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 6587891 : 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 6587891 : dy=degpol(y);
29 6587891 : dx=degpol(x);
30 6587891 : dz=dx-dy; if (dz<0) return NULL;
31 6586813 : z=cgetg(dz+3,t_POL); z[1] = x[1];
32 6586813 : x += 2; y += 2; z += 2;
33 6586813 : y_lead = gel(y,dy);
34 6586813 : if (equali1(y_lead)) y_lead = NULL;
35 :
36 6586813 : p1 = gel(x,dx);
37 6586813 : if (y_lead) {
38 : GEN r;
39 36295 : p1 = dvmdii(p1,y_lead, &r);
40 36295 : if (r != gen_0) return NULL;
41 : }
42 6550518 : else p1 = icopy(p1);
43 6582807 : gel(z,dz) = p1;
44 8280173 : for (i=dx-1; i>=dy; i--)
45 : {
46 1702645 : av = avma; p1 = gel(x,i);
47 5889937 : for (j=i-dy+1; j<=i && j<=dz; j++)
48 4187317 : p1 = subii(p1, mulii(gel(z,j),gel(y,i-j)));
49 1702620 : if (y_lead) {
50 : GEN r;
51 74431 : p1 = dvmdii(p1,y_lead, &r);
52 74431 : if (r != gen_0) return NULL;
53 : }
54 1701121 : if (B && abscmpii(p1, B) > 0) return NULL;
55 1697348 : p1 = gc_INT(av, p1);
56 1697366 : gel(z,i-dy) = p1;
57 : }
58 6577528 : av = avma;
59 17149139 : for (; i >= 0; i--)
60 : {
61 10614767 : p1 = gel(x,i);
62 : /* we always enter this loop at least once */
63 23906387 : for (j=0; j<=i && j<=dz; j++)
64 13291635 : p1 = subii(p1, mulii(gel(z,j),gel(y,i-j)));
65 10614752 : if (signe(p1)) return NULL;
66 10571609 : set_avma(av);
67 : }
68 6534372 : return z - 2;
69 : }
70 : static GEN
71 6557258 : 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 gc_INT(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 62421 : Mignotte_bound(GEN S)
110 : {
111 62421 : long i, d = degpol(S);
112 62421 : GEN C, N2, t, binlS, lS = leading_coeff(S), bin = vecbinomial(d-1);
113 :
114 62421 : N2 = sqrtr(RgX_fpnorml2(S,DEFAULTPREC));
115 62421 : binlS = is_pm1(lS)? bin: ZC_Z_mul(bin, lS);
116 :
117 : /* i = 0 */
118 62421 : C = gel(binlS,1);
119 : /* i = d */
120 62421 : t = N2; if (gcmp(C, t) < 0) C = t;
121 532050 : for (i = 1; i < d; i++)
122 : {
123 469629 : t = addri(mulir(gel(bin,i), N2), gel(binlS,i+1));
124 469629 : if (mpcmp(C, t) < 0) C = t;
125 : }
126 62421 : 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 62421 : Beauzamy_bound(GEN S)
132 : {
133 62421 : const long prec = DEFAULTPREC;
134 62421 : long i, d = degpol(S);
135 : GEN bin, lS, s, C;
136 62421 : bin = vecbinomial(d);
137 :
138 62421 : s = real_0(prec);
139 656888 : for (i=0; i<=d; i++)
140 : {
141 594467 : GEN c = gel(S,i+2);
142 594467 : if (gequal0(c)) continue;
143 : /* s += P_i^2 / binomial(d,i) */
144 498679 : s = addrr(s, divri(itor(sqri(c), prec), gel(bin,i+1)));
145 : }
146 : /* s = [S]_2^2 */
147 62421 : C = powruhalf(utor(3,prec), 3 + 2*d); /* 3^{3/2 + d} */
148 62421 : C = divrr(mulrr(C, s), mulur(4*d, mppi(prec)));
149 62421 : lS = absi_shallow(leading_coeff(S));
150 62421 : return mulir(lS, sqrtr(C));
151 : }
152 :
153 : static GEN
154 62421 : factor_bound(GEN S)
155 : {
156 62421 : pari_sp av = avma;
157 62421 : GEN a = Mignotte_bound(S);
158 62421 : GEN b = Beauzamy_bound(S);
159 62421 : if (DEBUGLEVEL>2)
160 : {
161 0 : err_printf("Mignotte bound: %Ps\n",a);
162 0 : err_printf("Beauzamy bound: %Ps\n",b);
163 : }
164 62421 : return gc_upto(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 51730 : cmbf(GEN pol, GEN famod, GEN bound, GEN p, long a, long b,
176 : long klim, long *pmaxK, int *done)
177 : {
178 51730 : long K = 1, cnt = 1, i,j,k, curdeg, lfamod = lg(famod)-1;
179 : ulong spa_b, spa_bs2, Sbound;
180 51730 : GEN lc, lcpol, pa = powiu(p,a), pas2 = shifti(pa,-1);
181 51730 : GEN trace1 = cgetg(lfamod+1, t_VECSMALL);
182 51730 : GEN trace2 = cgetg(lfamod+1, t_VECSMALL);
183 51730 : GEN ind = cgetg(lfamod+1, t_VECSMALL);
184 51730 : GEN deg = cgetg(lfamod+1, t_VECSMALL);
185 51730 : GEN degsofar = cgetg(lfamod+1, t_VECSMALL);
186 51730 : GEN listmod = cgetg(lfamod+1, t_VEC);
187 51730 : GEN fa = cgetg(lfamod+1, t_VEC);
188 :
189 51730 : *pmaxK = cmbf_maxK(lfamod);
190 51730 : lc = absi_shallow(leading_coeff(pol));
191 51730 : if (equali1(lc)) lc = NULL;
192 51730 : lcpol = lc? ZX_Z_mul(pol, lc): pol;
193 :
194 : {
195 51730 : GEN pa_b,pa_bs2,pb, lc2 = lc? sqri(lc): NULL;
196 :
197 51730 : pa_b = powiu(p, a-b); /* < 2^31 */
198 51730 : pa_bs2 = shifti(pa_b,-1);
199 51730 : pb= powiu(p, b);
200 182695 : for (i=1; i <= lfamod; i++)
201 : {
202 130965 : GEN T1,T2, P = gel(famod,i);
203 130965 : long d = degpol(P);
204 :
205 130965 : deg[i] = d; P += 2;
206 130965 : T1 = gel(P,d-1);/* = - S_1 */
207 130965 : T2 = sqri(T1);
208 130965 : if (d > 1) T2 = subii(T2, shifti(gel(P,d-2),1));
209 130965 : T2 = modii(T2, pa); /* = S_2 Newton sum */
210 130965 : if (lc)
211 : {
212 5075 : T1 = Fp_mul(lc, T1, pa);
213 5075 : T2 = Fp_mul(lc2,T2, pa);
214 : }
215 130965 : uel(trace1,i) = itou(diviiround(T1, pb));
216 130965 : uel(trace2,i) = itou(diviiround(T2, pb));
217 : }
218 51730 : spa_b = uel(pa_b,2); /* < 2^31 */
219 51730 : spa_bs2 = uel(pa_bs2,2); /* < 2^31 */
220 : }
221 51730 : degsofar[0] = 0; /* sentinel */
222 :
223 : /* ind runs through strictly increasing sequences of length K,
224 : * 1 <= ind[i] <= lfamod */
225 93214 : nextK:
226 93214 : if (K > *pmaxK || 2*K > lfamod) goto END;
227 57470 : if (DEBUGLEVEL > 3)
228 0 : err_printf("\n### K = %d, %Ps combinations\n", K,binomial(utoipos(lfamod), K));
229 57470 : setlg(ind, K+1); ind[1] = 1;
230 57470 : Sbound = (ulong) ((K+1)>>1);
231 57470 : i = 1; curdeg = deg[ind[1]];
232 : for(;;)
233 : { /* try all combinations of K factors */
234 642450 : for (j = i; j < K; j++)
235 : {
236 89957 : degsofar[j] = curdeg;
237 89957 : ind[j+1] = ind[j]+1; curdeg += deg[ind[j+1]];
238 : }
239 552493 : if (curdeg <= klim) /* trial divide */
240 10684 : {
241 : GEN y, q, list;
242 : pari_sp av;
243 : ulong t;
244 :
245 : /* d - 1 test */
246 1392290 : for (t=uel(trace1,ind[1]),i=2; i<=K; i++)
247 839797 : t = Fl_add(t, uel(trace1,ind[i]), spa_b);
248 552493 : if (t > spa_bs2) t = spa_b - t;
249 552493 : if (t > Sbound)
250 : {
251 439730 : if (DEBUGLEVEL>6) err_printf(".");
252 439730 : goto NEXT;
253 : }
