1 | /*****************************************************************************\ |
---|
2 | * Computer Algebra System SINGULAR |
---|
3 | \*****************************************************************************/ |
---|
4 | /** @file facHensel.cc |
---|
5 | * |
---|
6 | * This file implements functions to lift factors via Hensel lifting. |
---|
7 | * |
---|
8 | * ABSTRACT: Hensel lifting is described in "Efficient Multivariate |
---|
9 | * Factorization over Finite Fields" by L. Bernardin & M. Monagon. Division with |
---|
10 | * remainder is described in "Fast Recursive Division" by C. Burnikel and |
---|
11 | * J. Ziegler. Karatsuba multiplication is described in "Modern Computer |
---|
12 | * Algebra" by J. von zur Gathen and J. Gerhard. |
---|
13 | * |
---|
14 | * @author Martin Lee |
---|
15 | * |
---|
16 | * @internal @version \$Id$ |
---|
17 | * |
---|
18 | **/ |
---|
19 | /*****************************************************************************/ |
---|
20 | |
---|
21 | #include "config.h" |
---|
22 | |
---|
23 | #include "cf_assert.h" |
---|
24 | #include "debug.h" |
---|
25 | #include "timing.h" |
---|
26 | |
---|
27 | #include "algext.h" |
---|
28 | #include "facHensel.h" |
---|
29 | #include "facMul.h" |
---|
30 | #include "fac_util.h" |
---|
31 | #include "cf_algorithm.h" |
---|
32 | #include "cf_primes.h" |
---|
33 | |
---|
34 | #ifdef HAVE_NTL |
---|
35 | #include <NTL/lzz_pEX.h> |
---|
36 | #include "NTLconvert.h" |
---|
37 | |
---|
38 | static |
---|
39 | CFList productsNTL (const CFList& factors, const CanonicalForm& M) |
---|
40 | { |
---|
41 | zz_p::init (getCharacteristic()); |
---|
42 | zz_pX NTLMipo= convertFacCF2NTLzzpX (M); |
---|
43 | zz_pE::init (NTLMipo); |
---|
44 | zz_pEX prod; |
---|
45 | vec_zz_pEX v; |
---|
46 | v.SetLength (factors.length()); |
---|
47 | int j= 0; |
---|
48 | for (CFListIterator i= factors; i.hasItem(); i++, j++) |
---|
49 | { |
---|
50 | if (i.getItem().inCoeffDomain()) |
---|
51 | v[j]= to_zz_pEX (to_zz_pE (convertFacCF2NTLzzpX (i.getItem()))); |
---|
52 | else |
---|
53 | v[j]= convertFacCF2NTLzz_pEX (i.getItem(), NTLMipo); |
---|
54 | } |
---|
55 | CFList result; |
---|
56 | Variable x= Variable (1); |
---|
57 | for (int j= 0; j < factors.length(); j++) |
---|
58 | { |
---|
59 | int k= 0; |
---|
60 | set(prod); |
---|
61 | for (int i= 0; i < factors.length(); i++, k++) |
---|
62 | { |
---|
63 | if (k == j) |
---|
64 | continue; |
---|
65 | prod *= v[i]; |
---|
66 | } |
---|
67 | result.append (convertNTLzz_pEX2CF (prod, x, M.mvar())); |
---|
68 | } |
---|
69 | return result; |
---|
70 | } |
---|
71 | |
---|
72 | static |
---|
73 | void tryDiophantine (CFList& result, const CanonicalForm& F, |
---|
74 | const CFList& factors, const CanonicalForm& M, bool& fail) |
---|
75 | { |
---|
76 | ASSERT (M.isUnivariate(), "expected univariate poly"); |
---|
77 | |
---|
78 | CFList bufFactors= factors; |
---|
79 | bufFactors.removeFirst(); |
---|
80 | bufFactors.insert (factors.getFirst () (0,2)); |
---|
81 | CanonicalForm inv, leadingCoeff= Lc (F); |
---|
82 | CFListIterator i= bufFactors; |
---|
83 | if (bufFactors.getFirst().inCoeffDomain()) |
---|
84 | { |
---|
85 | if (i.hasItem()) |
---|
86 | i++; |
---|
87 | } |
---|
88 | for (; i.hasItem(); i++) |
---|
89 | { |
---|
90 | tryInvert (Lc (i.getItem()), M, inv ,fail); |
---|
91 | if (fail) |
---|
92 | return; |
---|
93 | i.getItem()= reduce (i.getItem()*inv, M); |
---|
94 | } |
---|
95 | bufFactors= productsNTL (bufFactors, M); |
---|
96 | |
---|
97 | CanonicalForm buf1, buf2, buf3, S, T; |
---|
98 | i= bufFactors; |
---|
99 | if (i.hasItem()) |
---|
100 | i++; |
---|
101 | buf1= bufFactors.getFirst(); |
---|
102 | buf2= i.getItem(); |
---|
103 | tryExtgcd (buf1, buf2, M, buf3, S, T, fail); |
---|
104 | if (fail) |
---|
105 | return; |
---|
106 | result.append (S); |
---|
107 | result.append (T); |
---|
108 | if (i.hasItem()) |
---|
109 | i++; |
---|
110 | for (; i.hasItem(); i++) |
---|
111 | { |
---|
112 | buf1= i.getItem(); |
---|
113 | tryExtgcd (buf3, buf1, M, buf3, S, T, fail); |
---|
114 | if (fail) |
---|
115 | return; |
---|
116 | CFListIterator k= factors; |
---|
117 | for (CFListIterator j= result; j.hasItem(); j++, k++) |
---|
118 | { |
---|
119 | j.getItem() *= S; |
---|
120 | j.getItem()= mod (j.getItem(), k.getItem()); |
---|
121 | j.getItem()= reduce (j.getItem(), M); |
---|
122 | } |
---|
123 | result.append (T); |
---|
124 | } |
---|
125 | } |
---|
126 | |
---|
127 | static |
---|
128 | CFList mapinto (const CFList& L) |
---|
129 | { |
---|
130 | CFList result; |
---|
131 | for (CFListIterator i= L; i.hasItem(); i++) |
---|
132 | result.append (mapinto (i.getItem())); |
---|
133 | return result; |
---|
134 | } |
---|
135 | |
---|
136 | static |
---|
137 | int mod (const CFList& L, const CanonicalForm& p) |
---|
138 | { |
---|
139 | for (CFListIterator i= L; i.hasItem(); i++) |
---|
140 | { |
---|
141 | if (mod (i.getItem(), p) == 0) |
---|
142 | return 0; |
---|
143 | } |
---|
144 | return 1; |
---|
145 | } |
---|
146 | |
---|
147 | |
---|
148 | static void |
---|
149 | chineseRemainder (const CFList & x1, const CanonicalForm & q1, |
---|
150 | const CFList & x2, const CanonicalForm & q2, |
---|
151 | CFList & xnew, CanonicalForm & qnew) |
---|
152 | { |
---|
153 | ASSERT (x1.length() == x2.length(), "expected lists of equal length"); |
---|
154 | CanonicalForm tmp1, tmp2; |
---|
155 | CFListIterator j= x2; |
---|
156 | for (CFListIterator i= x1; i.hasItem() && j.hasItem(); i++, j++) |
---|
157 | { |
---|
158 | chineseRemainder (i.getItem(), q1, j.getItem(), q2, tmp1, tmp2); |
---|
159 | xnew.append (tmp1); |
---|
160 | } |
---|
161 | qnew= tmp2; |
---|
162 | } |
---|
163 | |
---|
164 | static |
---|
165 | CFList Farey (const CFList& L, const CanonicalForm& q) |
---|
166 | { |
---|
167 | CFList result; |
---|
168 | for (CFListIterator i= L; i.hasItem(); i++) |
---|
169 | result.append (Farey (i.getItem(), q)); |
---|
170 | return result; |
---|
171 | } |
---|
172 | |
---|
173 | static |
---|
174 | CFList replacevar (const CFList& L, const Variable& a, const Variable& b) |
---|
175 | { |
---|
176 | CFList result; |
---|
177 | for (CFListIterator i= L; i.hasItem(); i++) |
---|
178 | result.append (replacevar (i.getItem(), a, b)); |
---|
179 | return result; |
---|
180 | } |
---|
181 | |
---|
182 | CFList |
---|
183 | modularDiophant (const CanonicalForm& f, const CFList& factors, |
---|
184 | const CanonicalForm& M) |
---|
185 | { |
---|
186 | bool save_rat=!isOn (SW_RATIONAL); |
---|
187 | On (SW_RATIONAL); |
---|
188 | CanonicalForm F= f*bCommonDen (f); |
---|
189 | CFList products= factors; |
---|
190 | for (CFListIterator i= products; i.hasItem(); i++) |
---|
191 | { |
---|
192 | if (products.getFirst().level() == 1) |
---|
193 | i.getItem() /= Lc (i.getItem()); |
---|
194 | i.getItem() *= bCommonDen (i.getItem()); |
---|
195 | } |
---|
196 | if (products.getFirst().level() == 1) |
---|
197 | products.insert (Lc (F)); |
---|
198 | CanonicalForm bound= maxNorm (F); |
---|
199 | CFList leadingCoeffs; |
---|
200 | leadingCoeffs.append (lc (F)); |
---|
201 | CanonicalForm dummy; |
---|
202 | for (CFListIterator i= products; i.hasItem(); i++) |
---|
203 | { |
---|
204 | leadingCoeffs.append (lc (i.getItem())); |
---|
205 | dummy= maxNorm (i.getItem()); |
---|
206 | bound= (dummy > bound) ? dummy : bound; |
---|
207 | } |
---|
208 | bound *= maxNorm (Lc (F))*maxNorm (Lc(F))*bound; |
---|
209 | bound *= bound*bound; |
---|
210 | bound= power (bound, degree (M)); |
---|
211 | bound *= power (CanonicalForm (2),degree (f)); |
---|
212 | CanonicalForm bufBound= bound; |
---|
213 | int i = cf_getNumBigPrimes() - 1; |
---|
214 | int p; |
---|
215 | CFList resultModP, result, newResult; |
---|
216 | CanonicalForm q (0), newQ; |
---|
217 | bool fail= false; |
---|
218 | Variable a= M.mvar(); |
---|
219 | Variable b= Variable (2); |
---|
220 | setReduce (M.mvar(), false); |
---|
221 | CanonicalForm mipo= bCommonDen (M)*M; |
---|
222 | Off (SW_RATIONAL); |
---|
223 | CanonicalForm modMipo; |
---|
224 | leadingCoeffs.append (lc (mipo)); |
---|
225 | CFList tmp1, tmp2; |
---|
226 | bool equal= false; |
---|
227 | int count= 0; |
---|
228 | do |
---|
229 | { |
---|
230 | p = cf_getBigPrime( i ); |
---|
231 | i--; |
---|
232 | while ( i >= 0 && mod( leadingCoeffs, p ) == 0) |
---|
233 | { |
---|
234 | p = cf_getBigPrime( i ); |
---|
235 | i--; |
---|
236 | } |
---|
237 | |
---|
238 | ASSERT (i >= 0, "ran out of primes"); //sic |
---|
239 | |
---|
240 | setCharacteristic (p); |
---|
241 | modMipo= mapinto (mipo); |
---|
242 | modMipo /= lc (modMipo); |
---|
243 | resultModP= CFList(); |
---|
244 | tryDiophantine (resultModP, mapinto (F), mapinto (products), modMipo, fail); |
---|
245 | setCharacteristic (0); |
---|
246 | if (fail) |
---|
247 | { |
---|
248 | fail= false; |
---|
249 | continue; |
---|
250 | } |
---|
251 | |
---|
252 | if ( q.isZero() ) |
---|
253 | { |
---|
254 | result= replacevar (mapinto(resultModP), a, b); |
---|
255 | q= p; |
---|
256 | } |
---|
257 | else |
---|
258 | { |
---|
259 | result= replacevar (result, a, b); |
---|
260 | newResult= CFList(); |
---|
261 | chineseRemainder( result, q, replacevar (mapinto (resultModP), a, b), |
---|
262 | p, newResult, newQ ); |
---|
263 | q= newQ; |
---|
264 | result= newResult; |
---|
265 | if (newQ > bound) |
---|
266 | { |
---|
267 | count++; |
---|
268 | tmp1= replacevar (Farey (result, q), b, a); |
---|
269 | if (tmp2.isEmpty()) |
---|
270 | tmp2= tmp1; |
---|
271 | else |
---|
272 | { |
---|
273 | equal= true; |
---|
274 | CFListIterator k= tmp1; |
---|
275 | for (CFListIterator j= tmp2; j.hasItem(); j++, k++) |
---|
276 | { |
---|
277 | if (j.getItem() != k.getItem()) |
---|
278 | equal= false; |
---|
279 | } |
---|
280 | if (!equal) |
---|
281 | tmp2= tmp1; |
---|
282 | } |
---|
283 | if (count > 2) |
---|
284 | { |
---|
285 | bound *= bufBound; |
---|
286 | equal= false; |
---|
287 | count= 0; |
---|
288 | } |
---|
289 | } |
---|
290 | if (newQ > bound && equal) |
---|
291 | { |
---|
292 | On( SW_RATIONAL ); |
---|
293 | CFList bufResult= result; |
---|
294 | result= tmp2; |
---|
295 | setReduce (M.mvar(), true); |
---|
296 | if (factors.getFirst().level() == 1) |
---|
297 | { |
---|
298 | result.removeFirst(); |
---|
299 | CFListIterator j= factors; |
---|
300 | CanonicalForm denf= bCommonDen (f); |
---|
