Line data Source code
1 : /* Copyright (C) 2000 The PARI group.
2 :
3 : This file is part of the PARI/GP package.
4 :
5 : PARI/GP is free software; you can redistribute it and/or modify it under the
6 : terms of the GNU General Public License as published by the Free Software
7 : Foundation; either version 2 of the License, or (at your option) any later
8 : version. It is distributed in the hope that it will be useful, but WITHOUT
9 : ANY WARRANTY WHATSOEVER.
10 :
11 : Check the License for details. You should have received a copy of it, along
12 : with the package; see the file 'COPYING'. If not, write to the Free Software
13 : Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. */
14 : #include "pari.h"
15 : #include "paripriv.h"
16 :
17 : #define DEBUGLEVEL DEBUGLEVEL_factor
18 :
19 : /* x,y two ZX, y non constant. Return q = x/y if y divides x in Z[X] and NULL
20 : * otherwise. If not NULL, B is a t_INT upper bound for ||q||_oo. */
21 : static GEN
22 6516730 : ZX_divides_i(GEN x, GEN y, GEN B)
23 : {
24 : long dx, dy, dz, i, j;
25 : pari_sp av;
26 : GEN z,p1,y_lead;
27 :
28 6516730 : dy=degpol(y);
29 6516730 : dx=degpol(x);
30 6516730 : dz=dx-dy; if (dz<0) return NULL;
31 6515666 : z=cgetg(dz+3,t_POL); z[1] = x[1];
32 6515664 : x += 2; y += 2; z += 2;
33 6515664 : y_lead = gel(y,dy);
34 6515664 : if (equali1(y_lead)) y_lead = NULL;
35 :
36 6515664 : p1 = gel(x,dx);
37 6515664 : if (y_lead) {
38 : GEN r;
39 16058 : p1 = dvmdii(p1,y_lead, &r);
40 16058 : if (r != gen_0) return NULL;
41 : }
42 6499606 : else p1 = icopy(p1);
43 6511504 : gel(z,dz) = p1;
44 8060613 : for (i=dx-1; i>=dy; i--)
45 : {
46 1554246 : av = avma; p1 = gel(x,i);
47 5423828 : for (j=i-dy+1; j<=i && j<=dz; j++)
48 3869646 : p1 = subii(p1, mulii(gel(z,j),gel(y,i-j)));
49 1554182 : if (y_lead) {
50 : GEN r;
51 33887 : p1 = dvmdii(p1,y_lead, &r);
52 33887 : if (r != gen_0) return NULL;
53 : }
54 1552774 : if (B && abscmpii(p1, B) > 0) return NULL;
55 1549053 : p1 = gerepileuptoint(av, p1);
56 1549109 : gel(z,i-dy) = p1;
57 : }
58 6506367 : av = avma;
59 16920193 : for (; i >= 0; i--)
60 : {
61 10451884 : p1 = gel(x,i);
62 : /* we always enter this loop at least once */
63 23463105 : for (j=0; j<=i && j<=dz; j++)
64 13011238 : p1 = subii(p1, mulii(gel(z,j),gel(y,i-j)));
65 10451867 : if (signe(p1)) return NULL;
66 10413828 : set_avma(av);
67 : }
68 6468309 : return z - 2;
69 : }
70 : static GEN
71 6490420 : ZX_divides(GEN x, GEN y) { return ZX_divides_i(x,y,NULL); }
72 :
73 : #if 0
74 : /* cf Beauzamy et al: upper bound for
75 : * lc(x) * [2^(5/8) / pi^(3/8)] e^(1/4n) 2^(n/2) sqrt([x]_2)/ n^(3/8)
76 : * where [x]_2 = sqrt(\sum_i=0^n x[i]^2 / binomial(n,i)). One factor has
77 : * all coeffs less than then bound */
78 : static GEN
79 : two_factor_bound(GEN x)
80 : {
81 : long i, j, n = lg(x) - 3;
82 : pari_sp av = avma;
83 : GEN *invbin, c, r = cgetr(3), z;
84 :
85 : x += 2; invbin = (GEN*)new_chunk(n+1);
86 : z = real_1(LOWDEFAULTPREC); /* invbin[i] = 1 / binomial(n, i) */
87 : for (i=0,j=n; j >= i; i++,j--)
88 : {
89 : invbin[i] = invbin[j] = z;
90 : z = divru(mulru(z, i+1), n-i);
91 : }
92 : z = invbin[0]; /* = 1 */
93 : for (i=0; i<=n; i++)
94 : {
95 : c = gel(x,i); if (!signe(c)) continue;
96 : affir(c, r);
97 : z = addrr(z, mulrr(sqrr(r), invbin[i]));
98 : }
99 : z = shiftr(sqrtr(z), n);
100 : z = divrr(z, dbltor(pow((double)n, 0.75)));
101 : z = roundr_safe(sqrtr(z));
102 : z = mulii(z, absi_shallow(gel(x,n)));
103 : return gerepileuptoint(av, shifti(z, 1));
104 : }
105 : #endif
106 :
107 : /* A | S ==> |a_i| <= binom(d-1, i-1) || S ||_2 + binom(d-1, i) lc(S) */
108 : static GEN
109 57282 : Mignotte_bound(GEN S)
110 : {
111 57282 : long i, d = degpol(S);
112 57282 : GEN C, N2, t, binlS, lS = leading_coeff(S), bin = vecbinomial(d-1);
113 :
114 57282 : N2 = sqrtr(RgX_fpnorml2(S,DEFAULTPREC));
115 57282 : binlS = is_pm1(lS)? bin: ZC_Z_mul(bin, lS);
116 :
117 : /* i = 0 */
118 57282 : C = gel(binlS,1);
119 : /* i = d */
120 57282 : t = N2; if (gcmp(C, t) < 0) C = t;
121 511395 : for (i = 1; i < d; i++)
122 : {
123 454116 : t = addri(mulir(gel(bin,i), N2), gel(binlS,i+1));
124 454111 : if (mpcmp(C, t) < 0) C = t;
125 : }
126 57279 : return C;
127 : }
128 : /* A | S ==> |a_i|^2 <= 3^{3/2 + d} / (4 \pi d) [P]_2^2,
129 : * where [P]_2 is Bombieri's 2-norm */
130 : static GEN
131 57282 : Beauzamy_bound(GEN S)
132 : {
133 57282 : const long prec = DEFAULTPREC;
134 57282 : long i, d = degpol(S);
135 : GEN bin, lS, s, C;
136 57282 : bin = vecbinomial(d);
137 :
138 57282 : s = real_0(prec);
139 625963 : for (i=0; i<=d; i++)
140 : {
141 568681 : GEN c = gel(S,i+2);
142 568681 : if (gequal0(c)) continue;
143 : /* s += P_i^2 / binomial(d,i) */
144 479331 : s = addrr(s, divri(itor(sqri(c), prec), gel(bin,i+1)));
145 : }
146 : /* s = [S]_2^2 */
147 57282 : C = powruhalf(utor(3,prec), 3 + 2*d); /* 3^{3/2 + d} */
148 57282 : C = divrr(mulrr(C, s), mulur(4*d, mppi(prec)));
149 57282 : lS = absi_shallow(leading_coeff(S));
150 57282 : return mulir(lS, sqrtr(C));
151 : }
152 :
153 : static GEN
154 57282 : factor_bound(GEN S)
155 : {
156 57282 : pari_sp av = avma;
157 57282 : GEN a = Mignotte_bound(S);
158 57282 : GEN b = Beauzamy_bound(S);
159 57282 : if (DEBUGLEVEL>2)
160 : {
161 0 : err_printf("Mignotte bound: %Ps\n",a);
162 0 : err_printf("Beauzamy bound: %Ps\n",b);
163 : }
164 57282 : return gerepileupto(av, ceil_safe(gmin_shallow(a, b)));
165 : }
166 :
167 : /* Naive recombination of modular factors: combine up to maxK modular
168 : * factors, degree <= klim
169 : *
170 : * target = polynomial we want to factor
171 : * famod = array of modular factors. Product should be congruent to
172 : * target/lc(target) modulo p^a
173 : * For true factors: S1,S2 <= p^b, with b <= a and p^(b-a) < 2^31 */
174 : static GEN
175 47007 : cmbf(GEN pol, GEN famod, GEN bound, GEN p, long a, long b,
176 : long klim, long *pmaxK, int *done)
177 : {
178 47007 : long K = 1, cnt = 1, i,j,k, curdeg, lfamod = lg(famod)-1;