254 : /* d - 2 test */
255 242095 : for (t=uel(trace2,ind[1]),i=2; i<=K; i++)
256 129332 : t = Fl_add(t, uel(trace2,ind[i]), spa_b);
257 112763 : if (t > spa_bs2) t = spa_b - t;
258 112763 : if (t > Sbound)
259 : {
260 59654 : if (DEBUGLEVEL>6) err_printf("|");
261 59654 : goto NEXT;
262 : }
263 :
264 53109 : av = avma;
265 : /* check trailing coeff */
266 53109 : y = lc;
267 154630 : for (i=1; i<=K; i++)
268 : {
269 101521 : GEN q = constant_coeff(gel(famod,ind[i]));
270 101521 : if (y) q = mulii(y, q);
271 101521 : y = centermodii(q, pa, pas2);
272 : }
273 53109 : if (!signe(y) || !dvdii(constant_coeff(lcpol), y))
274 : {
275 22637 : if (DEBUGLEVEL>3) err_printf("T");
276 22637 : set_avma(av); goto NEXT;
277 : }
278 30472 : y = lc; /* full computation */
279 67825 : for (i=1; i<=K; i++)
280 : {
281 37353 : GEN q = gel(famod,ind[i]);
282 37353 : if (y) q = gmul(y, q);
283 37353 : y = centermod_i(q, pa, pas2);
284 : }
285 :
286 : /* y is the candidate factor */
287 30472 : if (! (q = ZX_divides_i(lcpol,y,bound)) )
288 : {
289 3802 : if (DEBUGLEVEL>3) err_printf("*");
290 3802 : set_avma(av); goto NEXT;
291 : }
292 : /* found a factor */
293 26670 : list = cgetg(K+1, t_VEC);
294 26670 : gel(listmod,cnt) = list;
295 54075 : for (i=1; i<=K; i++) list[i] = famod[ind[i]];
296 :
297 26670 : y = Q_primpart(y);
298 26670 : gel(fa,cnt++) = y;
299 : /* fix up pol */
300 26670 : pol = q;
301 26670 : if (lc) pol = Q_div_to_int(pol, leading_coeff(y));
302 98912 : for (i=j=k=1; i <= lfamod; i++)
303 : { /* remove used factors */
304 72242 : if (j <= K && i == ind[j]) j++;
305 : else
306 : {
307 44837 : gel(famod,k) = gel(famod,i);
308 44837 : uel(trace1,k) = uel(trace1,i);
309 44837 : uel(trace2,k) = uel(trace2,i);
310 44837 : deg[k] = deg[i]; k++;
311 : }
312 : }
313 26670 : lfamod -= K;
314 26670 : *pmaxK = cmbf_maxK(lfamod);
315 26670 : if (lfamod < 2*K) goto END;
316 10684 : i = 1; curdeg = deg[ind[1]];
317 10684 : bound = factor_bound(pol);
318 10684 : if (lc) lc = absi_shallow(leading_coeff(pol));
319 10684 : lcpol = lc? ZX_Z_mul(pol, lc): pol;
320 10684 : if (DEBUGLEVEL>3)
321 0 : err_printf("\nfound factor %Ps\nremaining modular factor(s): %ld\n",
322 : y, lfamod);
323 10684 : continue;
324 : }
325 :
326 0 : NEXT:
327 525823 : for (i = K+1;;)
328 : {
329 656529 : if (--i == 0) { K++; goto nextK; }
330 615045 : if (++ind[i] <= lfamod - K + i)
331 : {
332 484339 : curdeg = degsofar[i-1] + deg[ind[i]];
333 484339 : if (curdeg <= klim) break;
334 : }
335 : }
336 : }
337 51730 : END:
338 51730 : *done = 1;
339 51730 : if (degpol(pol) > 0)
340 : { /* leftover factor */
341 51730 : if (signe(leading_coeff(pol)) < 0) pol = ZX_neg(pol);
342 51730 : if (lfamod >= 2*K) *done = 0;
343 :
344 51730 : setlg(famod, lfamod+1);
345 51730 : gel(listmod,cnt) = leafcopy(famod);
346 51730 : gel(fa,cnt++) = pol;
347 : }
348 51730 : if (DEBUGLEVEL>6) err_printf("\n");
349 51730 : setlg(listmod, cnt);
350 51730 : 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 186379 : shifteval(GEN Q, long n)
358 : {
359 186379 : pari_sp av = avma;
360 186379 : long i, l = lg(Q);
361 : GEN s;
362 :
363 186379 : if (!signe(Q)) return gen_0;
364 186379 : s = gel(Q,l-1);
365 985871 : for (i = l-2; i > 1; i--)
366 : {
367 799492 : s = addii(gel(Q,i), shifti(s, n));
368 799492 : if (gc_needed(av,1)) s = gc_INT(av, s);
369 : }
370 186379 : return s;
371 : }
372 :
373 : /* return integer y such that all |a| <= y if P(a) = 0 */
374 : static GEN
375 112919 : root_bound(GEN P0)
376 : {
377 112919 : GEN Q = leafcopy(P0), lP = absi_shallow(leading_coeff(Q)), x,y,z;
378 112919 : long k, d = degpol(Q);
379 :
380 : /* P0 = lP x^d + Q, deg Q < d */
381 112919 : Q = normalizepol_lg(Q, d+2);
382 705062 : for (k=lg(Q)-1; k>1; k--) gel(Q,k) = absi_shallow(gel(Q,k));
383 112919 : k = (long)(fujiwara_bound(P0));
384 187905 : for ( ; k >= 0; k--)
385 : {
386 186379 : pari_sp av = avma;
387 : /* y = 2^k; Q(y) >= lP y^d ? */
388 186379 : if (cmpii(shifteval(Q,k), shifti(lP, d*k)) >= 0) break;
389 74986 : set_avma(av);
390 : }
391 112919 : if (k < 0) k = 0;
392 112919 : y = int2n(k+1);
393 112919 : if (d > 2000) return y; /* likely to be expensive, don't bother */
394 112919 : x = int2n(k);
395 112919 : for(k=0; ; k++)
396 : {
397 604348 : z = shifti(addii(x,y), -1);
398 604348 : if (equalii(x,z) || k > 5) break;
399 491429 : if (cmpii(ZX_Z_eval(Q,z), mulii(lP, powiu(z, d))) < 0)
400 263443 : y = z;
401 : else
402 227986 : x = z;
403 : }
404 112919 : 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 1694 : 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 1694 : if (DEBUGLEVEL>2) timer_start(&T);
466 1694 : u = ZM_lll_norms(m, final? 0.999: 0.75, LLL_INPLACE | LLL_NOFLATTER, &norm);
467 1694 : if (DEBUGLEVEL>2) *ti_LLL += timer_delay(&T);
468 11571 : for (R=lg(m)-1; R > 0; R--)
469 11571 : if (cmprr(gel(norm,R), Bnorm) < 0) break;
470 17661 : for (i=1; i<=R; i++) setlg(u[i], n0+1);
471 1694 : if (R <= 1)
472 : {
473 126 : if (!R) pari_err_BUG("LLL_cmbf [no factor]");
474 126 : return NULL; /* irreducible */
475 : }
476 1568 : 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
495 147 : LLL_cmbf(GEN P, GEN famod, GEN p, GEN pa, GEN bound, long a, long rec)
496 : {
497 147 : const long N0 = 1; /* # of traces added at each step */
498 147 : double BitPerFactor = 0.4; /* nb bits in p^(a-b) / modular factor */
499 147 : long i,j,tmax,n0,C, dP = degpol(P);
500 147 : double logp = log((double)itos(p)), LOGp2 = M_LN2/logp;
501 147 : 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;
504 147 : long ti_LLL = 0, ti_CF = 0;
505 :
506 147 : lP = absi_shallow(leading_coeff(P));
507 147 : if (equali1(lP)) lP = NULL;
508 147 : Br = root_bound(P);
509 147 : if (lP) Br = mulii(lP, Br);
510 147 : logBr = dbllog2(Br) * LOGp2; /* log_p Br */
511 :
512 147 : n0 = lg(famod) - 1;
513 147 : C = (long)ceil( sqrt(N0 * n0 / 4.) ); /* > 1 */
514 147 : Bnorm = dbltor(n0 * (C*C + N0*n0/4.) * 1.00001);
515 147 : ZERO = zeromat(n0, N0);
516 :
517 147 : av = avma;
518 147 : TT = cgetg(n0+1, t_VEC);
519 147 : Tra = cgetg(n0+1, t_MAT);