301 | for (CFListIterator i= result; i.hasItem(); i++, j++) |
---|
302 | i.getItem() *= Lc (j.getItem())*denf; |
---|
303 | } |
---|
304 | if (factors.getFirst().level() != 1 && |
---|
305 | !bCommonDen (factors.getFirst()).isOne()) |
---|
306 | { |
---|
307 | CanonicalForm denFirst= bCommonDen (factors.getFirst()); |
---|
308 | for (CFListIterator i= result; i.hasItem(); i++) |
---|
309 | i.getItem() *= denFirst; |
---|
310 | } |
---|
311 | |
---|
312 | CanonicalForm test= 0; |
---|
313 | CFListIterator jj= factors; |
---|
314 | for (CFListIterator ii= result; ii.hasItem(); ii++, jj++) |
---|
315 | test += ii.getItem()*(f/jj.getItem()); |
---|
316 | if (!test.isOne()) |
---|
317 | { |
---|
318 | bound *= bufBound; |
---|
319 | equal= false; |
---|
320 | count= 0; |
---|
321 | setReduce (M.mvar(), false); |
---|
322 | result= bufResult; |
---|
323 | Off (SW_RATIONAL); |
---|
324 | } |
---|
325 | else |
---|
326 | break; |
---|
327 | } |
---|
328 | } |
---|
329 | } while (1); |
---|
330 | if (save_rat) Off(SW_RATIONAL); |
---|
331 | return result; |
---|
332 | } |
---|
333 | |
---|
334 | void sortList (CFList& list, const Variable& x) |
---|
335 | { |
---|
336 | int l= 1; |
---|
337 | int k= 1; |
---|
338 | CanonicalForm buf; |
---|
339 | CFListIterator m; |
---|
340 | for (CFListIterator i= list; l <= list.length(); i++, l++) |
---|
341 | { |
---|
342 | for (CFListIterator j= list; k <= list.length() - l; k++) |
---|
343 | { |
---|
344 | m= j; |
---|
345 | m++; |
---|
346 | if (degree (j.getItem(), x) > degree (m.getItem(), x)) |
---|
347 | { |
---|
348 | buf= m.getItem(); |
---|
349 | m.getItem()= j.getItem(); |
---|
350 | j.getItem()= buf; |
---|
351 | j++; |
---|
352 | j.getItem()= m.getItem(); |
---|
353 | } |
---|
354 | else |
---|
355 | j++; |
---|
356 | } |
---|
357 | k= 1; |
---|
358 | } |
---|
359 | } |
---|
360 | |
---|
361 | CFList diophantine (const CanonicalForm& F, const CFList& factors) |
---|
362 | { |
---|
363 | if (getCharacteristic() == 0) |
---|
364 | { |
---|
365 | Variable v; |
---|
366 | bool hasAlgVar= hasFirstAlgVar (F, v); |
---|
367 | for (CFListIterator i= factors; i.hasItem() && !hasAlgVar; i++) |
---|
368 | hasAlgVar= hasFirstAlgVar (i.getItem(), v); |
---|
369 | if (hasAlgVar) |
---|
370 | { |
---|
371 | CFList result= modularDiophant (F, factors, getMipo (v)); |
---|
372 | return result; |
---|
373 | } |
---|
374 | } |
---|
375 | |
---|
376 | CanonicalForm buf1, buf2, buf3, S, T; |
---|
377 | CFListIterator i= factors; |
---|
378 | CFList result; |
---|
379 | if (i.hasItem()) |
---|
380 | i++; |
---|
381 | buf1= F/factors.getFirst(); |
---|
382 | buf2= divNTL (F, i.getItem()); |
---|
383 | buf3= extgcd (buf1, buf2, S, T); |
---|
384 | result.append (S); |
---|
385 | result.append (T); |
---|
386 | if (i.hasItem()) |
---|
387 | i++; |
---|
388 | for (; i.hasItem(); i++) |
---|
389 | { |
---|
390 | buf1= divNTL (F, i.getItem()); |
---|
391 | buf3= extgcd (buf3, buf1, S, T); |
---|
392 | CFListIterator k= factors; |
---|
393 | for (CFListIterator j= result; j.hasItem(); j++, k++) |
---|
394 | { |
---|
395 | j.getItem()= mulNTL (j.getItem(), S); |
---|
396 | j.getItem()= modNTL (j.getItem(), k.getItem()); |
---|
397 | } |
---|
398 | result.append (T); |
---|
399 | } |
---|
400 | return result; |
---|
401 | } |
---|
402 | |
---|
403 | void |
---|
404 | henselStep12 (const CanonicalForm& F, const CFList& factors, |
---|
405 | CFArray& bufFactors, const CFList& diophant, CFMatrix& M, |
---|
406 | CFArray& Pi, int j) |
---|
407 | { |
---|
408 | CanonicalForm E; |
---|
409 | CanonicalForm xToJ= power (F.mvar(), j); |
---|
410 | Variable x= F.mvar(); |
---|
411 | // compute the error |
---|
412 | if (j == 1) |
---|
413 | E= F[j]; |
---|
414 | else |
---|
415 | { |
---|
416 | if (degree (Pi [factors.length() - 2], x) > 0) |
---|
417 | E= F[j] - Pi [factors.length() - 2] [j]; |
---|
418 | else |
---|
419 | E= F[j]; |
---|
420 | } |
---|
421 | |
---|
422 | CFArray buf= CFArray (diophant.length()); |
---|
423 | bufFactors[0]= mod (factors.getFirst(), power (F.mvar(), j + 1)); |
---|
424 | int k= 0; |
---|
425 | CanonicalForm remainder; |
---|
426 | // actual lifting |
---|
427 | for (CFListIterator i= diophant; i.hasItem(); i++, k++) |
---|
428 | { |
---|
429 | if (degree (bufFactors[k], x) > 0) |
---|
430 | { |
---|
431 | if (k > 0) |
---|
432 | remainder= modNTL (E, bufFactors[k] [0]); |
---|
433 | else |
---|
434 | remainder= E; |
---|
435 | } |
---|
436 | else |
---|
437 | remainder= modNTL (E, bufFactors[k]); |
---|
438 | |
---|
439 | buf[k]= mulNTL (i.getItem(), remainder); |
---|
440 | if (degree (bufFactors[k], x) > 0) |
---|
441 | buf[k]= modNTL (buf[k], bufFactors[k] [0]); |
---|
442 | else |
---|
443 | buf[k]= modNTL (buf[k], bufFactors[k]); |
---|
444 | } |
---|
445 | for (k= 1; k < factors.length(); k++) |
---|
446 | bufFactors[k] += xToJ*buf[k]; |
---|
447 | |
---|
448 | // update Pi [0] |
---|
449 | int degBuf0= degree (bufFactors[0], x); |
---|
450 | int degBuf1= degree (bufFactors[1], x); |
---|
451 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
452 | M (j + 1, 1)= mulNTL (bufFactors[0] [j], bufFactors[1] [j]); |
---|
453 | CanonicalForm uIZeroJ; |
---|
454 | if (j == 1) |
---|
455 | { |
---|
456 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
457 | uIZeroJ= mulNTL ((bufFactors[0] [0] + bufFactors[0] [j]), |
---|
458 | (bufFactors[1] [0] + buf[1])) - M(1, 1) - M(j + 1, 1); |
---|
459 | else if (degBuf0 > 0) |
---|
460 | uIZeroJ= mulNTL (bufFactors[0] [j], bufFactors[1]); |
---|
461 | else if (degBuf1 > 0) |
---|
462 | uIZeroJ= mulNTL (bufFactors[0], buf[1]); |
---|
463 | else |
---|
464 | uIZeroJ= 0; |
---|
465 | Pi [0] += xToJ*uIZeroJ; |
---|
466 | } |
---|
467 | else |
---|
468 | { |
---|
469 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
470 | uIZeroJ= mulNTL ((bufFactors[0] [0] + bufFactors[0] [j]), |
---|
471 | (bufFactors[1] [0] + buf[1])) - M(1, 1) - M(j + 1, 1); |
---|
472 | else if (degBuf0 > 0) |
---|
473 | uIZeroJ= mulNTL (bufFactors[0] [j], bufFactors[1]); |
---|
474 | else if (degBuf1 > 0) |
---|
475 | uIZeroJ= mulNTL (bufFactors[0], buf[1]); |
---|
476 | else |
---|
477 | uIZeroJ= 0; |
---|
478 | Pi [0] += xToJ*uIZeroJ; |
---|
479 | } |
---|
480 | CFArray tmp= CFArray (factors.length() - 1); |
---|
481 | for (k= 0; k < factors.length() - 1; k++) |
---|
482 | tmp[k]= 0; |
---|
483 | CFIterator one, two; |
---|
484 | one= bufFactors [0]; |
---|
485 | two= bufFactors [1]; |
---|
486 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
487 | { |
---|
488 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
489 | { |
---|
490 | if (k != j - k + 1) |
---|
491 | { |
---|
492 | if ((one.hasTerms() && one.exp() == j - k + 1) |
---|
493 | && (two.hasTerms() && two.exp() == j - k + 1)) |
---|
494 | { |
---|
495 | tmp[0] += mulNTL ((bufFactors[0][k]+one.coeff()), (bufFactors[1][k]+ |
---|
496 | two.coeff())) - M (k + 1, 1) - M (j - k + 2, 1); |
---|
497 | one++; |
---|
498 | two++; |
---|
499 | } |
---|
500 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
501 | { |
---|
502 | tmp[0] += mulNTL ((bufFactors[0][k]+one.coeff()), bufFactors[1][k])- |
---|
503 | M (k + 1, 1); |
---|
504 | one++; |
---|
505 | } |
---|
506 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
507 | { |
---|
508 | tmp[0] += mulNTL (bufFactors[0][k], (bufFactors[1][k]+two.coeff()))- |
---|
509 | M (k + 1, 1); |
---|
510 | two++; |
---|
511 | } |
---|
512 | } |
---|
513 | else |
---|
514 | { |
---|
515 | tmp[0] += M (k + 1, 1); |
---|
516 | } |
---|
517 | } |
---|
518 | } |
---|
519 | Pi [0] += tmp[0]*xToJ*F.mvar(); |
---|
520 | |
---|
521 | // update Pi [l] |
---|
522 | int degPi, degBuf; |
---|
523 | for (int l= 1; l < factors.length() - 1; l++) |
---|
524 | { |
---|
525 | degPi= degree (Pi [l - 1], x); |
---|
526 | degBuf= degree (bufFactors[l + 1], x); |
---|
527 | if (degPi > 0 && degBuf > 0) |
---|
528 | M (j + 1, l + 1)= mulNTL (Pi [l - 1] [j], bufFactors[l + 1] [j]); |
---|
529 | if (j == 1) |
---|
530 | { |
---|
531 | if (degPi > 0 && degBuf > 0) |
---|
532 | Pi [l] += xToJ*(mulNTL (Pi [l - 1] [0] + Pi [l - 1] [j], |
---|
533 | bufFactors[l + 1] [0] + buf[l + 1]) - M (j + 1, l +1) - |
---|
534 | M (1, l + 1)); |
---|
535 | else if (degPi > 0) |
---|
536 | Pi [l] += xToJ*(mulNTL (Pi [l - 1] [j], bufFactors[l + 1])); |
---|
537 | else if (degBuf > 0) |
---|
538 | Pi [l] += xToJ*(mulNTL (Pi [l - 1], buf[l + 1])); |
---|
539 | } |
---|
540 | else |
---|
541 | { |
---|
542 | if (degPi > 0 && degBuf > 0) |
---|
543 | { |
---|
544 | uIZeroJ= mulNTL (uIZeroJ, bufFactors [l + 1] [0]); |
---|
545 | uIZeroJ += mulNTL (Pi [l - 1] [0], buf [l + 1]); |
---|
546 | } |
---|
547 | else if (degPi > 0) |
---|
548 | uIZeroJ= mulNTL (uIZeroJ, bufFactors [l + 1]); |
---|
549 | else if (degBuf > 0) |
---|
550 | { |
---|
551 | uIZeroJ= mulNTL (uIZeroJ, bufFactors [l + 1] [0]); |
---|
552 | uIZeroJ += mulNTL (Pi [l - 1], buf[l + 1]); |
---|
553 | } |
---|
554 | Pi[l] += xToJ*uIZeroJ; |
---|
555 | } |
---|
556 | one= bufFactors [l + 1]; |
---|
557 | two= Pi [l - 1]; |
---|
558 | if (two.hasTerms() && two.exp() == j + 1) |
---|
559 | { |
---|
560 | if (degBuf > 0 && degPi > 0) |
---|
561 | { |
---|
562 | tmp[l] += mulNTL (two.coeff(), bufFactors[l + 1][0]); |
---|
563 | two++; |
---|
564 | } |
---|
565 | else if (degPi > 0) |
---|
566 | { |
---|
567 | tmp[l] += mulNTL (two.coeff(), bufFactors[l + 1]); |
---|
568 | two++; |
---|
569 | } |
---|
570 | } |
---|
571 | if (degBuf > 0 && degPi > 0) |
---|
572 | { |
---|
573 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
574 | { |
---|
575 | if (k != j - k + 1) |
---|
576 | { |
---|
577 | if ((one.hasTerms() && one.exp() == j - k + 1) && |
---|
578 | (two.hasTerms() && two.exp() == j - k + 1)) |
---|
579 | { |
---|
580 | tmp[l] += mulNTL ((bufFactors[l+1][k] + one.coeff()), (Pi[l-1][k] + |
---|
581 | two.coeff())) - M (k + 1, l + 1) - M (j - k + 2, l + 1); |
---|
582 | one++; |
---|
583 | two++; |
---|
584 | } |
---|
585 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
586 | { |
---|
587 | tmp[l] += mulNTL ((bufFactors[l+1][k]+one.coeff()), Pi[l-1][k]) - |
---|
588 | M (k + 1, l + 1); |
---|
589 | one++; |
---|
590 | } |
---|
591 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
592 | { |
---|
593 | tmp[l] += mulNTL (bufFactors[l+1][k], (Pi[l-1][k] + two.coeff())) - |
---|
594 | M (k + 1, l + 1); |
---|
595 | two++; |
---|
596 | } |