179 : ulong spa_b, spa_bs2, Sbound;
180 47007 : GEN lc, lcpol, pa = powiu(p,a), pas2 = shifti(pa,-1);
181 47007 : GEN trace1 = cgetg(lfamod+1, t_VECSMALL);
182 47007 : GEN trace2 = cgetg(lfamod+1, t_VECSMALL);
183 47007 : GEN ind = cgetg(lfamod+1, t_VECSMALL);
184 47007 : GEN deg = cgetg(lfamod+1, t_VECSMALL);
185 47007 : GEN degsofar = cgetg(lfamod+1, t_VECSMALL);
186 47007 : GEN listmod = cgetg(lfamod+1, t_VEC);
187 47007 : GEN fa = cgetg(lfamod+1, t_VEC);
188 :
189 47007 : *pmaxK = cmbf_maxK(lfamod);
190 47007 : lc = absi_shallow(leading_coeff(pol));
191 47007 : if (equali1(lc)) lc = NULL;
192 47007 : lcpol = lc? ZX_Z_mul(pol, lc): pol;
193 :
194 : {
195 47007 : GEN pa_b,pa_bs2,pb, lc2 = lc? sqri(lc): NULL;
196 :
197 47007 : pa_b = powiu(p, a-b); /* < 2^31 */
198 47007 : pa_bs2 = shifti(pa_b,-1);
199 47007 : pb= powiu(p, b);
200 167534 : for (i=1; i <= lfamod; i++)
201 : {
202 120527 : GEN T1,T2, P = gel(famod,i);
203 120527 : long d = degpol(P);
204 :
205 120527 : deg[i] = d; P += 2;
206 120527 : T1 = gel(P,d-1);/* = - S_1 */
207 120527 : T2 = sqri(T1);
208 120527 : if (d > 1) T2 = subii(T2, shifti(gel(P,d-2),1));
209 120527 : T2 = modii(T2, pa); /* = S_2 Newton sum */
210 120527 : if (lc)
211 : {
212 3059 : T1 = Fp_mul(lc, T1, pa);
213 3059 : T2 = Fp_mul(lc2,T2, pa);
214 : }
215 120527 : uel(trace1,i) = itou(diviiround(T1, pb));
216 120527 : uel(trace2,i) = itou(diviiround(T2, pb));
217 : }
218 47007 : spa_b = uel(pa_b,2); /* < 2^31 */
219 47007 : spa_bs2 = uel(pa_bs2,2); /* < 2^31 */
220 : }
221 47007 : degsofar[0] = 0; /* sentinel */
222 :
223 : /* ind runs through strictly increasing sequences of length K,
224 : * 1 <= ind[i] <= lfamod */
225 87438 : nextK:
226 87438 : if (K > *pmaxK || 2*K > lfamod) goto END;
227 52564 : if (DEBUGLEVEL > 3)
228 0 : err_printf("\n### K = %d, %Ps combinations\n", K,binomial(utoipos(lfamod), K));
229 52564 : setlg(ind, K+1); ind[1] = 1;
230 52564 : Sbound = (ulong) ((K+1)>>1);
231 52564 : i = 1; curdeg = deg[ind[1]];
232 : for(;;)
233 : { /* try all combinations of K factors */
234 618752 : for (j = i; j < K; j++)
235 : {
236 85684 : degsofar[j] = curdeg;
237 85684 : ind[j+1] = ind[j]+1; curdeg += deg[ind[j+1]];
238 : }
239 533068 : if (curdeg <= klim) /* trial divide */
240 : {
241 : GEN y, q, list;
242 : pari_sp av;
243 : ulong t;
244 :
245 : /* d - 1 test */
246 1348709 : for (t=uel(trace1,ind[1]),i=2; i<=K; i++)
247 815641 : t = Fl_add(t, uel(trace1,ind[i]), spa_b);
248 533068 : if (t > spa_bs2) t = spa_b - t;
249 533068 : if (t > Sbound)
250 : {
251 427208 : if (DEBUGLEVEL>6) err_printf(".");
252 427208 : goto NEXT;
253 : }
254 : /* d - 2 test */
255 231092 : for (t=uel(trace2,ind[1]),i=2; i<=K; i++)
256 125232 : t = Fl_add(t, uel(trace2,ind[i]), spa_b);
257 105860 : if (t > spa_bs2) t = spa_b - t;
258 105860 : if (t > Sbound)
259 : {
260 57228 : if (DEBUGLEVEL>6) err_printf("|");
261 57228 : goto NEXT;
262 : }
263 :
264 48632 : av = avma;
265 : /* check trailing coeff */
266 48632 : y = lc;
267 145402 : for (i=1; i<=K; i++)
268 : {
269 96770 : GEN q = constant_coeff(gel(famod,ind[i]));
270 96770 : if (y) q = mulii(y, q);
271 96770 : y = centermodii(q, pa, pas2);
272 : }
273 48632 : if (!signe(y) || !dvdii(constant_coeff(lcpol), y))
274 : {
275 22482 : if (DEBUGLEVEL>3) err_printf("T");
276 22482 : set_avma(av); goto NEXT;
277 : }
278 26150 : y = lc; /* full computation */
279 58905 : for (i=1; i<=K; i++)
280 : {
281 32755 : GEN q = gel(famod,ind[i]);
282 32755 : if (y) q = gmul(y, q);
283 32755 : y = centermod_i(q, pa, pas2);
284 : }
285 :
286 : /* y is the candidate factor */
287 26150 : if (! (q = ZX_divides_i(lcpol,y,bound)) )
288 : {
289 3749 : if (DEBUGLEVEL>3) err_printf("*");
290 3749 : set_avma(av); goto NEXT;
291 : }
292 : /* found a factor */
293 22401 : list = cgetg(K+1, t_VEC);
294 22401 : gel(listmod,cnt) = list;
295 45509 : for (i=1; i<=K; i++) list[i] = famod[ind[i]];
296 :
297 22401 : y = Q_primpart(y);
298 22401 : gel(fa,cnt++) = y;
299 : /* fix up pol */
300 22401 : pol = q;
301 22401 : if (lc) pol = Q_div_to_int(pol, leading_coeff(y));
302 84338 : for (i=j=k=1; i <= lfamod; i++)
303 : { /* remove used factors */
304 61937 : if (j <= K && i == ind[j]) j++;
305 : else
306 : {
307 38829 : gel(famod,k) = gel(famod,i);
308 38829 : uel(trace1,k) = uel(trace1,i);
309 38829 : uel(trace2,k) = uel(trace2,i);
310 38829 : deg[k] = deg[i]; k++;
311 : }
312 : }
313 22401 : lfamod -= K;
314 22401 : *pmaxK = cmbf_maxK(lfamod);
315 22401 : if (lfamod < 2*K) goto END;
316 10268 : i = 1; curdeg = deg[ind[1]];
317 10268 : bound = factor_bound(pol);
318 10268 : if (lc) lc = absi_shallow(leading_coeff(pol));
319 10268 : lcpol = lc? ZX_Z_mul(pol, lc): pol;
320 10268 : if (DEBUGLEVEL>3)
321 0 : err_printf("\nfound factor %Ps\nremaining modular factor(s): %ld\n",
322 : y, lfamod);
323 10268 : continue;
324 : }
325 :
326 0 : NEXT:
327 510667 : for (i = K+1;;)
328 : {
329 636075 : if (--i == 0) { K++; goto nextK; }
330 595644 : if (++ind[i] <= lfamod - K + i)
331 : {
332 470236 : curdeg = degsofar[i-1] + deg[ind[i]];
333 470236 : if (curdeg <= klim) break;
334 : }
335 : }
336 : }
337 47007 : END:
338 47007 : *done = 1;
339 47007 : if (degpol(pol) > 0)
340 : { /* leftover factor */
341 47007 : if (signe(leading_coeff(pol)) < 0) pol = ZX_neg(pol);
342 47007 : if (lfamod >= 2*K) *done = 0;
343 :
344 47007 : setlg(famod, lfamod+1);
345 47007 : gel(listmod,cnt) = leafcopy(famod);
346 47007 : gel(fa,cnt++) = pol;
347 : }
348 47007 : if (DEBUGLEVEL>6) err_printf("\n");
349 47007 : setlg(listmod, cnt);
350 47007 : setlg(fa, cnt); return mkvec2(fa, listmod);
351 : }
352 :
353 : /* recombination of modular factors: van Hoeij's algorithm */
354 :
355 : /* Q in Z[X], return Q(2^n) */
356 : static GEN
357 161548 : shifteval(GEN Q, long n)
358 : {
359 161548 : pari_sp av = avma;
360 161548 : long i, l = lg(Q);
361 : GEN s;
362 :
363 161548 : if (!signe(Q)) return gen_0;
364 161548 : s = gel(Q,l-1);
