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 6570004 : 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 6570004 : dy=degpol(y);
29 6570004 : dx=degpol(x);
30 6570004 : dz=dx-dy; if (dz<0) return NULL;
31 6568940 : z=cgetg(dz+3,t_POL); z[1] = x[1];
32 6568940 : x += 2; y += 2; z += 2;
33 6568940 : y_lead = gel(y,dy);
34 6568940 : if (equali1(y_lead)) y_lead = NULL;
35 :
36 6568940 : p1 = gel(x,dx);
37 6568940 : if (y_lead) {
38 : GEN r;
39 30135 : p1 = dvmdii(p1,y_lead, &r);
40 30135 : if (r != gen_0) return NULL;
41 : }
42 6538805 : else p1 = icopy(p1);
43 6564779 : gel(z,dz) = p1;
44 8228035 : for (i=dx-1; i>=dy; i--)
45 : {
46 1668433 : av = avma; p1 = gel(x,i);
47 5790464 : for (j=i-dy+1; j<=i && j<=dz; j++)
48 4122082 : p1 = subii(p1, mulii(gel(z,j),gel(y,i-j)));
49 1668382 : if (y_lead) {
50 : GEN r;
51 52822 : p1 = dvmdii(p1,y_lead, &r);
52 52822 : if (r != gen_0) return NULL;
53 : }
54 1666883 : if (B && abscmpii(p1, B) > 0) return NULL;
55 1663208 : p1 = gerepileuptoint(av, p1);
56 1663256 : gel(z,i-dy) = p1;
57 : }
58 6559602 : av = avma;
59 17095551 : for (; i >= 0; i--)
60 : {
61 10579109 : p1 = gel(x,i);
62 : /* we always enter this loop at least once */
63 23806132 : for (j=0; j<=i && j<=dz; j++)
64 13227042 : p1 = subii(p1, mulii(gel(z,j),gel(y,i-j)));
65 10579090 : if (signe(p1)) return NULL;
66 10535948 : set_avma(av);
67 : }
68 6516442 : return z - 2;
69 : }
70 : static GEN
71 6541351 : 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(LOWDEFAULTPREC), 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 60135 : Mignotte_bound(GEN S)
110 : {
111 60135 : long i, d = degpol(S);
112 60135 : GEN C, N2, t, binlS, lS = leading_coeff(S), bin = vecbinomial(d-1);
113 :
114 60135 : N2 = sqrtr(RgX_fpnorml2(S,DEFAULTPREC));
115 60135 : binlS = is_pm1(lS)? bin: ZC_Z_mul(bin, lS);
116 :
117 : /* i = 0 */
118 60135 : C = gel(binlS,1);
119 : /* i = d */
120 60135 : t = N2; if (gcmp(C, t) < 0) C = t;
121 520509 : for (i = 1; i < d; i++)
122 : {
123 460377 : t = addri(mulir(gel(bin,i), N2), gel(binlS,i+1));
124 460373 : if (mpcmp(C, t) < 0) C = t;
125 : }
126 60132 : 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 60135 : Beauzamy_bound(GEN S)
132 : {
133 60135 : const long prec = DEFAULTPREC;
134 60135 : long i, d = degpol(S);
135 : GEN bin, lS, s, C;
136 60135 : bin = vecbinomial(d);
137 :
138 60135 : s = real_0(prec);
139 640767 : for (i=0; i<=d; i++)
140 : {
141 580632 : GEN c = gel(S,i+2);
142 580632 : if (gequal0(c)) continue;
143 : /* s += P_i^2 / binomial(d,i) */
144 488601 : s = addrr(s, divri(itor(sqri(c), prec), gel(bin,i+1)));
145 : }
146 : /* s = [S]_2^2 */
147 60135 : C = powruhalf(utor(3,prec), 3 + 2*d); /* 3^{3/2 + d} */
148 60135 : C = divrr(mulrr(C, s), mulur(4*d, mppi(prec)));
149 60135 : lS = absi_shallow(leading_coeff(S));
150 60135 : return mulir(lS, sqrtr(C));
151 : }
152 :
153 : static GEN
154 60135 : factor_bound(GEN S)
155 : {
156 60135 : pari_sp av = avma;
157 60135 : GEN a = Mignotte_bound(S);
158 60135 : GEN b = Beauzamy_bound(S);
159 60135 : if (DEBUGLEVEL>2)
160 : {
161 0 : err_printf("Mignotte bound: %Ps\n",a);
162 0 : err_printf("Beauzamy bound: %Ps\n",b);
163 : }
164 60135 : 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 49623 : cmbf(GEN pol, GEN famod, GEN bound, GEN p, long a, long b,
176 : long klim, long *pmaxK, int *done)
177 : {
178 49623 : long K = 1, cnt = 1, i,j,k, curdeg, lfamod = lg(famod)-1;
179 : ulong spa_b, spa_bs2, Sbound;
180 49623 : GEN lc, lcpol, pa = powiu(p,a), pas2 = shifti(pa,-1);
181 49623 : GEN trace1 = cgetg(lfamod+1, t_VECSMALL);
182 49623 : GEN trace2 = cgetg(lfamod+1, t_VECSMALL);
183 49623 : GEN ind = cgetg(lfamod+1, t_VECSMALL);
184 49623 : GEN deg = cgetg(lfamod+1, t_VECSMALL);
185 49623 : GEN degsofar = cgetg(lfamod+1, t_VECSMALL);
186 49623 : GEN listmod = cgetg(lfamod+1, t_VEC);
187 49623 : GEN fa = cgetg(lfamod+1, t_VEC);
188 :
189 49623 : *pmaxK = cmbf_maxK(lfamod);
190 49623 : lc = absi_shallow(leading_coeff(pol));
191 49623 : if (equali1(lc)) lc = NULL;
192 49623 : lcpol = lc? ZX_Z_mul(pol, lc): pol;
193 :
194 : {
195 49623 : GEN pa_b,pa_bs2,pb, lc2 = lc? sqri(lc): NULL;
196 :
197 49623 : pa_b = powiu(p, a-b); /* < 2^31 */
198 49623 : pa_bs2 = shifti(pa_b,-1);
199 49623 : pb= powiu(p, b);
200 175700 : for (i=1; i <= lfamod; i++)
201 : {
202 126077 : GEN T1,T2, P = gel(famod,i);
203 126077 : long d = degpol(P);
204 :
205 126077 : deg[i] = d; P += 2;
206 126077 : T1 = gel(P,d-1);/* = - S_1 */
207 126077 : T2 = sqri(T1);
208 126078 : if (d > 1) T2 = subii(T2, shifti(gel(P,d-2),1));
