Line data Source code
1 : /* Copyright (C) 2000-2005 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 :
15 : /***********************************************************************/
16 : /** **/
17 : /** ARITHMETIC OPERATIONS ON POLYNOMIALS **/
18 : /** (third part) **/
19 : /** **/
20 : /***********************************************************************/
21 : #include "pari.h"
22 : #include "paripriv.h"
23 :
24 : #define DEBUGLEVEL DEBUGLEVEL_pol
25 :
26 : /************************************************************************
27 : ** **
28 : ** Ring membership **
29 : ** **
30 : ************************************************************************/
31 : struct charact {
32 : GEN q;
33 : int isprime;
34 : };
35 : static void
36 1239 : char_update_prime(struct charact *S, GEN p)
37 : {
38 1239 : if (!S->isprime) { S->isprime = 1; S->q = p; }
39 1239 : if (!equalii(p, S->q)) pari_err_MODULUS("characteristic", S->q, p);
40 1232 : }
41 : static void
42 6573 : char_update_int(struct charact *S, GEN n)
43 : {
44 6573 : if (S->isprime)
45 : {
46 7 : if (dvdii(n, S->q)) return;
47 7 : pari_err_MODULUS("characteristic", S->q, n);
48 : }
49 6566 : S->q = gcdii(S->q, n);
50 : }
51 : static void
52 163394 : charact(struct charact *S, GEN x)
53 : {
54 163394 : const long tx = typ(x);
55 : long i, l;
56 163394 : switch(tx)
57 : {
58 5124 : case t_INTMOD:char_update_int(S, gel(x,1)); break;
59 1148 : case t_FFELT: char_update_prime(S, gel(x,4)); break;
60 26334 : case t_COMPLEX: case t_QUAD:
61 : case t_POLMOD: case t_POL: case t_SER: case t_RFRAC:
62 : case t_VEC: case t_COL: case t_MAT:
63 26334 : l = lg(x);
64 174223 : for (i=lontyp[tx]; i < l; i++) charact(S,gel(x,i));
65 26320 : break;
66 7 : case t_LIST:
67 7 : x = list_data(x);
68 7 : if (x) charact(S, x);
69 7 : break;
70 : }
71 163366 : }
72 : static void
73 4634 : charact_res(struct charact *S, GEN x)
74 : {
75 4634 : const long tx = typ(x);
76 : long i, l;
77 4634 : switch(tx)
78 : {
79 1449 : case t_INTMOD:char_update_int(S, gel(x,1)); break;
80 0 : case t_FFELT: char_update_prime(S, gel(x,4)); break;
81 91 : case t_PADIC: char_update_prime(S, gel(x,2)); break;
82 1617 : case t_COMPLEX: case t_QUAD:
83 : case t_POLMOD: case t_POL: case t_SER: case t_RFRAC:
84 : case t_VEC: case t_COL: case t_MAT:
85 1617 : l = lg(x);
86 5922 : for (i=lontyp[tx]; i < l; i++) charact_res(S,gel(x,i));
87 1617 : break;
88 0 : case t_LIST:
89 0 : x = list_data(x);
90 0 : if (x) charact_res(S, x);
91 0 : break;
92 : }
93 4634 : }
94 : GEN
95 15491 : characteristic(GEN x)
96 : {
97 : struct charact S;
98 15491 : S.q = gen_0; S.isprime = 0;
99 15491 : charact(&S, x); return S.q;
100 : }
101 : GEN
102 329 : residual_characteristic(GEN x)
103 : {
104 : struct charact S;
105 329 : S.q = gen_0; S.isprime = 0;
106 329 : charact_res(&S, x); return S.q;
107 : }
108 :
109 : int
110 65779123 : Rg_is_Fp(GEN x, GEN *pp)
111 : {
112 : GEN mod;
113 65779123 : switch(typ(x))
114 : {
115 3202780 : case t_INTMOD:
116 3202780 : mod = gel(x,1);
117 3202780 : if (!*pp) *pp = mod;
118 2953608 : else if (mod != *pp && !equalii(mod, *pp))
119 : {
120 0 : if (DEBUGLEVEL) pari_warn(warner,"different moduli in Rg_is_Fp");
121 0 : return 0;
122 : }
123 3202780 : return 1;
124 54577938 : case t_INT:
125 54577938 : return 1;
126 7998405 : default: return 0;
127 : }
128 : }
129 :
130 : int
131 24566012 : RgX_is_FpX(GEN x, GEN *pp)
132 : {
133 24566012 : long i, lx = lg(x);
134 82320629 : for (i=2; i<lx; i++)
135 65753014 : if (!Rg_is_Fp(gel(x, i), pp))
136 7998415 : return 0;
137 16567615 : return 1;
138 : }
139 :
140 : int
141 0 : RgV_is_FpV(GEN x, GEN *pp)
142 : {
143 0 : long i, lx = lg(x);
144 0 : for (i=1; i<lx; i++)
145 0 : if (!Rg_is_Fp(gel(x,i), pp)) return 0;
146 0 : return 1;
147 : }
148 :
149 : int
150 0 : RgM_is_FpM(GEN x, GEN *pp)
151 : {
152 0 : long i, lx = lg(x);
153 0 : for (i=1; i<lx; i++)
154 0 : if (!RgV_is_FpV(gel(x, i), pp)) return 0;
155 0 : return 1;
156 : }
157 :
158 : int
159 59304 : Rg_is_FpXQ(GEN x, GEN *pT, GEN *pp)
160 : {
161 : GEN pol, mod, p;
162 59304 : switch(typ(x))
163 : {
164 26089 : case t_INTMOD:
165 26089 : return Rg_is_Fp(x, pp);
166 7105 : case t_INT:
167 7105 : return 1;
168 21 : case t_POL:
169 21 : return RgX_is_FpX(x, pp);
170 21350 : case t_FFELT:
171 21350 : mod = x; p = FF_p_i(x);
172 21350 : if (!*pp) *pp = p;
173 21350 : if (!*pT) *pT = mod;
174 19824 : else if (typ(*pT)!=t_FFELT || !FF_samefield(*pT,mod))
175 : {
176 42 : if (DEBUGLEVEL) pari_warn(warner,"different moduli in Rg_is_FpXQ");
177 42 : return 0;
178 : }
179 21308 : return 1;
180 4585 : case t_POLMOD:
181 4585 : mod = gel(x,1); pol = gel(x, 2);
182 4585 : if (!RgX_is_FpX(mod, pp)) return 0;
183 4585 : if (typ(pol)==t_POL)
184 : {
185 4578 : if (!RgX_is_FpX(pol, pp)) return 0;
186 : }
187 7 : else if (!Rg_is_Fp(pol, pp)) return 0;
188 4585 : if (!*pT) *pT = mod;
189 4431 : else if (mod != *pT && !gequal(mod, *pT))
190 : {
191 0 : if (DEBUGLEVEL) pari_warn(warner,"different moduli in Rg_is_FpXQ");
192 0 : return 0;
193 : }
194 4585 : return 1;
195 :
196 154 : default: return 0;
197 : }
198 : }
199 :
200 : int
201 3206 : RgX_is_FpXQX(GEN x, GEN *pT, GEN *pp)
202 : {
203 3206 : long i, lx = lg(x);
204 61754 : for (i = 2; i < lx; i++)
205 58646 : if (!Rg_is_FpXQ(gel(x,i), pT, pp)) return 0;
206 3108 : return 1;
207 : }
208 :
209 : /************************************************************************
210 : ** **
211 : ** Ring conversion **
212 : ** **
213 : ************************************************************************/
214 :
215 : /* p > 0 a t_INT, return lift(x * Mod(1,p)).
216 : * If x is an INTMOD, assume modulus is a multiple of p. */
217 : GEN
218 32518315 : Rg_to_Fp(GEN x, GEN p)
219 : {
220 32518315 : if (lgefint(p) == 3) return utoi(Rg_to_Fl(x, uel(p,2)));
221 4556480 : switch(typ(x))
222 : {
223 288552 : case t_INT: return modii(x, p);
224 18790 : case t_FRAC: {
225 18790 : pari_sp av = avma;
226 18790 : GEN z = modii(gel(x,1), p);
227 18790 : if (z == gen_0) return gen_0;
228 18785 : return gerepileuptoint(av, remii(mulii(z, Fp_inv(gel(x,2), p)), p));
229 : }
230 70 : case t_PADIC: return padic_to_Fp(x, p);
231 4249082 : case t_INTMOD: {
232 4249082 : GEN q = gel(x,1), a = gel(x,2);
233 4249082 : if (equalii(q, p)) return icopy(a);
234 14 : if (!dvdii(q,p)) pari_err_MODULUS("Rg_to_Fp", q, p);
235 0 : return remii(a, p);
236 : }
237 0 : default: pari_err_TYPE("Rg_to_Fp",x);
238 : return NULL; /* LCOV_EXCL_LINE */
239 : }
240 : }
241 : /* If x is a POLMOD, assume modulus is a multiple of T. */
242 : GEN
243 1291496 : Rg_to_FpXQ(GEN x, GEN T, GEN p)
244 : {
245 1291496 : long ta, tx = typ(x), v = get_FpX_var(T);
246 : GEN a, b;
247 1291496 : if (is_const_t(tx))
248 : {
249 58531 : if (tx == t_FFELT)
250 : {
251 17355 : GEN z = FF_to_FpXQ(x);
252 17355 : setvarn(z, v);
253 17355 : return z;
254 : }
255 41176 : return scalar_ZX(degpol(T)? Rg_to_Fp(x, p): gen_0, v);
256 : }
257 1232965 : switch(tx)
258 : {
259 1230886 : case t_POLMOD:
260 1230886 : b = gel(x,1);
261 1230886 : a = gel(x,2); ta = typ(a);
262 1230886 : if (is_const_t(ta))
263 4095 : return scalar_ZX(degpol(T)? Rg_to_Fp(a, p): gen_0, v);
264 1226791 : b = RgX_to_FpX(b, p); if (varn(b) != v) break;
265 1226791 : a = RgX_to_FpX(a, p);
266 1226791 : if (ZX_equal(b,get_FpX_mod(T)) || signe(FpX_rem(b,T,p))==0)
267 1226791 : return FpX_rem(a, T, p);
268 0 : break;
269 2079 : case t_POL:
270 2079 : if (varn(x) != v) break;
271 2079 : return FpX_rem(RgX_to_FpX(x,p), T, p);
272 0 : case t_RFRAC:
273 0 : a = Rg_to_FpXQ(gel(x,1), T,p);
274 0 : b = Rg_to_FpXQ(gel(x,2), T,p);
275 0 : return FpXQ_div(a,b, T,p);
276 : }
277 0 : pari_err_TYPE("Rg_to_FpXQ",x);
278 : return NULL; /* LCOV_EXCL_LINE */
279 : }
280 : GEN
281 3552023 : RgX_to_FpX(GEN x, GEN p)
282 : {
283 : long i, l;
284 3552023 : GEN z = cgetg_copy(x, &l); z[1] = x[1];
285 15793752 : for (i = 2; i < l; i++) gel(z,i) = Rg_to_Fp(gel(x,i), p);
286 3552023 : return FpX_renormalize(z, l);
287 : }
288 :
289 : GEN
290 140 : RgV_to_FpV(GEN x, GEN p)
291 483 : { pari_APPLY_type(t_VEC, Rg_to_Fp(gel(x,i), p)) }
292 :
293 : GEN
294 933010 : RgC_to_FpC(GEN x, GEN p)
295 11541908 : { pari_APPLY_type(t_COL, Rg_to_Fp(gel(x,i), p)) }
296 :
297 : GEN
298 133805 : RgM_to_FpM(GEN x, GEN p)
299 1066773 : { pari_APPLY_same(RgC_to_FpC(gel(x,i), p)) }
300 :
301 : GEN
302 285890 : RgV_to_Flv(GEN x, ulong p)
303 1172341 : { pari_APPLY_ulong(Rg_to_Fl(gel(x,i), p)) }
304 :
305 : GEN
306 114124 : RgM_to_Flm(GEN x, ulong p)
307 392639 : { pari_APPLY_same(RgV_to_Flv(gel(x,i), p)) }
308 :
309 : GEN
310 5014 : RgX_to_FpXQX(GEN x, GEN T, GEN p)
311 : {
312 5014 : long i, l = lg(x);
313 5014 : GEN z = cgetg(l, t_POL); z[1] = x[1];
314 42911 : for (i = 2; i < l; i++) gel(z,i) = Rg_to_FpXQ(gel(x,i), T,p);
315 5014 : return FpXQX_renormalize(z, l);
316 : }
317 : GEN
318 48741 : RgX_to_FqX(GEN x, GEN T, GEN p)
319 : {
320 48741 : long i, l = lg(x);
321 48741 : GEN z = cgetg(l, t_POL); z[1] = x[1];
322 48741 : if (T)
323 10990 : for (i = 2; i < l; i++) gel(z,i) = Rg_to_FpXQ(gel(x,i), T, p);
324 : else
325 787026 : for (i = 2; i < l; i++) gel(z,i) = Rg_to_Fp(gel(x,i), p);
326 48741 : return FpXQX_renormalize(z, l);
327 : }
328 :
329 : GEN
330 219128 : RgC_to_FqC(GEN x, GEN T, GEN p)
331 : {
332 219128 : long i, l = lg(x);
333 219128 : GEN z = cgetg(l, t_COL);
334 219128 : if (T)
335 1430310 : for (i = 1; i < l; i++) gel(z,i) = Rg_to_FpXQ(gel(x,i), T, p);
336 : else
337 0 : for (i = 1; i < l; i++) gel(z,i) = Rg_to_Fp(gel(x,i), p);
338 219128 : return z;
339 : }
340 :
341 : GEN
342 52325 : RgM_to_FqM(GEN x, GEN T, GEN p)
343 271418 : { pari_APPLY_same(RgC_to_FqC(gel(x, i), T, p)) }
344 :
345 : /* lg(V) > 1 */
346 : GEN
347 851487 : FpXV_FpC_mul(GEN V, GEN W, GEN p)
348 : {
349 851487 : pari_sp av = avma;
350 851487 : long i, l = lg(V);
351 851487 : GEN z = ZX_Z_mul(gel(V,1),gel(W,1));
352 4201029 : for(i=2; i<l; i++)
353 : {
354 3349542 : z = ZX_add(z, ZX_Z_mul(gel(V,i),gel(W,i)));
355 3349542 : if ((i & 7) == 0) z = gerepileupto(av, z);
356 : }
357 851487 : return gerepileupto(av, FpX_red(z,p));
358 : }
359 :
360 : GEN
361 55832 : FqX_Fq_add(GEN y, GEN x, GEN T, GEN p)
362 : {
363 55832 : long i, lz = lg(y);
364 : GEN z;
365 55832 : if (!T) return FpX_Fp_add(y, x, p);
366 8666 : if (lz == 2) return scalarpol(x, varn(y));
367 7952 : z = cgetg(lz,t_POL); z[1] = y[1];
368 7952 : gel(z,2) = Fq_add(gel(y,2),x, T, p);
369 7952 : if (lz == 3) z = FpXX_renormalize(z,lz);
370 : else