520 2702 : for (i=1; i<=n0; i++)
521 : {
522 2555 : gel(TT,i) = NULL;
523 2555 : gel(Tra,i) = cgetg(N0+1, t_COL);
524 : }
525 147 : CM_L = scalarmat_s(C, n0);
526 : /* tmax = current number of traces used (and computed so far) */
527 147 : for (tmax = 0;; tmax += N0)
528 518 : {
529 665 : long b, bmin, delta, tnew = tmax + N0, r = lg(CM_L)-1;
530 : GEN M_L, CM_Lp, oldCM_L;
531 665 : int first = 1;
532 : pari_timer ti2, TI;
533 :
534 665 : bmin = (long)ceil(b0 + tnew*logBr);
535 665 : 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) */
540 665 : 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++) gel(TT,i) = NULL;
548 : }
549 12537 : for (i=1; i<=n0; i++)
550 : {
551 11872 : GEN p1 = gel(Tra,i);
552 11872 : GEN p2 = polsym_gen(gel(famod,i), gel(TT,i), tnew, NULL, pa);
553 11872 : gel(TT,i) = p2;
554 11872 : p2 += 1+tmax; /* ignore traces number 0...tmax */
555 23744 : for (j=1; j<=N0; j++) gel(p1,j) = gel(p2,j);
556 11872 : 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 */
568 665 : if (DEBUGLEVEL>2) { timer_start(&ti2); timer_start(&TI); }
569 665 : oldCM_L = CM_L;
570 665 : av2 = avma;
571 665 : delta = b = 0; /* -Wall */
572 1218 : AGAIN:
573 1218 : M_L = Q_div_to_int(CM_L, utoipos(C));
574 1218 : T2 = centermod( ZM_mul(Tra, M_L), pa );
575 1218 : if (first)
576 : { /* initialize lattice, using few p-adic digits for traces */
577 665 : double t = gexpo(T2) - maxdd(32.0, BitPerFactor*r);
578 665 : b = maxss(bmin, (long)(t * LOGp2));
579 665 : delta = a - b; first = 0;
580 : }
581 : else
582 : { /* add more p-adic digits and continue reduction */
583 553 : long b0 = (long)(gexpo(T2) * LOGp2);
584 553 : if (b0 < b) b = b0;
585 553 : b = maxss(bmin, b - delta);
586 553 : if (b - delta/2 < bmin) b = bmin; /* near goal. Go all the way */
587 : }
588 1218 : m = vconcat(CM_L, ZM_mul(gdivround(Tra, powiu(p, b)), M_L));
589 1218 : m = shallowconcat( m, vconcat(ZERO, scalarmat(powiu(p, a-b), N0)) );
590 : /* [ C M_L 0 ]
591 : * m = [ ] square matrix
592 : * [ T2' p^(a-b) I_N0 ] T2' = Tra * M_L truncated */
593 1218 : CM_L = LLL_check_progress(Bnorm, n0, m, b == bmin, /*dbg:*/ &ti_LLL);
594 1218 : if (DEBUGLEVEL>2)
595 0 : err_printf("LLL_cmbf: (a,b) =%4ld,%4ld; r =%3ld -->%3ld, time = %ld\n",
596 0 : a,b, lg(m)-1, CM_L? lg(CM_L)-1: 1, timer_delay(&TI));
597 1218 : if (!CM_L) { list = mkvec(P); break; }
598 1113 : if (b > bmin)
599 : {
600 553 : CM_L = gc_GEN(av2, CM_L);
601 553 : goto AGAIN;
602 : }
603 560 : if (DEBUGLEVEL>2) timer_printf(&ti2, "for this block of traces");
604 :
605 560 : i = lg(CM_L) - 1;
606 560 : if (i == r && ZM_equal(CM_L, oldCM_L))
607 : {
608 56 : CM_L = oldCM_L;
609 56 : set_avma(av2); continue;
610 : }
611 :
612 504 : CM_Lp = FpM_image(CM_L, utoipos(27449)); /* inexpensive test */
613 504 : if (lg(CM_Lp) != lg(CM_L))
614 : {
615 7 : if (DEBUGLEVEL>2) err_printf("LLL_cmbf: rank decrease\n");
616 7 : CM_L = ZM_hnf(CM_L);
617 : }
618 :
619 504 : if (i <= r && i*rec < n0)
620 : {
621 : pari_timer ti;
622 140 : if (DEBUGLEVEL>2) timer_start(&ti);
623 140 : list = chk_factors(P, Q_div_to_int(CM_L,utoipos(C)), bound, famod, pa);
624 140 : if (DEBUGLEVEL>2) ti_CF += timer_delay(&ti);
625 140 : if (list) break;
626 98 : if (DEBUGLEVEL>2) err_printf("LLL_cmbf: chk_factors failed");
627 : }
628 462 : CM_L = gc_GEN(av2, CM_L);
629 462 : if (gc_needed(av,1))
630 : {
631 0 : if(DEBUGMEM>1) pari_warn(warnmem,"LLL_cmbf");
632 0 : (void)gc_all(av, 5, &CM_L, &TT, &Tra, &famod, &pa);
633 : }
634 : }
635 147 : if (DEBUGLEVEL>2)
636 0 : err_printf("* Time LLL: %ld\n* Time Check Factor: %ld\n",ti_LLL,ti_CF);
637 147 : return list;
638 : }
639 :
640 : /* Find a,b minimal such that A < q^a, B < q^b, 1 << q^(a-b) < 2^31 */
641 : static int
642 51730 : cmbf_precs(GEN q, GEN A, GEN B, long *pta, long *ptb, GEN *qa, GEN *qb)
643 : {
644 51730 : long a, b, amin, d = (long)(31 / dbllog2(q) - 1e-5);
645 51730 : int fl = 0;
646 :
647 51730 : b = logintall(B, q, qb) + 1;
648 51730 : *qb = mulii(*qb, q);
649 51730 : amin = b + d;
650 51730 : if (gcmp(powiu(q, amin), A) <= 0)
651 : {
652 14287 : a = logintall(A, q, qa) + 1;
653 14287 : *qa = mulii(*qa, q);
654 14287 : b = a - d; *qb = powiu(q, b);
655 : }
656 : else
657 : { /* not enough room */
658 37443 : a = amin; *qa = powiu(q, a);
659 37443 : fl = 1;
660 : }
661 51730 : if (DEBUGLEVEL > 3) {
662 0 : err_printf("S_2 bound: %Ps^%ld\n", q,b);
663 0 : err_printf("coeff bound: %Ps^%ld\n", q,a);
664 : }
665 51730 : *pta = a;
666 51730 : *ptb = b; return fl;
667 : }
668 :
669 : /* use van Hoeij's knapsack algorithm */
670 : static GEN
671 51730 : combine_factors(GEN target, GEN famod, GEN p, long klim)
672 : {
673 : GEN la, B, A, res, L, pa, pb, listmod;
674 51730 : long a,b, l, maxK, n = degpol(target);
675 : int done;
676 : pari_timer T;
677 :
678 51730 : A = factor_bound(target);
679 :
680 51730 : la = absi_shallow(leading_coeff(target));
681 51730 : B = mului(n, sqri(mulii(la, root_bound(target)))); /* = bound for S_2 */
682 :
683 51730 : (void)cmbf_precs(p, A, B, &a, &b, &pa, &pb);
684 :
685 51730 : if (DEBUGLEVEL>2) timer_start(&T);
686 51730 : famod = ZpX_liftfact(target, famod, pa, p, a);
687 51730 : if (DEBUGLEVEL>2) timer_printf(&T, "Hensel lift (mod %Ps^%ld)", p,a);
688 51730 : L = cmbf(target, famod, A, p, a, b, klim, &maxK, &done);
689 51730 : if (DEBUGLEVEL>2) timer_printf(&T, "Naive recombination");
690 :
691 51730 : res = gel(L,1);
692 51730 : listmod = gel(L,2); l = lg(listmod)-1;
693 51730 : famod = gel(listmod,l);
694 51730 : if (maxK > 0 && lg(famod)-1 > 2*maxK)
695 : {
696 147 : if (l!=1) A = factor_bound(gel(res,l));
697 147 : if (DEBUGLEVEL > 4) err_printf("last factor still to be checked\n");
698 147 : L = LLL_cmbf(gel(res,l), famod, p, pa, A, a, maxK);
699 147 : if (DEBUGLEVEL>2) timer_printf(&T,"Knapsack");
700 : /* remove last elt, possibly unfactored. Add all new ones. */
701 147 : setlg(res, l); res = shallowconcat(res, L);
702 : }
703 51730 : return res;
704 : }
705 :
706 : /* Assume 'a' a squarefree ZX; return 0 if no root (fl=1) / irreducible (fl=0).