---|
597 | } |
---|
598 | else |
---|
599 | tmp[l] += M (k + 1, l + 1); |
---|
600 | } |
---|
601 | } |
---|
602 | Pi[l] += tmp[l]*xToJ*F.mvar(); |
---|
603 | } |
---|
604 | return; |
---|
605 | } |
---|
606 | |
---|
607 | void |
---|
608 | henselLift12 (const CanonicalForm& F, CFList& factors, int l, CFArray& Pi, |
---|
609 | CFList& diophant, CFMatrix& M, bool sort) |
---|
610 | { |
---|
611 | if (sort) |
---|
612 | sortList (factors, Variable (1)); |
---|
613 | Pi= CFArray (factors.length() - 1); |
---|
614 | CFListIterator j= factors; |
---|
615 | diophant= diophantine (F[0], factors); |
---|
616 | DEBOUTLN (cerr, "diophant= " << diophant); |
---|
617 | j++; |
---|
618 | Pi [0]= mulNTL (j.getItem(), mod (factors.getFirst(), F.mvar())); |
---|
619 | M (1, 1)= Pi [0]; |
---|
620 | int i= 1; |
---|
621 | if (j.hasItem()) |
---|
622 | j++; |
---|
623 | for (; j.hasItem(); j++, i++) |
---|
624 | { |
---|
625 | Pi [i]= mulNTL (Pi [i - 1], j.getItem()); |
---|
626 | M (1, i + 1)= Pi [i]; |
---|
627 | } |
---|
628 | CFArray bufFactors= CFArray (factors.length()); |
---|
629 | i= 0; |
---|
630 | for (CFListIterator k= factors; k.hasItem(); i++, k++) |
---|
631 | { |
---|
632 | if (i == 0) |
---|
633 | bufFactors[i]= mod (k.getItem(), F.mvar()); |
---|
634 | else |
---|
635 | bufFactors[i]= k.getItem(); |
---|
636 | } |
---|
637 | for (i= 1; i < l; i++) |
---|
638 | henselStep12 (F, factors, bufFactors, diophant, M, Pi, i); |
---|
639 | |
---|
640 | CFListIterator k= factors; |
---|
641 | for (i= 0; i < factors.length (); i++, k++) |
---|
642 | k.getItem()= bufFactors[i]; |
---|
643 | factors.removeFirst(); |
---|
644 | return; |
---|
645 | } |
---|
646 | |
---|
647 | void |
---|
648 | henselLiftResume12 (const CanonicalForm& F, CFList& factors, int start, int |
---|
649 | end, CFArray& Pi, const CFList& diophant, CFMatrix& M) |
---|
650 | { |
---|
651 | CFArray bufFactors= CFArray (factors.length()); |
---|
652 | int i= 0; |
---|
653 | CanonicalForm xToStart= power (F.mvar(), start); |
---|
654 | for (CFListIterator k= factors; k.hasItem(); k++, i++) |
---|
655 | { |
---|
656 | if (i == 0) |
---|
657 | bufFactors[i]= mod (k.getItem(), xToStart); |
---|
658 | else |
---|
659 | bufFactors[i]= k.getItem(); |
---|
660 | } |
---|
661 | for (i= start; i < end; i++) |
---|
662 | henselStep12 (F, factors, bufFactors, diophant, M, Pi, i); |
---|
663 | |
---|
664 | CFListIterator k= factors; |
---|
665 | for (i= 0; i < factors.length(); k++, i++) |
---|
666 | k.getItem()= bufFactors [i]; |
---|
667 | factors.removeFirst(); |
---|
668 | return; |
---|
669 | } |
---|
670 | |
---|
671 | CFList |
---|
672 | biDiophantine (const CanonicalForm& F, const CFList& factors, int d) |
---|
673 | { |
---|
674 | Variable y= F.mvar(); |
---|
675 | CFList result; |
---|
676 | if (y.level() == 1) |
---|
677 | { |
---|
678 | result= diophantine (F, factors); |
---|
679 | return result; |
---|
680 | } |
---|
681 | else |
---|
682 | { |
---|
683 | CFList buf= factors; |
---|
684 | for (CFListIterator i= buf; i.hasItem(); i++) |
---|
685 | i.getItem()= mod (i.getItem(), y); |
---|
686 | CanonicalForm A= mod (F, y); |
---|
687 | int bufD= 1; |
---|
688 | CFList recResult= biDiophantine (A, buf, bufD); |
---|
689 | CanonicalForm e= 1; |
---|
690 | CFList p; |
---|
691 | CFArray bufFactors= CFArray (factors.length()); |
---|
692 | CanonicalForm yToD= power (y, d); |
---|
693 | int k= 0; |
---|
694 | for (CFListIterator i= factors; i.hasItem(); i++, k++) |
---|
695 | { |
---|
696 | bufFactors [k]= i.getItem(); |
---|
697 | } |
---|
698 | CanonicalForm b, quot; |
---|
699 | for (k= 0; k < factors.length(); k++) //TODO compute b's faster |
---|
700 | { |
---|
701 | b= 1; |
---|
702 | if (fdivides (bufFactors[k], F, quot)) |
---|
703 | b= quot; |
---|
704 | else |
---|
705 | { |
---|
706 | for (int l= 0; l < factors.length(); l++) |
---|
707 | { |
---|
708 | if (l == k) |
---|
709 | continue; |
---|
710 | else |
---|
711 | { |
---|
712 | b= mulMod2 (b, bufFactors[l], yToD); |
---|
713 | } |
---|
714 | } |
---|
715 | } |
---|
716 | p.append (b); |
---|
717 | } |
---|
718 | |
---|
719 | CFListIterator j= p; |
---|
720 | for (CFListIterator i= recResult; i.hasItem(); i++, j++) |
---|
721 | e -= i.getItem()*j.getItem(); |
---|
722 | |
---|
723 | if (e.isZero()) |
---|
724 | return recResult; |
---|
725 | CanonicalForm coeffE; |
---|
726 | CFList s; |
---|
727 | result= recResult; |
---|
728 | CanonicalForm g; |
---|
729 | for (int i= 1; i < d; i++) |
---|
730 | { |
---|
731 | if (degree (e, y) > 0) |
---|
732 | coeffE= e[i]; |
---|
733 | else |
---|
734 | coeffE= 0; |
---|
735 | if (!coeffE.isZero()) |
---|
736 | { |
---|
737 | CFListIterator k= result; |
---|
738 | CFListIterator l= p; |
---|
739 | int ii= 0; |
---|
740 | j= recResult; |
---|
741 | for (; j.hasItem(); j++, k++, l++, ii++) |
---|
742 | { |
---|
743 | g= coeffE*j.getItem(); |
---|
744 | if (degree (bufFactors[ii], y) <= 0) |
---|
745 | g= mod (g, bufFactors[ii]); |
---|
746 | else |
---|
747 | g= mod (g, bufFactors[ii][0]); |
---|
748 | k.getItem() += g*power (y, i); |
---|
749 | e -= mulMod2 (g*power(y, i), l.getItem(), yToD); |
---|
750 | DEBOUTLN (cerr, "mod (e, power (y, i + 1))= " << |
---|
751 | mod (e, power (y, i + 1))); |
---|
752 | } |
---|
753 | } |
---|
754 | if (e.isZero()) |
---|
755 | break; |
---|
756 | } |
---|
757 | |
---|
758 | DEBOUTLN (cerr, "mod (e, y)= " << mod (e, y)); |
---|
759 | |
---|
760 | #ifdef DEBUGOUTPUT |
---|
761 | CanonicalForm test= 0; |
---|
762 | j= p; |
---|
763 | for (CFListIterator i= result; i.hasItem(); i++, j++) |
---|
764 | test += mod (i.getItem()*j.getItem(), power (y, d)); |
---|
765 | DEBOUTLN (cerr, "test= " << test); |
---|
766 | #endif |
---|
767 | return result; |
---|
768 | } |
---|
769 | } |
---|
770 | |
---|
771 | CFList |
---|
772 | multiRecDiophantine (const CanonicalForm& F, const CFList& factors, |
---|
773 | const CFList& recResult, const CFList& M, int d) |
---|
774 | { |
---|
775 | Variable y= F.mvar(); |
---|
776 | CFList result; |
---|
777 | CFListIterator i; |
---|
778 | CanonicalForm e= 1; |
---|
779 | CFListIterator j= factors; |
---|
780 | CFList p; |
---|
781 | CFArray bufFactors= CFArray (factors.length()); |
---|
782 | CanonicalForm yToD= power (y, d); |
---|
783 | int k= 0; |
---|
784 | for (CFListIterator i= factors; i.hasItem(); i++, k++) |
---|
785 | bufFactors [k]= i.getItem(); |
---|
786 | CanonicalForm b, quot; |
---|
787 | CFList buf= M; |
---|
788 | buf.removeLast(); |
---|
789 | buf.append (yToD); |
---|
790 | for (k= 0; k < factors.length(); k++) //TODO compute b's faster |
---|
791 | { |
---|
792 | b= 1; |
---|
793 | if (fdivides (bufFactors[k], F, quot)) |
---|
794 | b= quot; |
---|
795 | else |
---|
796 | { |
---|
797 | for (int l= 0; l < factors.length(); l++) |
---|
798 | { |
---|
799 | if (l == k) |
---|
800 | continue; |
---|
801 | else |
---|
802 | { |
---|
803 | b= mulMod (b, bufFactors[l], buf); |
---|
804 | } |
---|
805 | } |
---|
806 | } |
---|
807 | p.append (b); |
---|
808 | } |
---|
809 | j= p; |
---|
810 | for (CFListIterator i= recResult; i.hasItem(); i++, j++) |
---|
811 | e -= mulMod (i.getItem(), j.getItem(), M); |
---|
812 | |
---|
813 | if (e.isZero()) |
---|
814 | return recResult; |
---|
815 | CanonicalForm coeffE; |
---|
816 | CFList s; |
---|
817 | result= recResult; |
---|
818 | CanonicalForm g; |
---|
819 | for (int i= 1; i < d; i++) |
---|
820 | { |
---|
821 | if (degree (e, y) > 0) |
---|
822 | coeffE= e[i]; |
---|
823 | else |
---|
824 | coeffE= 0; |
---|
825 | if (!coeffE.isZero()) |
---|
826 | { |
---|
827 | CFListIterator k= result; |
---|
828 | CFListIterator l= p; |
---|
829 | j= recResult; |
---|
830 | int ii= 0; |
---|
831 | CanonicalForm dummy; |
---|
832 | for (; j.hasItem(); j++, k++, l++, ii++) |
---|
833 | { |
---|
834 | g= mulMod (coeffE, j.getItem(), M); |
---|
835 | if (degree (bufFactors[ii], y) <= 0) |
---|
836 | divrem (g, mod (bufFactors[ii], Variable (y.level() - 1)), dummy, |
---|
837 | g, M); |
---|
838 | else |
---|
839 | divrem (g, bufFactors[ii][0], dummy, g, M); |
---|
840 | k.getItem() += g*power (y, i); |
---|
841 | e -= mulMod (g*power (y, i), l.getItem(), M); |
---|
842 | } |
---|
843 | } |
---|
844 | |
---|
845 | if (e.isZero()) |
---|
846 | break; |
---|
847 | } |
---|
848 | |
---|
849 | #ifdef DEBUGOUTPUT |
---|
850 | CanonicalForm test= 0; |
---|
851 | j= p; |
---|
852 | for (CFListIterator i= result; i.hasItem(); i++, j++) |
---|
853 | test += mod (i.getItem()*j.getItem(), power (y, d)); |
---|
854 | DEBOUTLN (cerr, "test= " << test); |
---|
855 | #endif |
---|
856 | return result; |
---|
857 | } |
---|
858 | |
---|
859 | static |
---|
860 | void |
---|
861 | henselStep (const CanonicalForm& F, const CFList& factors, CFArray& bufFactors, |
---|
862 | const CFList& diophant, CFMatrix& M, CFArray& Pi, int j, |
---|
863 | const CFList& MOD) |
---|
864 | { |
---|
865 | CanonicalForm E; |
---|
866 | CanonicalForm xToJ= power (F.mvar(), j); |
---|
867 | Variable x= F.mvar(); |
---|
868 | // compute the error |
---|
869 | if (j == 1) |
---|
870 | { |
---|
871 | E= F[j]; |
---|
872 | #ifdef DEBUGOUTPUT |
---|
873 | CanonicalForm test= 1; |
---|
874 | for (int i= 0; i < factors.length(); i++) |
---|
875 | { |
---|
876 | if (i == 0) |
---|
877 | test= mulMod (test, mod (bufFactors [i], xToJ), MOD); |
---|
878 | else |
---|
879 | test= mulMod (test, bufFactors[i], MOD); |
---|
880 | } |
---|
881 | CanonicalForm test2= mod (F-test, xToJ); |
---|
882 | |
---|
883 | test2= mod (test2, MOD); |
---|
884 | DEBOUTLN (cerr, "test= " << test2); |
---|
885 | #endif |
---|
886 | } |
---|
887 | else |
---|
888 | { |
---|
889 | #ifdef DEBUGOUTPUT |
---|
890 | CanonicalForm test= 1; |
---|
891 | for (int i= 0; i < factors.length(); i++) |
---|
892 | { |
---|
893 | if (i == 0) |
---|
894 | test *= mod (bufFactors [i], power (x, j)); |
---|
895 | else |
---|
896 | test *= bufFactors[i]; |
---|
897 | } |
---|
898 | test= mod (test, power (x, j)); |
---|
899 | test= mod (test, MOD); |
---|
900 | CanonicalForm test2= mod (F, power (x, j - 1)) - mod (test, power (x, j-1)); |
---|
901 | DEBOUTLN (cerr, "test= " << test2); |
---|
902 | #endif |
---|
903 | |
---|
904 | if (degree (Pi [factors.length() - 2], x) > 0) |
---|
905 | E= F[j] - Pi [factors.length() - 2] [j]; |
---|
906 | else |
---|
907 | E= F[j]; |
---|
908 | } |
---|
909 | |
---|