371 33145 : for(i=3;i<lz;i++) gel(z,i) = gcopy(gel(y,i));
372 7952 : return z;
373 : }
374 :
375 : GEN
376 1094 : FqX_Fq_sub(GEN y, GEN x, GEN T, GEN p)
377 : {
378 1094 : long i, lz = lg(y);
379 : GEN z;
380 1094 : if (!T) return FpX_Fp_sub(y, x, p);
381 1094 : if (lz == 2) return scalarpol(x, varn(y));
382 1094 : z = cgetg(lz,t_POL); z[1] = y[1];
383 1094 : gel(z,2) = Fq_sub(gel(y,2), x, T, p);
384 1094 : if (lz == 3) z = FpXX_renormalize(z,lz);
385 : else
386 10303 : for(i=3;i<lz;i++) gel(z,i) = gcopy(gel(y,i));
387 1094 : return z;
388 : }
389 :
390 : GEN
391 149023 : FqX_Fq_mul_to_monic(GEN P, GEN U, GEN T, GEN p)
392 : {
393 : long i, lP;
394 149023 : GEN res = cgetg_copy(P, &lP); res[1] = P[1];
395 918544 : for(i=2; i<lP-1; i++) gel(res,i) = Fq_mul(U,gel(P,i), T,p);
396 149023 : gel(res,lP-1) = gen_1; return res;
397 : }
398 :
399 : GEN
400 38145 : FpXQX_normalize(GEN z, GEN T, GEN p)
401 : {
402 : GEN lc;
403 38145 : if (lg(z) == 2) return z;
404 38131 : lc = leading_coeff(z);
405 38131 : if (typ(lc) == t_POL)
406 : {
407 2152 : if (lg(lc) > 3) /* nonconstant */
408 1880 : return FqX_Fq_mul_to_monic(z, Fq_inv(lc,T,p), T,p);
409 : /* constant */
410 272 : lc = gel(lc,2);
411 272 : z = shallowcopy(z);
412 272 : gel(z, lg(z)-1) = lc;
413 : }
414 : /* lc a t_INT */
415 36251 : if (equali1(lc)) return z;
416 80 : return FqX_Fq_mul_to_monic(z, Fp_inv(lc,p), T,p);
417 : }
418 :
419 : GEN
420 398816 : FqX_eval(GEN x, GEN y, GEN T, GEN p)
421 : {
422 : pari_sp av;
423 : GEN p1, r;
424 398816 : long j, i=lg(x)-1;
425 398816 : if (i<=2)
426 45957 : return (i==2)? Fq_red(gel(x,2), T, p): gen_0;
427 352859 : av=avma; p1=gel(x,i);
428 : /* specific attention to sparse polynomials (see poleval)*/
429 : /*You've guessed it! It's a copy-paste(tm)*/
430 1173793 : for (i--; i>=2; i=j-1)
431 : {
432 821637 : for (j=i; !signe(gel(x,j)); j--)
433 700 : if (j==2)
434 : {
435 490 : if (i!=j) y = Fq_pow(y,utoipos(i-j+1), T, p);
436 490 : return gerepileupto(av, Fq_mul(p1,y, T, p));
437 : }
438 820937 : r = (i==j)? y: Fq_pow(y, utoipos(i-j+1), T, p);
439 820937 : p1 = Fq_add(Fq_mul(p1,r,T,p), gel(x,j), T, p);
440 : }
441 352366 : return gerepileupto(av, p1);
442 : }
443 :
444 : GEN
445 99679 : FqXY_evalx(GEN Q, GEN x, GEN T, GEN p)
446 : {
447 99679 : long i, lb = lg(Q);
448 : GEN z;
449 99679 : if (!T) return FpXY_evalx(Q, x, p);
450 89319 : z = cgetg(lb, t_POL); z[1] = Q[1];
451 474735 : for (i=2; i<lb; i++)
452 : {
453 385416 : GEN q = gel(Q,i);
454 385416 : gel(z,i) = typ(q) == t_INT? modii(q,p): FqX_eval(q, x, T, p);
455 : }
456 89319 : return FpXQX_renormalize(z, lb);
457 : }
458 :
459 : /* Q an FpXY, evaluate at (X,Y) = (x,y) */
460 : GEN
461 14623 : FqXY_eval(GEN Q, GEN y, GEN x, GEN T, GEN p)
462 : {
463 14623 : pari_sp av = avma;
464 14623 : if (!T) return FpXY_eval(Q, y, x, p);
465 966 : return gerepileupto(av, FqX_eval(FqXY_evalx(Q, x, T, p), y, T, p));
466 : }
467 :
468 : /* a X^d */
469 : GEN
470 10042614 : monomial(GEN a, long d, long v)
471 : {
472 : long i, n;
473 : GEN P;
474 10042614 : if (d < 0) {
475 14 : if (isrationalzero(a)) return pol_0(v);
476 14 : retmkrfrac(a, pol_xn(-d, v));
477 : }
478 10042600 : if (gequal0(a))
479 : {
480 93275 : if (isexactzero(a)) return scalarpol_shallow(a,v);
481 0 : n = d+2; P = cgetg(n+1, t_POL);
482 0 : P[1] = evalsigne(0) | evalvarn(v);
483 : }
484 : else
485 : {
486 9949328 : n = d+2; P = cgetg(n+1, t_POL);
487 9949329 : P[1] = evalsigne(1) | evalvarn(v);
488 : }
489 29075420 : for (i = 2; i < n; i++) gel(P,i) = gen_0;
490 9949329 : gel(P,i) = a; return P;
491 : }
492 : GEN
493 1863380 : monomialcopy(GEN a, long d, long v)
494 : {
495 : long i, n;
496 : GEN P;
497 1863380 : if (d < 0) {
498 14 : if (isrationalzero(a)) return pol_0(v);
499 14 : retmkrfrac(gcopy(a), pol_xn(-d, v));
500 : }
501 1863366 : if (gequal0(a))
502 : {
503 14 : if (isexactzero(a)) return scalarpol(a,v);
504 7 : n = d+2; P = cgetg(n+1, t_POL);
505 7 : P[1] = evalsigne(0) | evalvarn(v);
506 : }
507 : else
508 : {
509 1863352 : n = d+2; P = cgetg(n+1, t_POL);
510 1863352 : P[1] = evalsigne(1) | evalvarn(v);
511 : }
512 3510654 : for (i = 2; i < n; i++) gel(P,i) = gen_0;
513 1863359 : gel(P,i) = gcopy(a); return P;
514 : }
515 : GEN
516 324781 : pol_x_powers(long N, long v)
517 : {
518 324781 : GEN L = cgetg(N+1,t_VEC);
519 : long i;
520 784734 : for (i=1; i<=N; i++) gel(L,i) = pol_xn(i-1, v);
521 324789 : return L;
522 : }
523 :
524 : GEN
525 0 : FqXQ_powers(GEN x, long l, GEN S, GEN T, GEN p)
526 : {
527 0 : return T ? FpXQXQ_powers(x, l, S, T, p): FpXQ_powers(x, l, S, p);
528 : }
529 :
530 : GEN
531 0 : FqXQ_matrix_pow(GEN y, long n, long m, GEN S, GEN T, GEN p)
532 : {
533 0 : return T ? FpXQXQ_matrix_pow(y, n, m, S, T, p): FpXQ_matrix_pow(y, n, m, S, p);
534 : }
535 :
536 : /*******************************************************************/
537 : /* */
538 : /* Fq */
539 : /* */
540 : /*******************************************************************/
541 :
542 : GEN
543 11010745 : Fq_add(GEN x, GEN y, GEN T/*unused*/, GEN p)
544 : {
545 : (void)T;
546 11010745 : switch((typ(x)==t_POL)|((typ(y)==t_POL)<<1))
547 : {
548 544907 : case 0: return Fp_add(x,y,p);
549 764607 : case 1: return FpX_Fp_add(x,y,p);
550 92056 : case 2: return FpX_Fp_add(y,x,p);
551 9609175 : case 3: return FpX_add(x,y,p);
552 : }
553 : return NULL;/*LCOV_EXCL_LINE*/
554 : }
555 :
556 : GEN
557 8337689 : Fq_sub(GEN x, GEN y, GEN T/*unused*/, GEN p)
558 : {
559 : (void)T;
560 8337689 : switch((typ(x)==t_POL)|((typ(y)==t_POL)<<1))
561 : {
562 243999 : case 0: return Fp_sub(x,y,p);
563 24480 : case 1: return FpX_Fp_sub(x,y,p);
564 23908 : case 2: return Fp_FpX_sub(x,y,p);
565 8045302 : case 3: return FpX_sub(x,y,p);
566 : }
567 : return NULL;/*LCOV_EXCL_LINE*/
568 : }
569 :
570 : GEN
571 891504 : Fq_neg(GEN x, GEN T/*unused*/, GEN p)
572 : {
573 : (void)T;
574 891504 : return (typ(x)==t_POL)? FpX_neg(x,p): Fp_neg(x,p);
575 : }
576 :
577 : GEN
578 83614 : Fq_halve(GEN x, GEN T/*unused*/, GEN p)
579 : {
580 : (void)T;
581 83614 : return (typ(x)==t_POL)? FpX_halve(x,p): Fp_halve(x,p);
582 : }
583 :
584 : /* If T==NULL do not reduce*/
585 : GEN
586 8008127 : Fq_mul(GEN x, GEN y, GEN T, GEN p)
587 : {
588 8008127 : switch((typ(x)==t_POL)|((typ(y)==t_POL)<<1))
589 : {
590 668961 : case 0: return Fp_mul(x,y,p);
591 128947 : case 1: return FpX_Fp_mul(x,y,p);
592 402199 : case 2: return FpX_Fp_mul(y,x,p);
593 6808023 : case 3: if (T) return FpXQ_mul(x,y,T,p);
594 4232038 : else return FpX_mul(x,y,p);
595 : }
596 : return NULL;/*LCOV_EXCL_LINE*/
597 : }
598 :
599 : /* If T==NULL do not reduce*/
600 : GEN
601 492787 : Fq_mulu(GEN x, ulong y, /*unused*/GEN T, GEN p)
602 : {
603 : (void) T;
604 492787 : return typ(x)==t_POL ? FpX_Fp_mul(x,utoi(y),p): Fp_mulu(x, y, p);
605 : }
606 :
607 : /* y t_INT */
608 : GEN
609 39037 : Fq_Fp_mul(GEN x, GEN y, GEN T/*unused*/, GEN p)
610 : {
611 : (void)T;
612 4933 : return (typ(x) == t_POL)? FpX_Fp_mul(x,y,p)
613 43970 : : Fp_mul(x,y,p);
614 : }
615 : /* If T==NULL do not reduce*/
616 : GEN
617 369590 : Fq_sqr(GEN x, GEN T, GEN p)
618 : {
619 369590 : if (typ(x) == t_POL)
620 : {
621 72844 : if (T) return FpXQ_sqr(x,T,p);
622 0 : else return FpX_sqr(x,p);
623 : }
624 : else
625 296746 : return Fp_sqr(x,p);
626 : }
627 :
628 : GEN
629 0 : Fq_neg_inv(GEN x, GEN T, GEN p)
630 : {
631 0 : if (typ(x) == t_INT) return Fp_inv(Fp_neg(x,p),p);
632 0 : return FpXQ_inv(FpX_neg(x,p),T,p);
633 : }
634 :
635 : GEN
636 0 : Fq_invsafe(GEN x, GEN pol, GEN p)
637 : {
638 0 : if (typ(x) == t_INT) return Fp_invsafe(x,p);
639 0 : return FpXQ_invsafe(x,pol,p);
640 : }
641 :
642 : GEN
643 89325 : Fq_inv(GEN x, GEN pol, GEN p)
644 : {
645 89325 : if (typ(x) == t_INT) return Fp_inv(x,p);
646 81559 : return FpXQ_inv(x,pol,p);
647 : }
648 :
649 : GEN
650 308357 : Fq_div(GEN x, GEN y, GEN pol, GEN p)
651 : {
652 308357 : switch((typ(x)==t_POL)|((typ(y)==t_POL)<<1))
653 : {
654 283038 : case 0: return Fp_div(x,y,p);
655 16702 : case 1: return FpX_Fp_div(x,y,p);
656 406 : case 2: return FpX_Fp_mul(FpXQ_inv(y,pol,p),x,p);
657 8211 : case 3: return FpXQ_div(x,y,pol,p);
658 : }
659 : return NULL;/*LCOV_EXCL_LINE*/
660 : }
661 :
662 : GEN
663 792230 : Fq_pow(GEN x, GEN n, GEN pol, GEN p)
664 : {
665 792230 : if (typ(x) == t_INT) return Fp_pow(x,n,p);
666 136648 : return FpXQ_pow(x,n,pol,p);
667 : }
668 :
669 : GEN
670 15050 : Fq_powu(GEN x, ulong n, GEN pol, GEN p)
671 : {
672 15050 : if (typ(x) == t_INT) return Fp_powu(x,n,p);
673 1267 : return FpXQ_powu(x,n,pol,p);
674 : }
675 :
676 : GEN
677 780730 : Fq_sqrt(GEN x, GEN T, GEN p)
678 : {
679 780730 : if (typ(x) == t_INT)
680 : {
681 710590 : if (!T || odd(get_FpX_degree(T))) return Fp_sqrt(x,p);
682 9603 : x = scalarpol_shallow(x, get_FpX_var(T));
683 : }
684 79743 : return FpXQ_sqrt(x,T,p);
685 : }
686 : GEN
687 61670 : Fq_sqrtn(GEN x, GEN n, GEN T, GEN p, GEN *zeta)
688 : {
689 61670 : if (typ(x) == t_INT)
690 : {
691 : long d;
692 61334 : if (!T) return Fp_sqrtn(x,n,p,zeta);
693 119 : d = get_FpX_degree(T);
694 119 : if (ugcdiu(n,d) == 1)
695 : {
696 56 : if (!zeta) return Fp_sqrtn(x,n,p,NULL);
697 : /* gcd(n,p-1)=gcd(n,q-1): same number of solutions in Fp and F_q */
698 21 : if (equalii(gcdii(subiu(p,1),n), gcdii(subiu(Fp_powu(p,d,n), 1), n)))
699 14 : return Fp_sqrtn(x,n,p,zeta);
700 : }
701 70 : x = scalarpol(x, get_FpX_var(T)); /* left on stack */
702 : }
703 406 : return FpXQ_sqrtn(x,n,T,p,zeta);
704 : }
705 :
706 : struct _Fq_field
707 : {
708 : GEN T, p;
709 : };
710 :
711 : static GEN
712 302701 : _Fq_red(void *E, GEN x)
713 302701 : { struct _Fq_field *s = (struct _Fq_field *)E;
714 302701 : return Fq_red(x, s->T, s->p);
715 : }
716 :
717 : static GEN
718 362523 : _Fq_add(void *E, GEN x, GEN y)
719 : {
720 : (void) E;
721 362523 : switch((typ(x)==t_POL)|((typ(y)==t_POL)<<1))
722 : {
723 14 : case 0: return addii(x,y);
724 0 : case 1: return ZX_Z_add(x,y);
725 15918 : case 2: return ZX_Z_add(y,x);
726 346591 : default: return ZX_add(x,y);
727 : }
728 : }
729 :
730 : static GEN
731 315028 : _Fq_neg(void *E, GEN x) { (void) E; return typ(x)==t_POL?ZX_neg(x):negi(x); }
732 :
733 : static GEN
734 242795 : _Fq_mul(void *E, GEN x, GEN y)
735 : {
736 : (void) E;
737 242795 : switch((typ(x)==t_POL)|((typ(y)==t_POL)<<1))
738 : {
739 133 : case 0: return mulii(x,y);
740 1085 : case 1: return ZX_Z_mul(x,y);
741 1043 : case 2: return ZX_Z_mul(y,x);
742 240534 : default: return ZX_mul(x,y);
743 : }
744 : }
745 :
746 : static GEN
747 9331 : _Fq_inv(void *E, GEN x)