209 126077 : T2 = modii(T2, pa); /* = S_2 Newton sum */
210 126076 : if (lc)
211 : {
212 4193 : T1 = Fp_mul(lc, T1, pa);
213 4193 : T2 = Fp_mul(lc2,T2, pa);
214 : }
215 126076 : uel(trace1,i) = itou(diviiround(T1, pb));
216 126078 : uel(trace2,i) = itou(diviiround(T2, pb));
217 : }
218 49623 : spa_b = uel(pa_b,2); /* < 2^31 */
219 49623 : spa_bs2 = uel(pa_bs2,2); /* < 2^31 */
220 : }
221 49623 : degsofar[0] = 0; /* sentinel */
222 :
223 : /* ind runs through strictly increasing sequences of length K,
224 : * 1 <= ind[i] <= lfamod */
225 90541 : nextK:
226 90541 : if (K > *pmaxK || 2*K > lfamod) goto END;
227 55200 : if (DEBUGLEVEL > 3)
228 0 : err_printf("\n### K = %d, %Ps combinations\n", K,binomial(utoipos(lfamod), K));
229 55200 : setlg(ind, K+1); ind[1] = 1;
230 55200 : Sbound = (ulong) ((K+1)>>1);
231 55200 : i = 1; curdeg = deg[ind[1]];
232 : for(;;)
233 : { /* try all combinations of K factors */
234 622737 : for (j = i; j < K; j++)
235 : {
236 85886 : degsofar[j] = curdeg;
237 85886 : ind[j+1] = ind[j]+1; curdeg += deg[ind[j+1]];
238 : }
239 536851 : if (curdeg <= klim) /* trial divide */
240 10505 : {
241 : GEN y, q, list;
242 : pari_sp av;
243 : ulong t;
244 :
245 : /* d - 1 test */
246 1353093 : for (t=uel(trace1,ind[1]),i=2; i<=K; i++)
247 816242 : t = Fl_add(t, uel(trace1,ind[i]), spa_b);
248 536851 : if (t > spa_bs2) t = spa_b - t;
249 536851 : if (t > Sbound)
250 : {
251 428222 : if (DEBUGLEVEL>6) err_printf(".");
252 428222 : goto NEXT;
253 : }
254 : /* d - 2 test */
255 233890 : for (t=uel(trace2,ind[1]),i=2; i<=K; i++)
256 125261 : t = Fl_add(t, uel(trace2,ind[i]), spa_b);
257 108629 : if (t > spa_bs2) t = spa_b - t;
258 108629 : if (t > Sbound)
259 : {
260 57582 : if (DEBUGLEVEL>6) err_printf("|");
261 57582 : goto NEXT;
262 : }
263 :
264 51047 : av = avma;
265 : /* check trailing coeff */
266 51047 : y = lc;
267 150219 : for (i=1; i<=K; i++)
268 : {
269 99172 : GEN q = constant_coeff(gel(famod,ind[i]));
270 99172 : if (y) q = mulii(y, q);
271 99172 : y = centermodii(q, pa, pas2);
272 : }
273 51047 : if (!signe(y) || !dvdii(constant_coeff(lcpol), y))
274 : {
275 22557 : if (DEBUGLEVEL>3) err_printf("T");
276 22557 : set_avma(av); goto NEXT;
277 : }
278 28490 : y = lc; /* full computation */
279 63602 : for (i=1; i<=K; i++)
280 : {
281 35112 : GEN q = gel(famod,ind[i]);
282 35112 : if (y) q = gmul(y, q);
283 35112 : y = centermod_i(q, pa, pas2);
284 : }
285 :
286 : /* y is the candidate factor */
287 28490 : if (! (q = ZX_divides_i(lcpol,y,bound)) )
288 : {
289 3703 : if (DEBUGLEVEL>3) err_printf("*");
290 3703 : set_avma(av); goto NEXT;
291 : }
292 : /* found a factor */
293 24787 : list = cgetg(K+1, t_VEC);
294 24787 : gel(listmod,cnt) = list;
295 50288 : for (i=1; i<=K; i++) list[i] = famod[ind[i]];
296 :
297 24787 : y = Q_primpart(y);
298 24787 : gel(fa,cnt++) = y;
299 : /* fix up pol */
300 24787 : pol = q;
301 24787 : if (lc) pol = Q_div_to_int(pol, leading_coeff(y));
302 91943 : for (i=j=k=1; i <= lfamod; i++)
303 : { /* remove used factors */
304 67156 : if (j <= K && i == ind[j]) j++;
305 : else
306 : {
307 41655 : gel(famod,k) = gel(famod,i);
308 41655 : uel(trace1,k) = uel(trace1,i);
309 41655 : uel(trace2,k) = uel(trace2,i);
310 41655 : deg[k] = deg[i]; k++;
311 : }
312 : }
313 24787 : lfamod -= K;
314 24787 : *pmaxK = cmbf_maxK(lfamod);
315 24787 : if (lfamod < 2*K) goto END;
316 10505 : i = 1; curdeg = deg[ind[1]];
317 10505 : bound = factor_bound(pol);
318 10505 : if (lc) lc = absi_shallow(leading_coeff(pol));
319 10505 : lcpol = lc? ZX_Z_mul(pol, lc): pol;
320 10505 : if (DEBUGLEVEL>3)
321 0 : err_printf("\nfound factor %Ps\nremaining modular factor(s): %ld\n",
322 : y, lfamod);
323 10505 : continue;
324 : }
325 :
326 0 : NEXT:
327 512064 : for (i = K+1;;)
328 : {
329 638154 : if (--i == 0) { K++; goto nextK; }
330 597236 : if (++ind[i] <= lfamod - K + i)
331 : {
332 471146 : curdeg = degsofar[i-1] + deg[ind[i]];
333 471146 : if (curdeg <= klim) break;
334 : }
335 : }
336 : }
337 49623 : END:
338 49623 : *done = 1;
339 49623 : if (degpol(pol) > 0)
340 : { /* leftover factor */
341 49623 : if (signe(leading_coeff(pol)) < 0) pol = ZX_neg(pol);
342 49623 : if (lfamod >= 2*K) *done = 0;
343 :
344 49623 : setlg(famod, lfamod+1);
345 49623 : gel(listmod,cnt) = leafcopy(famod);
346 49623 : gel(fa,cnt++) = pol;
347 : }
348 49623 : if (DEBUGLEVEL>6) err_printf("\n");
349 49623 : setlg(listmod, cnt);
350 49623 : 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 183291 : shifteval(GEN Q, long n)
358 : {