910 | CFArray buf= CFArray (diophant.length()); |
---|
911 | bufFactors[0]= mod (factors.getFirst(), power (F.mvar(), j + 1)); |
---|
912 | int k= 0; |
---|
913 | // actual lifting |
---|
914 | CanonicalForm dummy, rest1; |
---|
915 | for (CFListIterator i= diophant; i.hasItem(); i++, k++) |
---|
916 | { |
---|
917 | if (degree (bufFactors[k], x) > 0) |
---|
918 | { |
---|
919 | if (k > 0) |
---|
920 | divrem (E, bufFactors[k] [0], dummy, rest1, MOD); |
---|
921 | else |
---|
922 | rest1= E; |
---|
923 | } |
---|
924 | else |
---|
925 | divrem (E, bufFactors[k], dummy, rest1, MOD); |
---|
926 | |
---|
927 | buf[k]= mulMod (i.getItem(), rest1, MOD); |
---|
928 | |
---|
929 | if (degree (bufFactors[k], x) > 0) |
---|
930 | divrem (buf[k], bufFactors[k] [0], dummy, buf[k], MOD); |
---|
931 | else |
---|
932 | divrem (buf[k], bufFactors[k], dummy, buf[k], MOD); |
---|
933 | } |
---|
934 | for (k= 1; k < factors.length(); k++) |
---|
935 | bufFactors[k] += xToJ*buf[k]; |
---|
936 | |
---|
937 | // update Pi [0] |
---|
938 | int degBuf0= degree (bufFactors[0], x); |
---|
939 | int degBuf1= degree (bufFactors[1], x); |
---|
940 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
941 | M (j + 1, 1)= mulMod (bufFactors[0] [j], bufFactors[1] [j], MOD); |
---|
942 | CanonicalForm uIZeroJ; |
---|
943 | if (j == 1) |
---|
944 | { |
---|
945 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
946 | uIZeroJ= mulMod ((bufFactors[0] [0] + bufFactors[0] [j]), |
---|
947 | (bufFactors[1] [0] + buf[1]), MOD) - M(1, 1) - M(j + 1, 1); |
---|
948 | else if (degBuf0 > 0) |
---|
949 | uIZeroJ= mulMod (bufFactors[0] [j], bufFactors[1], MOD); |
---|
950 | else if (degBuf1 > 0) |
---|
951 | uIZeroJ= mulMod (bufFactors[0], buf[1], MOD); |
---|
952 | else |
---|
953 | uIZeroJ= 0; |
---|
954 | Pi [0] += xToJ*uIZeroJ; |
---|
955 | } |
---|
956 | else |
---|
957 | { |
---|
958 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
959 | uIZeroJ= mulMod ((bufFactors[0] [0] + bufFactors[0] [j]), |
---|
960 | (bufFactors[1] [0] + buf[1]), MOD) - M(1, 1) - M(j + 1, 1); |
---|
961 | else if (degBuf0 > 0) |
---|
962 | uIZeroJ= mulMod (bufFactors[0] [j], bufFactors[1], MOD); |
---|
963 | else if (degBuf1 > 0) |
---|
964 | uIZeroJ= mulMod (bufFactors[0], buf[1], MOD); |
---|
965 | else |
---|
966 | uIZeroJ= 0; |
---|
967 | Pi [0] += xToJ*uIZeroJ; |
---|
968 | } |
---|
969 | |
---|
970 | CFArray tmp= CFArray (factors.length() - 1); |
---|
971 | for (k= 0; k < factors.length() - 1; k++) |
---|
972 | tmp[k]= 0; |
---|
973 | CFIterator one, two; |
---|
974 | one= bufFactors [0]; |
---|
975 | two= bufFactors [1]; |
---|
976 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
977 | { |
---|
978 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
979 | { |
---|
980 | if (k != j - k + 1) |
---|
981 | { |
---|
982 | if ((one.hasTerms() && one.exp() == j - k + 1) && |
---|
983 | (two.hasTerms() && two.exp() == j - k + 1)) |
---|
984 | { |
---|
985 | tmp[0] += mulMod ((bufFactors[0] [k] + one.coeff()), |
---|
986 | (bufFactors[1] [k] + two.coeff()), MOD) - M (k + 1, 1) - |
---|
987 | M (j - k + 2, 1); |
---|
988 | one++; |
---|
989 | two++; |
---|
990 | } |
---|
991 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
992 | { |
---|
993 | tmp[0] += mulMod ((bufFactors[0] [k] + one.coeff()), |
---|
994 | bufFactors[1] [k], MOD) - M (k + 1, 1); |
---|
995 | one++; |
---|
996 | } |
---|
997 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
998 | { |
---|
999 | tmp[0] += mulMod (bufFactors[0] [k], (bufFactors[1] [k] + |
---|
1000 | two.coeff()), MOD) - M (k + 1, 1); |
---|
1001 | two++; |
---|
1002 | } |
---|
1003 | } |
---|
1004 | else |
---|
1005 | { |
---|
1006 | tmp[0] += M (k + 1, 1); |
---|
1007 | } |
---|
1008 | } |
---|
1009 | } |
---|
1010 | Pi [0] += tmp[0]*xToJ*F.mvar(); |
---|
1011 | |
---|
1012 | // update Pi [l] |
---|
1013 | int degPi, degBuf; |
---|
1014 | for (int l= 1; l < factors.length() - 1; l++) |
---|
1015 | { |
---|
1016 | degPi= degree (Pi [l - 1], x); |
---|
1017 | degBuf= degree (bufFactors[l + 1], x); |
---|
1018 | if (degPi > 0 && degBuf > 0) |
---|
1019 | M (j + 1, l + 1)= mulMod (Pi [l - 1] [j], bufFactors[l + 1] [j], MOD); |
---|
1020 | if (j == 1) |
---|
1021 | { |
---|
1022 | if (degPi > 0 && degBuf > 0) |
---|
1023 | Pi [l] += xToJ*(mulMod ((Pi [l - 1] [0] + Pi [l - 1] [j]), |
---|
1024 | (bufFactors[l + 1] [0] + buf[l + 1]), MOD) - M (j + 1, l +1)- |
---|
1025 | M (1, l + 1)); |
---|
1026 | else if (degPi > 0) |
---|
1027 | Pi [l] += xToJ*(mulMod (Pi [l - 1] [j], bufFactors[l + 1], MOD)); |
---|
1028 | else if (degBuf > 0) |
---|
1029 | Pi [l] += xToJ*(mulMod (Pi [l - 1], buf[l + 1], MOD)); |
---|
1030 | } |
---|
1031 | else |
---|
1032 | { |
---|
1033 | if (degPi > 0 && degBuf > 0) |
---|
1034 | { |
---|
1035 | uIZeroJ= mulMod (uIZeroJ, bufFactors [l + 1] [0], MOD); |
---|
1036 | uIZeroJ += mulMod (Pi [l - 1] [0], buf [l + 1], MOD); |
---|
1037 | } |
---|
1038 | else if (degPi > 0) |
---|
1039 | uIZeroJ= mulMod (uIZeroJ, bufFactors [l + 1], MOD); |
---|
1040 | else if (degBuf > 0) |
---|
1041 | { |
---|
1042 | uIZeroJ= mulMod (uIZeroJ, bufFactors [l + 1] [0], MOD); |
---|
1043 | uIZeroJ += mulMod (Pi [l - 1], buf[l + 1], MOD); |
---|
1044 | } |
---|
1045 | Pi[l] += xToJ*uIZeroJ; |
---|
1046 | } |
---|
1047 | one= bufFactors [l + 1]; |
---|
1048 | two= Pi [l - 1]; |
---|
1049 | if (two.hasTerms() && two.exp() == j + 1) |
---|
1050 | { |
---|
1051 | if (degBuf > 0 && degPi > 0) |
---|
1052 | { |
---|
1053 | tmp[l] += mulMod (two.coeff(), bufFactors[l + 1][0], MOD); |
---|
1054 | two++; |
---|
1055 | } |
---|
1056 | else if (degPi > 0) |
---|
1057 | { |
---|
1058 | tmp[l] += mulMod (two.coeff(), bufFactors[l + 1], MOD); |
---|
1059 | two++; |
---|
1060 | } |
---|
1061 | } |
---|
1062 | if (degBuf > 0 && degPi > 0) |
---|
1063 | { |
---|
1064 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
1065 | { |
---|
1066 | if (k != j - k + 1) |
---|
1067 | { |
---|
1068 | if ((one.hasTerms() && one.exp() == j - k + 1) && |
---|
1069 | (two.hasTerms() && two.exp() == j - k + 1)) |
---|
1070 | { |
---|
1071 | tmp[l] += mulMod ((bufFactors[l + 1] [k] + one.coeff()), |
---|
1072 | (Pi[l - 1] [k] + two.coeff()), MOD) - M (k + 1, l + 1) - |
---|
1073 | M (j - k + 2, l + 1); |
---|
1074 | one++; |
---|
1075 | two++; |
---|
1076 | } |
---|
1077 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
1078 | { |
---|
1079 | tmp[l] += mulMod ((bufFactors[l + 1] [k] + one.coeff()), |
---|
1080 | Pi[l - 1] [k], MOD) - M (k + 1, l + 1); |
---|
1081 | one++; |
---|
1082 | } |
---|
1083 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
1084 | { |
---|
1085 | tmp[l] += mulMod (bufFactors[l + 1] [k], |
---|
1086 | (Pi[l - 1] [k] + two.coeff()), MOD) - M (k + 1, l + 1); |
---|
1087 | two++; |
---|
1088 | } |
---|
1089 | } |
---|
1090 | else |
---|
1091 | tmp[l] += M (k + 1, l + 1); |
---|
1092 | } |
---|
1093 | } |
---|
1094 | Pi[l] += tmp[l]*xToJ*F.mvar(); |
---|
1095 | } |
---|
1096 | |
---|
1097 | return; |
---|
1098 | } |
---|
1099 | |
---|
1100 | CFList |
---|
1101 | henselLift23 (const CFList& eval, const CFList& factors, int* l, CFList& |
---|
1102 | diophant, CFArray& Pi, CFMatrix& M) |
---|
1103 | { |
---|
1104 | CFList buf= factors; |
---|
1105 | int k= 0; |
---|
1106 | int liftBoundBivar= l[k]; |
---|
1107 | diophant= biDiophantine (eval.getFirst(), buf, liftBoundBivar); |
---|
1108 | CFList MOD; |
---|
1109 | MOD.append (power (Variable (2), liftBoundBivar)); |
---|
1110 | CFArray bufFactors= CFArray (factors.length()); |
---|
1111 | k= 0; |
---|
1112 | CFListIterator j= eval; |
---|
1113 | j++; |
---|
1114 | buf.removeFirst(); |
---|
1115 | buf.insert (LC (j.getItem(), 1)); |
---|
1116 | for (CFListIterator i= buf; i.hasItem(); i++, k++) |
---|
1117 | bufFactors[k]= i.getItem(); |
---|
1118 | Pi= CFArray (factors.length() - 1); |
---|
1119 | CFListIterator i= buf; |
---|
1120 | i++; |
---|
1121 | Variable y= j.getItem().mvar(); |
---|
1122 | Pi [0]= mulMod (i.getItem(), mod (buf.getFirst(), y), MOD); |
---|
1123 | M (1, 1)= Pi [0]; |
---|
1124 | k= 1; |
---|
1125 | if (i.hasItem()) |
---|
1126 | i++; |
---|
1127 | for (; i.hasItem(); i++, k++) |
---|
1128 | { |
---|
1129 | Pi [k]= mulMod (Pi [k - 1], i.getItem(), MOD); |
---|
1130 | M (1, k + 1)= Pi [k]; |
---|
1131 | } |
---|
1132 | |
---|
1133 | for (int d= 1; d < l[1]; d++) |
---|
1134 | henselStep (j.getItem(), buf, bufFactors, diophant, M, Pi, d, MOD); |
---|
1135 | CFList result; |
---|
1136 | for (k= 1; k < factors.length(); k++) |
---|
1137 | result.append (bufFactors[k]); |
---|
1138 | return result; |
---|
1139 | } |
---|
1140 | |
---|
1141 | void |
---|
1142 | henselLiftResume (const CanonicalForm& F, CFList& factors, int start, int end, |
---|
1143 | CFArray& Pi, const CFList& diophant, CFMatrix& M, |
---|
1144 | const CFList& MOD) |
---|
1145 | { |
---|
1146 | CFArray bufFactors= CFArray (factors.length()); |
---|
1147 | int i= 0; |
---|
1148 | CanonicalForm xToStart= power (F.mvar(), start); |
---|
1149 | for (CFListIterator k= factors; k.hasItem(); k++, i++) |
---|
1150 | { |
---|
1151 | if (i == 0) |
---|
1152 | bufFactors[i]= mod (k.getItem(), xToStart); |
---|
1153 | else |
---|
1154 | bufFactors[i]= k.getItem(); |
---|
1155 | } |
---|
1156 | for (i= start; i < end; i++) |
---|
1157 | henselStep (F, factors, bufFactors, diophant, M, Pi, i, MOD); |
---|
1158 | |
---|
1159 | CFListIterator k= factors; |
---|
1160 | for (i= 0; i < factors.length(); k++, i++) |
---|
1161 | k.getItem()= bufFactors [i]; |
---|
1162 | factors.removeFirst(); |
---|
1163 | return; |
---|
1164 | } |
---|
1165 | |
---|
1166 | CFList |
---|
1167 | henselLift (const CFList& F, const CFList& factors, const CFList& MOD, CFList& |
---|
1168 | diophant, CFArray& Pi, CFMatrix& M, int lOld, int lNew) |
---|
1169 | { |
---|
1170 | diophant= multiRecDiophantine (F.getFirst(), factors, diophant, MOD, lOld); |
---|
1171 | int k= 0; |
---|
1172 | CFArray bufFactors= CFArray (factors.length()); |
---|
1173 | for (CFListIterator i= factors; i.hasItem(); i++, k++) |
---|
1174 | { |
---|
1175 | if (k == 0) |
---|
1176 | bufFactors[k]= LC (F.getLast(), 1); |
---|
1177 | else |
---|
1178 | bufFactors[k]= i.getItem(); |
---|
1179 | } |
---|
1180 | CFList buf= factors; |
---|
1181 | buf.removeFirst(); |
---|
1182 | buf.insert (LC (F.getLast(), 1)); |
---|
1183 | CFListIterator i= buf; |
---|
1184 | i++; |
---|