707 : * Otherwise return prime p such that a mod p has fewest roots / factors */
708 : static ulong
709 2088568 : pick_prime(GEN a, long fl, pari_timer *T)
710 : {
711 2088568 : pari_sp av = avma, av1;
712 2088568 : const long MAXNP = 7, da = degpol(a);
713 2088568 : long nmax = da+1, np;
714 2088568 : ulong chosenp = 0;
715 2088568 : GEN lead = gel(a,da+2);
716 : forprime_t S;
717 2088568 : if (equali1(lead)) lead = NULL;
718 2088569 : u_forprime_init(&S, 2, ULONG_MAX);
719 2088582 : av1 = avma;
720 8067061 : for (np = 0; np < MAXNP; set_avma(av1))
721 : {
722 7954282 : ulong p = u_forprime_next(&S);
723 : long nfacp;
724 : GEN z;
725 :
726 7954278 : if (!p) pari_err_OVERFLOW("DDF [out of small primes]");
727 7954278 : if (lead && !umodiu(lead,p)) continue;
728 7885054 : z = ZX_to_Flx(a, p);
729 7885038 : if (!Flx_is_squarefree(z, p)) continue;
730 :
731 4892081 : if (fl==1)
732 : {
733 4285873 : nfacp = Flx_nbroots(z, p);
734 4285871 : if (!nfacp) { chosenp = 0; break; } /* no root */
735 : }
736 606208 : else if(fl==0)
737 : {
738 605459 : nfacp = Flx_nbfact(z, p);
739 605453 : if (nfacp == 1) { chosenp = 0; break; } /* irreducible */
740 : } else
741 : {
742 749 : GEN f = gel(Flx_degfact(z, p),1);
743 749 : nfacp = lg(f)-1;
744 749 : if (f[1] > fl) { chosenp = 0; break; } /* no small factors */
745 : }
746 2916273 : if (DEBUGLEVEL>4)
747 0 : err_printf("...tried prime %3lu (%-3ld %s). Time = %ld\n",
748 : p, nfacp, fl==1? "roots": "factors", timer_delay(T));
749 2916283 : if (nfacp < nmax)
750 : {
751 1090234 : nmax = nfacp; chosenp = p;
752 1090234 : if (da > 100 && nmax < 5) break; /* large degree, few factors. Enough */
753 : }
754 2916276 : np++;
755 : }
756 2088580 : return gc_ulong(av, chosenp);
757 : }
758 :
759 : /* Assume A a squarefree ZX; return the vector of its rational roots */
760 : static GEN
761 1878324 : DDF_roots(GEN A)
762 : {
763 : GEN p, lc, lcpol, z, pe, pes2, bound;
764 : long i, m, e, lz;
765 : ulong pp;
766 : pari_sp av;
767 : pari_timer T;
768 :
769 1878324 : if (DEBUGLEVEL>2) timer_start(&T);
770 1878324 : pp = pick_prime(A, 1, &T);
771 1878325 : if (!pp) return cgetg(1,t_COL); /* no root */
772 61042 : p = utoipos(pp);
773 61042 : lc = leading_coeff(A);
774 61042 : if (is_pm1(lc))
775 52711 : { lc = NULL; lcpol = A; }
776 : else
777 8331 : { lc = absi_shallow(lc); lcpol = ZX_Z_mul(A, lc); }
778 61042 : bound = root_bound(A); if (lc) bound = mulii(lc, bound);
779 61042 : e = logintall(addiu(shifti(bound, 1), 1), p, &pe) + 1;
780 61042 : pe = mulii(pe, p);
781 61042 : pes2 = shifti(pe, -1);
782 61042 : if (DEBUGLEVEL>2) timer_printf(&T, "Root bound");
783 61042 : av = avma;
784 61042 : z = ZpX_roots(A, p, e); lz = lg(z);
785 61042 : z = deg1_from_roots(z, varn(A));
786 61042 : if (DEBUGLEVEL>2) timer_printf(&T, "Hensel lift (mod %lu^%ld)", pp,e);
787 133067 : for (m=1, i=1; i < lz; i++)
788 : {
789 72025 : GEN q, r, y = gel(z,i);
790 72025 : if (lc) y = ZX_Z_mul(y, lc);
791 72025 : y = centermod_i(y, pe, pes2);
792 72025 : if (! (q = ZX_divides(lcpol, y)) ) continue;
793 :
794 27535 : lcpol = q;
795 27535 : r = negi( constant_coeff(y) );
796 27535 : if (lc) {
797 9649 : r = gdiv(r,lc);
798 9649 : lcpol = Q_primpart(lcpol);
799 9649 : lc = absi_shallow( leading_coeff(lcpol) );
800 9649 : if (is_pm1(lc)) lc = NULL; else lcpol = ZX_Z_mul(lcpol, lc);
801 : }
802 27535 : gel(z,m++) = r;
803 27535 : if (gc_needed(av,2))
804 : {
805 0 : if (DEBUGMEM>1) pari_warn(warnmem,"DDF_roots, m = %ld", m);
806 0 : (void)gc_all(av, lc? 3:2, &z, &lcpol, &lc);
807 : }
808 : }
809 61042 : if (DEBUGLEVEL>2) timer_printf(&T, "Recombination");
810 61042 : setlg(z, m); return z;
811 : }
812 :
813 : /* Assume a squarefree ZX, deg(a) > 0, return rational factors.
814 : * In fact, a(0) != 0 but we don't use this
815 : * if dmax>0, Only look for factor of degree at most dmax */
816 : GEN
817 210246 : ZX_DDF_max(GEN a, long dmax)
818 : {
819 : GEN ap, prime, famod, z;
820 210246 : long ti = 0;
821 210246 : ulong p = 0;
822 210246 : pari_sp av = avma;
823 : pari_timer T, T2;
824 :
825 210246 : if (DEBUGLEVEL>2) { timer_start(&T); timer_start(&T2); }
826 210246 : p = pick_prime(a, dmax, &T2);
827 210247 : if (!p) return mkvec(a);
828 51730 : prime = utoipos(p);
829 51730 : ap = Flx_normalize(ZX_to_Flx(a, p), p);
830 51730 : famod = gel(Flx_factor(ap, p), 1);
831 51730 : if (DEBUGLEVEL>2)
832 : {
833 0 : if (DEBUGLEVEL>4) timer_printf(&T2, "splitting mod p = %lu", p);
834 0 : ti = timer_delay(&T);
835 0 : err_printf("Time setup: %ld\n", ti);
836 : }
837 51730 : z = combine_factors(a, FlxV_to_ZXV(famod), prime, degpol(a)-1);
838 51730 : if (DEBUGLEVEL>2)
839 0 : err_printf("Total Time: %ld\n===========\n", ti + timer_delay(&T));
840 51730 : return gc_GEN(av, z);
841 : }
842 :
843 : /* Distinct Degree Factorization (deflating first)
844 : * Assume x squarefree, degree(x) > 0, x(0) != 0 */
845 : GEN
846 155281 : ZX_DDF(GEN x)
847 : {
848 : GEN L;
849 : long m;
850 155281 : if (DEBUGLEVEL>2)
851 0 : err_printf("ZX_DDF: factoring pol of deg %ld, %ld bits\n",degpol(x),gexpo(x));
852 155281 : x = ZX_deflate_max(x, &m);
853 155277 : L = ZX_DDF_max(x,0);
854 155279 : if (m > 1)
855 : {
856 52432 : GEN e, v, fa = factoru(m);
857 : long i,j,k, l;
858 :
859 52432 : e = gel(fa,2); k = 0;
860 52432 : fa= gel(fa,1); l = lg(fa);
861 105186 : for (i=1; i<l; i++) k += e[i];
862 52432 : v = cgetg(k+1, t_VECSMALL); k = 1;
863 105186 : for (i=1; i<l; i++)
864 107094 : for (j=1; j<=e[i]; j++) v[k++] = fa[i];
865 106773 : for (k--; k; k--)
866 : {
867 54340 : GEN L2 = cgetg(1,t_VEC);
868 109136 : for (i=1; i < lg(L); i++)
869 54795 : L2 = shallowconcat(L2, ZX_DDF_max(RgX_inflate(gel(L,i), v[k]),0));
870 54341 : L = L2;
871 : }
872 : }
873 155280 : return L;
874 : }
875 :
876 : /* SquareFree Factorization in Z[X] (char 0 is enough, if ZX_gcd -> RgX_gcd)
877 : * f = prod Q[i]^E[i], E[1] < E[2] < ..., and Q[i] squarefree and coprime.