748 9331 : { struct _Fq_field *s = (struct _Fq_field *)E;
749 9331 : return Fq_inv(x,s->T,s->p);
750 : }
751 :
752 : static int
753 38388 : _Fq_equal0(GEN x) { return signe(x)==0; }
754 :
755 : static GEN
756 13965 : _Fq_s(void *E, long x) { (void) E; return stoi(x); }
757 :
758 : static const struct bb_field Fq_field={_Fq_red,_Fq_add,_Fq_mul,_Fq_neg,
759 : _Fq_inv,_Fq_equal0,_Fq_s};
760 :
761 4179 : const struct bb_field *get_Fq_field(void **E, GEN T, GEN p)
762 : {
763 4179 : if (!T)
764 0 : return get_Fp_field(E, p);
765 : else
766 : {
767 4179 : GEN z = new_chunk(sizeof(struct _Fq_field));
768 4179 : struct _Fq_field *e = (struct _Fq_field *) z;
769 4179 : e->T = T; e->p = p; *E = (void*)e;
770 4179 : return &Fq_field;
771 : }
772 : }
773 :
774 : /*******************************************************************/
775 : /* */
776 : /* Fq[X] */
777 : /* */
778 : /*******************************************************************/
779 : /* P(X + c) */
780 : GEN
781 266 : FpX_translate(GEN P, GEN c, GEN p)
782 : {
783 266 : pari_sp av = avma;
784 : GEN Q, *R;
785 : long i, k, n;
786 :
787 266 : if (!signe(P) || !signe(c)) return ZX_copy(P);
788 266 : Q = leafcopy(P);
789 266 : R = (GEN*)(Q+2); n = degpol(P);
790 3738 : for (i=1; i<=n; i++)
791 : {
792 118153 : for (k=n-i; k<n; k++)
793 114681 : R[k] = Fp_add(R[k], Fp_mul(c, R[k+1], p), p);
794 :
795 3472 : if (gc_needed(av,2))
796 : {
797 0 : if(DEBUGMEM>1) pari_warn(warnmem,"FpX_translate, i = %ld/%ld", i,n);
798 0 : Q = gerepilecopy(av, Q); R = (GEN*)Q+2;
799 : }
800 : }
801 266 : return gerepilecopy(av, FpX_renormalize(Q, lg(Q)));
802 : }
803 : /* P(X + c), c an Fq */
804 : GEN
805 33880 : FqX_translate(GEN P, GEN c, GEN T, GEN p)
806 : {
807 33880 : pari_sp av = avma;
808 : GEN Q, *R;
809 : long i, k, n;
810 :
811 : /* signe works for t_(INT|POL) */
812 33880 : if (!signe(P) || !signe(c)) return RgX_copy(P);
813 33880 : Q = leafcopy(P);
814 33880 : R = (GEN*)(Q+2); n = degpol(P);
815 150059 : for (i=1; i<=n; i++)
816 : {
817 433559 : for (k=n-i; k<n; k++)
818 317380 : R[k] = Fq_add(R[k], Fq_mul(c, R[k+1], T, p), T, p);
819 :
820 116179 : if (gc_needed(av,2))
821 : {
822 0 : if(DEBUGMEM>1) pari_warn(warnmem,"FqX_translate, i = %ld/%ld", i,n);
823 0 : Q = gerepilecopy(av, Q); R = (GEN*)Q+2;
824 : }
825 : }
826 33880 : return gerepilecopy(av, FpXQX_renormalize(Q, lg(Q)));
827 : }
828 :
829 : GEN
830 40474 : FqV_roots_to_pol(GEN V, GEN T, GEN p, long v)
831 : {
832 40474 : pari_sp ltop = avma;
833 : long k;
834 : GEN W;
835 40474 : if (lgefint(p) == 3)
836 : {
837 31772 : ulong pp = p[2];
838 31772 : GEN Tl = ZX_to_Flx(T, pp);
839 31771 : GEN Vl = ZXC_to_FlxC(V, pp, get_Flx_var(Tl));
840 31772 : Tl = FlxqV_roots_to_pol(Vl, Tl, pp, v);
841 31772 : return gerepileupto(ltop, FlxX_to_ZXX(Tl));
842 : }
843 8702 : W = cgetg(lg(V),t_VEC);
844 78018 : for(k=1; k < lg(V); k++)
845 69316 : gel(W,k) = deg1pol_shallow(gen_1,Fq_neg(gel(V,k),T,p),v);
846 8702 : return gerepileupto(ltop, FpXQXV_prod(W, T, p));
847 : }
848 :
849 : GEN
850 155442 : FqV_red(GEN x, GEN T, GEN p)
851 1240186 : { pari_APPLY_same(Fq_red(gel(x,i), T, p)) }
852 :
853 : GEN
854 0 : FqC_add(GEN x, GEN y, GEN T, GEN p)
855 : {
856 0 : if (!T) return FpC_add(x, y, p);
857 0 : pari_APPLY_type(t_COL, Fq_add(gel(x,i), gel(y,i), T, p))
858 : }
859 :
860 : GEN
861 0 : FqC_sub(GEN x, GEN y, GEN T, GEN p)
862 : {
863 0 : if (!T) return FpC_sub(x, y, p);
864 0 : pari_APPLY_type(t_COL, Fq_sub(gel(x,i), gel(y,i), T, p))
865 : }
866 :
867 : GEN
868 0 : FqC_Fq_mul(GEN x, GEN y, GEN T, GEN p)
869 : {
870 0 : if (!T) return FpC_Fp_mul(x, y, p);
871 0 : pari_APPLY_type(t_COL, Fq_mul(gel(x,i),y,T,p))
872 : }
873 :
874 : GEN
875 105 : FqC_FqV_mul(GEN x, GEN y, GEN T, GEN p)
876 : {
877 105 : long i,j, lx=lg(x), ly=lg(y);
878 : GEN z;
879 105 : if (ly==1) return cgetg(1,t_MAT);
880 105 : z = cgetg(ly,t_MAT);
881 819 : for (j=1; j < ly; j++)
882 : {
883 714 : GEN zj = cgetg(lx,t_COL);
884 4200 : for (i=1; i<lx; i++) gel(zj,i) = Fq_mul(gel(x,i),gel(y,j), T, p);
885 714 : gel(z, j) = zj;
886 : }
887 105 : return z;
888 : }
889 :
890 : GEN
891 5271 : FpXC_center(GEN x, GEN p, GEN pov2)
892 40524 : { pari_APPLY_type(t_COL, FpX_center(gel(x,i), p, pov2)) }
893 :
894 : GEN
895 1730 : FpXM_center(GEN x, GEN p, GEN pov2)
896 7001 : { pari_APPLY_same(FpXC_center(gel(x,i), p, pov2)) }
897 :
898 : /*******************************************************************/
899 : /* */
900 : /* GENERIC CRT */
901 : /* */
902 : /*******************************************************************/
903 : static GEN
904 8034164 : primelist(forprime_t *S, long n, GEN dB)
905 : {
906 8034164 : GEN P = cgetg(n+1, t_VECSMALL);
907 8034119 : long i = 1;
908 : ulong p;
909 19186519 : while (i <= n && (p = u_forprime_next(S)))
910 11152400 : if (!dB || umodiu(dB, p)) P[i++] = p;
911 8034117 : return P;
912 : }
913 :
914 : void
915 7520465 : gen_inccrt_i(const char *str, GEN worker, GEN dB, long n, long mmin,
916 : forprime_t *S, GEN *pH, GEN *pmod, GEN crt(GEN, GEN, GEN*),
917 : GEN center(GEN, GEN, GEN))
918 : {
919 7520465 : long m = mmin? minss(mmin, n): usqrt(n);
920 : GEN H, P, mod;
921 : pari_timer ti;
922 7520473 : if (DEBUGLEVEL > 4)
923 : {
924 0 : timer_start(&ti);
925 0 : err_printf("%s: nb primes: %ld\n",str, n);
926 : }
927 7520473 : if (m == 1)
928 : {
929 7251986 : GEN P = primelist(S, n, dB);
930 7251917 : GEN done = closure_callgen1(worker, P);
931 7251917 : H = gel(done,1);
932 7251917 : mod = gel(done,2);
933 7251917 : if (!*pH && center) H = center(H, mod, shifti(mod,-1));
934 7251879 : if (DEBUGLEVEL>4) timer_printf(&ti,"%s: modular", str);
935 : }
936 : else
937 : {
938 268487 : long i, s = (n+m-1)/m, r = m - (m*s-n), di = 0;
939 : struct pari_mt pt;
940 268487 : long pending = 0;
941 268487 : H = cgetg(m+1, t_VEC); P = cgetg(m+1, t_VEC);
942 268487 : mt_queue_start_lim(&pt, worker, m);
943 1107359 : for (i=1; i<=m || pending; i++)
944 : {
945 : GEN done;
946 838872 : GEN pr = i <= m ? mkvec(primelist(S, i<=r ? s: s-1, dB)): NULL;
947 838871 : mt_queue_submit(&pt, i, pr);
948 838871 : done = mt_queue_get(&pt, NULL, &pending);
949 838871 : if (done)
950 : {
951 782187 : di++;
952 782187 : gel(H, di) = gel(done,1);
953 782187 : gel(P, di) = gel(done,2);
954 782187 : if (DEBUGLEVEL>5) err_printf("%ld%% ",100*di/m);
955 : }
956 : }
957 268487 : mt_queue_end(&pt);
958 268487 : if (DEBUGLEVEL>5) err_printf("\n");
959 268487 : if (DEBUGLEVEL>4) timer_printf(&ti,"%s: modular", str);
960 268487 : H = crt(H, P, &mod);
961 268487 : if (DEBUGLEVEL>4) timer_printf(&ti,"%s: chinese", str);
962 : }
963 7520366 : if (*pH) H = crt(mkvec2(*pH, H), mkvec2(*pmod, mod), &mod);
964 7520366 : *pH = H; *pmod = mod;
965 7520366 : }
966 : void
967 2154686 : gen_inccrt(const char *str, GEN worker, GEN dB, long n, long mmin,
968 : forprime_t *S, GEN *pH, GEN *pmod, GEN crt(GEN, GEN, GEN*),
969 : GEN center(GEN, GEN, GEN))
970 : {
971 2154686 : pari_sp av = avma;
972 2154686 : gen_inccrt_i(str, worker, dB, n, mmin, S, pH, pmod, crt, center);
973 2154625 : gerepileall(av, 2, pH, pmod);
974 2154764 : }
975 :
976 : GEN
977 1391049 : gen_crt(const char *str, GEN worker, forprime_t *S, GEN dB, ulong bound, long mmin, GEN *pmod,
978 : GEN crt(GEN, GEN, GEN*), GEN center(GEN, GEN, GEN))
979 : {
980 1391049 : GEN mod = gen_1, H = NULL;
981 : ulong e;
982 :
983 1391049 : bound++;
984 2782161 : while (bound > (e = expi(mod)))
985 : {
986 1391006 : long n = (bound - e) / expu(S->p) + 1;
987 1391044 : gen_inccrt(str, worker, dB, n, mmin, S, &H, &mod, crt, center);
988 : }
989 1391069 : if (pmod) *pmod = mod;
990 1391069 : return H;
991 : }
992 :
993 : /*******************************************************************/
994 : /* */
995 : /* MODULAR GCD */
996 : /* */
997 : /*******************************************************************/
998 : /* return z = a mod q, b mod p (p,q) = 1; qinv = 1/q mod p; a in ]-q,q] */
999 : static GEN
1000 5112772 : Fl_chinese_coprime(GEN a, ulong b, GEN q, ulong p, ulong qinv, GEN pq, GEN pq2)
1001 : {
1002 5112772 : ulong d, amod = umodiu(a, p);
1003 5112917 : pari_sp av = avma;
1004 : GEN ax;
1005 :
1006 5112917 : if (b == amod) return NULL;
1007 2104689 : d = Fl_mul(Fl_sub(b, amod, p), qinv, p); /* != 0 */
1008 2105242 : if (d >= 1 + (p>>1))
1009 1026924 : ax = subii(a, mului(p-d, q));
1010 : else
1011 : {
1012 1078318 : ax = addii(a, mului(d, q)); /* in ]0, pq[ assuming a in ]-q,q[ */
1013 1077859 : if (cmpii(ax,pq2) > 0) ax = subii(ax,pq);
1014 : }
1015 2104476 : return gerepileuptoint(av, ax);
1016 : }
1017 : GEN
1018 378 : Z_init_CRT(ulong Hp, ulong p) { return stoi(Fl_center(Hp, p, p>>1)); }
1019 : GEN
1020 31542 : ZX_init_CRT(GEN Hp, ulong p, long v)
1021 : {
1022 31542 : long i, l = lg(Hp), lim = (long)(p>>1);
1023 31542 : GEN H = cgetg(l, t_POL);
1024 31542 : H[1] = evalsigne(1) | evalvarn(v);
1025 794458 : for (i=2; i<l; i++)
1026 762916 : gel(H,i) = stoi(Fl_center(Hp[i], p, lim));
1027 31542 : return ZX_renormalize(H,l);
1028 : }
1029 :
1030 : GEN
1031 3633 : ZM_init_CRT(GEN Hp, ulong p)
1032 : {
1033 3633 : long i,j, m, l = lg(Hp), lim = (long)(p>>1);
1034 3633 : GEN c, cp, H = cgetg(l, t_MAT);
1035 3633 : if (l==1) return H;
1036 3633 : m = lgcols(Hp);
1037 12544 : for (j=1; j<l; j++)
1038 : {
1039 8911 : cp = gel(Hp,j);
1040 8911 : c = cgetg(m, t_COL);
1041 8911 : gel(H,j) = c;
1042 87983 : for (i=1; i<m; i++) gel(c,i) = stoi(Fl_center(cp[i],p, lim));
1043 : }
1044 3633 : return H;
1045 : }
1046 :
1047 : int
1048 7616 : Z_incremental_CRT(GEN *H, ulong Hp, GEN *ptq, ulong p)
1049 : {
1050 7616 : GEN h, q = *ptq, qp = muliu(q,p);
1051 7616 : ulong qinv = Fl_inv(umodiu(q,p), p);
1052 7616 : int stable = 1;
1053 7616 : h = Fl_chinese_coprime(*H,Hp,q,p,qinv,qp,shifti(qp,-1));
1054 7616 : if (h) { *H = h; stable = 0; }
1055 7616 : *ptq = qp; return stable;
1056 : }
1057 :
1058 : static int
1059 147351 : ZX_incremental_CRT_raw(GEN *ptH, GEN Hp, GEN q, GEN qp, ulong p)
1060 : {
1061 147351 : GEN H = *ptH, h, qp2 = shifti(qp,-1);
1062 147345 : ulong qinv = Fl_inv(umodiu(q,p), p);
1063 147353 : long i, l = lg(H), lp = lg(Hp);
1064 147353 : int stable = 1;
1065 :
1066 147353 : if (l < lp)
1067 : { /* degree increases */
1068 0 : GEN x = cgetg(lp, t_POL);
1069 0 : for (i=1; i<l; i++) x[i] = H[i];
1070 0 : for ( ; i<lp; i++) gel(x,i) = gen_0;
1071 0 : *ptH = H = x;
1072 0 : stable = 0;
1073 147353 : } else if (l > lp)
1074 : { /* degree decreases */
1075 0 : GEN x = cgetg(l, t_VECSMALL);
1076 0 : for (i=1; i<lp; i++) x[i] = Hp[i];