1185 | Variable y= F.getLast().mvar(); |
---|
1186 | Variable x= F.getFirst().mvar(); |
---|
1187 | CanonicalForm xToLOld= power (x, lOld); |
---|
1188 | Pi [0]= mod (Pi[0], xToLOld); |
---|
1189 | M (1, 1)= Pi [0]; |
---|
1190 | k= 1; |
---|
1191 | if (i.hasItem()) |
---|
1192 | i++; |
---|
1193 | for (; i.hasItem(); i++, k++) |
---|
1194 | { |
---|
1195 | Pi [k]= mod (Pi [k], xToLOld); |
---|
1196 | M (1, k + 1)= Pi [k]; |
---|
1197 | } |
---|
1198 | |
---|
1199 | for (int d= 1; d < lNew; d++) |
---|
1200 | henselStep (F.getLast(), buf, bufFactors, diophant, M, Pi, d, MOD); |
---|
1201 | CFList result; |
---|
1202 | for (k= 1; k < factors.length(); k++) |
---|
1203 | result.append (bufFactors[k]); |
---|
1204 | return result; |
---|
1205 | } |
---|
1206 | |
---|
1207 | CFList |
---|
1208 | henselLift (const CFList& eval, const CFList& factors, int* l, int lLength, |
---|
1209 | bool sort) |
---|
1210 | { |
---|
1211 | CFList diophant; |
---|
1212 | CFList buf= factors; |
---|
1213 | buf.insert (LC (eval.getFirst(), 1)); |
---|
1214 | if (sort) |
---|
1215 | sortList (buf, Variable (1)); |
---|
1216 | CFArray Pi; |
---|
1217 | CFMatrix M= CFMatrix (l[1], factors.length()); |
---|
1218 | CFList result= henselLift23 (eval, buf, l, diophant, Pi, M); |
---|
1219 | if (eval.length() == 2) |
---|
1220 | return result; |
---|
1221 | CFList MOD; |
---|
1222 | for (int i= 0; i < 2; i++) |
---|
1223 | MOD.append (power (Variable (i + 2), l[i])); |
---|
1224 | CFListIterator j= eval; |
---|
1225 | j++; |
---|
1226 | CFList bufEval; |
---|
1227 | bufEval.append (j.getItem()); |
---|
1228 | j++; |
---|
1229 | |
---|
1230 | for (int i= 2; i < lLength && j.hasItem(); i++, j++) |
---|
1231 | { |
---|
1232 | result.insert (LC (bufEval.getFirst(), 1)); |
---|
1233 | bufEval.append (j.getItem()); |
---|
1234 | M= CFMatrix (l[i], factors.length()); |
---|
1235 | result= henselLift (bufEval, result, MOD, diophant, Pi, M, l[i - 1], l[i]); |
---|
1236 | MOD.append (power (Variable (i + 2), l[i])); |
---|
1237 | bufEval.removeFirst(); |
---|
1238 | } |
---|
1239 | return result; |
---|
1240 | } |
---|
1241 | |
---|
1242 | // nonmonic |
---|
1243 | |
---|
1244 | void |
---|
1245 | nonMonicHenselStep12 (const CanonicalForm& F, const CFList& factors, |
---|
1246 | CFArray& bufFactors, const CFList& diophant, CFMatrix& M, |
---|
1247 | CFArray& Pi, int j, const CFArray& /*LCs*/) |
---|
1248 | { |
---|
1249 | Variable x= F.mvar(); |
---|
1250 | CanonicalForm xToJ= power (x, j); |
---|
1251 | |
---|
1252 | CanonicalForm E; |
---|
1253 | // compute the error |
---|
1254 | if (degree (Pi [factors.length() - 2], x) > 0) |
---|
1255 | E= F[j] - Pi [factors.length() - 2] [j]; |
---|
1256 | else |
---|
1257 | E= F[j]; |
---|
1258 | |
---|
1259 | CFArray buf= CFArray (diophant.length()); |
---|
1260 | |
---|
1261 | int k= 0; |
---|
1262 | CanonicalForm remainder; |
---|
1263 | // actual lifting |
---|
1264 | for (CFListIterator i= diophant; i.hasItem(); i++, k++) |
---|
1265 | { |
---|
1266 | if (degree (bufFactors[k], x) > 0) |
---|
1267 | remainder= modNTL (E, bufFactors[k] [0]); |
---|
1268 | else |
---|
1269 | remainder= modNTL (E, bufFactors[k]); |
---|
1270 | buf[k]= mulNTL (i.getItem(), remainder); |
---|
1271 | if (degree (bufFactors[k], x) > 0) |
---|
1272 | buf[k]= modNTL (buf[k], bufFactors[k] [0]); |
---|
1273 | else |
---|
1274 | buf[k]= modNTL (buf[k], bufFactors[k]); |
---|
1275 | } |
---|
1276 | |
---|
1277 | for (k= 0; k < factors.length(); k++) |
---|
1278 | bufFactors[k] += xToJ*buf[k]; |
---|
1279 | |
---|
1280 | // update Pi [0] |
---|
1281 | int degBuf0= degree (bufFactors[0], x); |
---|
1282 | int degBuf1= degree (bufFactors[1], x); |
---|
1283 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
1284 | { |
---|
1285 | M (j + 1, 1)= mulNTL (bufFactors[0] [j], bufFactors[1] [j]); |
---|
1286 | if (j + 2 <= M.rows()) |
---|
1287 | M (j + 2, 1)= mulNTL (bufFactors[0] [j + 1], bufFactors[1] [j + 1]); |
---|
1288 | } |
---|
1289 | CanonicalForm uIZeroJ; |
---|
1290 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
1291 | uIZeroJ= mulNTL(bufFactors[0][0],buf[1])+mulNTL (bufFactors[1][0], buf[0]); |
---|
1292 | else if (degBuf0 > 0) |
---|
1293 | uIZeroJ= mulNTL (buf[0], bufFactors[1]); |
---|
1294 | else if (degBuf1 > 0) |
---|
1295 | uIZeroJ= mulNTL (bufFactors[0], buf [1]); |
---|
1296 | else |
---|
1297 | uIZeroJ= 0; |
---|
1298 | Pi [0] += xToJ*uIZeroJ; |
---|
1299 | |
---|
1300 | CFArray tmp= CFArray (factors.length() - 1); |
---|
1301 | for (k= 0; k < factors.length() - 1; k++) |
---|
1302 | tmp[k]= 0; |
---|
1303 | CFIterator one, two; |
---|
1304 | one= bufFactors [0]; |
---|
1305 | two= bufFactors [1]; |
---|
1306 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
1307 | { |
---|
1308 | while (one.hasTerms() && one.exp() > j) one++; |
---|
1309 | while (two.hasTerms() && two.exp() > j) two++; |
---|
1310 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
1311 | { |
---|
1312 | if (one.hasTerms() && two.hasTerms()) |
---|
1313 | { |
---|
1314 | if (k != j - k + 1) |
---|
1315 | { |
---|
1316 | if ((one.hasTerms() && one.exp() == j - k + 1) && + |
---|
1317 | (two.hasTerms() && two.exp() == j - k + 1)) |
---|
1318 | { |
---|
1319 | tmp[0] += mulNTL ((bufFactors[0][k]+one.coeff()),(bufFactors[1][k] + |
---|
1320 | two.coeff())) - M (k + 1, 1) - M (j - k + 2, 1); |
---|
1321 | one++; |
---|
1322 | two++; |
---|
1323 | } |
---|
1324 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
1325 | { |
---|
1326 | tmp[0] += mulNTL ((bufFactors[0][k]+one.coeff()), bufFactors[1] [k]) - |
---|
1327 | M (k + 1, 1); |
---|
1328 | one++; |
---|
1329 | } |
---|
1330 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
1331 | { |
---|
1332 | tmp[0] += mulNTL (bufFactors[0][k],(bufFactors[1][k] + two.coeff())) - |
---|
1333 | M (k + 1, 1); |
---|
1334 | two++; |
---|
1335 | } |
---|
1336 | } |
---|
1337 | else |
---|
1338 | tmp[0] += M (k + 1, 1); |
---|
1339 | } |
---|
1340 | } |
---|
1341 | } |
---|
1342 | |
---|
1343 | if (degBuf0 >= j + 1 && degBuf1 >= j + 1) |
---|
1344 | { |
---|
1345 | if (j + 2 <= M.rows()) |
---|
1346 | tmp [0] += mulNTL ((bufFactors [0] [j + 1]+ bufFactors [0] [0]), |
---|
1347 | (bufFactors [1] [j + 1] + bufFactors [1] [0])) |
---|
1348 | - M(1,1) - M (j + 2,1); |
---|
1349 | } |
---|
1350 | else if (degBuf0 >= j + 1) |
---|
1351 | { |
---|
1352 | if (degBuf1 > 0) |
---|
1353 | tmp[0] += mulNTL (bufFactors [0] [j+1], bufFactors [1] [0]); |
---|
1354 | else |
---|
1355 | tmp[0] += mulNTL (bufFactors [0] [j+1], bufFactors [1]); |
---|
1356 | } |
---|
1357 | else if (degBuf1 >= j + 1) |
---|
1358 | { |
---|
1359 | if (degBuf0 > 0) |
---|
1360 | tmp[0] += mulNTL (bufFactors [0] [0], bufFactors [1] [j + 1]); |
---|
1361 | else |
---|
1362 | tmp[0] += mulNTL (bufFactors [0], bufFactors [1] [j + 1]); |
---|
1363 | } |
---|
1364 | |
---|
1365 | Pi [0] += tmp[0]*xToJ*F.mvar(); |
---|
1366 | |
---|
1367 | int degPi, degBuf; |
---|
1368 | for (int l= 1; l < factors.length() - 1; l++) |
---|
1369 | { |
---|
1370 | degPi= degree (Pi [l - 1], x); |
---|
1371 | degBuf= degree (bufFactors[l + 1], x); |
---|
1372 | if (degPi > 0 && degBuf > 0) |
---|
1373 | { |
---|
1374 | M (j + 1, l + 1)= mulNTL (Pi [l - 1] [j], bufFactors[l + 1] [j]); |
---|
1375 | if (j + 2 <= M.rows()) |
---|
1376 | M (j + 2, l + 1)= mulNTL (Pi [l - 1][j + 1], bufFactors[l + 1] [j + 1]); |
---|
1377 | } |
---|
1378 | |
---|
1379 | if (degPi > 0 && degBuf > 0) |
---|
1380 | uIZeroJ= mulNTL (Pi[l -1] [0], buf[l + 1]) + |
---|
1381 | mulNTL (uIZeroJ, bufFactors[l+1] [0]); |
---|
1382 | else if (degPi > 0) |
---|
1383 | uIZeroJ= mulNTL (uIZeroJ, bufFactors[l + 1]); |
---|
1384 | else if (degBuf > 0) |
---|
1385 | uIZeroJ= mulNTL (Pi[l - 1], buf[1]); |
---|
1386 | else |
---|
1387 | uIZeroJ= 0; |
---|
1388 | |
---|
1389 | Pi [l] += xToJ*uIZeroJ; |
---|
1390 | |
---|
1391 | one= bufFactors [l + 1]; |
---|
1392 | two= Pi [l - 1]; |
---|
1393 | if (degBuf > 0 && degPi > 0) |
---|
1394 | { |
---|
1395 | while (one.hasTerms() && one.exp() > j) one++; |
---|
1396 | while (two.hasTerms() && two.exp() > j) two++; |
---|
1397 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
1398 | { |
---|
1399 | if (k != j - k + 1) |
---|
1400 | { |
---|
1401 | if ((one.hasTerms() && one.exp() == j - k + 1) && |
---|
1402 | (two.hasTerms() && two.exp() == j - k + 1)) |
---|
1403 | { |
---|
1404 | tmp[l] += mulNTL ((bufFactors[l + 1] [k] + one.coeff()), |
---|
1405 | (Pi[l - 1] [k] + two.coeff())) - M (k + 1, l + 1) - |
---|
1406 | M (j - k + 2, l + 1); |
---|
1407 | one++; |
---|
1408 | two++; |
---|
1409 | } |
---|
1410 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
1411 | { |
---|
1412 | tmp[l] += mulNTL ((bufFactors[l + 1] [k] + one.coeff()), |
---|
1413 | Pi[l - 1] [k]) - M (k + 1, l + 1); |
---|
1414 | one++; |
---|
1415 | } |
---|
1416 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
1417 | { |
---|
1418 | tmp[l] += mulNTL (bufFactors[l + 1] [k], |
---|
1419 | (Pi[l - 1] [k] + two.coeff())) - M (k + 1, l + 1); |
---|
1420 | two++; |
---|
1421 | } |
---|
1422 | } |
---|
1423 | else |
---|
1424 | tmp[l] += M (k + 1, l + 1); |
---|
1425 | } |
---|
1426 | } |
---|
1427 | |
---|
1428 | if (degPi >= j + 1 && degBuf >= j + 1) |
---|
1429 | { |
---|
1430 | if (j + 2 <= M.rows()) |
---|
1431 | tmp [l] += mulNTL ((Pi [l - 1] [j + 1]+ Pi [l - 1] [0]), |
---|
1432 | (bufFactors [l + 1] [j + 1] + bufFactors [l + 1] [0]) |
---|
1433 | ) - M(1,l+1) - M (j + 2,l+1); |
---|
1434 | } |
---|
1435 | else if (degPi >= j + 1) |
---|
1436 | { |
---|
1437 | if (degBuf > 0) |
---|
1438 | tmp[l] += mulNTL (Pi [l - 1] [j+1], bufFactors [l + 1] [0]); |
---|
1439 | else |
---|
1440 | tmp[l] += mulNTL (Pi [l - 1] [j+1], bufFactors [l + 1]); |
---|
1441 | } |
---|
1442 | else if (degBuf >= j + 1) |
---|
1443 | { |
---|
1444 | if (degPi > 0) |
---|
1445 | tmp[l] += mulNTL (Pi [l - 1] [0], bufFactors [l + 1] [j + 1]); |
---|
1446 | else |
---|
1447 | tmp[l] += mulNTL (Pi [l - 1], bufFactors [l + 1] [j + 1]); |
---|
1448 | } |
---|
1449 | |
---|
1450 | Pi[l] += tmp[l]*xToJ*F.mvar(); |
---|
1451 | } |
---|
1452 | return; |
---|
1453 | } |
---|
1454 | |
---|
1455 | void |
---|
1456 | nonMonicHenselLift12 (const CanonicalForm& F, CFList& factors, int l, |
---|
1457 | CFArray& Pi, CFList& diophant, CFMatrix& M, |
---|
1458 | const CFArray& LCs, bool sort) |
---|
1459 | { |
---|