878 : * Return Q, set *pE = E. For efficiency, caller should have used ZX_valrem
879 : * so that f(0) != 0 */
880 : GEN
881 322307 : ZX_squff(GEN f, GEN *pE)
882 : {
883 : GEN T, V, P, E;
884 322307 : long i, k, n = 1 + degpol(f);
885 :
886 322307 : if (signe(leading_coeff(f)) < 0) f = ZX_neg(f);
887 322307 : E = cgetg(n, t_VECSMALL);
888 322307 : P = cgetg(n, t_COL);
889 322307 : f = Q_primpart(f); /* FIXME: caller could ensure this */
890 322307 : T = ZX_gcd_all(f, ZX_deriv(f), &V);
891 322307 : for (k = i = 1;; k++)
892 3501 : { /* T, V are primitive */
893 325808 : GEN W = ZX_gcd_all(T,V, &T); /* V and W are squarefree */
894 325808 : long dW = degpol(W), dV = degpol(V);
895 : /* T, W are primitive */
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 325808 : if (!dW)
901 : {
902 322307 : if (dV) { gel(P,i) = V; E[i] = k; i++; }
903 322307 : break;
904 : }
905 3501 : if (dW == dV)
906 : {
907 : GEN U;
908 1659 : while ( (U = ZX_divides(T, V)) ) { k++; T = U; }
909 : }
910 : else
911 : {
912 2388 : gel(P,i) = RgX_div(V,W);
913 2388 : E[i] = k; i++; V = W;
914 : }
915 : }
916 322307 : setlg(P,i);
917 322307 : setlg(E,i); *pE = E; return P;
918 : }
919 :
920 : static GEN
921 39555 : fact_from_DDF(GEN Q, GEN E, long n)
922 : {
923 39555 : GEN v,w, y = cgetg(3, t_MAT);
924 39555 : long i,j,k, l = lg(Q);
925 :
926 39555 : v = cgetg(n+1, t_COL); gel(y,1) = v;
927 39555 : w = cgetg(n+1, t_COL); gel(y,2) = w;
928 80468 : for (k = i = 1; i < l; i++)
929 : {
930 40913 : GEN L = gel(Q,i), e = utoipos(E[i]);
931 40913 : long J = lg(L);
932 95007 : for (j = 1; j < J; j++,k++)
933 : {
934 54094 : gel(v,k) = ZX_copy(gel(L,j));
935 54094 : gel(w,k) = e;
936 : }
937 : }
938 39555 : return y;
939 : }
940 :
941 : /* Factor T in Z[x] */
942 : static GEN
943 39562 : ZX_factor_i(GEN T)
944 : {
945 : GEN Q, E, y;
946 : long n, i, l, v;
947 :
948 39562 : if (!signe(T)) return prime_fact(T);
949 39555 : v = ZX_valrem(T, &T);
950 39555 : Q = ZX_squff(T, &E); l = lg(Q);
951 79355 : for (i = 1, n = 0; i < l; i++)
952 : {
953 39800 : gel(Q,i) = ZX_DDF(gel(Q,i));
954 39800 : n += lg(gel(Q,i)) - 1;
955 : }
956 39555 : if (v)
957 : {
958 1113 : Q = vec_append(Q, mkvec(pol_x(varn(T))));
959 1113 : E = vecsmall_append(E, v); n++;
960 : }
961 39555 : y = fact_from_DDF(Q, E, n);
962 39555 : return sort_factor_pol(y, cmpii);
963 : }
964 : GEN
965 38757 : ZX_factor(GEN x)
966 : {
967 38757 : pari_sp av = avma;
968 38757 : return gc_upto(av, ZX_factor_i(x));
969 : }
970 : GEN
971 805 : QX_factor(GEN x)
972 : {
973 805 : pari_sp av = avma;
974 805 : return gc_upto(av, ZX_factor_i(Q_primpart(x)));
975 : }
976 :
977 : long
978 103069 : ZX_is_irred(GEN x)
979 : {
980 103069 : pari_sp av = avma;
981 103069 : long l = lg(x);
982 : GEN y;
983 103069 : if (l <= 3) return 0; /* degree < 1 */
984 103069 : if (l == 4) return 1; /* degree 1 */
985 99482 : if (ZX_val(x)) return 0;
986 99258 : if (!ZX_is_squarefree(x)) return 0;
987 99108 : y = ZX_DDF(x); set_avma(av);
988 99107 : return (lg(y) == 2);
989 : }
990 :
991 : GEN
992 1878328 : nfrootsQ(GEN x)
993 : {
994 1878328 : pari_sp av = avma;
995 : GEN z;
996 : long val;
997 :
998 1878328 : if (typ(x)!=t_POL) pari_err_TYPE("nfrootsQ",x);
999 1878328 : if (!signe(x)) pari_err_ROOTS0("nfrootsQ");
1000 1878328 : x = Q_primpart(x);
1001 1878321 : RgX_check_ZX(x,"nfrootsQ");
1002 1878320 : val = ZX_valrem(x, &x);
1003 1878319 : z = DDF_roots( ZX_radical(x) );
1004 1878324 : if (val) z = vec_append(z, gen_0);
1005 1878324 : return gc_upto(av, sort(z));
1006 : }
1007 :
1008 : /************************************************************************
1009 : * GCD OVER Z[X] / Q[X] *
1010 : ************************************************************************/
1011 : int
1012 199443 : ZX_is_squarefree(GEN x)
1013 : {
1014 199443 : pari_sp av = avma;
1015 : GEN d;
1016 : long m;
1017 199443 : if (lg(x) == 2) return 0;
1018 199443 : m = ZX_deflate_order(x);
1019 199441 : if (m > 1)
1020 : {
1021 86804 : if (!signe(gel(x,2))) return 0;
1022 86566 : x = RgX_deflate(x, m);
1023 : }
1024 199204 : d = ZX_gcd(x,ZX_deriv(x));
1025 199225 : return gc_bool(av, lg(d) == 3);
1026 : }
1027 :
1028 : static int
1029 122808 : ZX_gcd_filter(GEN *pt_A, GEN *pt_P)
1030 : {
1031 122808 : GEN A = *pt_A, P = *pt_P;
1032 122808 : long i, j, l = lg(A), n = 1, d = degpol(gel(A,1));
1033 : GEN B, Q;
1034 251554 : for (i=2; i<l; i++)
1035 : {
1036 128746 : long di = degpol(gel(A,i));
1037 128746 : if (di==d) n++;
1038 36 : else if (d > di)
1039 36 : { n=1; d = di; }
1040 : }
1041 122808 : if (n == l-1)
1042 122772 : 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 3368485 : ZX_gcd_Flx(GEN a, GEN b, ulong g, ulong p)
1059 : {
1060 3368485 : GEN H = Flx_gcd(a, b, p);
1061 3368484 : if (!g)
1062 3335554 : return Flx_normalize(H, p);
1063 : else
1064 : {
1065 32930 : ulong t = Fl_mul(g, Fl_inv(Flx_lead(H), p), p);
1066 32930 : return Flx_Fl_mul(H, t, p);
1067 : }
1068 : }
1069 :
1070 : static GEN
1071 3356783 : ZX_gcd_slice(GEN A, GEN B, GEN g, GEN P, GEN *mod)
1072 : {
1073 3356783 : pari_sp av = avma;
1074 3356783 : long i, n = lg(P)-1;
1075 : GEN H, T;
1076 3356783 : if (n == 1)
1077 : {
1078 3350578 : ulong p = uel(P,1), gp = g ? umodiu(g, p): 0;
1079 3350578 : GEN a = ZX_to_Flx(A, p), b = ZX_to_Flx(B, p);
1080 3350579 : GEN Hp = ZX_gcd_Flx(a, b, gp, p);
1081 3350579 : H = gc_upto(av, Flx_to_ZX(Hp));
1082 3350579 : *mod = utoi(p);
1083 3350579 : return H;
1084 : }
1085 6205 : T = ZV_producttree(P);
1086 6205 : A = ZX_nv_mod_tree(A, P, T);
1087 6205 : B = ZX_nv_mod_tree(B, P, T);
1088 6205 : g = g ? Z_ZV_mod_tree(g, P, T): NULL;
1089 6205 : H = cgetg(n+1, t_VEC);
1090 24111 : for(i=1; i <= n; i++)
1091 : {
1092 17906 : ulong p = P[i];
1093 17906 : GEN a = gel(A,i), b = gel(B,i);
1094 17906 : gel(H,i) = ZX_gcd_Flx(a, b, g? g[i]: 0, p);
1095 : }
1096 6205 : if (ZX_gcd_filter(&H, &P))
1097 12 : T = ZV_producttree(P);
1098 6205 : H = nxV_chinese_center_tree(H, P, T, ZV_chinesetree(P, T));
1099 6205 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
1100 : }
1101 :
1102 : GEN
1103 3356783 : ZX_gcd_worker(GEN P, GEN A, GEN B, GEN g)
1104 : {
1105 3356783 : GEN V = cgetg(3, t_VEC);
1106 3356783 : gel(V,1) = ZX_gcd_slice(A, B, equali1(g)? NULL: g, P, &gel(V,2));
1107 3356784 : return V;
1108 : }
1109 :
1110 : static GEN
1111 116603 : ZX_gcd_chinese(GEN A, GEN P, GEN *mod)
1112 : {
1113 116603 : ZX_gcd_filter(&A, &P);
1114 116603 : return nxV_chinese_center(A, P, mod);
1115 : }
1116 :
1117 : GEN
1118 14446251 : ZX_gcd_all(GEN A, GEN B, GEN *Anew)
1119 : {
1120 14446251 : pari_sp av = avma;
1121 14446251 : long k, valH, valA, valB, vA = varn(A), dA = degpol(A), dB = degpol(B);