359 183291 : pari_sp av = avma;
360 183291 : long i, l = lg(Q);
361 : GEN s;
362 :
363 183291 : if (!signe(Q)) return gen_0;
364 183291 : s = gel(Q,l-1);
365 970771 : for (i = l-2; i > 1; i--)
366 : {
367 787484 : s = addii(gel(Q,i), shifti(s, n));
368 787480 : if (gc_needed(av,1)) s = gerepileuptoint(av, s);
369 : }
370 183287 : return s;
371 : }
372 :
373 : /* return integer y such that all |a| <= y if P(a) = 0 */
374 : static GEN
375 109860 : root_bound(GEN P0)
376 : {
377 109860 : GEN Q = leafcopy(P0), lP = absi_shallow(leading_coeff(Q)), x,y,z;
378 109860 : long k, d = degpol(Q);
379 :
380 : /* P0 = lP x^d + Q, deg Q < d */
381 109860 : Q = normalizepol_lg(Q, d+2);
382 691755 : for (k=lg(Q)-1; k>1; k--) gel(Q,k) = absi_shallow(gel(Q,k));
383 109860 : k = (long)(fujiwara_bound(P0));
384 184117 : for ( ; k >= 0; k--)
385 : {
386 183291 : pari_sp av = avma;
387 : /* y = 2^k; Q(y) >= lP y^d ? */
388 183291 : if (cmpii(shifteval(Q,k), shifti(lP, d*k)) >= 0) break;
389 74257 : set_avma(av);
390 : }
391 109860 : if (k < 0) k = 0;
392 109860 : y = int2n(k+1);
393 109860 : if (d > 2000) return y; /* likely to be expensive, don't bother */
394 109860 : x = int2n(k);
395 109860 : for(k=0; ; k++)
396 : {
397 598704 : z = shifti(addii(x,y), -1);
398 598701 : if (equalii(x,z) || k > 5) break;
399 488843 : if (cmpii(ZX_Z_eval(Q,z), mulii(lP, powiu(z, d))) < 0)
400 261583 : y = z;
401 : else
402 227261 : x = z;
403 : }
404 109860 : 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 | LLL_NOFLATTER, &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 49623 : cmbf_precs(GEN q, GEN A, GEN B, long *pta, long *ptb, GEN *qa, GEN *qb)
653 : {
654 49623 : long a,b,amin,d = (long)(31 * M_LN2/gtodouble(glog(q,DEFAULTPREC)) - 1e-5);
655 49623 : int fl = 0;
656 :
657 49623 : b = logintall(B, q, qb) + 1;
658 49623 : *qb = mulii(*qb, q);
659 49623 : amin = b + d;
660 49623 : if (gcmp(powiu(q, amin), A) <= 0)
661 : {
662 14215 : a = logintall(A, q, qa) + 1;
663 14215 : *qa = mulii(*qa, q);
664 14215 : b = a - d; *qb = powiu(q, b);
665 : }
666 : else
667 : { /* not enough room */
668 35408 : a = amin; *qa = powiu(q, a);
669 35408 : fl = 1;
670 : }
671 49623 : 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 49623 : *pta = a;
676 49623 : *ptb = b; return fl;
677 : }
678 :
679 : /* use van Hoeij's knapsack algorithm */
680 : static GEN
681 49623 : combine_factors(GEN target, GEN famod, GEN p, long klim)
682 : {
683 : GEN la, B, A, res, L, pa, pb, listmod;
684 49623 : long a,b, l, maxK, n = degpol(target);
685 : int done;
686 : pari_timer T;
687 :
688 49623 : A = factor_bound(target);
689 :
690 49623 : la = absi_shallow(leading_coeff(target));
691 49623 : B = mului(n, sqri(mulii(la, root_bound(target)))); /* = bound for S_2 */
692 :
693 49623 : (void)cmbf_precs(p, A, B, &a, &b, &pa, &pb);
694 :
695 49623 : if (DEBUGLEVEL>2) timer_start(&T);
696 49623 : famod = ZpX_liftfact(target, famod, pa, p, a);
697 49623 : if (DEBUGLEVEL>2) timer_printf(&T, "Hensel lift (mod %Ps^%ld)", p,a);
698 49623 : L = cmbf(target, famod, A, p, a, b, klim, &maxK, &done);
699 49623 : if (DEBUGLEVEL>2) timer_printf(&T, "Naive recombination");
700 :
701 49623 : res = gel(L,1);
702 49623 : listmod = gel(L,2); l = lg(listmod)-1;
703 49623 : famod = gel(listmod,l);
704 49623 : 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 49623 : 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 1824064 : pick_prime(GEN a, long fl, pari_timer *T)
720 : {
721 1824064 : pari_sp av = avma, av1;
722 1824064 : const long MAXNP = 7, da = degpol(a);
723 1824064 : long nmax = da+1, np;
724 1824064 : ulong chosenp = 0;
725 1824064 : GEN lead = gel(a,da+2);
726 : forprime_t S;
727 1824064 : if (equali1(lead)) lead = NULL;
728 1824064 : u_forprime_init(&S, 2, ULONG_MAX);
729 1824070 : av1 = avma;
730 7402376 : for (np = 0; np < MAXNP; set_avma(av1))
731 : {
732 7292638 : ulong p = u_forprime_next(&S);
733 : long nfacp;
734 : GEN z;
735 :
736 7292631 : if (!p) pari_err_OVERFLOW("DDF [out of small primes]");
737 7292631 : if (lead && !umodiu(lead,p)) continue;
738 7226197 : z = ZX_to_Flx(a, p);
739 7226198 : if (!Flx_is_squarefree(z, p)) continue;
740 :
741 4494258 : if (fl==1)
742 : {
743 3909287 : nfacp = Flx_nbroots(z, p);
744 3909289 : if (!nfacp) { chosenp = 0; break; } /* no root */
745 : }
746 584971 : else if(fl==0)
747 : {
748 584223 : nfacp = Flx_nbfact(z, p);
749 584230 : if (nfacp == 1) { chosenp = 0; break; } /* irreducible */
750 : } else
751 : {
752 748 : GEN f = gel(Flx_degfact(z, p),1);
753 749 : nfacp = lg(f)-1;