1460 | if (sort) |
---|
1461 | sortList (factors, Variable (1)); |
---|
1462 | Pi= CFArray (factors.length() - 2); |
---|
1463 | CFList bufFactors2= factors; |
---|
1464 | bufFactors2.removeFirst(); |
---|
1465 | diophant= diophantine (F[0], bufFactors2); |
---|
1466 | DEBOUTLN (cerr, "diophant= " << diophant); |
---|
1467 | |
---|
1468 | CFArray bufFactors= CFArray (bufFactors2.length()); |
---|
1469 | int i= 0; |
---|
1470 | for (CFListIterator k= bufFactors2; k.hasItem(); i++, k++) |
---|
1471 | bufFactors[i]= replaceLc (k.getItem(), LCs [i]); |
---|
1472 | |
---|
1473 | Variable x= F.mvar(); |
---|
1474 | if (degree (bufFactors[0], x) > 0 && degree (bufFactors [1], x) > 0) |
---|
1475 | { |
---|
1476 | M (1, 1)= mulNTL (bufFactors [0] [0], bufFactors[1] [0]); |
---|
1477 | Pi [0]= M (1, 1) + (mulNTL (bufFactors [0] [1], bufFactors[1] [0]) + |
---|
1478 | mulNTL (bufFactors [0] [0], bufFactors [1] [1]))*x; |
---|
1479 | } |
---|
1480 | else if (degree (bufFactors[0], x) > 0) |
---|
1481 | { |
---|
1482 | M (1, 1)= mulNTL (bufFactors [0] [0], bufFactors[1]); |
---|
1483 | Pi [0]= M (1, 1) + |
---|
1484 | mulNTL (bufFactors [0] [1], bufFactors[1])*x; |
---|
1485 | } |
---|
1486 | else if (degree (bufFactors[1], x) > 0) |
---|
1487 | { |
---|
1488 | M (1, 1)= mulNTL (bufFactors [0], bufFactors[1] [0]); |
---|
1489 | Pi [0]= M (1, 1) + |
---|
1490 | mulNTL (bufFactors [0], bufFactors[1] [1])*x; |
---|
1491 | } |
---|
1492 | else |
---|
1493 | { |
---|
1494 | M (1, 1)= mulNTL (bufFactors [0], bufFactors[1]); |
---|
1495 | Pi [0]= M (1, 1); |
---|
1496 | } |
---|
1497 | |
---|
1498 | for (i= 1; i < Pi.size(); i++) |
---|
1499 | { |
---|
1500 | if (degree (Pi[i-1], x) > 0 && degree (bufFactors [i+1], x) > 0) |
---|
1501 | { |
---|
1502 | M (1,i+1)= mulNTL (Pi[i-1] [0], bufFactors[i+1] [0]); |
---|
1503 | Pi [i]= M (1,i+1) + (mulNTL (Pi[i-1] [1], bufFactors[i+1] [0]) + |
---|
1504 | mulNTL (Pi[i-1] [0], bufFactors [i+1] [1]))*x; |
---|
1505 | } |
---|
1506 | else if (degree (Pi[i-1], x) > 0) |
---|
1507 | { |
---|
1508 | M (1,i+1)= mulNTL (Pi[i-1] [0], bufFactors [i+1]); |
---|
1509 | Pi [i]= M(1,i+1) + mulNTL (Pi[i-1] [1], bufFactors[i+1])*x; |
---|
1510 | } |
---|
1511 | else if (degree (bufFactors[i+1], x) > 0) |
---|
1512 | { |
---|
1513 | M (1,i+1)= mulNTL (Pi[i-1], bufFactors [i+1] [0]); |
---|
1514 | Pi [i]= M (1,i+1) + mulNTL (Pi[i-1], bufFactors[i+1] [1])*x; |
---|
1515 | } |
---|
1516 | else |
---|
1517 | { |
---|
1518 | M (1,i+1)= mulNTL (Pi [i-1], bufFactors [i+1]); |
---|
1519 | Pi [i]= M (1,i+1); |
---|
1520 | } |
---|
1521 | } |
---|
1522 | |
---|
1523 | for (i= 1; i < l; i++) |
---|
1524 | nonMonicHenselStep12 (F, bufFactors2, bufFactors, diophant, M, Pi, i, LCs); |
---|
1525 | |
---|
1526 | factors= CFList(); |
---|
1527 | for (i= 0; i < bufFactors.size(); i++) |
---|
1528 | factors.append (bufFactors[i]); |
---|
1529 | return; |
---|
1530 | } |
---|
1531 | |
---|
1532 | |
---|
1533 | /// solve \f$ E=sum_{i= 1}^{r}{\sigma_{i}prod_{j=1, j\neq i}^{r}{f_{i}}}\f$ |
---|
1534 | /// mod M, products contains \f$ prod_{j=1, j\neq i}^{r}{f_{i}}} \f$ |
---|
1535 | CFList |
---|
1536 | diophantine (const CFList& recResult, const CFList& factors, |
---|
1537 | const CFList& products, const CFList& M, const CanonicalForm& E, |
---|
1538 | bool& bad) |
---|
1539 | { |
---|
1540 | if (M.isEmpty()) |
---|
1541 | { |
---|
1542 | CFList result; |
---|
1543 | CFListIterator j= factors; |
---|
1544 | CanonicalForm buf; |
---|
1545 | for (CFListIterator i= recResult; i.hasItem(); i++, j++) |
---|
1546 | { |
---|
1547 | ASSERT (E.isUnivariate() || E.inCoeffDomain(), |
---|
1548 | "constant or univariate poly expected"); |
---|
1549 | ASSERT (i.getItem().isUnivariate() || i.getItem().inCoeffDomain(), |
---|
1550 | "constant or univariate poly expected"); |
---|
1551 | ASSERT (j.getItem().isUnivariate() || j.getItem().inCoeffDomain(), |
---|
1552 | "constant or univariate poly expected"); |
---|
1553 | buf= mulNTL (E, i.getItem()); |
---|
1554 | result.append (modNTL (buf, j.getItem())); |
---|
1555 | } |
---|
1556 | return result; |
---|
1557 | } |
---|
1558 | Variable y= M.getLast().mvar(); |
---|
1559 | CFList bufFactors= factors; |
---|
1560 | for (CFListIterator i= bufFactors; i.hasItem(); i++) |
---|
1561 | i.getItem()= mod (i.getItem(), y); |
---|
1562 | CFList bufProducts= products; |
---|
1563 | for (CFListIterator i= bufProducts; i.hasItem(); i++) |
---|
1564 | i.getItem()= mod (i.getItem(), y); |
---|
1565 | CFList buf= M; |
---|
1566 | buf.removeLast(); |
---|
1567 | CanonicalForm bufE= mod (E, y); |
---|
1568 | CFList recDiophantine= diophantine (recResult, bufFactors, bufProducts, buf, |
---|
1569 | bufE, bad); |
---|
1570 | |
---|
1571 | if (bad) |
---|
1572 | return CFList(); |
---|
1573 | |
---|
1574 | CanonicalForm e= E; |
---|
1575 | CFListIterator j= products; |
---|
1576 | for (CFListIterator i= recDiophantine; i.hasItem(); i++, j++) |
---|
1577 | e -= i.getItem()*j.getItem(); |
---|
1578 | |
---|
1579 | CFList result= recDiophantine; |
---|
1580 | int d= degree (M.getLast()); |
---|
1581 | CanonicalForm coeffE; |
---|
1582 | for (int i= 1; i < d; i++) |
---|
1583 | { |
---|
1584 | if (degree (e, y) > 0) |
---|
1585 | coeffE= e[i]; |
---|
1586 | else |
---|
1587 | coeffE= 0; |
---|
1588 | if (!coeffE.isZero()) |
---|
1589 | { |
---|
1590 | CFListIterator k= result; |
---|
1591 | recDiophantine= diophantine (recResult, bufFactors, bufProducts, buf, |
---|
1592 | coeffE, bad); |
---|
1593 | if (bad) |
---|
1594 | return CFList(); |
---|
1595 | CFListIterator l= products; |
---|
1596 | for (j= recDiophantine; j.hasItem(); j++, k++, l++) |
---|
1597 | { |
---|
1598 | k.getItem() += j.getItem()*power (y, i); |
---|
1599 | e -= j.getItem()*power (y, i)*l.getItem(); |
---|
1600 | } |
---|
1601 | } |
---|
1602 | if (e.isZero()) |
---|
1603 | break; |
---|
1604 | } |
---|
1605 | if (!e.isZero()) |
---|
1606 | { |
---|
1607 | bad= true; |
---|
1608 | return CFList(); |
---|
1609 | } |
---|
1610 | return result; |
---|
1611 | } |
---|
1612 | |
---|
1613 | void |
---|
1614 | nonMonicHenselStep (const CanonicalForm& F, const CFList& factors, |
---|
1615 | CFArray& bufFactors, const CFList& diophant, CFMatrix& M, |
---|
1616 | CFArray& Pi, const CFList& products, int j, |
---|
1617 | const CFList& MOD, bool& noOneToOne) |
---|
1618 | { |
---|
1619 | CanonicalForm E; |
---|
1620 | CanonicalForm xToJ= power (F.mvar(), j); |
---|
1621 | Variable x= F.mvar(); |
---|
1622 | |
---|
1623 | // compute the error |
---|
1624 | #ifdef DEBUGOUTPUT |
---|
1625 | CanonicalForm test= 1; |
---|
1626 | for (int i= 0; i < factors.length(); i++) |
---|
1627 | { |
---|
1628 | if (i == 0) |
---|
1629 | test *= mod (bufFactors [i], power (x, j)); |
---|
1630 | else |
---|
1631 | test *= bufFactors[i]; |
---|
1632 | } |
---|
1633 | test= mod (test, power (x, j)); |
---|
1634 | test= mod (test, MOD); |
---|
1635 | CanonicalForm test2= mod (F, power (x, j - 1)) - mod (test, power (x, j-1)); |
---|
1636 | DEBOUTLN (cerr, "test= " << test2); |
---|
1637 | #endif |
---|
1638 | |
---|
1639 | if (degree (Pi [factors.length() - 2], x) > 0) |
---|
1640 | E= F[j] - Pi [factors.length() - 2] [j]; |
---|
1641 | else |
---|
1642 | E= F[j]; |
---|
1643 | |
---|
1644 | CFArray buf= CFArray (diophant.length()); |
---|
1645 | |
---|
1646 | // actual lifting |
---|
1647 | CFList diophantine2= diophantine (diophant, factors, products, MOD, E, |
---|
1648 | noOneToOne); |
---|
1649 | |
---|
1650 | if (noOneToOne) |
---|
1651 | return; |
---|
1652 | |
---|
1653 | int k= 0; |
---|
1654 | for (CFListIterator i= diophantine2; k < factors.length(); k++, i++) |
---|
1655 | { |
---|
1656 | buf[k]= i.getItem(); |
---|
1657 | bufFactors[k] += xToJ*i.getItem(); |
---|
1658 | } |
---|
1659 | |
---|
1660 | // update Pi [0] |
---|
1661 | int degBuf0= degree (bufFactors[0], x); |
---|
1662 | int degBuf1= degree (bufFactors[1], x); |
---|
1663 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
1664 | { |
---|
1665 | M (j + 1, 1)= mulMod (bufFactors[0] [j], bufFactors[1] [j], MOD); |
---|
1666 | if (j + 2 <= M.rows()) |
---|
1667 | M (j + 2, 1)= mulMod (bufFactors[0] [j + 1], bufFactors[1] [j + 1], MOD); |
---|
1668 | } |
---|
1669 | CanonicalForm uIZeroJ; |
---|
1670 | |
---|
1671 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
1672 | uIZeroJ= mulMod (bufFactors[0] [0], buf[1], MOD) + |
---|
1673 | mulMod (bufFactors[1] [0], buf[0], MOD); |
---|
1674 | else if (degBuf0 > 0) |
---|
1675 | uIZeroJ= mulMod (buf[0], bufFactors[1], MOD); |
---|
1676 | else if (degBuf1 > 0) |
---|
1677 | uIZeroJ= mulMod (bufFactors[0], buf[1], MOD); |
---|
1678 | else |
---|
1679 | uIZeroJ= 0; |
---|
1680 | Pi [0] += xToJ*uIZeroJ; |
---|
1681 | |
---|
1682 | CFArray tmp= CFArray (factors.length() - 1); |
---|
1683 | for (k= 0; k < factors.length() - 1; k++) |
---|
1684 | tmp[k]= 0; |
---|
1685 | CFIterator one, two; |
---|
1686 | one= bufFactors [0]; |
---|
1687 | two= bufFactors [1]; |
---|
1688 | if (degBuf0 > 0 && degBuf1 > 0) |
---|
1689 | { |
---|
1690 | while (one.hasTerms() && one.exp() > j) one++; |
---|
1691 | while (two.hasTerms() && two.exp() > j) two++; |
---|
1692 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
1693 | { |
---|
1694 | if (k != j - k + 1) |
---|
1695 | { |
---|
1696 | if ((one.hasTerms() && one.exp() == j - k + 1) && |
---|
1697 | (two.hasTerms() && two.exp() == j - k + 1)) |
---|
1698 | { |
---|
1699 | tmp[0] += mulMod ((bufFactors[0] [k] + one.coeff()), |
---|
1700 | (bufFactors[1] [k] + two.coeff()), MOD) - M (k + 1, 1) - |
---|
1701 | M (j - k + 2, 1); |
---|
1702 | one++; |
---|
1703 | two++; |
---|
1704 | } |
---|
1705 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
1706 | { |
---|
1707 | tmp[0] += mulMod ((bufFactors[0] [k] + one.coeff()), |
---|
1708 | bufFactors[1] [k], MOD) - M (k + 1, 1); |
---|
1709 | one++; |
---|
1710 | } |
---|
1711 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
1712 | { |
---|
1713 | tmp[0] += mulMod (bufFactors[0] [k], (bufFactors[1] [k] + |
---|
1714 | two.coeff()), MOD) - M (k + 1, 1); |
---|
1715 | two++; |
---|
1716 | } |
---|
1717 | } |
---|
1718 | else |
---|
1719 | { |
---|
1720 | tmp[0] += M (k + 1, 1); |
---|
1721 | } |
---|
1722 | } |
---|
1723 | } |
---|
1724 | |
---|
1725 | if (degBuf0 >= j + 1 && degBuf1 >= j + 1) |
---|
1726 | { |
---|
1727 | if (j + 2 <= M.rows()) |
---|