1122 14446219 : 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 14446219 : if (dA < 0) { if (Anew) *Anew = pol_0(vA); return ZX_copy(B); }
1127 14445897 : if (dB < 0) { if (Anew) *Anew = pol_1(vA); return ZX_copy(A); }
1128 14444672 : A = Q_primitive_part(A, &cA);
1129 14444791 : B = Q_primitive_part(B, &cB);
1130 14444744 : valA = ZX_valrem(A, &A); dA -= valA;
1131 14444739 : valB = ZX_valrem(B, &B); dB -= valB;
1132 14444783 : valH = minss(valA, valB);
1133 14444799 : valA -= valH; /* valuation(Anew) */
1134 14444799 : c = (cA && cB)? gcdii(cA, cB): NULL; /* content(gcd) */
1135 14444810 : if (!dA || !dB)
1136 : {
1137 7327735 : if (Anew) *Anew = RgX_shift_shallow(A, valA);
1138 7327735 : return monomial(c? c: gen_1, valH, vA);
1139 : }
1140 7117075 : g = gcdii(leading_coeff(A), leading_coeff(B)); /* multiple of lead(gcd) */
1141 7117015 : if (is_pm1(g)) {
1142 6906798 : g = NULL;
1143 6906798 : Ag = A;
1144 6906798 : Bg = B;
1145 : } else {
1146 210213 : Ag = ZX_Z_mul(A,g);
1147 210213 : Bg = ZX_Z_mul(B,g);
1148 : }
1149 7117011 : init_modular_big(&S);
1150 : do {
1151 7117091 : pp = u_forprime_next(&S);
1152 7117089 : Ap = ZX_to_Flx(Ag, pp);
1153 7117109 : Bp = ZX_to_Flx(Bg, pp);
1154 7117115 : } while (degpol(Ap) != dA || degpol(Bp) != dB);
1155 7117095 : if (degpol(Flx_gcd(Ap, Bp, pp)) == 0)
1156 : {
1157 3877320 : if (Anew) *Anew = RgX_shift_shallow(A, valA);
1158 3877322 : return monomial(c? c: gen_1, valH, vA);
1159 : }
1160 3239739 : worker = snm_closure(is_entry("_ZX_gcd_worker"), mkvec3(A, B, g? g: gen_1));
1161 3239739 : av = avma;
1162 3354373 : for (k = 1; ;k *= 2)
1163 : {
1164 3354373 : gen_inccrt_i("ZX_gcd", worker, g, (k+1)>>1, 0, &S, &H, &mod, ZX_gcd_chinese, NULL);
1165 3354371 : (void)gc_all(av, 2, &H, &mod);
1166 3354373 : Hp = ZX_to_Flx(H, pp);
1167 3354372 : if (lgpol(Flx_rem(Ap, Hp, pp)) || lgpol(Flx_rem(Bp, Hp, pp))) continue;
1168 3243805 : if (!ZX_divides(Bg, H)) continue;
1169 3239769 : R = ZX_divides(Ag, H);
1170 3239766 : if (R) break;
1171 : }
1172 : /* lead(H) = g */
1173 3239736 : if (g) H = Q_primpart(H);
1174 3239736 : if (c) H = ZX_Z_mul(H,c);
1175 3239736 : if (DEBUGLEVEL>5) err_printf("done\n");
1176 3239736 : if (Anew)
1177 : {
1178 89934 : if (g) R = Q_primpart(R);
1179 89934 : *Anew = RgX_shift_shallow(R, valA);
1180 : }
1181 3239736 : return valH? RgX_shift_shallow(H, valH): H;
1182 : }
1183 :
1184 : #if 0
1185 : /* ceil( || p ||_oo / lc(p) ) */
1186 : static GEN
1187 : maxnorm(GEN p)
1188 : {
1189 : long i, n = degpol(p), av = avma;
1190 : GEN x, m = gen_0;
1191 :
1192 : p += 2;
1193 : for (i=0; i<n; i++)
1194 : {
1195 : x = gel(p,i);
1196 : if (abscmpii(x,m) > 0) m = x;
1197 : }
1198 : m = divii(m, gel(p,n));
1199 : return gc_INT(av, addiu(absi_shallow(m),1));
1200 : }
1201 : #endif
1202 :
1203 : GEN
1204 10623691 : ZX_gcd(GEN A, GEN B)
1205 : {
1206 10623691 : pari_sp av = avma;
1207 10623691 : return gc_GEN(av, ZX_gcd_all(A,B,NULL));
1208 : }
1209 :
1210 : GEN
1211 3170174 : ZX_radical(GEN A) { GEN B; (void)ZX_gcd_all(A,ZX_deriv(A),&B); return B; }
1212 :
1213 : static GEN
1214 19558 : _gcd(GEN a, GEN b)
1215 : {
1216 19558 : if (!a) a = gen_1;
1217 19558 : if (!b) b = gen_1;
1218 19558 : return Q_gcd(a,b);
1219 : }
1220 : /* A0 and B0 in Q[X] */
1221 : GEN
1222 19369 : QX_gcd(GEN A0, GEN B0)
1223 : {
1224 : GEN a, b, D;
1225 19369 : pari_sp av = avma, av2;
1226 :
1227 19369 : D = ZX_gcd(Q_primitive_part(A0, &a), Q_primitive_part(B0, &b));
1228 19369 : av2 = avma; a = _gcd(a,b);
1229 19369 : if (isint1(a)) set_avma(av2); else D = ZX_Q_mul(D, a);
1230 19369 : return gc_upto(av, D);
1231 : }
1232 :
1233 : /***************************************************************************
1234 : *** ***
1235 : *** ZXk/QXk ***
1236 : *** ***
1237 : ***************************************************************************/
1238 :
1239 : /* ZXk/QXk: multivariate polynomials in Z[X_1,...,X_k] and Q[X_1,...,X_k] */
1240 :
1241 : INLINE GEN
1242 4958257 : ZXk_renormalize(GEN x, long lx) { return ZXX_renormalize(x,lx); }
1243 :
1244 : int
1245 0 : Rg_is_QXk(GEN z)
1246 : {
1247 0 : long i, t = typ(z), l = lg(z);
1248 0 : if (t==t_INT || t==t_FRAC) return 1;
1249 0 : if (t!=t_POL) return 0;
1250 0 : for (i = 2; i < l; i++)
1251 0 : if (!Rg_is_QXk(gel(z,i))) return 0;
1252 0 : return 1;
1253 : }
1254 :
1255 : static GEN
1256 3214092 : centeri2n(GEN z, long n, GEN N)
1257 : {
1258 3214092 : pari_sp av = avma;
1259 3214092 : z = remi2n(z, n);
1260 3214092 : if (expi(z)<n-1) return z;
1261 1253 : if (signe(z)<0) z = addii(z, N);
1262 763 : else z = subii(z, N);
1263 1253 : return gc_INT(av, z);
1264 : }
1265 :
1266 : static GEN
1267 4909872 : ZXk_center2n(GEN z, long n, GEN N)
1268 : {
1269 4909872 : if (typ(z) == t_INT)
1270 3214092 : return centeri2n(z, n, N);
1271 : else
1272 : {
1273 : long i,l;
1274 1695780 : GEN x = cgetg_copy(z, &l);
1275 1695780 : x[1] = z[1];
1276 3403852 : for (i = 2; i < l; i++)
1277 1708072 : gel(x,i) = ZXk_center2n(gel(z,i), n, N);
1278 1695780 : return ZXk_renormalize(x, l);
1279 : }
1280 : }
1281 :
1282 : static GEN ZXk_gcd_i(GEN A, GEN B);
1283 : static GEN
1284 3444534 : ZXk_content_shallow(GEN x)
1285 : {
1286 3444534 : long i, l = lg(x);
1287 : GEN c;
1288 3444534 : if (typ(x)==t_INT) return x;
1289 3444534 : if (!signe(x)) return gen_0;
1290 3444534 : c = gel(x, 2);
1291 3444534 : if (gequal1(c)) return gen_1;
1292 6098025 : for (i = 3; i < l; i++)
1293 : {
1294 3557357 : c = simplify_shallow(ZXk_gcd_i(c, gel(x,i)));
1295 3557357 : if (gequal1(c)) return gen_1;
1296 : }
1297 2540668 : return c;
1298 : }
1299 :
1300 : static GEN ZXk_divexact_s(GEN A, GEN B);
1301 :
1302 : static GEN
1303 2570537 : ZXkX_ZXk_divexact_s(GEN x, GEN B)
1304 7608844 : { pari_APPLY_ZX(ZXk_divexact_s(gel(x,i), B)); }
1305 :
1306 : static GEN
1307 2527823 : ZXkX_ZXk_divexact(GEN A, GEN B)
1308 : {
1309 2527823 : pari_sp av = avma;
1310 2527823 : return gc_upto(av, ZXkX_ZXk_divexact_s(A, simplify_shallow(B)));
1311 : }
1312 :
1313 : static GEN
1314 2529108 : ZXk_divexact_i(GEN x, GEN y)
1315 : {
1316 2529108 : long dx = degpol(x), dy = degpol(y), dz, i, j;
1317 2529108 : GEN z, y_lead = gel(y,dy+2);
1318 2529108 : if (dx < dy)
1319 0 : return gen_0;
1320 2529108 : dz = dx-dy;
1321 2529108 : z = cgetg(dz+3,t_POL); z[1] = x[1];
1322 2529108 : gel(z,dz+2) = ZXk_divexact_s(gel(x,dx+2), y_lead);
1323 2544641 : for (i=dx-1; i>=dy; i--)
1324 : {
1325 15533 : pari_sp btop = avma;
1326 15533 : GEN p1=gel(x,2+i);
1327 33978 : for (j=i-dy+1; j<=i && j<=dz; j++)
1328 18445 : p1 = gsub(p1, gmul(gel(z,2+j), gel(y,2+i-j)));
1329 15533 : gel(z,2+i-dy) = gc_upto(btop, ZXk_divexact_s(p1, y_lead));