365 910521 : for (i = l-2; i > 1; i--)
366 : {
367 748973 : s = addii(gel(Q,i), shifti(s, n));
368 748973 : if (gc_needed(av,1)) s = gerepileuptoint(av, s);
369 : }
370 161548 : return s;
371 : }
372 :
373 : /* return integer y such that all |a| <= y if P(a) = 0 */
374 : static GEN
375 97185 : root_bound(GEN P0)
376 : {
377 97185 : GEN Q = leafcopy(P0), lP = absi_shallow(leading_coeff(Q)), x,y,z;
378 97185 : long k, d = degpol(Q);
379 :
380 : /* P0 = lP x^d + Q, deg Q < d */
381 97185 : Q = normalizepol_lg(Q, d+2);
382 645939 : for (k=lg(Q)-1; k>1; k--) gel(Q,k) = absi_shallow(gel(Q,k));
383 97185 : k = (long)(fujiwara_bound(P0));
384 161878 : for ( ; k >= 0; k--)
385 : {
386 161549 : pari_sp av = avma;
387 : /* y = 2^k; Q(y) >= lP y^d ? */
388 161549 : if (cmpii(shifteval(Q,k), shifti(lP, d*k)) >= 0) break;
389 64693 : set_avma(av);
390 : }
391 97185 : if (k < 0) k = 0;
392 97185 : y = int2n(k+1);
393 97185 : if (d > 2000) return y; /* likely to be expensive, don't bother */
394 97185 : x = int2n(k);
395 97185 : for(k=0; ; k++)
396 : {
397 532159 : z = shifti(addii(x,y), -1);
398 532159 : if (equalii(x,z) || k > 5) break;
399 434974 : if (cmpii(ZX_Z_eval(Q,z), mulii(lP, powiu(z, d))) < 0)
400 232639 : y = z;
401 : else
402 202335 : x = z;
403 : }
404 97185 : return y;
405 : }
406 :
407 : GEN
408 350 : chk_factors_get(GEN lt, GEN famod, GEN c, GEN T, GEN N)
409 : {
410 350 : long i = 1, j, l = lg(famod);
411 350 : GEN V = cgetg(l, t_VEC);
412 8414 : for (j = 1; j < l; j++)
413 8064 : if (signe(gel(c,j))) gel(V,i++) = gel(famod,j);
414 350 : if (lt && i > 1) gel(V,1) = RgX_Rg_mul(gel(V,1), lt);
415 350 : setlg(V, i);
416 350 : return T? FpXQXV_prod(V, T, N): FpXV_prod(V,N);
417 : }
418 :
419 : static GEN
420 140 : chk_factors(GEN P, GEN M_L, GEN bound, GEN famod, GEN pa)
421 : {
422 : long i, r;
423 140 : GEN pol = P, list, piv, y, ltpol, lt, paov2;
424 :
425 140 : piv = ZM_hnf_knapsack(M_L);
426 140 : if (!piv) return NULL;
427 70 : if (DEBUGLEVEL>7) err_printf("ZM_hnf_knapsack output:\n%Ps\n",piv);
428 :
429 70 : r = lg(piv)-1;
430 70 : list = cgetg(r+1, t_VEC);
431 70 : lt = absi_shallow(leading_coeff(pol));
432 70 : if (equali1(lt)) lt = NULL;
433 70 : ltpol = lt? ZX_Z_mul(pol, lt): pol;
434 70 : paov2 = shifti(pa,-1);
435 70 : for (i = 1;;)
436 : {
437 161 : if (DEBUGLEVEL) err_printf("LLL_cmbf: checking factor %ld\n",i);
438 161 : y = chk_factors_get(lt, famod, gel(piv,i), NULL, pa);
439 161 : y = FpX_center_i(y, pa, paov2);
440 161 : if (! (pol = ZX_divides_i(ltpol,y,bound)) ) return NULL;
441 133 : if (lt) y = Q_primpart(y);
442 133 : gel(list,i) = y;
443 133 : if (++i >= r) break;
444 :
445 91 : if (lt)
446 : {
447 35 : pol = ZX_Z_divexact(pol, leading_coeff(y));
448 35 : lt = absi_shallow(leading_coeff(pol));
449 35 : ltpol = ZX_Z_mul(pol, lt);
450 : }
451 : else
452 56 : ltpol = pol;
453 : }
454 42 : y = Q_primpart(pol);
455 42 : gel(list,i) = y; return list;
456 : }
457 :
458 : GEN
459 1547 : LLL_check_progress(GEN Bnorm, long n0, GEN m, int final, long *ti_LLL)
460 : {
461 : GEN norm, u;
462 : long i, R;
463 : pari_timer T;
464 :
465 1547 : if (DEBUGLEVEL>2) timer_start(&T);
466 1547 : u = ZM_lll_norms(m, final? 0.999: 0.75, LLL_INPLACE, &norm);
467 1547 : if (DEBUGLEVEL>2) *ti_LLL += timer_delay(&T);
468 10542 : for (R=lg(m)-1; R > 0; R--)
469 10542 : if (cmprr(gel(norm,R), Bnorm) < 0) break;
470 16156 : for (i=1; i<=R; i++) setlg(u[i], n0+1);
471 1547 : if (R <= 1)
472 : {
473 105 : if (!R) pari_err_BUG("LLL_cmbf [no factor]");
474 105 : return NULL; /* irreducible */
475 : }
476 1442 : setlg(u, R+1); return u;
477 : }
478 :
479 : static ulong
480 14 : next2pow(ulong a)
481 : {
482 14 : ulong b = 1;
483 112 : while (b < a) b <<= 1;
484 14 : return b;
485 : }
486 :
487 : /* Recombination phase of Berlekamp-Zassenhaus algorithm using a variant of
488 : * van Hoeij's knapsack
489 : *
490 : * P = squarefree in Z[X].
491 : * famod = array of (lifted) modular factors mod p^a
492 : * bound = Mignotte bound for the size of divisors of P (for the sup norm)
493 : * previously recombined all set of factors with less than rec elts */
494 : static GEN
495 126 : LLL_cmbf(GEN P, GEN famod, GEN p, GEN pa, GEN bound, long a, long rec)
496 : {
497 126 : const long N0 = 1; /* # of traces added at each step */
498 126 : double BitPerFactor = 0.4; /* nb bits in p^(a-b) / modular factor */
499 126 : long i,j,tmax,n0,C, dP = degpol(P);
500 126 : double logp = log((double)itos(p)), LOGp2 = M_LN2/logp;
501 126 : double b0 = log((double)dP*2) / logp, logBr;
502 : GEN lP, Br, Bnorm, Tra, T2, TT, CM_L, m, list, ZERO;
503 : pari_sp av, av2;
504 126 : long ti_LLL = 0, ti_CF = 0;
505 :
506 126 : lP = absi_shallow(leading_coeff(P));
507 126 : if (equali1(lP)) lP = NULL;
508 126 : Br = root_bound(P);
509 126 : if (lP) Br = mulii(lP, Br);
510 126 : logBr = gtodouble(glog(Br, DEFAULTPREC)) / logp;
511 :
512 126 : n0 = lg(famod) - 1;
513 126 : C = (long)ceil( sqrt(N0 * n0 / 4.) ); /* > 1 */
514 126 : Bnorm = dbltor(n0 * (C*C + N0*n0/4.) * 1.00001);
515 126 : ZERO = zeromat(n0, N0);
516 :
517 126 : av = avma;
518 126 : TT = cgetg(n0+1, t_VEC);
519 126 : Tra = cgetg(n0+1, t_MAT);
520 2401 : for (i=1; i<=n0; i++)
521 : {
522 2275 : TT[i] = 0;
523 2275 : gel(Tra,i) = cgetg(N0+1, t_COL);
524 : }
525 126 : CM_L = scalarmat_s(C, n0);
526 : /* tmax = current number of traces used (and computed so far) */
527 126 : for (tmax = 0;; tmax += N0)
528 469 : {
529 595 : long b, bmin, bgood, delta, tnew = tmax + N0, r = lg(CM_L)-1;
530 : GEN M_L, q, CM_Lp, oldCM_L;
531 595 : int first = 1;
532 : pari_timer ti2, TI;
533 :
534 595 : bmin = (long)ceil(b0 + tnew*logBr);
535 595 : if (DEBUGLEVEL>2)
536 0 : err_printf("\nLLL_cmbf: %ld potential factors (tmax = %ld, bmin = %ld)\n",
537 : r, tmax, bmin);
538 :
539 : /* compute Newton sums (possibly relifting first) */
540 595 : if (a <= bmin)
541 : {
542 14 : a = (long)ceil(bmin + 3*N0*logBr) + 1; /* enough for 3 more rounds */