1077 0 : for ( ; i<l; i++) x[i] = 0;
1078 0 : Hp = x; lp = l;
1079 : }
1080 4932577 : for (i=2; i<lp; i++)
1081 : {
1082 4785338 : h = Fl_chinese_coprime(gel(H,i),Hp[i],q,p,qinv,qp,qp2);
1083 4785224 : if (h) { gel(H,i) = h; stable = 0; }
1084 : }
1085 147239 : (void)ZX_renormalize(H,lp);
1086 147352 : return stable;
1087 : }
1088 :
1089 : int
1090 0 : ZX_incremental_CRT(GEN *ptH, GEN Hp, GEN *ptq, ulong p)
1091 : {
1092 0 : GEN q = *ptq, qp = muliu(q,p);
1093 0 : int stable = ZX_incremental_CRT_raw(ptH, Hp, q, qp, p);
1094 0 : *ptq = qp; return stable;
1095 : }
1096 :
1097 : int
1098 5801 : ZM_incremental_CRT(GEN *pH, GEN Hp, GEN *ptq, ulong p)
1099 : {
1100 5801 : GEN h, H = *pH, q = *ptq, qp = muliu(q, p), qp2 = shifti(qp,-1);
1101 5801 : ulong qinv = Fl_inv(umodiu(q,p), p);
1102 5801 : long i,j, l = lg(H), m = lgcols(H);
1103 5801 : int stable = 1;
1104 20944 : for (j=1; j<l; j++)
1105 157160 : for (i=1; i<m; i++)
1106 : {
1107 142017 : h = Fl_chinese_coprime(gcoeff(H,i,j), coeff(Hp,i,j),q,p,qinv,qp,qp2);
1108 142017 : if (h) { gcoeff(H,i,j) = h; stable = 0; }
1109 : }
1110 5801 : *ptq = qp; return stable;
1111 : }
1112 :
1113 : GEN
1114 623 : ZXM_init_CRT(GEN Hp, long deg, ulong p)
1115 : {
1116 : long i, j, k;
1117 : GEN H;
1118 623 : long m, l = lg(Hp), lim = (long)(p>>1), n;
1119 623 : H = cgetg(l, t_MAT);
1120 623 : if (l==1) return H;
1121 623 : m = lgcols(Hp);
1122 623 : n = deg + 3;
1123 2114 : for (j=1; j<l; j++)
1124 : {
1125 1491 : GEN cp = gel(Hp,j);
1126 1491 : GEN c = cgetg(m, t_COL);
1127 1491 : gel(H,j) = c;
1128 23905 : for (i=1; i<m; i++)
1129 : {
1130 22414 : GEN dp = gel(cp, i);
1131 22414 : long l = lg(dp);
1132 22414 : GEN d = cgetg(n, t_POL);
1133 22414 : gel(c, i) = d;
1134 22414 : d[1] = dp[1] | evalsigne(1);
1135 45647 : for (k=2; k<l; k++)
1136 23233 : gel(d,k) = stoi(Fl_center(dp[k], p, lim));
1137 44457 : for ( ; k<n; k++)
1138 22043 : gel(d,k) = gen_0;
1139 : }
1140 : }
1141 623 : return H;
1142 : }
1143 :
1144 : int
1145 653 : ZXM_incremental_CRT(GEN *pH, GEN Hp, GEN *ptq, ulong p)
1146 : {
1147 653 : GEN v, H = *pH, q = *ptq, qp = muliu(q, p), qp2 = shifti(qp,-1);
1148 653 : ulong qinv = Fl_inv(umodiu(q,p), p);
1149 653 : long i,j,k, l = lg(H), m = lgcols(H), n = lg(gmael(H,1,1));
1150 653 : int stable = 1;
1151 2225 : for (j=1; j<l; j++)
1152 90418 : for (i=1; i<m; i++)
1153 : {
1154 88846 : GEN h = gmael(H,j,i), hp = gmael(Hp,j,i);
1155 88846 : long lh = lg(hp);
1156 246641 : for (k=2; k<lh; k++)
1157 : {
1158 157795 : v = Fl_chinese_coprime(gel(h,k),uel(hp,k),q,p,qinv,qp,qp2);
1159 157795 : if (v) { gel(h,k) = v; stable = 0; }
1160 : }
1161 108763 : for (; k<n; k++)
1162 : {
1163 19917 : v = Fl_chinese_coprime(gel(h,k),0,q,p,qinv,qp,qp2);
1164 19917 : if (v) { gel(h,k) = v; stable = 0; }
1165 : }
1166 : }
1167 653 : *ptq = qp; return stable;
1168 : }
1169 :
1170 : /* record the degrees of Euclidean remainders (make them as large as
1171 : * possible : smaller values correspond to a degenerate sequence) */
1172 : static void
1173 23160 : Flx_resultant_set_dglist(GEN a, GEN b, GEN dglist, ulong p)
1174 : {
1175 : long da,db,dc, ind;
1176 23160 : pari_sp av = avma;
1177 :
1178 23160 : if (lgpol(a)==0 || lgpol(b)==0) return;
1179 21893 : da = degpol(a);
1180 21893 : db = degpol(b);
1181 21893 : if (db > da)
1182 0 : { swapspec(a,b, da,db); }
1183 21893 : else if (!da) return;
1184 21893 : ind = 0;
1185 144011 : while (db)
1186 : {
1187 122119 : GEN c = Flx_rem(a,b, p);
1188 122118 : a = b; b = c; dc = degpol(c);
1189 122118 : if (dc < 0) break;
1190 :
1191 122118 : ind++;
1192 122118 : if (dc > dglist[ind]) dglist[ind] = dc;
1193 122118 : if (gc_needed(av,2))
1194 : {
1195 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flx_resultant_all");
1196 0 : gerepileall(av, 2, &a,&b);
1197 : }
1198 122118 : db = dc; /* = degpol(b) */
1199 : }
1200 21892 : if (ind+1 > lg(dglist)) setlg(dglist,ind+1);
1201 21892 : set_avma(av);
1202 : }
1203 : /* assuming the PRS finishes on a degree 1 polynomial C0 + C1X, with
1204 : * "generic" degree sequence as given by dglist, set *Ci and return
1205 : * resultant(a,b). Modular version of Collins's subresultant */
1206 : static ulong
1207 2083440 : Flx_resultant_all(GEN a, GEN b, long *C0, long *C1, GEN dglist, ulong p)
1208 : {
1209 : long da,db,dc, ind;
1210 2083440 : ulong lb, res, g = 1UL, h = 1UL, ca = 1UL, cb = 1UL;
1211 2083440 : int s = 1;
1212 2083440 : pari_sp av = avma;
1213 :
1214 2083440 : *C0 = 1; *C1 = 0;
1215 2083440 : if (lgpol(a)==0 || lgpol(b)==0) return 0;
1216 2074086 : da = degpol(a);
1217 2074084 : db = degpol(b);
1218 2074121 : if (db > da)
1219 : {
1220 0 : swapspec(a,b, da,db);
1221 0 : if (both_odd(da,db)) s = -s;
1222 : }
1223 2074121 : else if (!da) return 1; /* = a[2] ^ db, since 0 <= db <= da = 0 */
1224 2074121 : ind = 0;
1225 19791512 : while (db)
1226 : { /* sub-resultant algo., applied to ca * a and cb * b, ca,cb scalars,
1227 : * da = deg a, db = deg b */
1228 17722142 : GEN c = Flx_rem(a,b, p);
1229 17608099 : long delta = da - db;
1230 :
1231 17608099 : if (both_odd(da,db)) s = -s;
1232 17609470 : lb = Fl_mul(b[db+2], cb, p);
1233 17635538 : a = b; b = c; dc = degpol(c);
1234 17655478 : ind++;
1235 17655478 : if (dc != dglist[ind]) return gc_ulong(av,0); /* degenerates */
1236 17650587 : if (g == h)
1237 : { /* frequent */
1238 17590750 : ulong cc = Fl_mul(ca, Fl_powu(Fl_div(lb,g,p), delta+1, p), p);
1239 17658776 : ca = cb;
1240 17658776 : cb = cc;
1241 : }
1242 : else
1243 : {
1244 59837 : ulong cc = Fl_mul(ca, Fl_powu(lb, delta+1, p), p);
1245 59838 : ulong ghdelta = Fl_mul(g, Fl_powu(h, delta, p), p);
1246 59838 : ca = cb;
1247 59838 : cb = Fl_div(cc, ghdelta, p);
1248 : }
1249 17719442 : da = db; /* = degpol(a) */
1250 17719442 : db = dc; /* = degpol(b) */
1251 :
1252 17719442 : g = lb;
1253 17719442 : if (delta == 1)
1254 17620171 : h = g; /* frequent */
1255 : else
1256 99271 : h = Fl_mul(h, Fl_powu(Fl_div(g,h,p), delta, p), p);
1257 :
1258 17716415 : if (gc_needed(av,2))
1259 : {
1260 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flx_resultant_all");
1261 0 : gerepileall(av, 2, &a,&b);
1262 : }
1263 : }
1264 2069370 : if (da > 1) return 0; /* Failure */
1265 : /* last nonconstant polynomial has degree 1 */
1266 2069370 : *C0 = Fl_mul(ca, a[2], p);
1267 2069326 : *C1 = Fl_mul(ca, a[3], p);
1268 2069313 : res = Fl_mul(cb, b[2], p);
1269 2069350 : if (s == -1) res = p - res;
1270 2069350 : return gc_ulong(av,res);
1271 : }
1272 :
1273 : /* Q a vector of polynomials representing B in Fp[X][Y], evaluate at X = x,
1274 : * Return 0 in case of degree drop. */
1275 : static GEN
1276 2106577 : FlxY_evalx_drop(GEN Q, ulong x, ulong p)
1277 : {
1278 : GEN z;
1279 2106577 : long i, lb = lg(Q);
1280 2106577 : ulong leadz = Flx_eval(leading_coeff(Q), x, p);
1281 2106437 : long vs=mael(Q,2,1);
1282 2106437 : if (!leadz) return zero_Flx(vs);
1283 :
1284 2095777 : z = cgetg(lb, t_VECSMALL); z[1] = vs;
1285 20058951 : for (i=2; i<lb-1; i++) z[i] = Flx_eval(gel(Q,i), x, p);
1286 2093928 : z[i] = leadz; return z;
1287 : }
1288 :
1289 : GEN
1290 2072 : FpXY_FpXQ_evaly(GEN Q, GEN y, GEN T, GEN p, long vx)
1291 : {
1292 2072 : pari_sp av = avma;
1293 2072 : long i, lb = lg(Q);
1294 : GEN z;
1295 2072 : if (lb == 2) return pol_0(vx);
1296 2072 : z = gel(Q, lb-1);
1297 2072 : if (lb == 3 || !signe(y)) return typ(z)==t_INT? scalar_ZX(z, vx): ZX_copy(z);
1298 :
1299 2072 : if (typ(z) == t_INT) z = scalar_ZX_shallow(z, vx);
1300 48636 : for (i=lb-2; i>=2; i--)
1301 : {
1302 46564 : GEN c = gel(Q,i);
1303 46564 : z = FqX_Fq_mul(z, y, T, p);
1304 46564 : z = typ(c) == t_INT? FqX_Fq_add(z,c,T,p): FqX_add(z,c,T,p);
1305 : }
1306 2072 : return gerepileupto(av, z);
1307 : }
1308 :
1309 : static GEN
1310 272256 : ZX_norml1(GEN x)
1311 : {
1312 272256 : long i, l = lg(x);
1313 : GEN s;
1314 :
1315 272256 : if (l == 2) return gen_0;
1316 179702 : s = gel(x, l-1); /* != 0 */
1317 657877 : for (i = l-2; i > 1; i--) {
1318 478182 : GEN xi = gel(x,i);
1319 478182 : if (!signe(xi)) continue;
1320 239782 : s = addii_sign(s,1, xi,1);
1321 : }
1322 179695 : return s;
1323 : }
1324 : /* x >= 0, y != 0, return x + |y| */
1325 : static GEN
1326 25581 : addii_abs(GEN x, GEN y)
1327 : {
1328 25581 : if (!signe(x)) return absi_shallow(y);
1329 16046 : return addii_sign(x,1, y,1);
1330 : }
1331 :
1332 : /* x a ZX, return sum_{i >= k} |x[i]| binomial(i, k) */
1333 : static GEN
1334 31693 : ZX_norml1_1(GEN x, long k)
1335 : {
1336 31693 : long i, d = degpol(x);
1337 : GEN s, C; /* = binomial(i, k) */
1338 :
1339 31692 : if (!d || k > d) return gen_0;
1340 31694 : s = absi_shallow(gel(x, k+2)); /* may be 0 */
1341 31693 : C = gen_1;
1342 68140 : for (i = k+1; i <= d; i++) {
1343 36460 : GEN xi = gel(x,i+2);
1344 36460 : if (k) C = diviuexact(muliu(C, i), i-k);
1345 36458 : if (signe(xi)) s = addii_abs(s, mulii(C, xi));
1346 : }
1347 31680 : return s;
1348 : }
1349 : /* x has non-negative real coefficients */
1350 : static GEN
1351 3255 : RgX_norml1_1(GEN x, long k)
1352 : {
1353 3255 : long i, d = degpol(x);
1354 : GEN s, C; /* = binomial(i, k) */
1355 :
1356 3255 : if (!d || k > d) return gen_0;
1357 3255 : s = gel(x, k+2); /* may be 0 */
1358 3255 : C = gen_1;
1359 9128 : for (i = k+1; i <= d; i++) {
1360 5873 : GEN xi = gel(x,i+2);
1361 5873 : if (k) C = diviuexact(muliu(C, i), i-k);
1362 5873 : if (!gequal0(xi)) s = gadd(s, gmul(C, xi));
1363 : }
1364 3255 : return s;
1365 : }
1366 :
1367 : /* N_2(A)^2 */
1368 : static GEN
1369 7199 : sqrN2(GEN A, long prec)
1370 : {
1371 7199 : pari_sp av = avma;
1372 7199 : long i, l = lg(A);
1373 7199 : GEN a = gen_0;
1374 35453 : for (i = 2; i < l; i++)
1375 : {
1376 28254 : a = gadd(a, gabs(gnorm(gel(A,i)), prec));
1377 28254 : if (gc_needed(av,1))
1378 : {
1379 0 : if(DEBUGMEM>1) pari_warn(warnmem,"RgX_RgXY_ResBound i = %ld",i);
1380 0 : a = gerepileupto(av, a);
1381 : }
1382 : }
1383 7199 : return a;
1384 : }
1385 : /* Interpolate at roots of 1 and use Hadamard bound for univariate resultant:
1386 : * bound = N_2(A)^degpol B N_2(B)^degpol(A), where
1387 : * N_2(A) = sqrt(sum (N_1(Ai))^2)
1388 : * Return e such that Res(A, B) < 2^e */
1389 : static GEN
1390 6352 : RgX_RgXY_ResBound(GEN A, GEN B, long prec)
1391 : {
1392 6352 : pari_sp av = avma;
1393 6352 : GEN b = gen_0, bnd;
1394 6352 : long i, lB = lg(B);
1395 25314 : for (i=2; i<lB; i++)
1396 : {
1397 18962 : GEN t = gel(B,i);
1398 18962 : if (typ(t) == t_POL) t = gnorml1(t, prec);
1399 18962 : b = gadd(b, gabs(gsqr(t), prec));
