorry: need an ideal or a module for input"); |
---|
1498 | } |
---|
1499 | |
---|
1500 | def SS = SSinit(M); setring SS; |
---|
1501 | |
---|
1502 | if (l == 0) |
---|
1503 | { |
---|
1504 | l = nvars(basering) + 1; // not really an estimate...?! |
---|
1505 | } |
---|
1506 | |
---|
1507 | SSstep(); l = l - 1; |
---|
1508 | |
---|
1509 | SScontinue(l); |
---|
1510 | |
---|
1511 | return (SS); |
---|
1512 | } |
---|
1513 | example |
---|
1514 | { "EXAMPLE:"; echo = 2; |
---|
1515 | ring r; |
---|
1516 | module M = maxideal(1); M; |
---|
1517 | def S = SSres(M, 0); setring S; S; |
---|
1518 | RES; |
---|
1519 | MRES; |
---|
1520 | kill S; |
---|
1521 | setring r; kill M; |
---|
1522 | |
---|
1523 | def A = nc_algebra(-1,0); setring A; |
---|
1524 | ideal Q = var(1)^2, var(2)^2, var(3)^2; |
---|
1525 | qring SCA = twostd(Q); |
---|
1526 | basering; |
---|
1527 | |
---|
1528 | module M = maxideal(1); |
---|
1529 | def S = SSres(M, 2); setring S; S; |
---|
1530 | RES; |
---|
1531 | MRES; |
---|
1532 | } |
---|
1533 | |
---|
1534 | |
---|
1535 | |
---|
1536 | static proc loadme() |
---|
1537 | { |
---|
1538 | int @DEBUG = !system("with", "ndebug"); |
---|
1539 | |
---|
1540 | if( @DEBUG ) |
---|
1541 | { |
---|
1542 | |
---|
1543 | "ndebug?: ", system("with", "ndebug"); |
---|
1544 | "om_ndebug?: ", system("with", "om_ndebug"); |
---|
1545 | |
---|
1546 | listvar(Top); |
---|
1547 | listvar(Schreyer); |
---|
1548 | } |
---|
1549 | // listvar(Syzextra); listvar(Syzextra_g); |
---|
1550 | |
---|
1551 | if( !defined(DetailedPrint) ) |
---|
1552 | { |
---|
1553 | if( 1 ) |
---|
1554 | { |
---|
1555 | |
---|
1556 | if( @DEBUG ) |
---|
1557 | { |
---|
1558 | "Loading the Release version!"; |
---|
1559 | } |
---|
1560 | load("syzextra.so"); |
---|
1561 | |
---|
1562 | if( @DEBUG ) |
---|
1563 | { |
---|
1564 | listvar(Syzextra); |
---|
1565 | } |
---|
1566 | |
---|
1567 | exportto(Top, Syzextra::ClearContent); |
---|
1568 | exportto(Top, Syzextra::ClearDenominators); |
---|
1569 | |
---|
1570 | exportto(Schreyer, Syzextra::m2_end); |
---|
1571 | |
---|
1572 | // export Syzextra; |
---|
1573 | |
---|
1574 | // exportto(Schreyer, Syzextra::noop); |
---|
1575 | exportto(Schreyer, Syzextra::DetailedPrint); |
---|
1576 | exportto(Schreyer, Syzextra::leadmonomial); |
---|
1577 | exportto(Schreyer, Syzextra::leadcomp); |
---|
1578 | // exportto(Schreyer, Syzextra::leadrawexp); |
---|
1579 | // exportto(Schreyer, Syzextra::ISUpdateComponents); |
---|
1580 | exportto(Schreyer, Syzextra::SetInducedReferrence); |
---|
1581 | exportto(Schreyer, Syzextra::GetInducedData); |
---|
1582 | // exportto(Schreyer, Syzextra::GetAMData); |
---|
1583 | // exportto(Schreyer, Syzextra::SetSyzComp); |
---|
1584 | exportto(Schreyer, Syzextra::MakeInducedSchreyerOrdering); |
---|
1585 | // exportto(Schreyer, Syzextra::MakeSyzCompOrdering); |
---|
1586 | exportto(Schreyer, Syzextra::idPrepare); |
---|
1587 | // exportto(Schreyer, Syzextra::reduce_syz); |
---|
1588 | // exportto(Schreyer, Syzextra::p_Content); |
---|
1589 | |
---|
1590 | exportto(Schreyer, Syzextra::ProfilerStart); exportto(Schreyer, Syzextra::ProfilerStop); |
---|
1591 | |
---|
1592 | exportto(Schreyer, Syzextra::Tail); |
---|