543 14 : a = (long)next2pow((ulong)a);
544 :
545 14 : pa = powiu(p,a);
546 14 : famod = ZpX_liftfact(P, famod, pa, p, a);
547 266 : for (i=1; i<=n0; i++) TT[i] = 0;
548 : }
549 11557 : for (i=1; i<=n0; i++)
550 : {
551 10962 : GEN p1 = gel(Tra,i);
552 10962 : GEN p2 = polsym_gen(gel(famod,i), gel(TT,i), tnew, NULL, pa);
553 10962 : gel(TT,i) = p2;
554 10962 : p2 += 1+tmax; /* ignore traces number 0...tmax */
555 21924 : for (j=1; j<=N0; j++) gel(p1,j) = gel(p2,j);
556 10962 : if (lP)
557 : { /* make Newton sums integral */
558 1848 : GEN lPpow = powiu(lP, tmax);
559 3696 : for (j=1; j<=N0; j++)
560 : {
561 1848 : lPpow = mulii(lPpow,lP);
562 1848 : gel(p1,j) = mulii(gel(p1,j), lPpow);
563 : }
564 : }
565 : }
566 :
567 : /* compute truncation parameter */
568 595 : if (DEBUGLEVEL>2) { timer_start(&ti2); timer_start(&TI); }
569 595 : oldCM_L = CM_L;
570 595 : av2 = avma;
571 595 : delta = b = 0; /* -Wall */
572 1071 : AGAIN:
573 1071 : M_L = Q_div_to_int(CM_L, utoipos(C));
574 1071 : T2 = centermod( ZM_mul(Tra, M_L), pa );
575 1071 : if (first)
576 : { /* initialize lattice, using few p-adic digits for traces */
577 595 : double t = gexpo(T2) - maxdd(32.0, BitPerFactor*r);
578 595 : bgood = (long) (t * LOGp2);
579 595 : b = maxss(bmin, bgood);
580 595 : delta = a - b;
581 : }
582 : else
583 : { /* add more p-adic digits and continue reduction */
584 476 : long b0 = (long)(gexpo(T2) * LOGp2);
585 476 : if (b0 < b) b = b0;
586 476 : b = maxss(b-delta, bmin);
587 476 : if (b - delta/2 < bmin) b = bmin; /* near there. Go all the way */
588 : }
589 :
590 1071 : q = powiu(p, b);
591 1071 : m = vconcat( CM_L, gdivround(T2, q) );
592 1071 : if (first)
593 : {
594 595 : GEN P1 = scalarmat(powiu(p, a-b), N0);
595 595 : first = 0;
596 595 : m = shallowconcat( m, vconcat(ZERO, P1) );
597 : /* [ C M_L 0 ]
598 : * m = [ ] square matrix
599 : * [ T2' p^(a-b) I_N0 ] T2' = Tra * M_L truncated
600 : */
601 : }
602 :
603 1071 : CM_L = LLL_check_progress(Bnorm, n0, m, b == bmin, /*dbg:*/ &ti_LLL);
604 1071 : if (DEBUGLEVEL>2)
605 0 : err_printf("LLL_cmbf: (a,b) =%4ld,%4ld; r =%3ld -->%3ld, time = %ld\n",
606 0 : a,b, lg(m)-1, CM_L? lg(CM_L)-1: 1, timer_delay(&TI));
607 1071 : if (!CM_L) { list = mkvec(P); break; }
608 987 : if (b > bmin)
609 : {
610 476 : CM_L = gerepilecopy(av2, CM_L);
611 476 : goto AGAIN;
612 : }
613 511 : if (DEBUGLEVEL>2) timer_printf(&ti2, "for this block of traces");
614 :
615 511 : i = lg(CM_L) - 1;
616 511 : if (i == r && ZM_equal(CM_L, oldCM_L))
617 : {
618 63 : CM_L = oldCM_L;
619 63 : set_avma(av2); continue;
620 : }
621 :
622 448 : CM_Lp = FpM_image(CM_L, utoipos(27449)); /* inexpensive test */
623 448 : if (lg(CM_Lp) != lg(CM_L))
624 : {
625 7 : if (DEBUGLEVEL>2) err_printf("LLL_cmbf: rank decrease\n");
626 7 : CM_L = ZM_hnf(CM_L);
627 : }
628 :
629 448 : if (i <= r && i*rec < n0)
630 : {
631 : pari_timer ti;
632 140 : if (DEBUGLEVEL>2) timer_start(&ti);
633 140 : list = chk_factors(P, Q_div_to_int(CM_L,utoipos(C)), bound, famod, pa);
634 140 : if (DEBUGLEVEL>2) ti_CF += timer_delay(&ti);
635 140 : if (list) break;
636 98 : if (DEBUGLEVEL>2) err_printf("LLL_cmbf: chk_factors failed");
637 : }
638 406 : CM_L = gerepilecopy(av2, CM_L);
639 406 : if (gc_needed(av,1))
640 : {
641 0 : if(DEBUGMEM>1) pari_warn(warnmem,"LLL_cmbf");
642 0 : gerepileall(av, 5, &CM_L, &TT, &Tra, &famod, &pa);
643 : }
644 : }
645 126 : if (DEBUGLEVEL>2)
646 0 : err_printf("* Time LLL: %ld\n* Time Check Factor: %ld\n",ti_LLL,ti_CF);
647 126 : return list;
648 : }
649 :
650 : /* Find a,b minimal such that A < q^a, B < q^b, 1 << q^(a-b) < 2^31 */
651 : static int
652 47007 : cmbf_precs(GEN q, GEN A, GEN B, long *pta, long *ptb, GEN *qa, GEN *qb)
653 : {
654 47007 : long a,b,amin,d = (long)(31 * M_LN2/gtodouble(glog(q,DEFAULTPREC)) - 1e-5);
655 47007 : int fl = 0;
656 :
657 47007 : b = logintall(B, q, qb) + 1;
658 47007 : *qb = mulii(*qb, q);
659 47007 : amin = b + d;
660 47007 : if (gcmp(powiu(q, amin), A) <= 0)
661 : {
662 13902 : a = logintall(A, q, qa) + 1;
663 13902 : *qa = mulii(*qa, q);
664 13902 : b = a - d; *qb = powiu(q, b);
665 : }
666 : else
667 : { /* not enough room */
668 33105 : a = amin; *qa = powiu(q, a);
669 33105 : fl = 1;
670 : }
671 47007 : if (DEBUGLEVEL > 3) {
672 0 : err_printf("S_2 bound: %Ps^%ld\n", q,b);
673 0 : err_printf("coeff bound: %Ps^%ld\n", q,a);
674 : }
675 47007 : *pta = a;
676 47007 : *ptb = b; return fl;
677 : }
678 :
679 : /* use van Hoeij's knapsack algorithm */
680 : static GEN
681 47007 : combine_factors(GEN target, GEN famod, GEN p, long klim)
682 : {
683 : GEN la, B, A, res, L, pa, pb, listmod;
684 47007 : long a,b, l, maxK, n = degpol(target);
685 : int done;
686 : pari_timer T;
687 :
688 47007 : A = factor_bound(target);
689 :
690 47007 : la = absi_shallow(leading_coeff(target));
691 47007 : B = mului(n, sqri(mulii(la, root_bound(target)))); /* = bound for S_2 */
692 :
693 47007 : (void)cmbf_precs(p, A, B, &a, &b, &pa, &pb);
694 :
695 47007 : if (DEBUGLEVEL>2) timer_start(&T);
696 47007 : famod = ZpX_liftfact(target, famod, pa, p, a);
697 47007 : if (DEBUGLEVEL>2) timer_printf(&T, "Hensel lift (mod %Ps^%ld)", p,a);
698 47007 : L = cmbf(target, famod, A, p, a, b, klim, &maxK, &done);
699 47007 : if (DEBUGLEVEL>2) timer_printf(&T, "Naive recombination");
700 :
701 47007 : res = gel(L,1);
702 47007 : listmod = gel(L,2); l = lg(listmod)-1;
703 47007 : famod = gel(listmod,l);
704 47007 : if (maxK > 0 && lg(famod)-1 > 2*maxK)
705 : {
706 126 : if (l!=1) A = factor_bound(gel(res,l));
707 126 : if (DEBUGLEVEL > 4) err_printf("last factor still to be checked\n");
708 126 : L = LLL_cmbf(gel(res,l), famod, p, pa, A, a, maxK);
709 126 : if (DEBUGLEVEL>2) timer_printf(&T,"Knapsack");
710 : /* remove last elt, possibly unfactored. Add all new ones. */
711 126 : setlg(res, l); res = shallowconcat(res, L);
712 : }
713 47007 : return res;
714 : }
715 :
716 : /* Assume 'a' a squarefree ZX; return 0 if no root (fl=1) / irreducible (fl=0).