1400 18962 : if (gc_needed(av,1))
1401 : {
1402 0 : if(DEBUGMEM>1) pari_warn(warnmem,"RgX_RgXY_ResBound i = %ld",i);
1403 0 : b = gerepileupto(av, b);
1404 : }
1405 : }
1406 6352 : bnd = gsqrt(gmul(gpowgs(sqrN2(A,prec), degpol(B)),
1407 : gpowgs(b, degpol(A))), prec);
1408 6352 : return gerepileupto(av, bnd);
1409 : }
1410 : /* A,B in C[X] return RgX_RgXY_ResBound(A, B(x+y)) */
1411 : static GEN
1412 847 : RgX_RgXY_ResBound_1(GEN A, GEN B, long prec)
1413 : {
1414 847 : pari_sp av = avma, av2;
1415 847 : GEN b = gen_0, bnd;
1416 847 : long i, lB = lg(B);
1417 847 : B = shallowcopy(B);
1418 4102 : for (i=2; i<lB; i++) gel(B,i) = gabs(gel(B,i), prec);
1419 847 : av2 = avma;
1420 4102 : for (i=2; i<lB; i++)
1421 : {
1422 3255 : b = gadd(b, gsqr(RgX_norml1_1(B, i-2)));
1423 3255 : if (gc_needed(av2,1))
1424 : {
1425 0 : if(DEBUGMEM>1) pari_warn(warnmem,"RgX_RgXY_ResBound i = %ld",i);
1426 0 : b = gerepileupto(av2, b);
1427 : }
1428 : }
1429 847 : bnd = gsqrt(gmul(gpowgs(sqrN2(A,prec), degpol(B)),
1430 : gpowgs(b, degpol(A))), prec);
1431 847 : return gerepileupto(av, bnd);
1432 : }
1433 :
1434 : /* log2 N_2(A)^2 */
1435 : static double
1436 187806 : log2N2(GEN A)
1437 : {
1438 187806 : pari_sp av = avma;
1439 187806 : long i, l = lg(A);
1440 187806 : GEN a = gen_0;
1441 1132841 : for (i=2; i < l; i++)
1442 : {
1443 945044 : a = addii(a, sqri(gel(A,i)));
1444 945035 : if (gc_needed(av,1))
1445 : {
1446 0 : if(DEBUGMEM>1) pari_warn(warnmem,"ZX_ZXY_ResBound i = %ld",i);
1447 0 : a = gerepileupto(av, a);
1448 : }
1449 : }
1450 187797 : return gc_double(av, dbllog2(a));
1451 : }
1452 : /* Interpolate at roots of 1 and use Hadamard bound for univariate resultant:
1453 : * bound = N_2(A)^degpol B N_2(B)^degpol(A), where
1454 : * N_2(A) = sqrt(sum (N_1(Ai))^2)
1455 : * Return e such that Res(A, B) < 2^e */
1456 : ulong
1457 177714 : ZX_ZXY_ResBound(GEN A, GEN B, GEN dB)
1458 : {
1459 177714 : pari_sp av = avma;
1460 177714 : GEN b = gen_0;
1461 177714 : long i, lB = lg(B);
1462 : double logb;
1463 1031336 : for (i=2; i<lB; i++)
1464 : {
1465 853629 : GEN t = gel(B,i);
1466 853629 : if (typ(t) == t_POL) t = ZX_norml1(t);
1467 853628 : b = addii(b, sqri(t));
1468 853622 : if (gc_needed(av,1))
1469 : {
1470 0 : if(DEBUGMEM>1) pari_warn(warnmem,"ZX_ZXY_ResBound i = %ld",i);
1471 0 : b = gerepileupto(av, b);
1472 : }
1473 : }
1474 177707 : logb = dbllog2(b); if (dB) logb -= 2 * dbllog2(dB);
1475 177709 : i = (long)((degpol(B) * log2N2(A) + degpol(A) * logb) / 2);
1476 177712 : return gc_ulong(av, (i <= 0)? 1: 1 + (ulong)i);
1477 : }
1478 : /* A,B ZX. Return ZX_ZXY_ResBound(A(x), B(x+y)) */
1479 : static ulong
1480 10103 : ZX_ZXY_ResBound_1(GEN A, GEN B)
1481 : {
1482 10103 : pari_sp av = avma;
1483 10103 : GEN b = gen_0;
1484 10103 : long i, lB = lg(B);
1485 41795 : for (i=2; i<lB; i++)
1486 : {
1487 31692 : b = addii(b, sqri(ZX_norml1_1(B, i-2)));
1488 31692 : if (gc_needed(av,1))
1489 : {
1490 0 : if(DEBUGMEM>1) pari_warn(warnmem,"ZX_ZXY_ResBound i = %ld",i);
1491 0 : b = gerepileupto(av, b);
1492 : }
1493 : }
1494 10103 : i = (long)((degpol(B) * log2N2(A) + degpol(A) * dbllog2(b)) / 2);
1495 10102 : return gc_ulong(av, (i <= 0)? 1: 1 + (ulong)i);
1496 : }
1497 : /* special case B = A' */
1498 : static ulong
1499 1127681 : ZX_discbound(GEN A)
1500 : {
1501 1127681 : pari_sp av = avma;
1502 1127681 : GEN a = gen_0, b = gen_0;
1503 1127681 : long i , lA = lg(A), dA = degpol(A);
1504 : double loga, logb;
1505 6714516 : for (i = 2; i < lA; i++)
1506 : {
1507 5587122 : GEN c = sqri(gel(A,i));
1508 5586677 : a = addii(a, c);
1509 5586741 : if (i > 2) b = addii(b, mulii(c, sqru(i-2)));
1510 5586865 : if (gc_needed(av,1))
1511 : {
1512 0 : if(DEBUGMEM>1) pari_warn(warnmem,"ZX_discbound i = %ld",i);
1513 0 : gerepileall(av, 2, &a, &b);
1514 : }
1515 : }
1516 1127394 : loga = dbllog2(a);
1517 1127606 : logb = dbllog2(b); set_avma(av);
1518 1127639 : i = (long)(((dA-1) * loga + dA * logb) / 2);
1519 1127639 : return (i <= 0)? 1: 1 + (ulong)i;
1520 : }
1521 :
1522 : /* return Res(a(Y), b(n,Y)) over Fp. la = leading_coeff(a) [for efficiency] */
1523 : static ulong
1524 2268209 : Flx_FlxY_eval_resultant(GEN a, GEN b, ulong n, ulong p, ulong pi, ulong la)
1525 : {
1526 2268209 : GEN ev = FlxY_evalx_pre(b, n, p, pi);
1527 2268462 : long drop = lg(b) - lg(ev);
1528 2268462 : ulong r = Flx_resultant_pre(a, ev, p, pi);
1529 2268066 : if (drop && la != 1) r = Fl_mul(r, Fl_powu_pre(la, drop, p, pi), p);
1530 2268074 : return r;
1531 : }
1532 : static GEN
1533 284 : FpX_FpXY_eval_resultant(GEN a, GEN b, GEN n, GEN p, GEN la, long db, long vX)
1534 : {
1535 284 : GEN ev = FpXY_evaly(b, n, p, vX);
1536 284 : long drop = db-degpol(ev);
1537 284 : GEN r = FpX_resultant(a, ev, p);
1538 284 : if (drop && !gequal1(la)) r = Fp_mul(r, Fp_powu(la, drop,p),p);
1539 284 : return r;
1540 : }
1541 :
1542 : /* assume dres := deg(Res_X(a,b), Y) <= deg(a,X) * deg(b,Y) < p */
1543 : /* Return a Fly */
1544 : static GEN
1545 177132 : Flx_FlxY_resultant_polint(GEN a, GEN b, ulong p, ulong pi, long dres, long sx)
1546 : {
1547 : long i;
1548 177132 : ulong n, la = Flx_lead(a);
1549 177131 : GEN x = cgetg(dres+2, t_VECSMALL);
1550 177132 : GEN y = cgetg(dres+2, t_VECSMALL);
1551 : /* Evaluate at dres+ 1 points: 0 (if dres even) and +/- n, so that P_n(X) =
1552 : * P_{-n}(-X), where P_i is Lagrange polynomial: P_i(j) = delta_{i,j} */
1553 1225792 : for (i=0,n = 1; i < dres; n++)
1554 : {
1555 1048661 : x[++i] = n; y[i] = Flx_FlxY_eval_resultant(a,b, x[i], p,pi,la);
1556 1048606 : x[++i] = p-n; y[i] = Flx_FlxY_eval_resultant(a,b, x[i], p,pi,la);
1557 : }
1558 177131 : if (i == dres)
1559 : {
1560 171705 : x[++i] = 0; y[i] = Flx_FlxY_eval_resultant(a,b, x[i], p,pi,la);
1561 : }
1562 177130 : return Flv_polint(x,y, p, sx);
1563 : }
1564 :
1565 : static GEN
1566 8055 : FlxX_pseudorem(GEN x, GEN y, ulong p, ulong pi)
1567 : {
1568 8055 : long vx = varn(x), dx, dy, dz, i, lx, dp;
1569 8055 : pari_sp av = avma, av2;
1570 :
1571 8055 : if (!signe(y)) pari_err_INV("FlxX_pseudorem",y);
1572 8055 : (void)new_chunk(2);
1573 8055 : dx=degpol(x); x = RgX_recip_i(x)+2;
1574 8055 : dy=degpol(y); y = RgX_recip_i(y)+2; dz=dx-dy; dp = dz+1;
1575 8056 : av2 = avma;
1576 : for (;;)
1577 : {
1578 65777 : gel(x,0) = Flx_neg(gel(x,0), p); dp--;
1579 245700 : for (i=1; i<=dy; i++)
1580 178493 : gel(x,i) = Flx_add( Flx_mul_pre(gel(y,0), gel(x,i), p, pi),
1581 179846 : Flx_mul_pre(gel(x,0), gel(y,i), p, pi), p );
1582 1150973 : for ( ; i<=dx; i++)
1583 1086079 : gel(x,i) = Flx_mul_pre(gel(y,0), gel(x,i), p, pi);
1584 69862 : do { x++; dx--; } while (dx >= 0 && lg(gel(x,0))==2);
1585 64894 : if (dx < dy) break;
1586 56845 : if (gc_needed(av2,1))
1587 : {
1588 0 : if(DEBUGMEM>1) pari_warn(warnmem,"FlxX_pseudorem dx = %ld >= %ld",dx,dy);
1589 0 : gerepilecoeffs(av2,x,dx+1);
1590 : }
1591 : }
1592 8049 : if (dx < 0) return zero_Flx(0);
1593 8049 : lx = dx+3; x -= 2;
1594 8049 : x[0]=evaltyp(t_POL) | evallg(lx);
1595 8049 : x[1]=evalsigne(1) | evalvarn(vx);
1596 8049 : x = RgX_recip_i(x);
1597 8053 : if (dp)
1598 : { /* multiply by y[0]^dp [beware dummy vars from FpX_FpXY_resultant] */
1599 2107 : GEN t = Flx_powu_pre(gel(y,0), dp, p, pi);
1600 8424 : for (i=2; i<lx; i++) gel(x,i) = Flx_mul_pre(gel(x,i), t, p, pi);
1601 : }
1602 8058 : return gerepilecopy(av, x);
1603 : }
1604 :
1605 : /* return a Flx */
1606 : GEN
1607 2695 : FlxX_resultant(GEN u, GEN v, ulong p, long sx)
1608 : {
1609 2695 : pari_sp av = avma, av2;
1610 : long degq, dx, dy, du, dv, dr, signh;
1611 : ulong pi;
1612 : GEN z, g, h, r, p1;
1613 :
1614 2695 : dx = degpol(u); dy = degpol(v); signh = 1;
1615 2695 : if (dx < dy)
1616 : {
1617 7 : swap(u,v); lswap(dx,dy);
1618 7 : if (both_odd(dx, dy)) signh = -signh;
1619 : }
1620 2695 : if (dy < 0) return zero_Flx(sx);
1621 2695 : pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
1622 2695 : if (dy==0) return gerepileupto(av, Flx_powu_pre(gel(v,2),dx,p,pi));
1623 :
1624 2695 : g = h = pol1_Flx(sx); av2 = avma;
1625 : for(;;)
1626 : {
1627 8055 : r = FlxX_pseudorem(u,v,p,pi); dr = lg(r);
1628 8056 : if (dr == 2) { set_avma(av); return zero_Flx(sx); }
1629 8056 : du = degpol(u); dv = degpol(v); degq = du-dv;
1630 8056 : u = v; p1 = g; g = leading_coeff(u);
1631 8057 : switch(degq)
1632 : {
1633 0 : case 0: break;
1634 5936 : case 1:
1635 5936 : p1 = Flx_mul_pre(h,p1, p, pi); h = g; break;
1636 2121 : default:
1637 2121 : p1 = Flx_mul_pre(Flx_powu_pre(h,degq,p,pi), p1, p, pi);
1638 2120 : h = Flx_div_pre(Flx_powu_pre(g,degq,p,pi),
1639 2121 : Flx_powu_pre(h,degq-1,p,pi), p, pi);
1640 : }
1641 8047 : if (both_odd(du,dv)) signh = -signh;
1642 8043 : v = FlxY_Flx_div(r, p1, p);
1643 8051 : if (dr==3) break;
1644 5357 : if (gc_needed(av2,1))
1645 : {
1646 0 : if(DEBUGMEM>1) pari_warn(warnmem,"FlxX_resultant, dr = %ld",dr);
1647 0 : gerepileall(av2,4, &u, &v, &g, &h);
1648 : }
1649 : }
1650 2694 : z = gel(v,2);
1651 2694 : if (dv > 1) z = Flx_div_pre(Flx_powu_pre(z,dv,p,pi),
1652 0 : Flx_powu_pre(h,dv-1,p,pi), p, pi);
1653 2693 : if (signh < 0) z = Flx_neg(z,p);
1654 2693 : return gerepileupto(av, z);
1655 : }
1656 :
1657 : /* Warning:
1658 : * This function switches between valid and invalid variable ordering*/
1659 :
1660 : static GEN
1661 6312 : FlxY_to_FlyX(GEN b, long sv)
1662 : {
1663 6312 : long i, n=-1;
1664 6312 : long sw = b[1]&VARNBITS;
1665 21637 : for(i=2;i<lg(b);i++) n = maxss(n,lgpol(gel(b,i)));
1666 6312 : return Flm_to_FlxX(Flm_transpose(FlxX_to_Flm(b,n)),sv,sw);
1667 : }
1668 :
1669 : /* Return a Fly*/
1670 : GEN
1671 6312 : Flx_FlxY_resultant(GEN a, GEN b, ulong p)
1672 : {
1673 6312 : pari_sp ltop=avma;
1674 6312 : long dres = degpol(a)*degpol(b);
1675 6312 : long sx=a[1], sy=b[1]&VARNBITS;
1676 : GEN z;
1677 6312 : b = FlxY_to_FlyX(b,sx);
1678 6310 : if ((ulong)dres >= p)
1679 2692 : z = FlxX_resultant(Fly_to_FlxY(a, sy), b, p, sx);
1680 : else
1681 : {
1682 3618 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
1683 3618 : z = Flx_FlxY_resultant_polint(a, b, p, pi, (ulong)dres, sy);
1684 : }
1685 6313 : return gerepileupto(ltop,z);
1686 : }
1687 :
1688 : /* return a t_POL (in variable v >= 0) whose coeffs are the coeffs of b,
1689 : * in variable v. This is an incorrect PARI object if initially varn(b) << v.
1690 : * We could return a vector of coeffs, but it is convenient to have degpol()
1691 : * and friends available. Even in that case, it will behave nicely with all
1692 : * functions treating a polynomial as a vector of coeffs (eg poleval).