1330 : }
1331 2529108 : return z;
1332 : }
1333 :
1334 : static GEN
1335 7582948 : ZXk_divexact_s(GEN A, GEN B)
1336 : {
1337 7582948 : if (!signe(A)) return gen_0;
1338 5135859 : if (typ(A)==t_INT && typ(B)==t_INT)
1339 2564037 : return diviiexact(A, B);
1340 2571822 : else if (typ(B)==t_INT || varn(A)!=varn(B))
1341 42714 : return ZXkX_ZXk_divexact_s(A, B);
1342 : else
1343 2529108 : return ZXk_divexact_i(A, B);
1344 : }
1345 :
1346 : GEN
1347 0 : ZXk_divexact(GEN A, GEN B)
1348 : {
1349 0 : pari_sp av = avma;
1350 0 : return gc_upto(av, ZXk_divexact_s(A, simplify_shallow(B)));
1351 : }
1352 :
1353 : static GEN ZXk_divides_s(GEN A, GEN B);
1354 :
1355 : static GEN
1356 3262477 : ZXkX_ZXk_divides_s(GEN x, GEN B)
1357 : {
1358 3262477 : pari_sp av = avma;
1359 : long i, l;
1360 3262477 : GEN y = cgetg_copy(x, &l); y[1] = x[1];
1361 3262477 : if (l == 2) return y;
1362 6910239 : for (i=2; i<l; i++)
1363 : {
1364 3647762 : GEN c = ZXk_divides_s(gel(x,i), B);
1365 3647762 : if (!c) return gc_NULL(av);
1366 3647762 : gel(y, i) = c;
1367 : }
1368 3262477 : return ZXk_renormalize(y, l);
1369 : }
1370 :
1371 : static GEN
1372 2789701 : ZXk_divides_i(GEN x, GEN y)
1373 : {
1374 2789701 : pari_sp av = avma, av2;
1375 2789701 : long dx = degpol(x), dy = degpol(y), dz, i, j, c;
1376 2789701 : GEN z, y_lead = gel(y,dy+2);
1377 2789701 : if (dx < dy)
1378 0 : return gen_0;
1379 2789701 : dz = dx-dy;
1380 2789701 : z = cgetg(dz+3,t_POL); z[1] = x[1];
1381 2789701 : gel(z,dz+2) = ZXk_divides(gel(x,dx+2), y_lead);
1382 2789701 : if (!gel(z,dz+2)) return gc_NULL(av);
1383 3042765 : for (i=dx-1; i>=dy; i--)
1384 : {
1385 253078 : pari_sp btop = avma;
1386 253078 : GEN p1 = gel(x,2+i), c;
1387 514164 : for (j=i-dy+1; j<=i && j<=dz; j++)
1388 261086 : p1 = gsub(p1, gmul(gel(z,2+j), gel(y,2+i-j)));
1389 253078 : c = ZXk_divides_s(p1, y_lead);
1390 253078 : if (!c) return gc_NULL(av);
1391 253078 : gel(z,2+i-dy) = gc_upto(btop, c);
1392 : }
1393 2789687 : av2 = avma;
1394 2789687 : c = gc_bool(av2, gequal(gmul(z,y),x));
1395 2789687 : return c ? z: gc_NULL(av);
1396 : }
1397 :
1398 : static GEN
1399 3479688 : dividesii(GEN A, GEN B)
1400 : {
1401 3479688 : GEN r, q = dvmdii(A, B, &r);
1402 3479688 : return signe(r) ? NULL: q;
1403 : }
1404 :
1405 : static GEN
1406 10118541 : ZXk_divides_s(GEN A, GEN B)
1407 : {
1408 10118541 : if (!signe(A)) return gen_0;
1409 9531873 : if (typ(B)==t_INT)
1410 3479688 : return typ(A)==t_INT ? dividesii(A, B)
1411 10221195 : : ZXkX_ZXk_divides_s(A, B);
1412 2790366 : else if (typ(A)==t_INT) return NULL;
1413 : else
1414 : {
1415 2790359 : long c = varncmp(varn(A),varn(B));
1416 2790359 : if (c < 0)
1417 658 : return ZXkX_ZXk_divides_s(A, B);
1418 2789701 : else if (c>0)
1419 0 : return NULL;
1420 : else
1421 2789701 : return ZXk_divides_i(A, B);
1422 : }
1423 : }
1424 :
1425 : GEN
1426 6217701 : ZXk_divides(GEN A, GEN B)
1427 : {
1428 6217701 : pari_sp av = avma;
1429 6217701 : GEN z = ZXk_divides_s(A, simplify_shallow(B));
1430 6217701 : return z ? gc_upto(av, z): z;
1431 : }
1432 :
1433 : static GEN
1434 1714007 : rec(GEN g, long e, GEN N, long v)
1435 : {
1436 1714007 : pari_sp av = avma;
1437 1714007 : long i, d = (gexpo(g)+2*e-1)/e;
1438 1714007 : GEN s = cgetg(d+3,t_POL);
1439 1714007 : s[1] = evalvarn(v);
1440 3201800 : for (i = 0; i <= d; i++)
1441 : {
1442 3201800 : GEN c = ZXk_center2n(g, e, N);
1443 3201800 : gel(s,i+2) = c;
1444 3201800 : g = gmul2n(gsub(g,c),-e);
1445 3201800 : if (!signe(g)) break;
1446 : }
1447 1714007 : s = RgX_renormalize_lg(s,i+3);
1448 1714007 : return gc_GEN(av, s);
1449 : }
1450 :
1451 : static GEN
1452 8688143 : ZXk_gcd_i(GEN A, GEN B)
1453 : {
1454 : pari_sp av;
1455 : long e, v, vc;
1456 : GEN c, cA, cB;
1457 8688143 : if (signe(A)==0) return gcopy(B);
1458 5161421 : if (signe(B)==0) return gcopy(A);
1459 5159818 : if (typ(A) == t_INT) return gcdii(A, typ(B)==t_INT ? B: Q_content(B));
1460 4291561 : if (typ(B) == t_INT) return gcdii(Q_content(A), B);
1461 3439067 : v = varn(A); vc = varncmp(v, varn(B));
1462 3439067 : if (vc < 0) return ZXk_gcd_i(ZXk_content_shallow(A), B);
1463 3427531 : if (vc > 0) return ZXk_gcd_i(A, ZXk_content_shallow(B));
1464 3422519 : if (RgX_is_ZX(A) && RgX_is_ZX(B)) return ZX_gcd(A,B);
1465 1713993 : cA = ZXk_content_shallow(A); if (!gequal1(cA)) A = ZXkX_ZXk_divexact(A, cA);
1466 1713993 : cB = ZXk_content_shallow(B); if (!gequal1(cB)) B = ZXkX_ZXk_divexact(B, cB);
1467 1713993 : c = ZXk_gcd_i(cA, cB); av = avma;
1468 1713993 : e = maxss(3, minss(gexpo(A), gexpo(B)) + 2);
1469 14 : for ( ; ; e++, set_avma(av))
1470 14 : {
1471 1714007 : GEN N = int2n(e), G = ZXk_gcd_i(poleval(A,N), poleval(B,N));
1472 1714007 : GEN g = Q_primpart(rec(G, e, N, v));
1473 1714007 : if (ZXk_divides(A,g) && ZXk_divides(B,g))
1474 1713993 : return gmul(c,g);
1475 : }
1476 : }
1477 : GEN
1478 1686049 : ZXk_gcd(GEN A, GEN B)
1479 1686049 : { pari_sp av = avma; return gc_upto(av, ZXk_gcd_i(A, B)); }
1480 :
1481 : GEN
1482 189 : QXk_gcd(GEN A, GEN B)
1483 : {
1484 : GEN a, b, D;
1485 189 : pari_sp av = avma, av2;
1486 189 : D = ZXk_gcd_i(Q_primitive_part(A, &a), Q_primitive_part(B, &b));
1487 189 : av2 = avma; a = _gcd(a,b);
1488 189 : if (isint1(a)) set_avma(av2); else D = gmul(D, a);
1489 189 : return gc_upto(av, D);
1490 : }
1491 :
1492 : /*****************************************************************************
1493 : * Variants of the Bradford-Davenport algorithm: look for cyclotomic *
1494 : * factors, and decide whether a ZX is cyclotomic or a product of cyclotomic *
1495 : *****************************************************************************/
1496 : /* f of degree 1, return a cyclotomic factor (Phi_1 or Phi_2) or NULL */
1497 : static GEN
1498 0 : BD_deg1(GEN f)
1499 : {
1500 0 : GEN a = gel(f,3), b = gel(f,2); /* f = ax + b */
1501 0 : if (!absequalii(a,b)) return NULL;
1502 0 : return polcyclo((signe(a) == signe(b))? 2: 1, varn(f));
1503 : }
1504 :
1505 : /* f a squarefree ZX; not divisible by any Phi_n, n even */
1506 : static GEN
1507 420 : BD_odd(GEN f)
1508 : {
1509 427 : while(degpol(f) > 1)
1510 : {
1511 420 : GEN f1 = ZX_graeffe(f); /* contain all cyclotomic divisors of f */
1512 420 : if (ZX_equal(f1, f)) return f; /* product of cyclotomics */
1513 7 : f = ZX_gcd(f, f1);
1514 : }
1515 7 : if (degpol(f) == 1) return BD_deg1(f);
1516 7 : return NULL; /* no cyclotomic divisor */
1517 : }
1518 :
1519 : static GEN
1520 2317 : myconcat(GEN v, GEN x)
1521 : {
1522 2317 : if (typ(x) != t_VEC) x = mkvec(x);
1523 2317 : if (!v) return x;
1524 1470 : return shallowconcat(v, x);
1525 : }
1526 :
1527 : /* Bradford-Davenport algorithm.