717 : * Otherwise return prime p such that a mod p has fewest roots / factors */
718 : static ulong
719 838886 : pick_prime(GEN a, long fl, pari_timer *T)
720 : {
721 838886 : pari_sp av = avma, av1;
722 838886 : const long MAXNP = 7, da = degpol(a);
723 838886 : long nmax = da+1, np;
724 838886 : ulong chosenp = 0;
725 838886 : GEN lead = gel(a,da+2);
726 : forprime_t S;
727 838886 : if (equali1(lead)) lead = NULL;
728 838886 : u_forprime_init(&S, 2, ULONG_MAX);
729 838885 : av1 = avma;
730 4596164 : for (np = 0; np < MAXNP; set_avma(av1))
731 : {
732 4499100 : ulong p = u_forprime_next(&S);
733 : long nfacp;
734 : GEN z;
735 :
736 4499098 : if (!p) pari_err_OVERFLOW("DDF [out of small primes]");
737 4499098 : if (lead && !umodiu(lead,p)) continue;
738 4443544 : z = ZX_to_Flx(a, p);
739 4443541 : if (!Flx_is_squarefree(z, p)) continue;
740 :
741 2808920 : if (fl)
742 : {
743 2248034 : nfacp = Flx_nbroots(z, p);
744 2248034 : if (!nfacp) { chosenp = 0; break; } /* no root */
745 : }
746 : else
747 : {
748 560886 : nfacp = Flx_nbfact(z, p);
749 560892 : if (nfacp == 1) { chosenp = 0; break; } /* irreducible */
750 : }
751 2067098 : if (DEBUGLEVEL>4)
752 0 : err_printf("...tried prime %3lu (%-3ld %s). Time = %ld\n",
753 : p, nfacp, fl? "roots": "factors", timer_delay(T));
754 2067104 : if (nfacp < nmax)
755 : {
756 666212 : nmax = nfacp; chosenp = p;
757 666212 : if (da > 100 && nmax < 5) break; /* large degree, few factors. Enough */
758 : }
759 2067097 : np++;
760 : }
761 838897 : return gc_ulong(av, chosenp);
762 : }
763 :
764 : /* Assume A a squarefree ZX; return the vector of its rational roots */
765 : static GEN
766 643586 : DDF_roots(GEN A)
767 : {
768 : GEN p, lc, lcpol, z, pe, pes2, bound;
769 : long i, m, e, lz;
770 : ulong pp;
771 : pari_sp av;
772 : pari_timer T;
773 :
774 643586 : if (DEBUGLEVEL>2) timer_start(&T);
775 643586 : pp = pick_prime(A, 1, &T);
776 643586 : if (!pp) return cgetg(1,t_VEC); /* no root */
777 50052 : p = utoipos(pp);
778 50052 : lc = leading_coeff(A);
779 50052 : if (is_pm1(lc))
780 46586 : { lc = NULL; lcpol = A; }
781 : else
782 3466 : { lc = absi_shallow(lc); lcpol = ZX_Z_mul(A, lc); }
783 50052 : bound = root_bound(A); if (lc) bound = mulii(lc, bound);
784 50052 : e = logintall(addiu(shifti(bound, 1), 1), p, &pe) + 1;
785 50052 : pe = mulii(pe, p);
786 50052 : pes2 = shifti(pe, -1);
787 50052 : if (DEBUGLEVEL>2) timer_printf(&T, "Root bound");
788 50052 : av = avma;
789 50052 : z = ZpX_roots(A, p, e); lz = lg(z);
790 50052 : z = deg1_from_roots(z, varn(A));
791 50052 : if (DEBUGLEVEL>2) timer_printf(&T, "Hensel lift (mod %lu^%ld)", pp,e);
792 107923 : for (m=1, i=1; i < lz; i++)
793 : {
794 57871 : GEN q, r, y = gel(z,i);
795 57871 : if (lc) y = ZX_Z_mul(y, lc);
796 57871 : y = centermod_i(y, pe, pes2);
797 57871 : if (! (q = ZX_divides(lcpol, y)) ) continue;
798 :
799 18575 : lcpol = q;
800 18575 : r = negi( constant_coeff(y) );
801 18575 : if (lc) {
802 2754 : r = gdiv(r,lc);
803 2754 : lcpol = Q_primpart(lcpol);
804 2754 : lc = absi_shallow( leading_coeff(lcpol) );
805 2754 : if (is_pm1(lc)) lc = NULL; else lcpol = ZX_Z_mul(lcpol, lc);
806 : }
807 18575 : gel(z,m++) = r;
808 18575 : if (gc_needed(av,2))
809 : {
810 0 : if (DEBUGMEM>1) pari_warn(warnmem,"DDF_roots, m = %ld", m);
811 0 : gerepileall(av, lc? 3:2, &z, &lcpol, &lc);
812 : }
813 : }
814 50052 : if (DEBUGLEVEL>2) timer_printf(&T, "Recombination");
815 50052 : z[0] = evaltyp(t_VEC) | evallg(m); return z;
816 : }
817 :
818 : /* Assume a squarefree ZX, deg(a) > 0, return rational factors.
819 : * In fact, a(0) != 0 but we don't use this */
820 : static GEN
821 195301 : DDF(GEN a)
822 : {
823 : GEN ap, prime, famod, z;
824 195301 : long ti = 0;
825 195301 : ulong p = 0;
826 195301 : pari_sp av = avma;
827 : pari_timer T, T2;
828 :
829 195301 : if (DEBUGLEVEL>2) { timer_start(&T); timer_start(&T2); }
830 195301 : p = pick_prime(a, 0, &T2);
831 195301 : if (!p) return mkvec(a);
832 47007 : prime = utoipos(p);
833 47007 : ap = Flx_normalize(ZX_to_Flx(a, p), p);
834 47007 : famod = gel(Flx_factor(ap, p), 1);
835 47007 : if (DEBUGLEVEL>2)
836 : {
837 0 : if (DEBUGLEVEL>4) timer_printf(&T2, "splitting mod p = %lu", p);
838 0 : ti = timer_delay(&T);
839 0 : err_printf("Time setup: %ld\n", ti);
840 : }
841 47007 : z = combine_factors(a, FlxV_to_ZXV(famod), prime, degpol(a)-1);
842 47007 : if (DEBUGLEVEL>2)
843 0 : err_printf("Total Time: %ld\n===========\n", ti + timer_delay(&T));
844 47007 : return gerepilecopy(av, z);
845 : }
846 :
847 : /* Distinct Degree Factorization (deflating first)
848 : * Assume x squarefree, degree(x) > 0, x(0) != 0 */
849 : GEN
850 147165 : ZX_DDF(GEN x)
851 : {
852 : GEN L;
853 : long m;
854 147165 : x = ZX_deflate_max(x, &m);
855 147167 : L = DDF(x);
856 147168 : if (m > 1)
857 : {
858 46103 : GEN e, v, fa = factoru(m);
859 : long i,j,k, l;
860 :
861 46103 : e = gel(fa,2); k = 0;
862 46103 : fa= gel(fa,1); l = lg(fa);
863 92529 : for (i=1; i<l; i++) k += e[i];
864 46103 : v = cgetg(k+1, t_VECSMALL); k = 1;
865 92529 : for (i=1; i<l; i++)
866 94189 : for (j=1; j<=e[i]; j++) v[k++] = fa[i];
867 93866 : for (k--; k; k--)
868 : {
869 47763 : GEN L2 = cgetg(1,t_VEC);
870 95897 : for (i=1; i < lg(L); i++)
871 48134 : L2 = shallowconcat(L2, DDF(RgX_inflate(gel(L,i), v[k])));
872 47763 : L = L2;
873 : }
874 : }
875 147168 : return L;
876 : }
877 :
878 : /* SquareFree Factorization in Z[X] (char 0 is enough, if ZX_gcd -> RgX_gcd)
879 : * f = prod Q[i]^E[i], E[1] < E[2] < ..., and Q[i] squarefree and coprime.