754 749 : if (f[1] > fl) { chosenp = 0; break; } /* no small factors */
755 : }
756 2779942 : if (DEBUGLEVEL>4)
757 0 : err_printf("...tried prime %3lu (%-3ld %s). Time = %ld\n",
758 : p, nfacp, fl==1? "roots": "factors", timer_delay(T));
759 2779952 : if (nfacp < nmax)
760 : {
761 992480 : nmax = nfacp; chosenp = p;
762 992480 : if (da > 100 && nmax < 5) break; /* large degree, few factors. Enough */
763 : }
764 2779945 : np++;
765 : }
766 1824069 : return gc_ulong(av, chosenp);
767 : }
768 :
769 : /* Assume A a squarefree ZX; return the vector of its rational roots */
770 : static GEN
771 1621089 : DDF_roots(GEN A)
772 : {
773 : GEN p, lc, lcpol, z, pe, pes2, bound;
774 : long i, m, e, lz;
775 : ulong pp;
776 : pari_sp av;
777 : pari_timer T;
778 :
779 1621089 : if (DEBUGLEVEL>2) timer_start(&T);
780 1621089 : pp = pick_prime(A, 1, &T);
781 1621086 : if (!pp) return cgetg(1,t_COL); /* no root */
782 60111 : p = utoipos(pp);
783 60111 : lc = leading_coeff(A);
784 60111 : if (is_pm1(lc))
785 52599 : { lc = NULL; lcpol = A; }
786 : else
787 7512 : { lc = absi_shallow(lc); lcpol = ZX_Z_mul(A, lc); }
788 60111 : bound = root_bound(A); if (lc) bound = mulii(lc, bound);
789 60111 : e = logintall(addiu(shifti(bound, 1), 1), p, &pe) + 1;
790 60111 : pe = mulii(pe, p);
791 60111 : pes2 = shifti(pe, -1);
792 60111 : if (DEBUGLEVEL>2) timer_printf(&T, "Root bound");
793 60111 : av = avma;
794 60111 : z = ZpX_roots(A, p, e); lz = lg(z);
795 60111 : z = deg1_from_roots(z, varn(A));
796 60111 : if (DEBUGLEVEL>2) timer_printf(&T, "Hensel lift (mod %lu^%ld)", pp,e);
797 130547 : for (m=1, i=1; i < lz; i++)
798 : {
799 70436 : GEN q, r, y = gel(z,i);
800 70436 : if (lc) y = ZX_Z_mul(y, lc);
801 70436 : y = centermod_i(y, pe, pes2);
802 70436 : if (! (q = ZX_divides(lcpol, y)) ) continue;
803 :
804 25946 : lcpol = q;
805 25946 : r = negi( constant_coeff(y) );
806 25946 : if (lc) {
807 8277 : r = gdiv(r,lc);
808 8277 : lcpol = Q_primpart(lcpol);
809 8277 : lc = absi_shallow( leading_coeff(lcpol) );
810 8277 : if (is_pm1(lc)) lc = NULL; else lcpol = ZX_Z_mul(lcpol, lc);
811 : }
812 25946 : gel(z,m++) = r;
813 25946 : if (gc_needed(av,2))
814 : {
815 0 : if (DEBUGMEM>1) pari_warn(warnmem,"DDF_roots, m = %ld", m);
816 0 : gerepileall(av, lc? 3:2, &z, &lcpol, &lc);
817 : }
818 : }
819 60111 : if (DEBUGLEVEL>2) timer_printf(&T, "Recombination");
820 60111 : setlg(z, m); return z;
821 : }
822 :
823 : /* Assume a squarefree ZX, deg(a) > 0, return rational factors.
824 : * In fact, a(0) != 0 but we don't use this
825 : * if dmax>0, Only look for factor of degree at most dmax */
826 : GEN
827 202975 : ZX_DDF_max(GEN a, long dmax)
828 : {
829 : GEN ap, prime, famod, z;
830 202975 : long ti = 0;
831 202975 : ulong p = 0;
832 202975 : pari_sp av = avma;
833 : pari_timer T, T2;
834 :
835 202975 : if (DEBUGLEVEL>2) { timer_start(&T); timer_start(&T2); }
836 202975 : p = pick_prime(a, dmax, &T2);
837 202974 : if (!p) return mkvec(a);
838 49623 : prime = utoipos(p);
839 49623 : ap = Flx_normalize(ZX_to_Flx(a, p), p);
840 49623 : famod = gel(Flx_factor(ap, p), 1);
841 49623 : if (DEBUGLEVEL>2)
842 : {
843 0 : if (DEBUGLEVEL>4) timer_printf(&T2, "splitting mod p = %lu", p);
844 0 : ti = timer_delay(&T);
845 0 : err_printf("Time setup: %ld\n", ti);
846 : }
847 49623 : z = combine_factors(a, FlxV_to_ZXV(famod), prime, degpol(a)-1);
848 49623 : if (DEBUGLEVEL>2)
849 0 : err_printf("Total Time: %ld\n===========\n", ti + timer_delay(&T));
850 49623 : return gerepilecopy(av, z);
851 : }
852 :
853 : /* Distinct Degree Factorization (deflating first)
854 : * Assume x squarefree, degree(x) > 0, x(0) != 0 */
855 : GEN
856 151487 : ZX_DDF(GEN x)
857 : {
858 : GEN L;
859 : long m;
860 151487 : x = ZX_deflate_max(x, &m);
861 151487 : L = ZX_DDF_max(x,0);
862 151486 : if (m > 1)
863 : {
864 49189 : GEN e, v, fa = factoru(m);
865 : long i,j,k, l;
866 :
867 49189 : e = gel(fa,2); k = 0;
868 49189 : fa= gel(fa,1); l = lg(fa);
869 98721 : for (i=1; i<l; i++) k += e[i];
870 49189 : v = cgetg(k+1, t_VECSMALL); k = 1;
871 98721 : for (i=1; i<l; i++)
872 100489 : for (j=1; j<=e[i]; j++) v[k++] = fa[i];
873 100145 : for (k--; k; k--)
874 : {
875 50957 : GEN L2 = cgetg(1,t_VEC);
876 102277 : for (i=1; i < lg(L); i++)
877 51321 : L2 = shallowconcat(L2, ZX_DDF_max(RgX_inflate(gel(L,i), v[k]),0));
878 50956 : L = L2;
879 : }
880 : }
881 151485 : return L;
882 : }
883 :
884 : /* SquareFree Factorization in Z[X] (char 0 is enough, if ZX_gcd -> RgX_gcd)
885 : * f = prod Q[i]^E[i], E[1] < E[2] < ..., and Q[i] squarefree and coprime.