1728 | tmp [0] += mulMod ((bufFactors [0] [j + 1]+ bufFactors [0] [0]), |
---|
1729 | (bufFactors [1] [j + 1] + bufFactors [1] [0]), MOD) |
---|
1730 | - M(1,1) - M (j + 2,1); |
---|
1731 | } |
---|
1732 | else if (degBuf0 >= j + 1) |
---|
1733 | { |
---|
1734 | if (degBuf1 > 0) |
---|
1735 | tmp[0] += mulMod (bufFactors [0] [j+1], bufFactors [1] [0], MOD); |
---|
1736 | else |
---|
1737 | tmp[0] += mulMod (bufFactors [0] [j+1], bufFactors [1], MOD); |
---|
1738 | } |
---|
1739 | else if (degBuf1 >= j + 1) |
---|
1740 | { |
---|
1741 | if (degBuf0 > 0) |
---|
1742 | tmp[0] += mulMod (bufFactors [0] [0], bufFactors [1] [j + 1], MOD); |
---|
1743 | else |
---|
1744 | tmp[0] += mulMod (bufFactors [0], bufFactors [1] [j + 1], MOD); |
---|
1745 | } |
---|
1746 | Pi [0] += tmp[0]*xToJ*F.mvar(); |
---|
1747 | |
---|
1748 | // update Pi [l] |
---|
1749 | int degPi, degBuf; |
---|
1750 | for (int l= 1; l < factors.length() - 1; l++) |
---|
1751 | { |
---|
1752 | degPi= degree (Pi [l - 1], x); |
---|
1753 | degBuf= degree (bufFactors[l + 1], x); |
---|
1754 | if (degPi > 0 && degBuf > 0) |
---|
1755 | { |
---|
1756 | M (j + 1, l + 1)= mulMod (Pi [l - 1] [j], bufFactors[l + 1] [j], MOD); |
---|
1757 | if (j + 2 <= M.rows()) |
---|
1758 | M (j + 2, l + 1)= mulMod (Pi [l - 1] [j + 1], bufFactors[l + 1] [j + 1], |
---|
1759 | MOD); |
---|
1760 | } |
---|
1761 | |
---|
1762 | if (degPi > 0 && degBuf > 0) |
---|
1763 | uIZeroJ= mulMod (Pi[l -1] [0], buf[l + 1], MOD) + |
---|
1764 | mulMod (uIZeroJ, bufFactors[l+1] [0], MOD); |
---|
1765 | else if (degPi > 0) |
---|
1766 | uIZeroJ= mulMod (uIZeroJ, bufFactors[l + 1], MOD); |
---|
1767 | else if (degBuf > 0) |
---|
1768 | uIZeroJ= mulMod (Pi[l - 1], buf[1], MOD); |
---|
1769 | else |
---|
1770 | uIZeroJ= 0; |
---|
1771 | |
---|
1772 | Pi [l] += xToJ*uIZeroJ; |
---|
1773 | |
---|
1774 | one= bufFactors [l + 1]; |
---|
1775 | two= Pi [l - 1]; |
---|
1776 | if (degBuf > 0 && degPi > 0) |
---|
1777 | { |
---|
1778 | while (one.hasTerms() && one.exp() > j) one++; |
---|
1779 | while (two.hasTerms() && two.exp() > j) two++; |
---|
1780 | for (k= 1; k <= (int) ceil (j/2.0); k++) |
---|
1781 | { |
---|
1782 | if (k != j - k + 1) |
---|
1783 | { |
---|
1784 | if ((one.hasTerms() && one.exp() == j - k + 1) && |
---|
1785 | (two.hasTerms() && two.exp() == j - k + 1)) |
---|
1786 | { |
---|
1787 | tmp[l] += mulMod ((bufFactors[l + 1] [k] + one.coeff()), |
---|
1788 | (Pi[l - 1] [k] + two.coeff()), MOD) - M (k + 1, l + 1) - |
---|
1789 | M (j - k + 2, l + 1); |
---|
1790 | one++; |
---|
1791 | two++; |
---|
1792 | } |
---|
1793 | else if (one.hasTerms() && one.exp() == j - k + 1) |
---|
1794 | { |
---|
1795 | tmp[l] += mulMod ((bufFactors[l + 1] [k] + one.coeff()), |
---|
1796 | Pi[l - 1] [k], MOD) - M (k + 1, l + 1); |
---|
1797 | one++; |
---|
1798 | } |
---|
1799 | else if (two.hasTerms() && two.exp() == j - k + 1) |
---|
1800 | { |
---|
1801 | tmp[l] += mulMod (bufFactors[l + 1] [k], |
---|
1802 | (Pi[l - 1] [k] + two.coeff()), MOD) - M (k + 1, l + 1); |
---|
1803 | two++; |
---|
1804 | } |
---|
1805 | } |
---|
1806 | else |
---|
1807 | tmp[l] += M (k + 1, l + 1); |
---|
1808 | } |
---|
1809 | } |
---|
1810 | |
---|
1811 | if (degPi >= j + 1 && degBuf >= j + 1) |
---|
1812 | { |
---|
1813 | if (j + 2 <= M.rows()) |
---|
1814 | tmp [l] += mulMod ((Pi [l - 1] [j + 1]+ Pi [l - 1] [0]), |
---|
1815 | (bufFactors [l + 1] [j + 1] + bufFactors [l + 1] [0]) |
---|
1816 | , MOD) - M(1,l+1) - M (j + 2,l+1); |
---|
1817 | } |
---|
1818 | else if (degPi >= j + 1) |
---|
1819 | { |
---|
1820 | if (degBuf > 0) |
---|
1821 | tmp[l] += mulMod (Pi [l - 1] [j+1], bufFactors [l + 1] [0], MOD); |
---|
1822 | else |
---|
1823 | tmp[l] += mulMod (Pi [l - 1] [j+1], bufFactors [l + 1], MOD); |
---|
1824 | } |
---|
1825 | else if (degBuf >= j + 1) |
---|
1826 | { |
---|
1827 | if (degPi > 0) |
---|
1828 | tmp[l] += mulMod (Pi [l - 1] [0], bufFactors [l + 1] [j + 1], MOD); |
---|
1829 | else |
---|
1830 | tmp[l] += mulMod (Pi [l - 1], bufFactors [l + 1] [j + 1], MOD); |
---|
1831 | } |
---|
1832 | |
---|
1833 | Pi[l] += tmp[l]*xToJ*F.mvar(); |
---|
1834 | } |
---|
1835 | return; |
---|
1836 | } |
---|
1837 | |
---|
1838 | // wrt. Variable (1) |
---|
1839 | CanonicalForm replaceLC (const CanonicalForm& F, const CanonicalForm& c) |
---|
1840 | { |
---|
1841 | if (degree (F, 1) <= 0) |
---|
1842 | return c; |
---|
1843 | else |
---|
1844 | { |
---|
1845 | CanonicalForm result= swapvar (F, Variable (F.level() + 1), Variable (1)); |
---|
1846 | result += (swapvar (c, Variable (F.level() + 1), Variable (1)) |
---|
1847 | - LC (result))*power (result.mvar(), degree (result)); |
---|
1848 | return swapvar (result, Variable (F.level() + 1), Variable (1)); |
---|
1849 | } |
---|
1850 | } |
---|
1851 | |
---|
1852 | CFList |
---|
1853 | nonMonicHenselLift232(const CFList& eval, const CFList& factors, int* l, CFList& |
---|
1854 | diophant, CFArray& Pi, CFMatrix& M, const CFList& LCs1, |
---|
1855 | const CFList& LCs2, bool& bad) |
---|
1856 | { |
---|
1857 | CFList buf= factors; |
---|
1858 | int k= 0; |
---|
1859 | int liftBoundBivar= l[k]; |
---|
1860 | CFList bufbuf= factors; |
---|
1861 | Variable v= Variable (2); |
---|
1862 | |
---|
1863 | CFList MOD; |
---|
1864 | MOD.append (power (Variable (2), liftBoundBivar)); |
---|
1865 | CFArray bufFactors= CFArray (factors.length()); |
---|
1866 | k= 0; |
---|
1867 | CFListIterator j= eval; |
---|
1868 | j++; |
---|
1869 | CFListIterator iter1= LCs1; |
---|
1870 | CFListIterator iter2= LCs2; |
---|
1871 | iter1++; |
---|
1872 | iter2++; |
---|
1873 | bufFactors[0]= replaceLC (buf.getFirst(), iter1.getItem()); |
---|
1874 | bufFactors[1]= replaceLC (buf.getLast(), iter2.getItem()); |
---|
1875 | |
---|
1876 | CFListIterator i= buf; |
---|
1877 | i++; |
---|
1878 | Variable y= j.getItem().mvar(); |
---|
1879 | if (y.level() != 3) |
---|
1880 | y= Variable (3); |
---|
1881 | |
---|
1882 | Pi[0]= mod (Pi[0], power (v, liftBoundBivar)); |
---|
1883 | M (1, 1)= Pi[0]; |
---|
1884 | if (degree (bufFactors[0], y) > 0 && degree (bufFactors [1], y) > 0) |
---|
1885 | Pi [0] += (mulMod (bufFactors [0] [1], bufFactors[1] [0], MOD) + |
---|
1886 | mulMod (bufFactors [0] [0], bufFactors [1] [1], MOD))*y; |
---|
1887 | else if (degree (bufFactors[0], y) > 0) |
---|
1888 | Pi [0] += mulMod (bufFactors [0] [1], bufFactors[1], MOD)*y; |
---|
1889 | else if (degree (bufFactors[1], y) > 0) |
---|
1890 | Pi [0] += mulMod (bufFactors [0], bufFactors[1] [1], MOD)*y; |
---|
1891 | |
---|
1892 | CFList products; |
---|
1893 | for (int i= 0; i < bufFactors.size(); i++) |
---|
1894 | { |
---|
1895 | if (degree (bufFactors[i], y) > 0) |
---|
1896 | products.append (eval.getFirst()/bufFactors[i] [0]); |
---|
1897 | else |
---|
1898 | products.append (eval.getFirst()/bufFactors[i]); |
---|
1899 | } |
---|
1900 | |
---|
1901 | for (int d= 1; d < l[1]; d++) |
---|
1902 | { |
---|
1903 | nonMonicHenselStep (j.getItem(), buf, bufFactors, diophant, M, Pi, products, |
---|
1904 | d, MOD, bad); |
---|
1905 | if (bad) |
---|
1906 | return CFList(); |
---|
1907 | } |
---|
1908 | CFList result; |
---|
1909 | for (k= 0; k < factors.length(); k++) |
---|
1910 | result.append (bufFactors[k]); |
---|
1911 | return result; |
---|
1912 | } |
---|
1913 | |
---|
1914 | |
---|
1915 | CFList |
---|
1916 | nonMonicHenselLift2 (const CFList& F, const CFList& factors, const CFList& MOD, |
---|
1917 | CFList& diophant, CFArray& Pi, CFMatrix& M, int lOld, |
---|
1918 | int& lNew, const CFList& LCs1, const CFList& LCs2, bool& bad |
---|
1919 | ) |
---|
1920 | { |
---|
1921 | int k= 0; |
---|
1922 | CFArray bufFactors= CFArray (factors.length()); |
---|
1923 | bufFactors[0]= replaceLC (factors.getFirst(), LCs1.getLast()); |
---|
1924 | bufFactors[1]= replaceLC (factors.getLast(), LCs2.getLast()); |
---|
1925 | CFList buf= factors; |
---|
1926 | Variable y= F.getLast().mvar(); |
---|
1927 | Variable x= F.getFirst().mvar(); |
---|
1928 | CanonicalForm xToLOld= power (x, lOld); |
---|
1929 | Pi [0]= mod (Pi[0], xToLOld); |
---|
1930 | M (1, 1)= Pi [0]; |
---|
1931 | |
---|
1932 | if (degree (bufFactors[0], y) > 0 && degree (bufFactors [1], y) > 0) |
---|
1933 | Pi [0] += (mulMod (bufFactors [0] [1], bufFactors[1] [0], MOD) + |
---|
1934 | mulMod (bufFactors [0] [0], bufFactors [1] [1], MOD))*y; |
---|
1935 | else if (degree (bufFactors[0], y) > 0) |
---|
1936 | Pi [0] += mulMod (bufFactors [0] [1], bufFactors[1], MOD)*y; |
---|
1937 | else if (degree (bufFactors[1], y) > 0) |
---|
1938 | Pi [0] += mulMod (bufFactors [0], bufFactors[1] [1], MOD)*y; |
---|
1939 | |
---|
1940 | CFList products; |
---|
1941 | CanonicalForm quot; |
---|
1942 | for (int i= 0; i < bufFactors.size(); i++) |
---|
1943 | { |
---|
1944 | if (degree (bufFactors[i], y) > 0) |
---|
1945 | { |
---|
1946 | if (!fdivides (bufFactors[i] [0], F.getFirst(), quot)) |
---|
1947 | { |
---|
1948 | bad= true; |
---|
1949 | return CFList(); |
---|
1950 | } |
---|
1951 | products.append (quot); |
---|
1952 | } |
---|
1953 | else |
---|
1954 | { |
---|
1955 | if (!fdivides (bufFactors[i], F.getFirst(), quot)) |
---|
1956 | { |
---|
1957 | bad= true; |
---|
1958 | return CFList(); |
---|
1959 | } |
---|
1960 | products.append (quot); |
---|
1961 | } |
---|
1962 | } |
---|
1963 | |
---|
1964 | for (int d= 1; d < lNew; d++) |
---|
1965 | { |
---|
1966 | nonMonicHenselStep (F.getLast(), buf, bufFactors, diophant, M, Pi, products, |
---|
1967 | d, MOD, bad); |
---|
1968 | if (bad) |
---|
1969 | return CFList(); |
---|
1970 | } |
---|
1971 | |
---|
1972 | CFList result; |
---|
1973 | for (k= 0; k < factors.length(); k++) |
---|
1974 | result.append (bufFactors[k]); |
---|
1975 | return result; |
---|
1976 | } |
---|
1977 | |
---|
1978 | CFList |
---|
1979 | nonMonicHenselLift2 (const CFList& eval, const CFList& factors, int* l, int |
---|
1980 | lLength, bool sort, const CFList& LCs1, const CFList& LCs2, |
---|
1981 | const CFArray& Pi, const CFList& diophant, bool& bad) |
---|
1982 | { |
---|
1983 | CFList bufDiophant= diophant; |
---|
1984 | CFList buf= factors; |
---|
1985 | if (sort) |
---|
1986 | sortList (buf, Variable (1)); |
---|
1987 | CFArray bufPi= Pi; |
---|
1988 | CFMatrix M= CFMatrix (l[1], factors.length()); |
---|
1989 | CFList result= |
---|
1990 | nonMonicHenselLift232(eval, buf, l, bufDiophant, bufPi, M, LCs1, LCs2, bad); |
---|
1991 | if (bad) |
---|