880 : * Return Q, set *pE = E. For efficiency, caller should have used ZX_valrem
881 : * so that f(0) != 0 */
882 : GEN
883 109742 : ZX_squff(GEN f, GEN *pE)
884 : {
885 : GEN T, V, P, E;
886 109742 : long i, k, n = 1 + degpol(f);
887 :
888 109742 : if (signe(leading_coeff(f)) < 0) f = ZX_neg(f);
889 109742 : E = cgetg(n, t_VECSMALL);
890 109742 : P = cgetg(n, t_COL);
891 109742 : T = ZX_gcd_all(f, ZX_deriv(f), &V);
892 109743 : for (k = i = 1;; k++)
893 3620 : {
894 113363 : GEN W = ZX_gcd_all(T,V, &T); /* V and W are squarefree */
895 113363 : long dW = degpol(W), dV = degpol(V);
896 : /* f = prod_i T_i^{e_i}
897 : * W = prod_{i: e_i > k} T_i,
898 : * V = prod_{i: e_i >= k} T_i,
899 : * T = prod_{i: e_i > k} T_i^{e_i - k} */
900 113363 : if (!dW)
901 : {
902 109743 : if (dV) { gel(P,i) = Q_primpart(V); E[i] = k; i++; }
903 109743 : break;
904 : }
905 3620 : if (dW == dV)
906 : {
907 : GEN U;
908 1659 : while ( (U = ZX_divides(T, V)) ) { k++; T = U; }
909 : }
910 : else
911 : {
912 2367 : gel(P,i) = Q_primpart(RgX_div(V,W));
913 2367 : E[i] = k; i++; V = W;
914 : }
915 : }
916 109743 : setlg(P,i);
917 109743 : setlg(E,i); *pE = E; return P;
918 : }
919 :
920 : static GEN
921 47510 : fact_from_DDF(GEN Q, GEN E, long n)
922 : {
923 47510 : GEN v,w, y = cgetg(3, t_MAT);
924 47510 : long i,j,k, l = lg(Q);
925 :
926 47510 : v = cgetg(n+1, t_COL); gel(y,1) = v;
927 47510 : w = cgetg(n+1, t_COL); gel(y,2) = w;
928 96805 : for (k = i = 1; i < l; i++)
929 : {
930 49295 : GEN L = gel(Q,i), e = utoipos(E[i]);
931 49295 : long J = lg(L);
932 120683 : for (j = 1; j < J; j++,k++)
933 : {
934 71388 : gel(v,k) = ZX_copy(gel(L,j));
935 71388 : gel(w,k) = e;
936 : }
937 : }
938 47510 : return y;
939 : }
940 :
941 : /* Factor T in Z[x] */
942 : static GEN
943 47516 : ZX_factor_i(GEN T)
944 : {
945 : GEN Q, E, y;
946 : long n, i, l, v;
947 :
948 47516 : if (!signe(T)) return prime_fact(T);
949 47509 : v = ZX_valrem(T, &T);
950 47509 : Q = ZX_squff(T, &E); l = lg(Q);
951 95265 : for (i = 1, n = 0; i < l; i++)
952 : {
953 47755 : gel(Q,i) = ZX_DDF(gel(Q,i));
954 47755 : n += lg(gel(Q,i)) - 1;
955 : }
956 47510 : if (v)
957 : {
958 1540 : Q = vec_append(Q, mkvec(pol_x(varn(T))));
959 1540 : E = vecsmall_append(E, v); n++;
960 : }
961 47510 : y = fact_from_DDF(Q, E, n);
962 47510 : return sort_factor_pol(y, cmpii);
963 : }
964 : GEN
965 46788 : ZX_factor(GEN x)
966 : {
967 46788 : pari_sp av = avma;
968 46788 : return gerepileupto(av, ZX_factor_i(x));
969 : }
970 : GEN
971 728 : QX_factor(GEN x)
972 : {
973 728 : pari_sp av = avma;
974 728 : return gerepileupto(av, ZX_factor_i(Q_primpart(x)));
975 : }
976 :
977 : long
978 99729 : ZX_is_irred(GEN x)
979 : {
980 99729 : pari_sp av = avma;
981 99729 : long l = lg(x);
982 : GEN y;
983 99729 : if (l <= 3) return 0; /* degree < 1 */
984 99729 : if (l == 4) return 1; /* degree 1 */
985 96534 : if (ZX_val(x)) return 0;
986 96338 : if (!ZX_is_squarefree(x)) return 0;
987 96268 : y = ZX_DDF(x); set_avma(av);
988 96271 : return (lg(y) == 2);
989 : }
990 :
991 : GEN
992 643586 : nfrootsQ(GEN x)
993 : {
994 643586 : pari_sp av = avma;
995 : GEN z;
996 : long val;
997 :
998 643586 : if (typ(x)!=t_POL) pari_err_TYPE("nfrootsQ",x);
999 643586 : if (!signe(x)) pari_err_ROOTS0("nfrootsQ");
1000 643586 : x = Q_primpart(x);
1001 643586 : RgX_check_ZX(x,"nfrootsQ");
1002 643586 : val = ZX_valrem(x, &x);
1003 643586 : z = DDF_roots( ZX_radical(x) );
1004 643586 : if (val) z = vec_append(z, gen_0);
1005 643586 : return gerepileupto(av, sort(z));
1006 : }
1007 :
1008 : /************************************************************************
1009 : * GCD OVER Z[X] / Q[X] *
1010 : ************************************************************************/
1011 : int
1012 203126 : ZX_is_squarefree(GEN x)
1013 : {
1014 203126 : pari_sp av = avma;
1015 : GEN d;
1016 : long m;
1017 203126 : if (lg(x) == 2) return 0;
1018 203126 : m = ZX_deflate_order(x);
1019 203129 : if (m > 1)
1020 : {
1021 85442 : if (!signe(gel(x,2))) return 0;
1022 85204 : x = RgX_deflate(x, m);
1023 : }
1024 202890 : d = ZX_gcd(x,ZX_deriv(x));
1025 202904 : return gc_bool(av, lg(d) == 3);
1026 : }
1027 :
1028 : static int
1029 102115 : ZX_gcd_filter(GEN *pt_A, GEN *pt_P)
1030 : {
1031 102115 : GEN A = *pt_A, P = *pt_P;
1032 102115 : long i, j, l = lg(A), n = 1, d = degpol(gel(A,1));
1033 : GEN B, Q;
1034 205236 : for (i=2; i<l; i++)
1035 : {
1036 103121 : long di = degpol(gel(A,i));
1037 103121 : if (di==d) n++;
1038 36 : else if (d > di)
1039 36 : { n=1; d = di; }
1040 : }
1041 102115 : if (n == l-1)
1042 102079 : return 0;
1043 36 : B = cgetg(n+1, t_VEC);
1044 36 : Q = cgetg(n+1, typ(P));
1045 156 : for (i=1, j=1; i<l; i++)
1046 : {
1047 120 : if (degpol(gel(A,i))==d)
1048 : {
1049 84 : gel(B,j) = gel(A,i);
1050 84 : Q[j] = P[i];
1051 84 : j++;
1052 : }
1053 : }
1054 36 : *pt_A = B; *pt_P = Q; return 1;
1055 : }
1056 :
1057 : static GEN
1058 3316518 : ZX_gcd_Flx(GEN a, GEN b, ulong g, ulong p)
1059 : {
1060 3316518 : GEN H = Flx_gcd(a, b, p);
1061 3316518 : if (!g)
1062 3300533 : return Flx_normalize(H, p);
1063 : else
1064 : {
1065 15985 : ulong t = Fl_mul(g, Fl_inv(Flx_lead(H), p), p);
1066 15985 : return Flx_Fl_mul(H, t, p);
1067 : }
1068 : }
1069 :
1070 : static GEN
1071 3310612 : ZX_gcd_slice(GEN A, GEN B, GEN g, GEN P, GEN *mod)
1072 : {
1073 3310612 : pari_sp av = avma;
1074 3310612 : long i, n = lg(P)-1;
1075 : GEN H, T;
1076 3310612 : if (n == 1)
1077 : {
1078 3305684 : ulong p = uel(P,1), gp = g ? umodiu(g, p): 0;
1079 3305684 : GEN a = ZX_to_Flx(A, p), b = ZX_to_Flx(B, p);
1080 3305682 : GEN Hp = ZX_gcd_Flx(a, b, gp, p);
1081 3305684 : H = gerepileupto(av, Flx_to_ZX(Hp));
1082 3305685 : *mod = utoi(p);
1083 3305685 : return H;
1084 : }
1085 4928 : T = ZV_producttree(P);
1086 4928 : A = ZX_nv_mod_tree(A, P, T);
1087 4928 : B = ZX_nv_mod_tree(B, P, T);
1088 4928 : g = g ? Z_ZV_mod_tree(g, P, T): NULL;
1089 4928 : H = cgetg(n+1, t_VEC);
1090 15764 : for(i=1; i <= n; i++)