886 : * Return Q, set *pE = E. For efficiency, caller should have used ZX_valrem
887 : * so that f(0) != 0 */
888 : GEN
889 332424 : ZX_squff(GEN f, GEN *pE)
890 : {
891 : GEN T, V, P, E;
892 332424 : long i, k, n = 1 + degpol(f);
893 :
894 332424 : if (signe(leading_coeff(f)) < 0) f = ZX_neg(f);
895 332424 : E = cgetg(n, t_VECSMALL);
896 332424 : P = cgetg(n, t_COL);
897 332424 : T = ZX_gcd_all(f, ZX_deriv(f), &V);
898 332424 : for (k = i = 1;; k++)
899 3641 : {
900 336065 : GEN W = ZX_gcd_all(T,V, &T); /* V and W are squarefree */
901 336065 : long dW = degpol(W), dV = degpol(V);
902 : /* f = prod_i T_i^{e_i}
903 : * W = prod_{i: e_i > k} T_i,
904 : * V = prod_{i: e_i >= k} T_i,
905 : * T = prod_{i: e_i > k} T_i^{e_i - k} */
906 336065 : if (!dW)
907 : {
908 332424 : if (dV) { gel(P,i) = Q_primpart(V); E[i] = k; i++; }
909 332424 : break;
910 : }
911 3641 : if (dW == dV)
912 : {
913 : GEN U;
914 1638 : while ( (U = ZX_divides(T, V)) ) { k++; T = U; }
915 : }
916 : else
917 : {
918 2388 : gel(P,i) = Q_primpart(RgX_div(V,W));
919 2388 : E[i] = k; i++; V = W;
920 : }
921 : }
922 332424 : setlg(P,i);
923 332424 : setlg(E,i); *pE = E; return P;
924 : }
925 :
926 : static GEN
927 50316 : fact_from_DDF(GEN Q, GEN E, long n)
928 : {
929 50316 : GEN v,w, y = cgetg(3, t_MAT);
930 50316 : long i,j,k, l = lg(Q);
931 :
932 50316 : v = cgetg(n+1, t_COL); gel(y,1) = v;
933 50316 : w = cgetg(n+1, t_COL); gel(y,2) = w;
934 102431 : for (k = i = 1; i < l; i++)
935 : {
936 52115 : GEN L = gel(Q,i), e = utoipos(E[i]);
937 52115 : long J = lg(L);
938 128702 : for (j = 1; j < J; j++,k++)
939 : {
940 76587 : gel(v,k) = ZX_copy(gel(L,j));
941 76587 : gel(w,k) = e;
942 : }
943 : }
944 50316 : return y;
945 : }
946 :
947 : /* Factor T in Z[x] */
948 : static GEN
949 50323 : ZX_factor_i(GEN T)
950 : {
951 : GEN Q, E, y;
952 : long n, i, l, v;
953 :
954 50323 : if (!signe(T)) return prime_fact(T);
955 50316 : v = ZX_valrem(T, &T);
956 50316 : Q = ZX_squff(T, &E); l = lg(Q);
957 100877 : for (i = 1, n = 0; i < l; i++)
958 : {
959 50561 : gel(Q,i) = ZX_DDF(gel(Q,i));
960 50561 : n += lg(gel(Q,i)) - 1;
961 : }
962 50316 : if (v)
963 : {
964 1554 : Q = vec_append(Q, mkvec(pol_x(varn(T))));
965 1554 : E = vecsmall_append(E, v); n++;
966 : }
967 50316 : y = fact_from_DDF(Q, E, n);
968 50316 : return sort_factor_pol(y, cmpii);
969 : }
970 : GEN
971 49553 : ZX_factor(GEN x)
972 : {
973 49553 : pari_sp av = avma;
974 49553 : return gerepileupto(av, ZX_factor_i(x));
975 : }
976 : GEN
977 770 : QX_factor(GEN x)
978 : {
979 770 : pari_sp av = avma;
980 770 : return gerepileupto(av, ZX_factor_i(Q_primpart(x)));
981 : }
982 :
983 : long
984 101747 : ZX_is_irred(GEN x)
985 : {
986 101747 : pari_sp av = avma;
987 101747 : long l = lg(x);
988 : GEN y;
989 101747 : if (l <= 3) return 0; /* degree < 1 */
990 101747 : if (l == 4) return 1; /* degree 1 */
991 98244 : if (ZX_val(x)) return 0;
992 98034 : if (!ZX_is_squarefree(x)) return 0;
993 97895 : y = ZX_DDF(x); set_avma(av);
994 97893 : return (lg(y) == 2);
995 : }
996 :
997 : GEN
998 1621077 : nfrootsQ(GEN x)
999 : {
1000 1621077 : pari_sp av = avma;
1001 : GEN z;
1002 : long val;
1003 :
1004 1621077 : if (typ(x)!=t_POL) pari_err_TYPE("nfrootsQ",x);
1005 1621077 : if (!signe(x)) pari_err_ROOTS0("nfrootsQ");
1006 1621077 : x = Q_primpart(x);
1007 1621079 : RgX_check_ZX(x,"nfrootsQ");
1008 1621079 : val = ZX_valrem(x, &x);
1009 1621081 : z = DDF_roots( ZX_radical(x) );
1010 1621087 : if (val) z = vec_append(z, gen_0);
1011 1621087 : return gerepileupto(av, sort(z));
1012 : }
1013 :
1014 : /************************************************************************
1015 : * GCD OVER Z[X] / Q[X] *
1016 : ************************************************************************/
1017 : int
1018 197105 : ZX_is_squarefree(GEN x)
1019 : {
1020 197105 : pari_sp av = avma;
1021 : GEN d;
1022 : long m;
1023 197105 : if (lg(x) == 2) return 0;
1024 197105 : m = ZX_deflate_order(x);
1025 197106 : if (m > 1)
1026 : {
1027 85202 : if (!signe(gel(x,2))) return 0;
1028 84964 : x = RgX_deflate(x, m);
1029 : }
1030 196866 : d = ZX_gcd(x,ZX_deriv(x));
1031 196877 : return gc_bool(av, lg(d) == 3);
1032 : }
1033 :
1034 : static int
1035 114558 : ZX_gcd_filter(GEN *pt_A, GEN *pt_P)
1036 : {
1037 114558 : GEN A = *pt_A, P = *pt_P;
1038 114558 : long i, j, l = lg(A), n = 1, d = degpol(gel(A,1));
1039 : GEN B, Q;
1040 230122 : for (i=2; i<l; i++)
1041 : {
1042 115564 : long di = degpol(gel(A,i));
1043 115564 : if (di==d) n++;
1044 36 : else if (d > di)
1045 36 : { n=1; d = di; }
1046 : }
1047 114558 : if (n == l-1)