1528 : * f a primitive squarefree ZX of degree > 0, return NULL or a vector of
1529 : * coprime cyclotomic factors of f [ possibly reducible ] */
1530 : static GEN
1531 2366 : BD(GEN f)
1532 : {
1533 2366 : GEN G = NULL, Gs = NULL, Gp = NULL, Gi = NULL;
1534 : GEN fs2, fp, f2, f1, fe, fo, fe1, fo1;
1535 2366 : RgX_even_odd(f, &fe, &fo);
1536 2366 : fe1 = ZX_eval1(fe);
1537 2366 : fo1 = ZX_eval1(fo);
1538 2366 : if (absequalii(fe1, fo1)) /* f(1) = 0 or f(-1) = 0 */
1539 : {
1540 1519 : long i, v = varn(f);
1541 1519 : if (!signe(fe1))
1542 371 : G = mkvec2(polcyclo(1, v), polcyclo(2, v)); /* both 0 */
1543 1148 : else if (signe(fe1) == signe(fo1))
1544 693 : G = mkvec(polcyclo(2, v)); /*f(-1) = 0*/
1545 : else
1546 455 : G = mkvec(polcyclo(1, v)); /*f(1) = 0*/
1547 3409 : for (i = lg(G)-1; i; i--) f = RgX_div(f, gel(G,i));
1548 : }
1549 : /* f no longer divisible by Phi_1 or Phi_2 */
1550 2366 : if (degpol(f) <= 1) return G;
1551 2065 : f1 = ZX_graeffe(f); /* primitive, has at most square factors */
1552 2065 : if (ZX_equal(f1, f)) return myconcat(G,f); /* f = product of Phi_n, n odd */
1553 :
1554 1190 : fs2 = ZX_gcd_all(f1, ZX_deriv(f1), &f2); /* fs2 squarefree primitive */
1555 1190 : if (degpol(fs2))
1556 : { /* fs contains all Phi_n | f, 4 | n; and only those */
1557 : /* In that case, Graeffe(Phi_n) = Phi_{n/2}^2, and Phi_n = Phi_{n/2}(x^2) */
1558 1029 : GEN fs = RgX_inflate(fs2, 2);
1559 1029 : (void)ZX_gcd_all(f, fs, &f); /* remove those Phi_n | f, 4 | n */
1560 1029 : Gs = BD(fs2);
1561 1029 : if (Gs)
1562 : {
1563 : long i;
1564 2555 : for (i = lg(Gs)-1; i; i--) gel(Gs,i) = RgX_inflate(gel(Gs,i), 2);
1565 : /* prod Gs[i] is the product of all Phi_n | f, 4 | n */
1566 1029 : G = myconcat(G, Gs);
1567 : }
1568 : /* f2 = f1 / fs2 */
1569 1029 : f1 = RgX_div(f2, fs2); /* f1 / fs2^2 */
1570 : }
1571 1190 : fp = ZX_gcd(f, f1); /* contains all Phi_n | f, n > 1 odd; and only those */
1572 1190 : if (degpol(fp))
1573 : {
1574 203 : Gp = BD_odd(fp);
1575 : /* Gp is the product of all Phi_n | f, n odd */
1576 203 : if (Gp) G = myconcat(G, Gp);
1577 203 : f = RgX_div(f, fp);
1578 : }
1579 1190 : if (degpol(f))
1580 : { /* contains all Phi_n originally dividing f, n = 2 mod 4, n > 2;
1581 : * and only those
1582 : * In that case, Graeffe(Phi_n) = Phi_{n/2}, and Phi_n = Phi_{n/2}(-x) */
1583 217 : Gi = BD_odd(ZX_z_unscale(f, -1));
1584 217 : if (Gi)
1585 : { /* N.B. Phi_2 does not divide f */
1586 210 : Gi = ZX_z_unscale(Gi, -1);
1587 : /* Gi is the product of all Phi_n | f, n = 2 mod 4 */
1588 210 : G = myconcat(G, Gi);
1589 : }
1590 : }
1591 1190 : return G;
1592 : }
1593 :
1594 : /* Let f be a nonzero QX, return the (squarefree) product of cyclotomic
1595 : * divisors of f */
1596 : GEN
1597 322 : polcyclofactors(GEN f)
1598 : {
1599 322 : pari_sp av = avma;
1600 322 : if (typ(f) != t_POL || !signe(f)) pari_err_TYPE("polcyclofactors",f);
1601 322 : (void)RgX_valrem(f, &f);
1602 322 : f = Q_primpart(f);
1603 322 : RgX_check_ZX(f,"polcyclofactors");
1604 322 : if (degpol(f))
1605 : {
1606 322 : f = BD(ZX_radical(f));
1607 322 : if (f) return gc_GEN(av, f);
1608 : }
1609 0 : retgc_const(av, cgetg(1, t_VEC));
1610 : }
1611 :
1612 : /* list of all squarefree odd x such that phi(x) = n, P^-(x) > m. Unsorted */
1613 : static GEN
1614 19655 : invphi(ulong n, ulong m)
1615 : {
1616 : GEN C, D;
1617 : long l, i;
1618 19655 : if (n == 1) return mkvecsmall(1);
1619 14381 : D = divisorsu(n); l = lg(D);
1620 14381 : C = cgetg(1, t_VECSMALL);
1621 39986 : for (i = 2; i < l; i++) /* skip 1 */
1622 : {
1623 25605 : ulong d = D[i], p;
1624 25605 : if (d < m) continue;
1625 20396 : p = d + 1; if (!uisprime(p)) continue;
1626 10594 : C = vecsmall_concat(C, zv_z_mul(invphi(D[l-i], p), p));
1627 : }
1628 14381 : return C;
1629 : }
1630 :
1631 : long
1632 99213 : poliscyclo(GEN f)
1633 : {
1634 99213 : const ulong p = 2147483647; /* prime */
1635 : pari_sp av;
1636 : long i, n, e, l;
1637 : ulong f3, fm3;
1638 : GEN D, fp, _3;
1639 99213 : if (typ(f) != t_POL) pari_err_TYPE("poliscyclo", f);
1640 99206 : n = degpol(f);
1641 99206 : if (n <= 0 || !RgX_is_ZX(f)) return 0;
1642 99200 : if (!equali1(gel(f,n+2)) || !is_pm1(gel(f,2))) return 0;
1643 9166 : if (n == 1) return signe(gel(f,2)) > 0? 2: 1;
1644 9061 : av = avma;
1645 9061 : f = ZX_deflate_max(f, &e); if (e != 1) n = degpol(f);
1646 9061 : D = invphi(n, 1); /* squareefree odd d s.t. phi(d) = n */
1647 9061 : l = lg(D); _3 = gmodulss(3, p);
1648 9061 : fp = ZX_to_Flx(f, p);
1649 9061 : f3 = Flx_eval(fp, 3, p);
1650 9061 : fm3 = Flx_eval(fp, p-3, p);
1651 : /* f(x^e) is cyclotomic (= Phi_{de}) iff f = Phi_d, where all prime dividing
1652 : * e also divide d. */
1653 11843 : for (i = 1; i < l; i++)
1654 : {
1655 5141 : long d = D[i]; /* squarefree odd */
1656 5141 : if (odd(e))
1657 : {
1658 4092 : if (e == 1 || u_ppo(e, d) == 1)
1659 : { /* early abort: check whether f(3) = Phi_d(3) or Phi_2d(3) = Phi_d(-3)
1660 : * mod p before checking in Z. N.B. phi(d) and value at 3 mod p
1661 : * determine Phi_d for all d <= 10^7 */
1662 3861 : ulong F3 = Rg_to_Fl(polcyclo_eval(d, _3), p);
1663 3861 : if (F3 == f3 && ZX_equal(f, polcyclo(d, varn(f))))
1664 1029 : return gc_long(av, d * e);
1665 2832 : if (F3 == fm3 && ZX_equal(f, polcyclo(2*d, varn(f))))
1666 749 : return gc_long(av, 2* d * e);
1667 : }
1668 : }
1669 : else
1670 : {
1671 1049 : if (u_ppo(e, 2*d) == 1)
1672 : { /* early abort: check whether f(3) = Phi_2d(3) mod p */
1673 1042 : ulong F3 = Rg_to_Fl(polcyclo_eval(2*d, _3), p);
1674 1042 : if (F3 == f3 && ZX_equal(f, polcyclo(2*d, varn(f))))
1675 581 : return gc_long(av, 2* d * e);
1676 : }
1677 : }
1678 : }
1679 6702 : return gc_long(av, 0);
1680 : }
1681 :
1682 : long
1683 1029 : poliscycloprod(GEN f)
1684 : {
1685 1029 : pari_sp av = avma;
1686 1029 : long i, d = degpol(f);
1687 1029 : if (typ(f) != t_POL) pari_err_TYPE("poliscycloprod",f);
1688 1029 : if (!RgX_is_ZX(f)) return 0;
1689 1029 : if (!ZX_is_monic(f) || !is_pm1(constant_coeff(f))) return 0;
1690 1029 : if (d < 2) return (d == 1);
1691 1022 : if ( degpol(ZX_gcd_all(f, ZX_deriv(f), &f)) )
1692 : {
1693 14 : d = degpol(f);
1694 14 : if (d == 1) return 1;
1695 : }
1696 1015 : f = BD(f); if (!f) return 0;
1697 3619 : for (i = lg(f)-1; i; i--) d -= degpol(gel(f,i));
1698 1015 : return gc_long(av, d == 0);
1699 : }
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