1693 : * FOR INTERNAL USE! */
1694 : GEN
1695 125938 : swap_vars(GEN b0, long v)
1696 : {
1697 125938 : long i, n = RgX_degree(b0, v);
1698 : GEN b, x;
1699 125938 : if (n < 0) return pol_0(v);
1700 125938 : b = cgetg(n+3, t_POL); x = b + 2;
1701 125938 : b[1] = evalsigne(1) | evalvarn(v);
1702 640190 : for (i=0; i<=n; i++) gel(x,i) = polcoef_i(b0, i, v);
1703 125937 : return b;
1704 : }
1705 :
1706 : /* assume varn(b) << varn(a) */
1707 : /* return a FpY*/
1708 : GEN
1709 15 : FpX_FpXY_resultant(GEN a, GEN b, GEN p)
1710 : {
1711 15 : long i,n,dres, db, vY = varn(b), vX = varn(a);
1712 : GEN la,x,y;
1713 :
1714 15 : if (lgefint(p) == 3)
1715 : {
1716 0 : ulong pp = uel(p,2);
1717 0 : b = ZXX_to_FlxX(b, pp, vX);
1718 0 : a = ZX_to_Flx(a, pp);
1719 0 : x = Flx_FlxY_resultant(a, b, pp);
1720 0 : return Flx_to_ZX(x);
1721 : }
1722 15 : db = RgXY_degreex(b);
1723 15 : dres = degpol(a)*degpol(b);
1724 15 : la = leading_coeff(a);
1725 15 : x = cgetg(dres+2, t_VEC);
1726 15 : y = cgetg(dres+2, t_VEC);
1727 : /* Evaluate at dres+ 1 points: 0 (if dres even) and +/- n, so that P_n(X) =
1728 : * P_{-n}(-X), where P_i is Lagrange polynomial: P_i(j) = delta_{i,j} */
1729 157 : for (i=0,n = 1; i < dres; n++)
1730 : {
1731 142 : gel(x,++i) = utoipos(n);
1732 142 : gel(y,i) = FpX_FpXY_eval_resultant(a,b,gel(x,i),p,la,db,vY);
1733 142 : gel(x,++i) = subiu(p,n);
1734 142 : gel(y,i) = FpX_FpXY_eval_resultant(a,b,gel(x,i),p,la,db,vY);
1735 : }
1736 15 : if (i == dres)
1737 : {
1738 0 : gel(x,++i) = gen_0;
1739 0 : gel(y,i) = FpX_FpXY_eval_resultant(a,b, gel(x,i), p,la,db,vY);
1740 : }
1741 15 : return FpV_polint(x,y, p, vY);
1742 : }
1743 :
1744 : GEN
1745 79 : FpX_composedsum(GEN P, GEN Q, GEN p)
1746 : {
1747 79 : pari_sp av = avma;
1748 79 : if (lgefint(p)==3)
1749 : {
1750 0 : ulong pp = p[2];
1751 0 : GEN z = Flx_composedsum(ZX_to_Flx(P, pp), ZX_to_Flx(Q, pp), pp);
1752 0 : return gerepileupto(av, Flx_to_ZX(z));
1753 : }
1754 : else
1755 : {
1756 79 : long n = 1+ degpol(P)*degpol(Q);
1757 79 : GEN Pl = FpX_invLaplace(FpX_Newton(P,n,p), p);
1758 79 : GEN Ql = FpX_invLaplace(FpX_Newton(Q,n,p), p);
1759 79 : GEN L = FpX_Laplace(FpXn_mul(Pl, Ql, n, p), p);
1760 79 : GEN lead = Fp_mul(Fp_powu(leading_coeff(P),degpol(Q), p),
1761 79 : Fp_powu(leading_coeff(Q),degpol(P), p), p);
1762 79 : GEN R = FpX_fromNewton(L, p);
1763 79 : return gerepileupto(av, FpX_Fp_mul(R, lead, p));
1764 : }
1765 : }
1766 :
1767 : GEN
1768 0 : FpX_composedprod(GEN P, GEN Q, GEN p)
1769 : {
1770 0 : pari_sp av = avma;
1771 0 : if (lgefint(p)==3)
1772 : {
1773 0 : ulong pp = p[2];
1774 0 : GEN z = Flx_composedprod(ZX_to_Flx(P, pp), ZX_to_Flx(Q, pp), pp);
1775 0 : return gerepileupto(av, Flx_to_ZX(z));
1776 : }
1777 : else
1778 : {
1779 0 : long n = 1+ degpol(P)*degpol(Q);
1780 0 : GEN L = FpX_convol(FpX_Newton(P,n,p), FpX_Newton(Q,n,p), p);
1781 0 : return gerepileupto(av,FpX_fromNewton(L, p));
1782 : }
1783 : }
1784 :
1785 : static GEN
1786 79 : _FpX_composedsum(void *E, GEN a, GEN b)
1787 79 : { return FpX_composedsum(a,b, (GEN)E); }
1788 :
1789 : GEN
1790 1574 : FpXV_composedsum(GEN V, GEN p)
1791 : {
1792 1574 : if (lgefint(p)==3)
1793 : {
1794 0 : ulong pp = p[2];
1795 0 : return Flx_to_ZX(FlxV_composedsum(ZXV_to_FlxV(V, pp), pp));
1796 : }
1797 1574 : return gen_product(V, (void *)p, &_FpX_composedsum);
1798 : }
1799 :
1800 : /* 0, 1, -1, 2, -2, ... */
1801 : #define next_lambda(a) (a>0 ? -a : 1-a)
1802 :
1803 : /* Assume A in Z[Y], B in Q[Y][X], B squarefree in (Q[Y]/(A))[X] and
1804 : * Res_Y(A, B) in Z[X]. Find a small lambda (start from *lambda, use
1805 : * next_lambda successively) such that C(X) = Res_Y(A(Y), B(X + lambda Y))
1806 : * is squarefree, reset *lambda to the chosen value and return C. Set LERS to
1807 : * the Last nonconstant polynomial in the Euclidean Remainder Sequence */
1808 : static GEN
1809 20993 : ZX_ZXY_resultant_LERS(GEN A, GEN B0, long *plambda, GEN *LERS)
1810 : {
1811 : ulong bound, dp;
1812 20993 : pari_sp av = avma, av2 = 0;
1813 20993 : long lambda = *plambda, degA = degpol(A), dres = degA*degpol(B0);
1814 : long stable, checksqfree, i,n, cnt, degB;
1815 20993 : long v, vX = varn(B0), vY = varn(A); /* vY < vX */
1816 : GEN x, y, dglist, B, q, a, b, ev, H, H0, H1, Hp, H0p, H1p, C0, C1;
1817 : forprime_t S;
1818 :
1819 20993 : if (degA == 1)
1820 : {
1821 1043 : GEN a1 = gel(A,3), a0 = gel(A,2);
1822 1043 : B = lambda? RgX_translate(B0, monomial(stoi(lambda), 1, vY)): B0;
1823 1043 : H = gsubst(B, vY, gdiv(gneg(a0),a1));
1824 1043 : if (!equali1(a1)) H = RgX_Rg_mul(H, powiu(a1, poldegree(B,vY)));
1825 1043 : *LERS = mkvec2(scalarpol_shallow(a0,vX), scalarpol_shallow(a1,vX));
1826 1043 : return gc_all(av, 2, &H, LERS);
1827 : }
1828 :
1829 19950 : dglist = Hp = H0p = H1p = C0 = C1 = NULL; /* gcc -Wall */
1830 19950 : C0 = cgetg(dres+2, t_VECSMALL);
1831 19950 : C1 = cgetg(dres+2, t_VECSMALL);
1832 19950 : dglist = cgetg(dres+1, t_VECSMALL);
1833 19950 : x = cgetg(dres+2, t_VECSMALL);
1834 19950 : y = cgetg(dres+2, t_VECSMALL);
1835 19950 : B0 = leafcopy(B0);
1836 19950 : A = leafcopy(A);
1837 19950 : B = B0;
1838 19950 : v = fetch_var_higher(); setvarn(A,v);
1839 : /* make sure p large enough */
1840 20612 : INIT:
1841 : /* always except the first time */
1842 20612 : if (av2) { set_avma(av2); lambda = next_lambda(lambda); }
1843 20612 : if (lambda) B = RgX_translate(B0, monomial(stoi(lambda), 1, vY));
1844 20612 : B = swap_vars(B, vY); setvarn(B,v);
1845 : /* B0(lambda v + x, v) */
1846 20612 : if (DEBUGLEVEL>4) err_printf("Trying lambda = %ld\n", lambda);
1847 20612 : av2 = avma;
1848 :
1849 20612 : if (degA <= 3)
1850 : { /* sub-resultant faster for small degrees */
1851 9989 : H = RgX_resultant_all(A,B,&q);
1852 9989 : if (typ(q) != t_POL || degpol(q)!=1) goto INIT;
1853 9478 : H0 = gel(q,2);
1854 9478 : if (typ(H0) == t_POL) setvarn(H0,vX); else H0 = scalarpol(H0,vX);
1855 9478 : H1 = gel(q,3);
1856 9478 : if (typ(H1) == t_POL) setvarn(H1,vX); else H1 = scalarpol(H1,vX);
1857 9477 : if (!ZX_is_squarefree(H)) goto INIT;
1858 9436 : goto END;
1859 : }
1860 :
1861 10623 : H = H0 = H1 = NULL;
1862 10623 : degB = degpol(B);
1863 10623 : bound = ZX_ZXY_ResBound(A, B, NULL);
1864 10623 : if (DEBUGLEVEL>4) err_printf("bound for resultant coeffs: 2^%ld\n",bound);
1865 10623 : dp = 1;
1866 10623 : init_modular_big(&S);
1867 10623 : for(cnt = 0, checksqfree = 1;;)
1868 49118 : {
1869 59741 : ulong p = u_forprime_next(&S);
1870 : GEN Hi;
1871 59741 : a = ZX_to_Flx(A, p);
1872 59740 : b = ZXX_to_FlxX(B, p, varn(A));
1873 59740 : if (degpol(a) < degA || degpol(b) < degB) continue; /* p | lc(A)lc(B) */
1874 59740 : if (checksqfree)
1875 : { /* find degree list for generic Euclidean Remainder Sequence */
1876 10623 : long goal = minss(degpol(a), degpol(b)); /* longest possible */
1877 72895 : for (n=1; n <= goal; n++) dglist[n] = 0;
1878 10623 : setlg(dglist, 1);
1879 23552 : for (n=0; n <= dres; n++)
1880 : {
1881 23160 : ev = FlxY_evalx_drop(b, n, p);
1882 23160 : Flx_resultant_set_dglist(a, ev, dglist, p);
1883 23160 : if (lg(dglist)-1 == goal) break;
1884 : }
1885 : /* last pol in ERS has degree > 1 ? */
1886 10623 : goal = lg(dglist)-1;
1887 10623 : if (degpol(B) == 1) { if (!goal) goto INIT; }
1888 : else
1889 : {
1890 10567 : if (goal <= 1) goto INIT;
1891 10511 : if (dglist[goal] != 0 || dglist[goal-1] != 1) goto INIT;
1892 : }
1893 10567 : if (DEBUGLEVEL>4)
1894 0 : err_printf("Degree list for ERS (trials: %ld) = %Ps\n",n+1,dglist);
1895 : }
1896 :
1897 2143115 : for (i=0,n = 0; i <= dres; n++)
1898 : {
1899 2083467 : ev = FlxY_evalx_drop(b, n, p);
1900 2083420 : x[++i] = n; y[i] = Flx_resultant_all(a, ev, C0+i, C1+i, dglist, p);
1901 2083431 : if (!C1[i]) i--; /* C1(i) = 0. No way to recover C0(i) */
1902 : }
1903 59648 : Hi = Flv_Flm_polint(x, mkvec3(y,C0,C1), p, 0);
1904 59685 : Hp = gel(Hi,1); H0p = gel(Hi,2); H1p = gel(Hi,3);
1905 59685 : if (!H && degpol(Hp) != dres) continue;
1906 59685 : if (dp != 1) Hp = Flx_Fl_mul(Hp, Fl_powu(Fl_inv(dp,p), degA, p), p);
1907 59685 : if (checksqfree) {
1908 10567 : if (!Flx_is_squarefree(Hp, p)) goto INIT;
1909 10514 : if (DEBUGLEVEL>4) err_printf("Final lambda = %ld\n", lambda);
1910 10514 : checksqfree = 0;
1911 : }
1912 :
1913 59632 : if (!H)
1914 : { /* initialize */
1915 10514 : q = utoipos(p); stable = 0;
1916 10514 : H = ZX_init_CRT(Hp, p,vX);
1917 10514 : H0= ZX_init_CRT(H0p, p,vX);
1918 10514 : H1= ZX_init_CRT(H1p, p,vX);
1919 : }
1920 : else
1921 : {
1922 49118 : GEN qp = muliu(q,p);
1923 49117 : stable = ZX_incremental_CRT_raw(&H, Hp, q,qp, p)
1924 49118 : & ZX_incremental_CRT_raw(&H0,H0p, q,qp, p)
1925 49117 : & ZX_incremental_CRT_raw(&H1,H1p, q,qp, p);
1926 49118 : q = qp;
1927 : }
1928 : /* could make it probabilistic for H ? [e.g if stable twice, etc]
1929 : * Probabilistic anyway for H0, H1 */
1930 59632 : if (DEBUGLEVEL>5 && (stable || ++cnt==100))
1931 0 : { cnt=0; err_printf("%ld%%%s ",100*expi(q)/bound,stable?"s":""); }
1932 59632 : if (stable && (ulong)expi(q) >= bound) break; /* DONE */
1933 49118 : if (gc_needed(av,2))
1934 : {
1935 0 : if (DEBUGMEM>1) pari_warn(warnmem,"ZX_ZXY_rnfequation");
1936 0 : gerepileall(av2, 4, &H, &q, &H0, &H1);
1937 : }
1938 : }
1939 19950 : END:
1940 19950 : if (DEBUGLEVEL>5) err_printf(" done\n");
1941 19950 : setvarn(H, vX); (void)delete_var();
1942 19950 : *LERS = mkvec2(H0,H1);
1943 19950 : *plambda = lambda; return gc_all(av, 2, &H, LERS);
1944 : }
1945 :
1946 : GEN
1947 58758 : ZX_ZXY_resultant_all(GEN A, GEN B, long *plambda, GEN *LERS)
1948 : {
1949 58758 : if (LERS)
1950 : {
1951 20993 : if (!plambda)
1952 0 : pari_err_BUG("ZX_ZXY_resultant_all [LERS != NULL needs lambda]");
1953 20993 : return ZX_ZXY_resultant_LERS(A, B, plambda, LERS);
1954 : }
1955 37765 : return ZX_ZXY_rnfequation(A, B, plambda);
1956 : }
1957 :
1958 : /* If lambda = NULL, return caract(Mod(A, T)), T,A in Z[X].