1048 114522 : return 0;
1049 36 : B = cgetg(n+1, t_VEC);
1050 36 : Q = cgetg(n+1, typ(P));
1051 156 : for (i=1, j=1; i<l; i++)
1052 : {
1053 120 : if (degpol(gel(A,i))==d)
1054 : {
1055 84 : gel(B,j) = gel(A,i);
1056 84 : Q[j] = P[i];
1057 84 : j++;
1058 : }
1059 : }
1060 36 : *pt_A = B; *pt_P = Q; return 1;
1061 : }
1062 :
1063 : static GEN
1064 3348157 : ZX_gcd_Flx(GEN a, GEN b, ulong g, ulong p)
1065 : {
1066 3348157 : GEN H = Flx_gcd(a, b, p);
1067 3348157 : if (!g)
1068 3326285 : return Flx_normalize(H, p);
1069 : else
1070 : {
1071 21872 : ulong t = Fl_mul(g, Fl_inv(Flx_lead(H), p), p);
1072 21872 : return Flx_Fl_mul(H, t, p);
1073 : }
1074 : }
1075 :
1076 : static GEN
1077 3342243 : ZX_gcd_slice(GEN A, GEN B, GEN g, GEN P, GEN *mod)
1078 : {
1079 3342243 : pari_sp av = avma;
1080 3342243 : long i, n = lg(P)-1;
1081 : GEN H, T;
1082 3342243 : if (n == 1)
1083 : {
1084 3337309 : ulong p = uel(P,1), gp = g ? umodiu(g, p): 0;
1085 3337309 : GEN a = ZX_to_Flx(A, p), b = ZX_to_Flx(B, p);
1086 3337309 : GEN Hp = ZX_gcd_Flx(a, b, gp, p);
1087 3337309 : H = gerepileupto(av, Flx_to_ZX(Hp));
1088 3337308 : *mod = utoi(p);
1089 3337308 : return H;
1090 : }
1091 4934 : T = ZV_producttree(P);
1092 4934 : A = ZX_nv_mod_tree(A, P, T);
1093 4934 : B = ZX_nv_mod_tree(B, P, T);
1094 4934 : g = g ? Z_ZV_mod_tree(g, P, T): NULL;
1095 4934 : H = cgetg(n+1, t_VEC);
1096 15782 : for(i=1; i <= n; i++)
1097 : {
1098 10848 : ulong p = P[i];
1099 10848 : GEN a = gel(A,i), b = gel(B,i);
1100 10848 : gel(H,i) = ZX_gcd_Flx(a, b, g? g[i]: 0, p);
1101 : }
1102 4934 : if (ZX_gcd_filter(&H, &P))
1103 12 : T = ZV_producttree(P);
1104 4934 : H = nxV_chinese_center_tree(H, P, T, ZV_chinesetree(P, T));
1105 4934 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
1106 : }
1107 :
1108 : GEN
1109 3342243 : ZX_gcd_worker(GEN P, GEN A, GEN B, GEN g)
1110 : {
1111 3342243 : GEN V = cgetg(3, t_VEC);
1112 3342243 : gel(V,1) = ZX_gcd_slice(A, B, equali1(g)? NULL: g , P, &gel(V,2));
1113 3342242 : return V;
1114 : }
1115 :
1116 : static GEN
1117 109624 : ZX_gcd_chinese(GEN A, GEN P, GEN *mod)
1118 : {
1119 109624 : ZX_gcd_filter(&A, &P);
1120 109624 : return nxV_chinese_center(A, P, mod);
1121 : }
1122 :
1123 : GEN
1124 13098888 : ZX_gcd_all(GEN A, GEN B, GEN *Anew)
1125 : {
1126 13098888 : pari_sp av = avma;
1127 13098888 : long k, valH, valA, valB, vA = varn(A), dA = degpol(A), dB = degpol(B);
1128 13098855 : GEN worker, c, cA, cB, g, Ag, Bg, H = NULL, mod = gen_1, R;
1129 : GEN Ap, Bp, Hp;
1130 : forprime_t S;
1131 : ulong pp;
1132 13098855 : if (dA < 0) { if (Anew) *Anew = pol_0(vA); return ZX_copy(B); }
1133 13098834 : if (dB < 0) { if (Anew) *Anew = pol_1(vA); return ZX_copy(A); }
1134 13097693 : A = Q_primitive_part(A, &cA);
1135 13097850 : B = Q_primitive_part(B, &cB);
1136 13097781 : valA = ZX_valrem(A, &A); dA -= valA;
1137 13097784 : valB = ZX_valrem(B, &B); dB -= valB;
1138 13097844 : valH = minss(valA, valB);
1139 13097847 : valA -= valH; /* valuation(Anew) */
1140 13097847 : c = (cA && cB)? gcdii(cA, cB): NULL; /* content(gcd) */
1141 13097853 : if (!dA || !dB)
1142 : {
1143 6481956 : if (Anew) *Anew = RgX_shift_shallow(A, valA);
1144 6481956 : return monomial(c? c: gen_1, valH, vA);
1145 : }
1146 6615897 : g = gcdii(leading_coeff(A), leading_coeff(B)); /* multiple of lead(gcd) */
1147 6615819 : if (is_pm1(g)) {
1148 6409495 : g = NULL;
1149 6409495 : Ag = A;
1150 6409495 : Bg = B;
1151 : } else {
1152 206321 : Ag = ZX_Z_mul(A,g);
1153 206321 : Bg = ZX_Z_mul(B,g);
1154 : }
1155 6615816 : init_modular_big(&S);
1156 : do {
1157 6615932 : pp = u_forprime_next(&S);
1158 6615928 : Ap = ZX_to_Flx(Ag, pp);
1159 6615933 : Bp = ZX_to_Flx(Bg, pp);
1160 6615931 : } while (degpol(Ap) != dA || degpol(Bp) != dB);
1161 6615917 : if (degpol(Flx_gcd(Ap, Bp, pp)) == 0)
1162 : {
1163 3383306 : if (Anew) *Anew = RgX_shift_shallow(A, valA);
1164 3383306 : return monomial(c? c: gen_1, valH, vA);
1165 : }
1166 3232592 : worker = snm_closure(is_entry("_ZX_gcd_worker"), mkvec3(A, B, g? g: gen_1));
1167 3232592 : av = avma;
1168 3340564 : for (k = 1; ;k *= 2)
1169 : {
1170 3340564 : gen_inccrt_i("ZX_gcd", worker, g, (k+1)>>1, 0, &S, &H, &mod, ZX_gcd_chinese, NULL);
1171 3340564 : gerepileall(av, 2, &H, &mod);
1172 3340564 : Hp = ZX_to_Flx(H, pp);
1173 3340564 : if (lgpol(Flx_rem(Ap, Hp, pp)) || lgpol(Flx_rem(Bp, Hp, pp))) continue;
1174 3236658 : if (!ZX_divides(Bg, H)) continue;
1175 3232622 : R = ZX_divides(Ag, H);
1176 3232622 : if (R) break;
1177 : }
1178 : /* lead(H) = g */
1179 3232592 : if (g) H = Q_primpart(H);
1180 3232592 : if (c) H = ZX_Z_mul(H,c);