1593 | exportto(Schreyer, Syzextra::ComputeLeadingSyzygyTerms); |
---|
1594 | exportto(Schreyer, Syzextra::Compute2LeadingSyzygyTerms); |
---|
1595 | exportto(Schreyer, Syzextra::Sort_c_ds); |
---|
1596 | } |
---|
1597 | /* |
---|
1598 | else |
---|
1599 | { |
---|
1600 | if( @DEBUG ) |
---|
1601 | { |
---|
1602 | "Loading the Debug version!"; |
---|
1603 | } |
---|
1604 | |
---|
1605 | load("syzextra.so"); |
---|
1606 | |
---|
1607 | if( @DEBUG ) |
---|
1608 | { |
---|
1609 | listvar(Syzextra_g); |
---|
1610 | } |
---|
1611 | |
---|
1612 | exportto(Top, Syzextra_g::ClearContent); |
---|
1613 | exportto(Top, Syzextra_g::ClearDenominators); |
---|
1614 | |
---|
1615 | exportto(Schreyer, Syzextra_g::m2_end); |
---|
1616 | |
---|
1617 | // export Syzextra_g; |
---|
1618 | // exportto(Schreyer, Syzextra_g::noop); |
---|
1619 | exportto(Schreyer, Syzextra_g::DetailedPrint); |
---|
1620 | exportto(Schreyer, Syzextra_g::leadmonomial); |
---|
1621 | exportto(Schreyer, Syzextra_g::leadcomp); |
---|
1622 | // exportto(Schreyer, Syzextra_g::leadrawexp); |
---|
1623 | // exportto(Schreyer, Syzextra_g::ISUpdateComponents); |
---|
1624 | exportto(Schreyer, Syzextra_g::SetInducedReferrence); |
---|
1625 | exportto(Schreyer, Syzextra_g::GetInducedData); |
---|
1626 | // exportto(Schreyer, Syzextra_g::GetAMData); |
---|
1627 | // exportto(Schreyer, Syzextra_g::SetSyzComp); |
---|
1628 | exportto(Schreyer, Syzextra_g::MakeInducedSchreyerOrdering); |
---|
1629 | // exportto(Schreyer, Syzextra_g::MakeSyzCompOrdering); |
---|
1630 | exportto(Schreyer, Syzextra_g::idPrepare); |
---|
1631 | // exportto(Schreyer, Syzextra_g::reduce_syz); |
---|
1632 | // exportto(Schreyer, Syzextra_g::p_Content); |
---|
1633 | |
---|
1634 | exportto(Schreyer, Syzextra_g::ProfilerStart); exportto(Schreyer, Syzextra_g::ProfilerStop); |
---|
1635 | |
---|
1636 | exportto(Schreyer, Syzextra_g::Tail); |
---|
1637 | exportto(Schreyer, Syzextra_g::ComputeLeadingSyzygyTerms); |
---|
1638 | exportto(Schreyer, Syzextra_g::Compute2LeadingSyzygyTerms); |
---|
1639 | exportto(Schreyer, Syzextra_g::Sort_c_ds); |
---|
1640 | |
---|
1641 | } |
---|
1642 | */ |
---|
1643 | |
---|
1644 | exportto(Top, DetailedPrint); |
---|
1645 | exportto(Top, GetInducedData); |
---|
1646 | |
---|
1647 | if( @DEBUG ) |
---|
1648 | { |
---|
1649 | listvar(Top); |
---|
1650 | listvar(Schreyer); |
---|
1651 | } |
---|
1652 | } |
---|
1653 | |
---|
1654 | if( !defined(GetInducedData) ) |
---|
1655 | { |
---|
1656 | ERROR("Sorry but we are missing the dynamic module (syzextra.so)..."); |
---|
1657 | } |
---|
1658 | |
---|
1659 | } |
---|
1660 | |
---|
1661 | static proc mod_init() |
---|
1662 | { |
---|
1663 | loadme(); |
---|
1664 | } |
---|
1665 | |
---|
1666 | |
---|
1667 | proc testallSexamples() |
---|
1668 | { |
---|
1669 | example Ssyz; |
---|
1670 | example Scontinue; |
---|
1671 | example Sres; |
---|
1672 | } |
---|
1673 | |
---|
1674 | proc testallSSexamples() |
---|
1675 | { |
---|
1676 | example SSsyz; |
---|
1677 | example SScontinue; |
---|
1678 | example SSres; |
---|
1679 | } |
---|
1680 | |
---|
1681 | example |
---|
1682 | { "EXAMPLE:"; echo = 2; |
---|
1683 | testallSexamples(); |
---|
1684 | testallSSexamples(); |
---|
1685 | } |
---|