1959 : * Otherwise find a small lambda such that caract (Mod(A + lambda X, T)) is
1960 : * squarefree */
1961 : GEN
1962 3476 : ZXQ_charpoly_sqf(GEN A, GEN T, long *lambda, long v)
1963 : {
1964 3476 : pari_sp av = avma;
1965 : GEN R, a;
1966 : long dA;
1967 : int delvar;
1968 :
1969 3476 : if (v < 0) v = 0;
1970 3476 : switch (typ(A))
1971 : {
1972 3476 : case t_POL: dA = degpol(A); if (dA > 0) break;
1973 0 : A = constant_coeff(A);
1974 0 : default:
1975 0 : if (lambda) { A = scalar_ZX_shallow(A,varn(T)); dA = 0; break;}
1976 0 : return gerepileupto(av, gpowgs(gsub(pol_x(v), A), degpol(T)));
1977 : }
1978 3476 : delvar = 0;
1979 3476 : if (varn(T) == 0)
1980 : {
1981 3275 : long v0 = fetch_var(); delvar = 1;
1982 3275 : T = leafcopy(T); setvarn(T,v0);
1983 3275 : A = leafcopy(A); setvarn(A,v0);
1984 : }
1985 3476 : R = ZX_ZXY_rnfequation(T, deg1pol_shallow(gen_1, gneg_i(A), 0), lambda);
1986 3476 : if (delvar) (void)delete_var();
1987 3476 : setvarn(R, v); a = leading_coeff(T);
1988 3476 : if (!gequal1(a)) R = gdiv(R, powiu(a, dA));
1989 3476 : return gerepileupto(av, R);
1990 : }
1991 :
1992 : /* charpoly(Mod(A,T)), A may be in Q[X], but assume T and result are integral */
1993 : GEN
1994 120742 : ZXQ_charpoly(GEN A, GEN T, long v)
1995 : {
1996 120742 : return (degpol(T) < 16) ? RgXQ_charpoly_i(A,T,v): ZXQ_charpoly_sqf(A,T, NULL, v);
1997 : }
1998 :
1999 : GEN
2000 9723 : QXQ_charpoly(GEN A, GEN T, long v)
2001 : {
2002 9723 : pari_sp av = avma;
2003 9723 : GEN den, B = Q_remove_denom(A, &den);
2004 9723 : GEN P = ZXQ_charpoly(B, T, v);
2005 9723 : return gerepilecopy(av, den ? RgX_rescale(P, ginv(den)): P);
2006 : }
2007 :
2008 : static ulong
2009 3966433 : ZX_resultant_prime(GEN a, GEN b, GEN dB, long degA, long degB, ulong p)
2010 : {
2011 3966433 : long dropa = degA - degpol(a), dropb = degB - degpol(b);
2012 : ulong H, dp;
2013 3966300 : if (dropa && dropb) return 0; /* p | lc(A), p | lc(B) */
2014 3966300 : H = Flx_resultant(a, b, p);
2015 3966135 : if (dropa)
2016 : { /* multiply by ((-1)^deg B lc(B))^(deg A - deg a) */
2017 0 : ulong c = b[degB+2]; /* lc(B) */
2018 0 : if (odd(degB)) c = p - c;
2019 0 : c = Fl_powu(c, dropa, p);
2020 0 : if (c != 1) H = Fl_mul(H, c, p);
2021 : }
2022 3966135 : else if (dropb)
2023 : { /* multiply by lc(A)^(deg B - deg b) */
2024 0 : ulong c = a[degA+2]; /* lc(A) */
2025 0 : c = Fl_powu(c, dropb, p);
2026 0 : if (c != 1) H = Fl_mul(H, c, p);
2027 : }
2028 3966136 : dp = dB ? umodiu(dB, p): 1;
2029 3966136 : if (dp != 1) H = Fl_mul(H, Fl_powu(Fl_inv(dp,p), degA, p), p);
2030 3966127 : return H;
2031 : }
2032 :
2033 : /* If B=NULL, assume B=A' */
2034 : static GEN
2035 1628666 : ZX_resultant_slice(GEN A, GEN B, GEN dB, GEN P, GEN *mod)
2036 : {
2037 1628666 : pari_sp av = avma, av2;
2038 1628666 : long degA, degB, i, n = lg(P)-1;
2039 : GEN H, T;
2040 :
2041 1628666 : degA = degpol(A);
2042 1628671 : degB = B? degpol(B): degA - 1;
2043 1628674 : if (n == 1)
2044 : {
2045 943056 : ulong Hp, p = uel(P,1);
2046 943056 : GEN a = ZX_to_Flx(A, p), b = B? ZX_to_Flx(B, p): Flx_deriv(a, p);
2047 943041 : Hp = ZX_resultant_prime(a, b, dB, degA, degB, p);
2048 943093 : set_avma(av); *mod = utoipos(p); return utoi(Hp);
2049 : }
2050 685618 : T = ZV_producttree(P);
2051 685616 : A = ZX_nv_mod_tree(A, P, T);
2052 685619 : if (B) B = ZX_nv_mod_tree(B, P, T);
2053 685619 : H = cgetg(n+1, t_VECSMALL); av2 = avma;
2054 3708707 : for(i=1; i <= n; i++, set_avma(av2))
2055 : {
2056 3023101 : ulong p = P[i];
2057 3023101 : GEN a = gel(A,i), b = B? gel(B,i): Flx_deriv(a, p);
2058 3023414 : H[i] = ZX_resultant_prime(a, b, dB, degA, degB, p);
2059 : }
2060 685606 : H = ZV_chinese_tree(H, P, T, ZV_chinesetree(P,T));
2061 685617 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2062 : }
2063 :
2064 : GEN
2065 1628708 : ZX_resultant_worker(GEN P, GEN A, GEN B, GEN dB)
2066 : {
2067 1628708 : GEN V = cgetg(3, t_VEC);
2068 1628669 : if (typ(B) == t_INT) B = NULL;
2069 1628669 : if (!signe(dB)) dB = NULL;
2070 1628669 : gel(V,1) = ZX_resultant_slice(A, B, dB, P, &gel(V,2));
2071 1628713 : return V;
2072 : }
2073 :
2074 : /* Compute Res(A, B/dB) in Z, assuming A,B in Z[X], dB in Z or NULL (= 1)
2075 : * If B=NULL, take B = A' and assume deg A > 1 and 'bound' is set */
2076 : GEN
2077 1298377 : ZX_resultant_all(GEN A, GEN B, GEN dB, ulong bound)
2078 : {
2079 1298377 : pari_sp av = avma;
2080 : forprime_t S;
2081 : GEN H, worker;
2082 1298377 : if (B)
2083 : {
2084 108824 : long a = degpol(A), b = degpol(B);
2085 108824 : if (a < 0 || b < 0) return gen_0;
2086 108794 : if (!a) return powiu(gel(A,2), b);
2087 108794 : if (!b) return powiu(gel(B,2), a);
2088 107049 : if (!bound) bound = ZX_ZXY_ResBound(A, B, dB);
2089 : }
2090 1296599 : worker = snm_closure(is_entry("_ZX_resultant_worker"),
2091 : mkvec3(A, B? B: gen_0, dB? dB: gen_0));
2092 1296679 : init_modular_big(&S);
2093 1296642 : H = gen_crt("ZX_resultant_all", worker, &S, dB, bound, 0, NULL,
2094 : ZV_chinese_center, Fp_center);
2095 1296646 : return gerepileuptoint(av, H);
2096 : }
2097 :
2098 : /* A0 and B0 in Q[X] */
2099 : GEN
2100 56 : QX_resultant(GEN A0, GEN B0)
2101 : {
2102 : GEN s, a, b, A, B;
2103 56 : pari_sp av = avma;
2104 :
2105 56 : A = Q_primitive_part(A0, &a);
2106 56 : B = Q_primitive_part(B0, &b);
2107 56 : s = ZX_resultant(A, B);
2108 56 : if (!signe(s)) { set_avma(av); return gen_0; }
2109 56 : if (a) s = gmul(s, gpowgs(a,degpol(B)));
2110 56 : if (b) s = gmul(s, gpowgs(b,degpol(A)));
2111 56 : return gerepileupto(av, s);
2112 : }
2113 :
2114 : GEN
2115 25312 : ZX_resultant(GEN A, GEN B) { return ZX_resultant_all(A,B,NULL,0); }
2116 :
2117 : GEN
2118 0 : QXQ_intnorm(GEN A, GEN B)
2119 : {
2120 : GEN c, n, R, lB;
2121 0 : long dA = degpol(A), dB = degpol(B);
2122 0 : pari_sp av = avma;
2123 0 : if (dA < 0) return gen_0;
2124 0 : A = Q_primitive_part(A, &c);
2125 0 : if (!c || typ(c) == t_INT) {
2126 0 : n = c;
2127 0 : R = ZX_resultant(B, A);
2128 : } else {
2129 0 : n = gel(c,1);
2130 0 : R = ZX_resultant_all(B, A, gel(c,2), 0);
2131 : }
2132 0 : if (n && !equali1(n)) R = mulii(R, powiu(n, dB));
2133 0 : lB = leading_coeff(B);
2134 0 : if (!equali1(lB)) R = diviiexact(R, powiu(lB, dA));
2135 0 : return gerepileuptoint(av, R);
2136 : }
2137 :
2138 : GEN
2139 18858 : QXQ_norm(GEN A, GEN B)
2140 : {
2141 : GEN c, R, lB;
2142 18858 : long dA = degpol(A), dB = degpol(B);
2143 18858 : pari_sp av = avma;
2144 18858 : if (dA < 0) return gen_0;
2145 18858 : A = Q_primitive_part(A, &c);
2146 18858 : R = ZX_resultant(B, A);
2147 18858 : if (c) R = gmul(R, gpowgs(c, dB));
2148 18858 : lB = leading_coeff(B);
2149 18858 : if (!equali1(lB)) R = gdiv(R, gpowgs(lB, dA));
2150 18858 : return gerepileupto(av, R);
2151 : }
2152 :
2153 : /* assume x has integral coefficients */
2154 : GEN
2155 1192656 : ZX_disc_all(GEN x, ulong bound)
2156 : {
2157 1192656 : pari_sp av = avma;
2158 1192656 : long s, d = degpol(x);
2159 : GEN l, R;
2160 :
2161 1192654 : if (d <= 1) return d == 1? gen_1: gen_0;
2162 1189599 : s = (d & 2) ? -1: 1;
2163 1189599 : l = leading_coeff(x);
2164 1189598 : if (!bound) bound = ZX_discbound(x);
2165 1189542 : R = ZX_resultant_all(x, NULL, NULL, bound);
2166 1189601 : if (is_pm1(l))
2167 1014312 : { if (signe(l) < 0) s = -s; }
2168 : else
2169 175278 : R = diviiexact(R,l);
2170 1189590 : if (s == -1) togglesign_safe(&R);
2171 1189578 : return gerepileuptoint(av,R);
2172 : }
2173 :
2174 : GEN
2175 1130693 : ZX_disc(GEN x) { return ZX_disc_all(x,0); }
2176 :
2177 : static GEN
2178 8557 : ZXQX_resultant_prime(GEN a, GEN b, GEN dB, long degA, long degB, GEN T, ulong p)
2179 : {
2180 8557 : long dropa = degA - degpol(a), dropb = degB - degpol(b);
2181 : GEN H, dp;
2182 8557 : if (dropa && dropb) return pol0_Flx(T[1]); /* p | lc(A), p | lc(B) */
2183 8557 : H = FlxqX_saferesultant(a, b, T, p);
2184 8556 : if (!H) return NULL;
2185 8556 : if (dropa)
2186 : { /* multiply by ((-1)^deg B lc(B))^(deg A - deg a) */
2187 0 : GEN c = gel(b,degB+2); /* lc(B) */
2188 0 : if (odd(degB)) c = Flx_neg(c, p);
2189 0 : c = Flxq_powu(c, dropa, T, p);
2190 0 : if (!Flx_equal1(c)) H = Flxq_mul(H, c, T, p);
2191 : }
2192 8556 : else if (dropb)
2193 : { /* multiply by lc(A)^(deg B - deg b) */
2194 0 : GEN c = gel(a,degA+2); /* lc(A) */
2195 0 : c = Flxq_powu(c, dropb, T, p);
2196 0 : if (!Flx_equal1(c)) H = Flxq_mul(H, c, T, p);
2197 : }
2198 8556 : dp = dB ? ZX_to_Flx(dB, p): pol1_Flx(T[1]);
2199 8556 : if (!Flx_equal1(dp))
2200 : {
2201 0 : GEN idp = Flxq_invsafe(dp, T, p);
2202 0 : if (!idp) return NULL;
2203 0 : H = Flxq_mul(H, Flxq_powu(idp, degA, T, p), T, p);
2204 : }
2205 8556 : return H;
2206 : }
2207 :
2208 : /* If B=NULL, assume B=A' */
2209 : static GEN
2210 3814 : ZXQX_resultant_slice(GEN A, GEN B, GEN U, GEN dB, GEN P, GEN *mod)
2211 : {
2212 3814 : pari_sp av = avma;
2213 3814 : long degA, degB, i, n = lg(P)-1;
2214 : GEN H, T;
2215 3814 : long v = varn(U), redo = 0;
2216 :
2217 3814 : degA = degpol(A);
2218 3814 : degB = B? degpol(B): degA - 1;
2219 3814 : if (n == 1)
2220 : {
2221 2311 : ulong p = uel(P,1);
2222 2311 : GEN a = ZXX_to_FlxX(A, p, v), b = B? ZXX_to_FlxX(B, p, v): FlxX_deriv(a, p);
2223 2311 : GEN u = ZX_to_Flx(U, p);
2224 2311 : GEN Hp = ZXQX_resultant_prime(a, b, dB, degA, degB, u, p);
2225 2311 : if (!Hp) { set_avma(av); *mod = gen_1; return pol_0(v); }
2226 2311 : Hp = gerepileupto(av, Flx_to_ZX(Hp)); *mod = utoipos(p); return Hp;
2227 : }
2228 1503 : T = ZV_producttree(P);
2229 1503 : A = ZXX_nv_mod_tree(A, P, T, v);
2230 1503 : if (B) B = ZXX_nv_mod_tree(B, P, T, v);
2231 1503 : U = ZX_nv_mod_tree(U, P, T);
2232 1503 : H = cgetg(n+1, t_VEC);
2233 7748 : for(i=1; i <= n; i++)
2234 : {
2235 6245 : ulong p = P[i];
2236 6245 : GEN a = gel(A,i), b = B? gel(B,i): FlxX_deriv(a, p), u = gel(U, i);
2237 6245 : GEN h = ZXQX_resultant_prime(a, b, dB, degA, degB, u, p);
2238 6245 : if (!h)
2239 : {
2240 0 : gel(H,i) = pol_0(v);
2241 0 : P[i] = 1; redo = 1;
2242 : }
2243 : else
2244 6245 : gel(H,i) = h;
2245 : }
2246 1503 : if (redo) T = ZV_producttree(P);
2247 1503 : H = nxV_chinese_center_tree(H, P, T, ZV_chinesetree(P, T));
2248 1503 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2249 : }
2250 :
2251 : GEN
2252 3814 : ZXQX_resultant_worker(GEN P, GEN A, GEN B, GEN T, GEN dB)
2253 : {
2254 3814 : GEN V = cgetg(3, t_VEC);
2255 3814 : if (isintzero(B)) B = NULL;
2256 3814 : if (!signe(dB)) dB = NULL;
2257 3814 : gel(V,1) = ZXQX_resultant_slice(A, B, T, dB, P, &gel(V,2));
2258 3814 : return V;
2259 : }
2260 :
2261 : static ulong
2262 3349 : ZXQX_resultant_bound_i(GEN nf, GEN A, GEN B, GEN (*f)(GEN,GEN,long))
2263 : {
2264 3349 : pari_sp av = avma;
2265 3349 : GEN r, M = nf_L2_bound(nf, NULL, &r);
2266 3349 : long v = nf_get_varn(nf), i, l = lg(r);
2267 3349 : GEN a = cgetg(l, t_COL);
2268 10548 : for (i = 1; i < l; i++)
2269 7199 : gel(a, i) = f(gsubst(A, v, gel(r,i)), gsubst(B, v, gel(r,i)), DEFAULTPREC);
2270 3349 : return gc_ulong(av, (ulong) dbllog2(gmul(M,RgC_fpnorml2(a, DEFAULTPREC))));
2271 : }
2272 : static ulong
2273 3041 : ZXQX_resultant_bound(GEN nf, GEN A, GEN B)
2274 3041 : { return ZXQX_resultant_bound_i(nf, A, B, &RgX_RgXY_ResBound); }
2275 :
2276 : static GEN
2277 56 : _ZXQ_powu(GEN x, ulong u, GEN T)
2278 56 : { return typ(x) == t_INT? powiu(x, u): ZXQ_powu(x, u, T); }
2279 :