1181 3232592 : if (DEBUGLEVEL>5) err_printf("done\n");
1182 3232592 : if (Anew) *Anew = RgX_shift_shallow(R, valA);
1183 3232592 : return valH? RgX_shift_shallow(H, valH): H;
1184 : }
1185 :
1186 : #if 0
1187 : /* ceil( || p ||_oo / lc(p) ) */
1188 : static GEN
1189 : maxnorm(GEN p)
1190 : {
1191 : long i, n = degpol(p), av = avma;
1192 : GEN x, m = gen_0;
1193 :
1194 : p += 2;
1195 : for (i=0; i<n; i++)
1196 : {
1197 : x = gel(p,i);
1198 : if (abscmpii(x,m) > 0) m = x;
1199 : }
1200 : m = divii(m, gel(p,n));
1201 : return gerepileuptoint(av, addiu(absi_shallow(m),1));
1202 : }
1203 : #endif
1204 :
1205 : GEN
1206 9766877 : ZX_gcd(GEN A, GEN B)
1207 : {
1208 9766877 : pari_sp av = avma;
1209 9766877 : return gerepilecopy(av, ZX_gcd_all(A,B,NULL));
1210 : }
1211 :
1212 : GEN
1213 2659282 : ZX_radical(GEN A) { GEN B; (void)ZX_gcd_all(A,ZX_deriv(A),&B); return B; }
1214 :
1215 : static GEN
1216 18629 : _gcd(GEN a, GEN b)
1217 : {
1218 18629 : if (!a) a = gen_1;
1219 18629 : if (!b) b = gen_1;
1220 18629 : return Q_gcd(a,b);
1221 : }
1222 : /* A0 and B0 in Q[X] */
1223 : GEN
1224 18629 : QX_gcd(GEN A0, GEN B0)
1225 : {
1226 : GEN a, b, D;
1227 18629 : pari_sp av = avma, av2;
1228 :
1229 18629 : D = ZX_gcd(Q_primitive_part(A0, &a), Q_primitive_part(B0, &b));
1230 18629 : av2 = avma; a = _gcd(a,b);
1231 18629 : if (isint1(a)) set_avma(av2); else D = ZX_Q_mul(D, a);
1232 18629 : return gerepileupto(av, D);
1233 : }
1234 :
1235 : /*****************************************************************************
1236 : * Variants of the Bradford-Davenport algorithm: look for cyclotomic *
1237 : * factors, and decide whether a ZX is cyclotomic or a product of cyclotomic *
1238 : *****************************************************************************/
1239 : /* f of degree 1, return a cyclotomic factor (Phi_1 or Phi_2) or NULL */
1240 : static GEN
1241 0 : BD_deg1(GEN f)
1242 : {
1243 0 : GEN a = gel(f,3), b = gel(f,2); /* f = ax + b */
1244 0 : if (!absequalii(a,b)) return NULL;
1245 0 : return polcyclo((signe(a) == signe(b))? 2: 1, varn(f));
1246 : }
1247 :
1248 : /* f a squarefree ZX; not divisible by any Phi_n, n even */
1249 : static GEN
1250 406 : BD_odd(GEN f)
1251 : {
1252 406 : while(degpol(f) > 1)
1253 : {
1254 406 : GEN f1 = ZX_graeffe(f); /* contain all cyclotomic divisors of f */
1255 406 : if (ZX_equal(f1, f)) return f; /* product of cyclotomics */
1256 0 : f = ZX_gcd(f, f1);
1257 : }
1258 0 : if (degpol(f) == 1) return BD_deg1(f);
1259 0 : return NULL; /* no cyclotomic divisor */
1260 : }
1261 :
1262 : static GEN
1263 2310 : myconcat(GEN v, GEN x)
1264 : {
1265 2310 : if (typ(x) != t_VEC) x = mkvec(x);
1266 2310 : if (!v) return x;
1267 1470 : return shallowconcat(v, x);
1268 : }
1269 :
1270 : /* Bradford-Davenport algorithm.
1271 : * f a squarefree ZX of degree > 0, return NULL or a vector of coprime
1272 : * cyclotomic factors of f [ possibly reducible ] */
1273 : static GEN
1274 2359 : BD(GEN f)
1275 : {
1276 2359 : GEN G = NULL, Gs = NULL, Gp = NULL, Gi = NULL;
1277 : GEN fs2, fp, f2, f1, fe, fo, fe1, fo1;
1278 2359 : RgX_even_odd(f, &fe, &fo);
1279 2359 : fe1 = ZX_eval1(fe);
1280 2359 : fo1 = ZX_eval1(fo);
1281 2359 : if (absequalii(fe1, fo1)) /* f(1) = 0 or f(-1) = 0 */
1282 : {
1283 1519 : long i, v = varn(f);
1284 1519 : if (!signe(fe1))
1285 371 : G = mkvec2(polcyclo(1, v), polcyclo(2, v)); /* both 0 */
1286 1148 : else if (signe(fe1) == signe(fo1))
1287 693 : G = mkvec(polcyclo(2, v)); /*f(-1) = 0*/
1288 : else
1289 455 : G = mkvec(polcyclo(1, v)); /*f(1) = 0*/
1290 3409 : for (i = lg(G)-1; i; i--) f = RgX_div(f, gel(G,i));
1291 : }
1292 : /* f no longer divisible by Phi_1 or Phi_2 */
1293 2359 : if (degpol(f) <= 1) return G;
1294 2058 : f1 = ZX_graeffe(f); /* has at most square factors */
1295 2058 : if (ZX_equal(f1, f)) return myconcat(G,f); /* f = product of Phi_n, n odd */
1296 :
1297 1183 : fs2 = ZX_gcd_all(f1, ZX_deriv(f1), &f2); /* fs2 squarefree */
1298 1183 : if (degpol(fs2))
1299 : { /* fs contains all Phi_n | f, 4 | n; and only those */
1300 : /* In that case, Graeffe(Phi_n) = Phi_{n/2}^2, and Phi_n = Phi_{n/2}(x^2) */
1301 1029 : GEN fs = RgX_inflate(fs2, 2);
1302 1029 : (void)ZX_gcd_all(f, fs, &f); /* remove those Phi_n | f, 4 | n */
1303 1029 : Gs = BD(fs2);
1304 1029 : if (Gs)
1305 : {
1306 : long i;
1307 2555 : for (i = lg(Gs)-1; i; i--) gel(Gs,i) = RgX_inflate(gel(Gs,i), 2);
1308 : /* prod Gs[i] is the product of all Phi_n | f, 4 | n */
1309 1029 : G = myconcat(G, Gs);
1310 : }
1311 : /* f2 = f1 / fs2 */
1312 1029 : f1 = RgX_div(f2, fs2); /* f1 / fs2^2 */