1992 | return CFList(); |
---|
1993 | |
---|
1994 | if (eval.length() == 2) |
---|
1995 | return result; |
---|
1996 | CFList MOD; |
---|
1997 | for (int i= 0; i < 2; i++) |
---|
1998 | MOD.append (power (Variable (i + 2), l[i])); |
---|
1999 | CFListIterator j= eval; |
---|
2000 | j++; |
---|
2001 | CFList bufEval; |
---|
2002 | bufEval.append (j.getItem()); |
---|
2003 | j++; |
---|
2004 | CFListIterator jj= LCs1; |
---|
2005 | CFListIterator jjj= LCs2; |
---|
2006 | CFList bufLCs1, bufLCs2; |
---|
2007 | jj++, jjj++; |
---|
2008 | bufLCs1.append (jj.getItem()); |
---|
2009 | bufLCs2.append (jjj.getItem()); |
---|
2010 | jj++, jjj++; |
---|
2011 | |
---|
2012 | for (int i= 2; i < lLength && j.hasItem(); i++, j++, jj++, jjj++) |
---|
2013 | { |
---|
2014 | bufEval.append (j.getItem()); |
---|
2015 | bufLCs1.append (jj.getItem()); |
---|
2016 | bufLCs2.append (jjj.getItem()); |
---|
2017 | M= CFMatrix (l[i], factors.length()); |
---|
2018 | result= nonMonicHenselLift2 (bufEval, result, MOD, bufDiophant, bufPi, M, |
---|
2019 | l[i - 1], l[i], bufLCs1, bufLCs2, bad); |
---|
2020 | if (bad) |
---|
2021 | return CFList(); |
---|
2022 | MOD.append (power (Variable (i + 2), l[i])); |
---|
2023 | bufEval.removeFirst(); |
---|
2024 | bufLCs1.removeFirst(); |
---|
2025 | bufLCs2.removeFirst(); |
---|
2026 | } |
---|
2027 | return result; |
---|
2028 | } |
---|
2029 | |
---|
2030 | CFList |
---|
2031 | nonMonicHenselLift23 (const CanonicalForm& F, const CFList& factors, const |
---|
2032 | CFList& LCs, CFList& diophant, CFArray& Pi, int liftBound, |
---|
2033 | int bivarLiftBound, bool& bad) |
---|
2034 | { |
---|
2035 | CFList bufFactors2= factors; |
---|
2036 | |
---|
2037 | Variable y= Variable (2); |
---|
2038 | for (CFListIterator i= bufFactors2; i.hasItem(); i++) |
---|
2039 | i.getItem()= mod (i.getItem(), y); |
---|
2040 | |
---|
2041 | CanonicalForm bufF= F; |
---|
2042 | bufF= mod (bufF, y); |
---|
2043 | bufF= mod (bufF, Variable (3)); |
---|
2044 | |
---|
2045 | diophant= diophantine (bufF, bufFactors2); |
---|
2046 | |
---|
2047 | CFMatrix M= CFMatrix (liftBound, bufFactors2.length() - 1); |
---|
2048 | |
---|
2049 | Pi= CFArray (bufFactors2.length() - 1); |
---|
2050 | |
---|
2051 | CFArray bufFactors= CFArray (bufFactors2.length()); |
---|
2052 | CFListIterator j= LCs; |
---|
2053 | int i= 0; |
---|
2054 | for (CFListIterator k= factors; k.hasItem(); j++, k++, i++) |
---|
2055 | bufFactors[i]= replaceLC (k.getItem(), j.getItem()); |
---|
2056 | |
---|
2057 | //initialise Pi |
---|
2058 | Variable v= Variable (3); |
---|
2059 | CanonicalForm yToL= power (y, bivarLiftBound); |
---|
2060 | if (degree (bufFactors[0], v) > 0 && degree (bufFactors [1], v) > 0) |
---|
2061 | { |
---|
2062 | M (1, 1)= mulMod2 (bufFactors [0] [0], bufFactors[1] [0], yToL); |
---|
2063 | Pi [0]= M (1,1) + (mulMod2 (bufFactors [0] [1], bufFactors[1] [0], yToL) + |
---|
2064 | mulMod2 (bufFactors [0] [0], bufFactors [1] [1], yToL))*v; |
---|
2065 | } |
---|
2066 | else if (degree (bufFactors[0], v) > 0) |
---|
2067 | { |
---|
2068 | M (1,1)= mulMod2 (bufFactors [0] [0], bufFactors [1], yToL); |
---|
2069 | Pi [0]= M(1,1) + mulMod2 (bufFactors [0] [1], bufFactors[1], yToL)*v; |
---|
2070 | } |
---|
2071 | else if (degree (bufFactors[1], v) > 0) |
---|
2072 | { |
---|
2073 | M (1,1)= mulMod2 (bufFactors [0], bufFactors [1] [0], yToL); |
---|
2074 | Pi [0]= M (1,1) + mulMod2 (bufFactors [0], bufFactors[1] [1], yToL)*v; |
---|
2075 | } |
---|
2076 | else |
---|
2077 | { |
---|
2078 | M (1,1)= mulMod2 (bufFactors [0], bufFactors [1], yToL); |
---|
2079 | Pi [0]= M (1,1); |
---|
2080 | } |
---|
2081 | |
---|
2082 | for (i= 1; i < Pi.size(); i++) |
---|
2083 | { |
---|
2084 | if (degree (Pi[i-1], v) > 0 && degree (bufFactors [i+1], v) > 0) |
---|
2085 | { |
---|
2086 | M (1,i+1)= mulMod2 (Pi[i-1] [0], bufFactors[i+1] [0], yToL); |
---|
2087 | Pi [i]= M (1,i+1) + (mulMod2 (Pi[i-1] [1], bufFactors[i+1] [0], yToL) + |
---|
2088 | mulMod2 (Pi[i-1] [0], bufFactors [i+1] [1], yToL))*v; |
---|
2089 | } |
---|
2090 | else if (degree (Pi[i-1], v) > 0) |
---|
2091 | { |
---|
2092 | M (1,i+1)= mulMod2 (Pi[i-1] [0], bufFactors [i+1], yToL); |
---|
2093 | Pi [i]= M(1,i+1) + mulMod2 (Pi[i-1] [1], bufFactors[i+1], yToL)*v; |
---|
2094 | } |
---|
2095 | else if (degree (bufFactors[i+1], v) > 0) |
---|
2096 | { |
---|
2097 | M (1,i+1)= mulMod2 (Pi[i-1], bufFactors [i+1] [0], yToL); |
---|
2098 | Pi [i]= M (1,i+1) + mulMod2 (Pi[i-1], bufFactors[i+1] [1], yToL)*v; |
---|
2099 | } |
---|
2100 | else |
---|
2101 | { |
---|
2102 | M (1,i+1)= mulMod2 (Pi [i-1], bufFactors [i+1], yToL); |
---|
2103 | Pi [i]= M (1,i+1); |
---|
2104 | } |
---|
2105 | } |
---|
2106 | |
---|
2107 | CFList products; |
---|
2108 | bufF= mod (F, Variable (3)); |
---|
2109 | for (CFListIterator k= factors; k.hasItem(); k++) |
---|
2110 | products.append (bufF/k.getItem()); |
---|
2111 | |
---|
2112 | CFList MOD= CFList (power (v, liftBound)); |
---|
2113 | MOD.insert (yToL); |
---|
2114 | for (int d= 1; d < liftBound; d++) |
---|
2115 | { |
---|
2116 | nonMonicHenselStep (F, factors, bufFactors, diophant, M, Pi, products, d, |
---|
2117 | MOD, bad); |
---|
2118 | if (bad) |
---|
2119 | return CFList(); |
---|
2120 | } |
---|
2121 | |
---|
2122 | CFList result; |
---|
2123 | for (i= 0; i < factors.length(); i++) |
---|
2124 | result.append (bufFactors[i]); |
---|
2125 | return result; |
---|
2126 | } |
---|
2127 | |
---|
2128 | CFList |
---|
2129 | nonMonicHenselLift (const CFList& F, const CFList& factors, const CFList& LCs, |
---|
2130 | CFList& diophant, CFArray& Pi, CFMatrix& M, int lOld, |
---|
2131 | int& lNew, const CFList& MOD, bool& noOneToOne |
---|
2132 | ) |
---|
2133 | { |
---|
2134 | |
---|
2135 | int k= 0; |
---|
2136 | CFArray bufFactors= CFArray (factors.length()); |
---|
2137 | CFListIterator j= LCs; |
---|
2138 | for (CFListIterator i= factors; i.hasItem(); i++, j++, k++) |
---|
2139 | bufFactors [k]= replaceLC (i.getItem(), j.getItem()); |
---|
2140 | |
---|
2141 | Variable y= F.getLast().mvar(); |
---|
2142 | Variable x= F.getFirst().mvar(); |
---|
2143 | CanonicalForm xToLOld= power (x, lOld); |
---|
2144 | |
---|
2145 | Pi [0]= mod (Pi[0], xToLOld); |
---|
2146 | M (1, 1)= Pi [0]; |
---|
2147 | |
---|
2148 | if (degree (bufFactors[0], y) > 0 && degree (bufFactors [1], y) > 0) |
---|
2149 | Pi [0] += (mulMod (bufFactors [0] [1], bufFactors[1] [0], MOD) + |
---|
2150 | mulMod (bufFactors [0] [0], bufFactors [1] [1], MOD))*y; |
---|
2151 | else if (degree (bufFactors[0], y) > 0) |
---|
2152 | Pi [0] += mulMod (bufFactors [0] [1], bufFactors[1], MOD)*y; |
---|
2153 | else if (degree (bufFactors[1], y) > 0) |
---|
2154 | Pi [0] += mulMod (bufFactors [0], bufFactors[1] [1], MOD)*y; |
---|
2155 | |
---|
2156 | for (int i= 1; i < Pi.size(); i++) |
---|
2157 | { |
---|
2158 | Pi [i]= mod (Pi [i], xToLOld); |
---|
2159 | M (1, i + 1)= Pi [i]; |
---|
2160 | |
---|
2161 | if (degree (Pi[i-1], y) > 0 && degree (bufFactors [i+1], y) > 0) |
---|
2162 | Pi [i] += (mulMod (Pi[i-1] [1], bufFactors[i+1] [0], MOD) + |
---|
2163 | mulMod (Pi[i-1] [0], bufFactors [i+1] [1], MOD))*y; |
---|
2164 | else if (degree (Pi[i-1], y) > 0) |
---|
2165 | Pi [i] += mulMod (Pi[i-1] [1], bufFactors[i+1], MOD)*y; |
---|
2166 | else if (degree (bufFactors[i+1], y) > 0) |
---|
2167 | Pi [i] += mulMod (Pi[i-1], bufFactors[i+1] [1], MOD)*y; |
---|
2168 | } |
---|
2169 | |
---|
2170 | CFList products; |
---|
2171 | CanonicalForm quot, bufF= F.getFirst(); |
---|
2172 | |
---|
2173 | for (int i= 0; i < bufFactors.size(); i++) |
---|
2174 | { |
---|
2175 | if (degree (bufFactors[i], y) > 0) |
---|
2176 | { |
---|
2177 | if (!fdivides (bufFactors[i] [0], bufF, quot)) |
---|
2178 | { |
---|
2179 | noOneToOne= true; |
---|
2180 | return factors; |
---|
2181 | } |
---|
2182 | products.append (quot); |
---|
2183 | } |
---|
2184 | else |
---|
2185 | { |
---|
2186 | if (!fdivides (bufFactors[i], bufF, quot)) |
---|
2187 | { |
---|
2188 | noOneToOne= true; |
---|
2189 | return factors; |
---|
2190 | } |
---|
2191 | products.append (quot); |
---|
2192 | } |
---|
2193 | } |
---|
2194 | |
---|
2195 | for (int d= 1; d < lNew; d++) |
---|
2196 | { |
---|
2197 | nonMonicHenselStep (F.getLast(), factors, bufFactors, diophant, M, Pi, |
---|
2198 | products, d, MOD, noOneToOne); |
---|
2199 | if (noOneToOne) |
---|
2200 | return CFList(); |
---|
2201 | } |
---|
2202 | |
---|
2203 | CFList result; |
---|
2204 | for (k= 0; k < factors.length(); k++) |
---|
2205 | result.append (bufFactors[k]); |
---|
2206 | return result; |
---|
2207 | } |
---|
2208 | |
---|
2209 | CFList |
---|
2210 | nonMonicHenselLift (const CFList& eval, const CFList& factors, |
---|
2211 | CFList* const& LCs, CFList& diophant, CFArray& Pi, |
---|
2212 | int* liftBound, int length, bool& noOneToOne |
---|
2213 | ) |
---|
2214 | { |
---|
2215 | CFList bufDiophant= diophant; |
---|
2216 | CFList buf= factors; |
---|
2217 | CFArray bufPi= Pi; |
---|
2218 | CFMatrix M= CFMatrix (liftBound[1], factors.length() - 1); |
---|
2219 | int k= 0; |
---|
2220 | |
---|
2221 | CFList result= |
---|
2222 | nonMonicHenselLift23 (eval.getFirst(), factors, LCs [0], diophant, bufPi, |
---|
2223 | liftBound[1], liftBound[0], noOneToOne); |
---|
2224 | |
---|
2225 | if (noOneToOne) |
---|
2226 | return CFList(); |
---|
2227 | |
---|
2228 | if (eval.length() == 1) |
---|
2229 | return result; |
---|
2230 | |
---|
2231 | k++; |
---|
2232 | CFList MOD; |
---|
2233 | for (int i= 0; i < 2; i++) |
---|
2234 | MOD.append (power (Variable (i + 2), liftBound[i])); |
---|
2235 | |
---|
2236 | CFListIterator j= eval; |
---|
2237 | CFList bufEval; |
---|
2238 | bufEval.append (j.getItem()); |
---|
2239 | j++; |
---|
2240 | |
---|
2241 | for (int i= 2; i <= length && j.hasItem(); i++, j++, k++) |
---|
2242 | { |
---|
2243 | bufEval.append (j.getItem()); |
---|
2244 | M= CFMatrix (liftBound[i], factors.length() - 1); |
---|
2245 | result= nonMonicHenselLift (bufEval, result, LCs [i-1], diophant, bufPi, M, |
---|
2246 | liftBound[i-1], liftBound[i], MOD, noOneToOne); |
---|
2247 | if (noOneToOne) |
---|
2248 | return result; |
---|
2249 | MOD.append (power (Variable (i + 2), liftBound[i])); |
---|
2250 | bufEval.removeFirst(); |
---|
2251 | } |
---|
2252 | |
---|
2253 | return result; |
---|
2254 | } |
---|
2255 | |
---|
2256 | #endif |
---|
2257 | /* HAVE_NTL */ |
---|
2258 | |
---|