1091 : {
1092 10836 : ulong p = P[i];
1093 10836 : GEN a = gel(A,i), b = gel(B,i);
1094 10836 : gel(H,i) = ZX_gcd_Flx(a, b, g? g[i]: 0, p);
1095 : }
1096 4928 : if (ZX_gcd_filter(&H, &P))
1097 12 : T = ZV_producttree(P);
1098 4928 : H = nxV_chinese_center_tree(H, P, T, ZV_chinesetree(P, T));
1099 4928 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
1100 : }
1101 :
1102 : GEN
1103 3310613 : ZX_gcd_worker(GEN P, GEN A, GEN B, GEN g)
1104 : {
1105 3310613 : GEN V = cgetg(3, t_VEC);
1106 3310612 : gel(V,1) = ZX_gcd_slice(A, B, equali1(g)? NULL: g , P, &gel(V,2));
1107 3310613 : return V;
1108 : }
1109 :
1110 : static GEN
1111 97187 : ZX_gcd_chinese(GEN A, GEN P, GEN *mod)
1112 : {
1113 97187 : ZX_gcd_filter(&A, &P);
1114 97187 : return nxV_chinese_center(A, P, mod);
1115 : }
1116 :
1117 : GEN
1118 10756424 : ZX_gcd_all(GEN A, GEN B, GEN *Anew)
1119 : {
1120 10756424 : pari_sp av = avma;
1121 10756424 : long k, valH, valA, valB, vA = varn(A), dA = degpol(A), dB = degpol(B);
1122 10756442 : GEN worker, c, cA, cB, g, Ag, Bg, H = NULL, mod = gen_1, R;
1123 : GEN Ap, Bp, Hp;
1124 : forprime_t S;
1125 : ulong pp;
1126 10756442 : if (dA < 0) { if (Anew) *Anew = pol_0(vA); return ZX_copy(B); }
1127 10756421 : if (dB < 0) { if (Anew) *Anew = pol_1(vA); return ZX_copy(A); }
1128 10755308 : A = Q_primitive_part(A, &cA);
1129 10755302 : B = Q_primitive_part(B, &cB);
1130 10755306 : valA = ZX_valrem(A, &A); dA -= valA;
1131 10755312 : valB = ZX_valrem(B, &B); dB -= valB;
1132 10755316 : valH = minss(valA, valB);
1133 10755314 : valA -= valH; /* valuation(Anew) */
1134 10755314 : c = (cA && cB)? gcdii(cA, cB): NULL; /* content(gcd) */
1135 10755318 : if (!dA || !dB)
1136 : {
1137 6205920 : if (Anew) *Anew = RgX_shift_shallow(A, valA);
1138 6205920 : return monomial(c? c: gen_1, valH, vA);
1139 : }
1140 4549398 : g = gcdii(leading_coeff(A), leading_coeff(B)); /* multiple of lead(gcd) */
1141 4549367 : if (is_pm1(g)) {
1142 4440267 : g = NULL;
1143 4440267 : Ag = A;
1144 4440267 : Bg = B;
1145 : } else {
1146 109104 : Ag = ZX_Z_mul(A,g);
1147 109104 : Bg = ZX_Z_mul(B,g);
1148 : }
1149 4549371 : init_modular_big(&S);
1150 : do {
1151 4549410 : pp = u_forprime_next(&S);
1152 4549410 : Ap = ZX_to_Flx(Ag, pp);
1153 4549414 : Bp = ZX_to_Flx(Bg, pp);
1154 4549407 : } while (degpol(Ap) != dA || degpol(Bp) != dB);
1155 4549389 : if (degpol(Flx_gcd(Ap, Bp, pp)) == 0)
1156 : {
1157 1335997 : if (Anew) *Anew = RgX_shift_shallow(A, valA);
1158 1335997 : return monomial(c? c: gen_1, valH, vA);
1159 : }
1160 3213398 : worker = snm_closure(is_entry("_ZX_gcd_worker"), mkvec3(A, B, g? g: gen_1));
1161 3213398 : av = avma;
1162 3213398 : for (k = 1; ;k *= 2)
1163 : {
1164 3308932 : gen_inccrt_i("ZX_gcd", worker, g, (k+1)>>1, 0, &S, &H, &mod, ZX_gcd_chinese, NULL);
1165 3308934 : gerepileall(av, 2, &H, &mod);
1166 3308932 : Hp = ZX_to_Flx(H, pp);
1167 3308934 : if (lgpol(Flx_rem(Ap, Hp, pp)) || lgpol(Flx_rem(Bp, Hp, pp))) continue;
1168 3217464 : if (!ZX_divides(Bg, H)) continue;
1169 3213428 : R = ZX_divides(Ag, H);
1170 3213426 : if (R) break;
1171 : }
1172 : /* lead(H) = g */
1173 3213396 : if (g) H = Q_primpart(H);
1174 3213396 : if (c) H = ZX_Z_mul(H,c);
1175 3213396 : if (DEBUGLEVEL>5) err_printf("done\n");
1176 3213397 : if (Anew) *Anew = RgX_shift_shallow(R, valA);
1177 3213397 : return valH? RgX_shift_shallow(H, valH): H;
1178 : }
1179 :
1180 : #if 0
1181 : /* ceil( || p ||_oo / lc(p) ) */
1182 : static GEN
1183 : maxnorm(GEN p)
1184 : {
1185 : long i, n = degpol(p), av = avma;
1186 : GEN x, m = gen_0;
1187 :
1188 : p += 2;
1189 : for (i=0; i<n; i++)
1190 : {
1191 : x = gel(p,i);
1192 : if (abscmpii(x,m) > 0) m = x;
1193 : }
1194 : m = divii(m, gel(p,n));
1195 : return gerepileuptoint(av, addiu(absi_shallow(m),1));
1196 : }
1197 : #endif
1198 :
1199 : GEN
1200 9724065 : ZX_gcd(GEN A, GEN B)
1201 : {
1202 9724065 : pari_sp av = avma;
1203 9724065 : return gerepilecopy(av, ZX_gcd_all(A,B,NULL));
1204 : }
1205 :
1206 : GEN
1207 804838 : ZX_radical(GEN A) { GEN B; (void)ZX_gcd_all(A,ZX_deriv(A),&B); return B; }
1208 :
1209 : static GEN
1210 12935 : _gcd(GEN a, GEN b)
1211 : {
1212 12935 : if (!a) a = gen_1;
1213 12935 : if (!b) b = gen_1;
1214 12935 : return Q_gcd(a,b);
1215 : }
1216 : /* A0 and B0 in Q[X] */
1217 : GEN
1218 12936 : QX_gcd(GEN A0, GEN B0)
1219 : {
1220 : GEN a, b, D;
1221 12936 : pari_sp av = avma, av2;
1222 :
1223 12936 : D = ZX_gcd(Q_primitive_part(A0, &a), Q_primitive_part(B0, &b));
1224 12935 : av2 = avma; a = _gcd(a,b);
1225 12934 : if (isint1(a)) set_avma(av2); else D = ZX_Q_mul(D, a);
1226 12934 : return gerepileupto(av, D);
1227 : }
1228 :
1229 : /*****************************************************************************
1230 : * Variants of the Bradford-Davenport algorithm: look for cyclotomic *
1231 : * factors, and decide whether a ZX is cyclotomic or a product of cyclotomic *
1232 : *****************************************************************************/
1233 : /* f of degree 1, return a cyclotomic factor (Phi_1 or Phi_2) or NULL */
1234 : static GEN
1235 0 : BD_deg1(GEN f)
1236 : {
1237 0 : GEN a = gel(f,3), b = gel(f,2); /* f = ax + b */
1238 0 : if (!absequalii(a,b)) return NULL;
1239 0 : return polcyclo((signe(a) == signe(b))? 2: 1, varn(f));
1240 : }
1241 :
1242 : /* f a squarefree ZX; not divisible by any Phi_n, n even */
1243 : static GEN
1244 406 : BD_odd(GEN f)
1245 : {
1246 406 : while(degpol(f) > 1)
1247 : {
1248 406 : GEN f1 = ZX_graeffe(f); /* contain all cyclotomic divisors of f */
1249 406 : if (ZX_equal(f1, f)) return f; /* product of cyclotomics */
1250 0 : f = ZX_gcd(f, f1);
1251 : }
1252 0 : if (degpol(f) == 1) return BD_deg1(f);
1253 0 : return NULL; /* no cyclotomic divisor */
1254 : }
1255 :
1256 : static GEN
1257 2310 : myconcat(GEN v, GEN x)
1258 : {
1259 2310 : if (typ(x) != t_VEC) x = mkvec(x);
1260 2310 : if (!v) return x;
1261 1470 : return shallowconcat(v, x);
1262 : }
1263 :
1264 : /* Bradford-Davenport algorithm.