1313 : }
1314 1183 : fp = ZX_gcd(f, f1); /* contains all Phi_n | f, n > 1 odd; and only those */
1315 1183 : if (degpol(fp))
1316 : {
1317 196 : Gp = BD_odd(fp);
1318 : /* Gp is the product of all Phi_n | f, n odd */
1319 196 : if (Gp) G = myconcat(G, Gp);
1320 196 : f = RgX_div(f, fp);
1321 : }
1322 1183 : if (degpol(f))
1323 : { /* contains all Phi_n originally dividing f, n = 2 mod 4, n > 2;
1324 : * and only those
1325 : * In that case, Graeffe(Phi_n) = Phi_{n/2}, and Phi_n = Phi_{n/2}(-x) */
1326 210 : Gi = BD_odd(ZX_z_unscale(f, -1));
1327 210 : if (Gi)
1328 : { /* N.B. Phi_2 does not divide f */
1329 210 : Gi = ZX_z_unscale(Gi, -1);
1330 : /* Gi is the product of all Phi_n | f, n = 2 mod 4 */
1331 210 : G = myconcat(G, Gi);
1332 : }
1333 : }
1334 1183 : return G;
1335 : }
1336 :
1337 : /* Let f be a nonzero QX, return the (squarefree) product of cyclotomic
1338 : * divisors of f */
1339 : GEN
1340 315 : polcyclofactors(GEN f)
1341 : {
1342 315 : pari_sp av = avma;
1343 315 : if (typ(f) != t_POL || !signe(f)) pari_err_TYPE("polcyclofactors",f);
1344 315 : (void)RgX_valrem(f, &f);
1345 315 : f = Q_primpart(f);
1346 315 : RgX_check_ZX(f,"polcyclofactors");
1347 315 : if (degpol(f))
1348 : {
1349 315 : f = BD(ZX_radical(f));
1350 315 : if (f) return gerepilecopy(av, f);
1351 : }
1352 0 : set_avma(av); return cgetg(1,t_VEC);
1353 : }
1354 :
1355 : /* list of all squarefree odd x such that phi(x) = n, P^-(x) > m. Unsorted */
1356 : static GEN
1357 19544 : invphi(ulong n, ulong m)
1358 : {
1359 : GEN C, D;
1360 : long l, i;
1361 19544 : if (n == 1) return mkvecsmall(1);
1362 14278 : D = divisorsu(n); l = lg(D);
1363 14278 : C = cgetg(1, t_VECSMALL);
1364 39670 : for (i = 2; i < l; i++) /* skip 1 */
1365 : {
1366 25392 : ulong d = D[i], p;
1367 25392 : if (d < m) continue;
1368 20235 : p = d + 1; if (!uisprime(p)) continue;
1369 10505 : C = vecsmall_concat(C, zv_z_mul(invphi(D[l-i], p), p));
1370 : }
1371 14278 : return C;
1372 : }
1373 :
1374 : long
1375 98901 : poliscyclo(GEN f)
1376 : {
1377 98901 : const ulong p = 2147483647; /* prime */
1378 : pari_sp av;
1379 : long i, n, e, l;
1380 : ulong f3, fm3;
1381 : GEN D, fp, _3;
1382 98901 : if (typ(f) != t_POL) pari_err_TYPE("poliscyclo", f);
1383 98894 : n = degpol(f);
1384 98894 : if (n <= 0 || !RgX_is_ZX(f)) return 0;
1385 98887 : if (!equali1(gel(f,n+2)) || !is_pm1(gel(f,2))) return 0;
1386 9137 : if (n == 1) return signe(gel(f,2)) > 0? 2: 1;
1387 9039 : av = avma;
1388 9039 : f = ZX_deflate_max(f, &e); if (e != 1) n = degpol(f);
1389 9039 : D = invphi(n, 1); /* squareefree odd d s.t. phi(d) = n */
1390 9039 : l = lg(D); _3 = gmodulss(3, p);
1391 9039 : fp = ZX_to_Flx(f, p);
1392 9039 : f3 = Flx_eval(fp, 3, p);
1393 9039 : fm3 = Flx_eval(fp, p-3, p);
1394 : /* f(x^e) is cyclotomic (= Phi_{de}) iff f = Phi_d, where all prime dividing
1395 : * e also divide d. */
1396 11885 : for (i = 1; i < l; i++)
1397 : {
1398 5134 : long d = D[i]; /* squarefree odd */
1399 5134 : if (odd(e))
1400 : {
1401 4075 : if (e == 1 || u_ppo(e, d) == 1)
1402 : { /* early abort: check whether f(3) = Phi_d(3) or Phi_2d(3) = Phi_d(-3)
1403 : * mod p before checking in Z. N.B. phi(d) and value at 3 mod p
1404 : * determine Phi_d for all d <= 10^7 */
1405 3833 : ulong F3 = Rg_to_Fl(polcyclo_eval(d, _3), p);
1406 3833 : if (F3 == f3 && ZX_equal(f, polcyclo(d, varn(f))))
1407 973 : return gc_long(av, d * e);
1408 2860 : if (F3 == fm3 && ZX_equal(f, polcyclo(2*d, varn(f))))
1409 749 : return gc_long(av, 2* d * e);
1410 : }
1411 : }
1412 : else
1413 : {
1414 1059 : if (u_ppo(e, 2*d) == 1)
1415 : { /* early abort: check whether f(3) = Phi_2d(3) mod p */
1416 1052 : ulong F3 = Rg_to_Fl(polcyclo_eval(2*d, _3), p);
1417 1052 : if (F3 == f3 && ZX_equal(f, polcyclo(2*d, varn(f))))
1418 567 : return gc_long(av, 2* d * e);
1419 : }
1420 : }
1421 : }
1422 6751 : return gc_long(av, 0);
1423 : }
1424 :
1425 : long
1426 1029 : poliscycloprod(GEN f)
1427 : {
1428 1029 : pari_sp av = avma;
1429 1029 : long i, d = degpol(f);
1430 1029 : if (typ(f) != t_POL) pari_err_TYPE("poliscycloprod",f);
1431 1029 : if (!RgX_is_ZX(f)) return 0;
1432 1029 : if (!ZX_is_monic(f) || !is_pm1(constant_coeff(f))) return 0;
1433 1029 : if (d < 2) return (d == 1);
1434 1022 : if ( degpol(ZX_gcd_all(f, ZX_deriv(f), &f)) )
1435 : {
1436 14 : d = degpol(f);
1437 14 : if (d == 1) return 1;
1438 : }
1439 1015 : f = BD(f); if (!f) return 0;
1440 3619 : for (i = lg(f)-1; i; i--) d -= degpol(gel(f,i));
1441 1015 : return gc_long(av, d == 0);
1442 : }
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