1265 : * f a squarefree ZX of degree > 0, return NULL or a vector of coprime
1266 : * cyclotomic factors of f [ possibly reducible ] */
1267 : static GEN
1268 2359 : BD(GEN f)
1269 : {
1270 2359 : GEN G = NULL, Gs = NULL, Gp = NULL, Gi = NULL;
1271 : GEN fs2, fp, f2, f1, fe, fo, fe1, fo1;
1272 2359 : RgX_even_odd(f, &fe, &fo);
1273 2359 : fe1 = ZX_eval1(fe);
1274 2359 : fo1 = ZX_eval1(fo);
1275 2359 : if (absequalii(fe1, fo1)) /* f(1) = 0 or f(-1) = 0 */
1276 : {
1277 1519 : long i, v = varn(f);
1278 1519 : if (!signe(fe1))
1279 371 : G = mkvec2(polcyclo(1, v), polcyclo(2, v)); /* both 0 */
1280 1148 : else if (signe(fe1) == signe(fo1))
1281 693 : G = mkvec(polcyclo(2, v)); /*f(-1) = 0*/
1282 : else
1283 455 : G = mkvec(polcyclo(1, v)); /*f(1) = 0*/
1284 3409 : for (i = lg(G)-1; i; i--) f = RgX_div(f, gel(G,i));
1285 : }
1286 : /* f no longer divisible by Phi_1 or Phi_2 */
1287 2359 : if (degpol(f) <= 1) return G;
1288 2058 : f1 = ZX_graeffe(f); /* has at most square factors */
1289 2058 : if (ZX_equal(f1, f)) return myconcat(G,f); /* f = product of Phi_n, n odd */
1290 :
1291 1183 : fs2 = ZX_gcd_all(f1, ZX_deriv(f1), &f2); /* fs2 squarefree */
1292 1183 : if (degpol(fs2))
1293 : { /* fs contains all Phi_n | f, 4 | n; and only those */
1294 : /* In that case, Graeffe(Phi_n) = Phi_{n/2}^2, and Phi_n = Phi_{n/2}(x^2) */
1295 1029 : GEN fs = RgX_inflate(fs2, 2);
1296 1029 : (void)ZX_gcd_all(f, fs, &f); /* remove those Phi_n | f, 4 | n */
1297 1029 : Gs = BD(fs2);
1298 1029 : if (Gs)
1299 : {
1300 : long i;
1301 2555 : for (i = lg(Gs)-1; i; i--) gel(Gs,i) = RgX_inflate(gel(Gs,i), 2);
1302 : /* prod Gs[i] is the product of all Phi_n | f, 4 | n */
1303 1029 : G = myconcat(G, Gs);
1304 : }
1305 : /* f2 = f1 / fs2 */
1306 1029 : f1 = RgX_div(f2, fs2); /* f1 / fs2^2 */
1307 : }
1308 1183 : fp = ZX_gcd(f, f1); /* contains all Phi_n | f, n > 1 odd; and only those */
1309 1183 : if (degpol(fp))
1310 : {
1311 196 : Gp = BD_odd(fp);
1312 : /* Gp is the product of all Phi_n | f, n odd */
1313 196 : if (Gp) G = myconcat(G, Gp);
1314 196 : f = RgX_div(f, fp);
1315 : }
1316 1183 : if (degpol(f))
1317 : { /* contains all Phi_n originally dividing f, n = 2 mod 4, n > 2;
1318 : * and only those
1319 : * In that case, Graeffe(Phi_n) = Phi_{n/2}, and Phi_n = Phi_{n/2}(-x) */
1320 210 : Gi = BD_odd(ZX_z_unscale(f, -1));
1321 210 : if (Gi)
1322 : { /* N.B. Phi_2 does not divide f */
1323 210 : Gi = ZX_z_unscale(Gi, -1);
1324 : /* Gi is the product of all Phi_n | f, n = 2 mod 4 */
1325 210 : G = myconcat(G, Gi);
1326 : }
1327 : }
1328 1183 : return G;
1329 : }
1330 :
1331 : /* Let f be a nonzero QX, return the (squarefree) product of cyclotomic
1332 : * divisors of f */
1333 : GEN
1334 315 : polcyclofactors(GEN f)
1335 : {
1336 315 : pari_sp av = avma;
1337 315 : if (typ(f) != t_POL || !signe(f)) pari_err_TYPE("polcyclofactors",f);
1338 315 : (void)RgX_valrem(f, &f);
1339 315 : f = Q_primpart(f);
1340 315 : RgX_check_ZX(f,"polcyclofactors");
1341 315 : if (degpol(f))
1342 : {
1343 315 : f = BD(ZX_radical(f));
1344 315 : if (f) return gerepilecopy(av, f);
1345 : }
1346 0 : set_avma(av); return cgetg(1,t_VEC);
1347 : }
1348 :
1349 : /* return t*x mod T(x), T a monic ZX. Assume deg(t) < deg(T) */
1350 : static GEN
1351 50105 : ZXQ_mul_by_X(GEN t, GEN T)
1352 : {
1353 : GEN lt;
1354 50105 : t = RgX_shift_shallow(t, 1);
1355 50105 : if (degpol(t) < degpol(T)) return t;
1356 7363 : lt = leading_coeff(t);
1357 7363 : if (is_pm1(lt)) return signe(lt) > 0 ? ZX_sub(t, T): ZX_add(t, T);
1358 217 : return ZX_sub(t, ZX_Z_mul(T, leading_coeff(t)));
1359 : }
1360 : /* f a product of Phi_n, all n odd; deg f > 1. Is it irreducible ? */
1361 : static long
1362 2078 : BD_odd_iscyclo(GEN f)
1363 : {
1364 : pari_sp av;
1365 : long d, e, n, bound;
1366 : GEN t;
1367 2078 : f = ZX_deflate_max(f, &e);
1368 2078 : av = avma;
1369 : /* The original f is cyclotomic (= Phi_{ne}) iff the present one is Phi_n,
1370 : * where all prime dividing e also divide n. If current f is Phi_n,
1371 : * then n is odd and squarefree */
1372 2078 : d = degpol(f); /* = phi(n) */
1373 : /* Let e > 0, g multiplicative such that
1374 : g(p) = p / (p-1)^(1+e) < 1 iff p < (p-1)^(1+e)
1375 : For all squarefree odd n, we have g(n) < C, hence n < C phi(n)^(1+e), where
1376 : C = \prod_{p odd | p > (p-1)^(1+e)} g(p)
1377 : For e = 1/10, we obtain p = 3, 5 and C < 1.523
1378 : For e = 1/100, we obtain p = 3, 5, ..., 29 and C < 2.573
1379 : In fact, for n <= 10^7 odd & squarefree, we have n < 2.92 * phi(n)
1380 : By the above, n<10^7 covers all d <= (10^7/2.573)^(1/(1+1/100)) < 3344391.
1381 : */
1382 2078 : if (d <= 3344391)
1383 2078 : bound = (long)(2.92 * d);
1384 : else
1385 0 : bound = (long)(2.573 * pow(d,1.01));
1386 : /* IF f = Phi_n, n squarefree odd, then n <= bound */
1387 2078 : t = pol_xn(d-1, varn(f));
1388 50140 : for (n = d; n <= bound; n++)
1389 : {
1390 50105 : t = ZXQ_mul_by_X(t, f);
1391 : /* t = (X mod f(X))^d */
1392 50106 : if (degpol(t) == 0) break;
1393 48062 : if (gc_needed(av,1))
1394 : {
1395 461 : if(DEBUGMEM>1) pari_warn(warnmem,"BD_odd_iscyclo");
1396 461 : t = gerepilecopy(av, t);
1397 : }
1398 : }
1399 2079 : if (n > bound || eulerphiu(n) != (ulong)d) return 0;
1400 :
1401 2016 : if (e > 1) return (u_ppo(e, n) == 1)? e * n : 0;
1402 1848 : return n;
1403 : }
1404 :
1405 : /* Checks if f, monic squarefree ZX with |constant coeff| = 1, is a cyclotomic
1406 : * polynomial. Returns n if f = Phi_n, and 0 otherwise */
1407 : static long
1408 10332 : BD_iscyclo(GEN f)
1409 : {
1410 10332 : pari_sp av = avma;
1411 : GEN f2, fn, f1;
1412 :
1413 10332 : if (degpol(f) == 1) return isint1(gel(f,2))? 2: 1;
1414 10003 : f1 = ZX_graeffe(f);
1415 : /* f = product of Phi_n, n odd */
1416 10002 : if (ZX_equal(f, f1)) return gc_long(av, BD_odd_iscyclo(f));
1417 :
1418 8980 : fn = ZX_z_unscale(f, -1); /* f(-x) */
1419 : /* f = product of Phi_n, n = 2 mod 4 */
1420 8980 : if (ZX_equal(f1, fn)) return gc_long(av, 2*BD_odd_iscyclo(fn));
1421 :
1422 7924 : if (issquareall(f1, &f2))
1423 : {
1424 1344 : GEN lt = leading_coeff(f2);
1425 : long c;
1426 1344 : if (signe(lt) < 0) f2 = ZX_neg(f2);
1427 1344 : c = BD_iscyclo(f2);
1428 1344 : return odd(c)? 0: 2*c;
1429 : }
1430 6580 : return gc_long(av, 0);
1431 : }
1432 : long
1433 98723 : poliscyclo(GEN f)
1434 : {
1435 : long d;
1436 98723 : if (typ(f) != t_POL) pari_err_TYPE("poliscyclo", f);
1437 98716 : d = degpol(f);
1438 98716 : if (d <= 0 || !RgX_is_ZX(f)) return 0;
1439 98711 : if (!equali1(gel(f,d+2)) || !is_pm1(gel(f,2))) return 0;
1440 9051 : if (d == 1) return signe(gel(f,2)) > 0? 2: 1;
1441 8988 : return ZX_is_squarefree(f)? BD_iscyclo(f): 0;
1442 : }
1443 :
1444 : long
1445 1029 : poliscycloprod(GEN f)
1446 : {
1447 1029 : pari_sp av = avma;
1448 1029 : long i, d = degpol(f);
1449 1029 : if (typ(f) != t_POL) pari_err_TYPE("poliscycloprod",f);
1450 1029 : if (!RgX_is_ZX(f)) return 0;
1451 1029 : if (!ZX_is_monic(f) || !is_pm1(constant_coeff(f))) return 0;
1452 1029 : if (d < 2) return (d == 1);
1453 1022 : if ( degpol(ZX_gcd_all(f, ZX_deriv(f), &f)) )
1454 : {
1455 14 : d = degpol(f);
1456 14 : if (d == 1) return 1;
1457 : }
1458 1015 : f = BD(f); if (!f) return 0;
1459 3619 : for (i = lg(f)-1; i; i--) d -= degpol(gel(f,i));
1460 1015 : return gc_long(av, d == 0);
1461 : }
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