2280 : /* Compute Res(A, B/dB) in Z[X]/T, assuming A,B in Z[X,Y], dB in Z or NULL (= 1)
2281 : * If B=NULL, take B = A' and assume deg A > 1 */
2282 : static GEN
2283 3038 : ZXQX_resultant_all(GEN A, GEN B, GEN T, GEN dB, ulong bound)
2284 : {
2285 3038 : pari_sp av = avma;
2286 : forprime_t S;
2287 : GEN H, worker;
2288 3038 : if (B)
2289 : {
2290 49 : long a = degpol(A), b = degpol(B);
2291 49 : if (a < 0 || b < 0) return gen_0;
2292 49 : if (!a) return _ZXQ_powu(gel(A,2), b, T);
2293 49 : if (!b) return _ZXQ_powu(gel(B,2), a, T);
2294 : } else
2295 2989 : if (!bound) B = RgX_deriv(A);
2296 3038 : if (!bound) bound = ZXQX_resultant_bound(nfinit(T, DEFAULTPREC), A, B);
2297 3038 : worker = snm_closure(is_entry("_ZXQX_resultant_worker"),
2298 : mkvec4(A, B? B: gen_0, T, dB? dB: gen_0));
2299 3038 : init_modular_big(&S);
2300 3038 : H = gen_crt("ZXQX_resultant_all", worker, &S, dB, bound, 0, NULL,
2301 : nxV_chinese_center, FpX_center);
2302 3038 : if (DEBUGLEVEL)
2303 0 : err_printf("ZXQX_resultant_all: a priori bound: %lu, a posteriori: %lu\n",
2304 : bound, expi(gsupnorm(H, DEFAULTPREC)));
2305 3038 : return gerepileupto(av, H);
2306 : }
2307 :
2308 : GEN
2309 105 : nfX_resultant(GEN nf, GEN x, GEN y)
2310 : {
2311 105 : pari_sp av = avma;
2312 105 : GEN cx, cy, D, T = nf_get_pol(nf);
2313 105 : long dx = degpol(x), dy = degpol(y);
2314 105 : if (dx < 0 || dy < 0) return gen_0;
2315 105 : x = Q_primitive_part(x, &cx); if (cx) cx = gpowgs(cx, dy);
2316 105 : y = Q_primitive_part(y, &cy); if (cy) cy = gpowgs(cy, dx);
2317 105 : if (!dx) D = _ZXQ_powu(gel(x,2), dy, T);
2318 105 : else if (!dy) D = _ZXQ_powu(gel(y,2), dx, T);
2319 : else
2320 : {
2321 49 : ulong bound = ZXQX_resultant_bound(nf, x, y);
2322 49 : D = ZXQX_resultant_all(x, y, T, NULL, bound);
2323 : }
2324 105 : cx = mul_content(cx, cy); if (cx) D = gmul(D, cx);
2325 105 : return gerepileupto(av, D);
2326 : }
2327 :
2328 : static GEN
2329 217 : to_ZX(GEN a, long v) { return typ(a)==t_INT? scalarpol(a,v): a; }
2330 :
2331 : static GEN
2332 2989 : ZXQX_disc_all(GEN x, GEN T, ulong bound)
2333 : {
2334 2989 : pari_sp av = avma;
2335 2989 : long s, d = degpol(x), v = varn(T);
2336 : GEN l, R;
2337 :
2338 2989 : if (d <= 1) return d == 1? pol_1(v): pol_0(v);
2339 2989 : s = (d & 2) ? -1: 1;
2340 2989 : l = leading_coeff(x);
2341 2989 : R = ZXQX_resultant_all(x, NULL, T, NULL, bound);
2342 2989 : if (!gequal1(l)) R = QXQ_div(R, to_ZX(l,v), T);
2343 2989 : if (s == -1) R = RgX_neg(R);
2344 2989 : return gerepileupto(av, R);
2345 : }
2346 :
2347 : GEN
2348 7 : QX_disc(GEN x)
2349 : {
2350 7 : pari_sp av = avma;
2351 7 : GEN c, d = ZX_disc( Q_primitive_part(x, &c) );
2352 7 : if (c) d = gmul(d, gpowgs(c, 2*degpol(x) - 2));
2353 7 : return gerepileupto(av, d);
2354 : }
2355 :
2356 : GEN
2357 3150 : nfX_disc(GEN nf, GEN x)
2358 : {
2359 3150 : pari_sp av = avma;
2360 3150 : GEN c, D, T = nf_get_pol(nf);
2361 : ulong bound;
2362 3150 : long d = degpol(x), v = varn(T);
2363 3150 : if (d <= 1) return d == 1? pol_1(v): pol_0(v);
2364 2989 : x = Q_primitive_part(x, &c);
2365 2989 : bound = ZXQX_resultant_bound(nf, x, RgX_deriv(x));
2366 2989 : D = ZXQX_disc_all(x, T, bound);
2367 2989 : if (c) D = gmul(D, gpowgs(c, 2*d - 2));
2368 2989 : return gerepileupto(av, D);
2369 : }
2370 :
2371 : GEN
2372 559723 : QXQ_mul(GEN x, GEN y, GEN T)
2373 : {
2374 559723 : GEN dx, nx = Q_primitive_part(x, &dx);
2375 559724 : GEN dy, ny = Q_primitive_part(y, &dy);
2376 559722 : GEN z = ZXQ_mul(nx, ny, T);
2377 559723 : if (dx || dy)
2378 : {
2379 556664 : GEN d = dx ? dy ? gmul(dx, dy): dx : dy;
2380 556664 : if (!gequal1(d)) z = ZX_Q_mul(z, d);
2381 : }
2382 559725 : return z;
2383 : }
2384 :
2385 : GEN
2386 93204 : QXQ_sqr(GEN x, GEN T)
2387 : {
2388 93204 : GEN dx, nx = Q_primitive_part(x, &dx);
2389 93204 : GEN z = ZXQ_sqr(nx, T);
2390 93204 : if (dx)
2391 91377 : z = ZX_Q_mul(z, gsqr(dx));
2392 93204 : return z;
2393 : }
2394 :
2395 : static GEN
2396 102266 : QXQ_inv_slice(GEN A, GEN B, GEN P, GEN *mod)
2397 : {
2398 102266 : pari_sp av = avma;
2399 102266 : long i, n = lg(P)-1, v = varn(A), redo = 0;
2400 : GEN H, T;
2401 102266 : if (n == 1)
2402 : {
2403 57002 : ulong p = uel(P,1);
2404 57002 : GEN a = ZX_to_Flx(A, p), b = ZX_to_Flx(B, p);
2405 57002 : GEN U = Flxq_invsafe(a, b, p);
2406 57002 : if (!U)
2407 : {
2408 24 : set_avma(av);
2409 24 : *mod = gen_1; return pol_0(v);
2410 : }
2411 56978 : H = gerepilecopy(av, Flx_to_ZX(U));
2412 56978 : *mod = utoipos(p); return H;
2413 : }
2414 45264 : T = ZV_producttree(P);
2415 45264 : A = ZX_nv_mod_tree(A, P, T);
2416 45264 : B = ZX_nv_mod_tree(B, P, T);
2417 45264 : H = cgetg(n+1, t_VEC);
2418 224993 : for(i=1; i <= n; i++)
2419 : {
2420 179729 : ulong p = P[i];
2421 179729 : GEN a = gel(A,i), b = gel(B,i);
2422 179729 : GEN U = Flxq_invsafe(a, b, p);
2423 179728 : if (!U)
2424 : {
2425 601 : gel(H,i) = pol_0(v);
2426 601 : P[i] = 1; redo = 1;
2427 : }
2428 : else
2429 179127 : gel(H,i) = U;
2430 : }
2431 45264 : if (redo) T = ZV_producttree(P);
2432 45264 : H = nxV_chinese_center_tree(H, P, T, ZV_chinesetree(P, T));
2433 45264 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2434 : }
2435 :
2436 : GEN
2437 102266 : QXQ_inv_worker(GEN P, GEN A, GEN B)
2438 : {
2439 102266 : GEN V = cgetg(3, t_VEC);
2440 102266 : gel(V,1) = QXQ_inv_slice(A, B, P, &gel(V,2));
2441 102266 : return V;
2442 : }
2443 :
2444 : /* lift(1 / Mod(A,B)). B a ZX, A a scalar or a QX */
2445 : GEN
2446 38008 : QXQ_inv(GEN A, GEN B)
2447 : {
2448 : GEN D, Ap, Bp;
2449 : ulong pp;
2450 38008 : pari_sp av2, av = avma;
2451 : forprime_t S;
2452 38008 : GEN worker, U, H = NULL, mod = gen_1;
2453 : pari_timer ti;
2454 : long k, dA, dB;
2455 38008 : if (is_scalar_t(typ(A))) return scalarpol(ginv(A), varn(B));
2456 : /* A a QX, B a ZX */
2457 38008 : A = Q_primitive_part(A, &D);
2458 38008 : dA = degpol(A); dB= degpol(B);
2459 : /* A, B in Z[X] */
2460 38008 : init_modular_small(&S);
2461 : do {
2462 38008 : pp = u_forprime_next(&S);
2463 38008 : Ap = ZX_to_Flx(A, pp);
2464 38008 : Bp = ZX_to_Flx(B, pp);
2465 38008 : } while (degpol(Ap) != dA || degpol(Bp) != dB);
2466 38008 : if (degpol(Flx_gcd(Ap, Bp, pp)) != 0 && degpol(ZX_gcd(A,B))!=0)
2467 14 : pari_err_INV("QXQ_inv",mkpolmod(A,B));
2468 37994 : worker = snm_closure(is_entry("_QXQ_inv_worker"), mkvec2(A, B));
2469 37994 : av2 = avma;
2470 37994 : for (k = 1; ;k *= 2)
2471 41011 : {
2472 : GEN res, b, N, den;
2473 79005 : gen_inccrt_i("QXQ_inv", worker, NULL, (k+1)>>1, 0, &S, &H, &mod,
2474 : nxV_chinese_center, FpX_center);
2475 79004 : gerepileall(av2, 2, &H, &mod);
2476 79005 : b = sqrti(shifti(mod,-1));
2477 79003 : if (DEBUGLEVEL>5) timer_start(&ti);
2478 79003 : U = FpX_ratlift(H, mod, b, b, NULL);
2479 79005 : if (DEBUGLEVEL>5) timer_printf(&ti,"QXQ_inv: ratlift");
2480 84573 : if (!U) continue;
2481 43562 : N = Q_remove_denom(U, &den); if (!den) den = gen_1;
2482 43562 : res = Flx_rem(Flx_Fl_sub(Flx_mul(Ap, ZX_to_Flx(N,pp), pp),
2483 : umodiu(den, pp), pp), Bp, pp);
2484 43561 : if (degpol(res) >= 0) continue;
2485 37993 : res = ZX_Z_sub(ZX_mul(A, N), den);
2486 37994 : res = ZX_is_monic(B) ? ZX_rem(res, B): RgX_pseudorem(res, B);
2487 37994 : if (DEBUGLEVEL>5) timer_printf(&ti,"QXQ_inv: final check");
2488 37994 : if (degpol(res)<0)
2489 : {
2490 37994 : if (D) U = RgX_Rg_div(U, D);
2491 37994 : return gerepilecopy(av, U);
2492 : }
2493 : }
2494 : }
2495 :
2496 : static GEN
2497 117202 : QXQ_div_slice(GEN A, GEN B, GEN C, GEN P, GEN *mod)
2498 : {
2499 117202 : pari_sp av = avma;
2500 117202 : long i, n = lg(P)-1, v = varn(A), redo = 0;
2501 : GEN H, T;
2502 117202 : if (n == 1)
2503 : {
2504 42885 : ulong p = uel(P,1);
2505 42885 : GEN a = ZX_to_Flx(A, p), b = ZX_to_Flx(B, p), c = ZX_to_Flx(C, p);
2506 42885 : GEN bi = Flxq_invsafe(b, c, p), U;
2507 42885 : if (!bi)
2508 : {
2509 0 : set_avma(av);
2510 0 : *mod = gen_1; return pol_0(v);
2511 : }
2512 42885 : U = Flxq_mul(a, bi, c, p);
2513 42885 : H = gerepilecopy(av, Flx_to_ZX(U));
2514 42885 : *mod = utoipos(p); return H;
2515 : }
2516 74317 : T = ZV_producttree(P);
2517 74317 : A = ZX_nv_mod_tree(A, P, T);
2518 74317 : B = ZX_nv_mod_tree(B, P, T);
2519 74317 : C = ZX_nv_mod_tree(C, P, T);
2520 74317 : H = cgetg(n+1, t_VEC);
2521 327089 : for(i=1; i <= n; i++)
2522 : {
2523 252772 : ulong p = P[i];
2524 252772 : GEN a = gel(A,i), b = gel(B,i), c = gel(C, i);
2525 252772 : GEN bi = Flxq_invsafe(b, c, p);
2526 252772 : if (!bi)
2527 : {
2528 0 : gel(H,i) = pol_0(v);
2529 0 : P[i] = 1; redo = 1;
2530 : }
2531 : else
2532 252772 : gel(H,i) = Flxq_mul(a, bi, c, p);
2533 : }
2534 74317 : if (redo) T = ZV_producttree(P);
2535 74317 : H = nxV_chinese_center_tree(H, P, T, ZV_chinesetree(P, T));
2536 74317 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2537 : }
2538 :
2539 : GEN
2540 117202 : QXQ_div_worker(GEN P, GEN A, GEN B, GEN C)
2541 : {
2542 117202 : GEN V = cgetg(3, t_VEC);
2543 117202 : gel(V,1) = QXQ_div_slice(A, B, C, P, &gel(V,2));
2544 117202 : return V;
2545 : }
2546 :
2547 : /* lift(Mod(A/B, C)). C a ZX, A, B a scalar or a QX */
2548 : GEN
2549 31758 : QXQ_div(GEN A, GEN B, GEN C)
2550 : {
2551 : GEN DA, DB, Ap, Bp, Cp;
2552 : ulong pp;
2553 31758 : pari_sp av2, av = avma;
2554 : forprime_t S;
2555 31758 : GEN worker, U, H = NULL, mod = gen_1;
2556 : pari_timer ti;
2557 : long k, dA, dB, dC;
2558 31758 : if (is_scalar_t(typ(A))) return scalarpol(ginv(A), varn(B));
2559 : /* A a QX, B a ZX */
2560 31758 : A = Q_primitive_part(A, &DA);
2561 31758 : B = Q_primitive_part(B, &DB);
2562 31757 : dA = degpol(A); dB = degpol(B); dC = degpol(C);
2563 : /* A, B in Z[X] */
2564 31758 : init_modular_small(&S);
2565 : do {
2566 31758 : pp = u_forprime_next(&S);
2567 31758 : Ap = ZX_to_Flx(A, pp);
2568 31758 : Bp = ZX_to_Flx(B, pp);
2569 31758 : Cp = ZX_to_Flx(C, pp);
2570 31758 : } while (degpol(Ap) != dA || degpol(Bp) != dB || degpol(Cp) != dC);
2571 31758 : if (degpol(Flx_gcd(Bp, Cp, pp)) != 0 && degpol(ZX_gcd(B,C))!=0)
2572 0 : pari_err_INV("QXQ_div",mkpolmod(B,C));
2573 31758 : worker = snm_closure(is_entry("_QXQ_div_worker"), mkvec3(A, B, C));
2574 31758 : av2 = avma;
2575 31758 : for (k = 1; ;k *= 2)
2576 45440 : {
2577 : GEN res, b, N, den;
2578 77198 : gen_inccrt_i("QXQ_div", worker, NULL, (k+1)>>1, 0, &S, &H, &mod,
2579 : nxV_chinese_center, FpX_center);
2580 77198 : gerepileall(av2, 2, &H, &mod);
2581 77198 : b = sqrti(shifti(mod,-1));
2582 77198 : if (DEBUGLEVEL>5) timer_start(&ti);
2583 77198 : U = FpX_ratlift(H, mod, b, b, NULL);
2584 77198 : if (DEBUGLEVEL>5) timer_printf(&ti,"QXQ_div: ratlift");
2585 87609 : if (!U) continue;
2586 42169 : N = Q_remove_denom(U, &den); if (!den) den = gen_1;
2587 42169 : res = Flx_rem(Flx_sub(Flx_mul(Bp, ZX_to_Flx(N,pp), pp),
2588 : Flx_Fl_mul(Ap, umodiu(den, pp), pp), pp), Cp, pp);
2589 42169 : if (degpol(res) >= 0) continue;
2590 31758 : res = ZX_sub(ZX_mul(B, N), ZX_Z_mul(A,den));
2591 31758 : res = ZX_is_monic(C) ? ZX_rem(res, C): RgX_pseudorem(res, C);
2592 31758 : if (DEBUGLEVEL>5) timer_printf(&ti,"QXQ_div: final check");
2593 31758 : if (degpol(res)<0)
2594 : {
2595 31758 : if (DA && DB) U = RgX_Rg_mul(U, gdiv(DA,DB));
2596 26977 : else if (DA) U = RgX_Rg_mul(U, DA);
2597 15183 : else if (DB) U = RgX_Rg_div(U, DB);
2598 31758 : return gerepilecopy(av, U);
2599 : }
2600 : }
2601 : }
2602 :
2603 : /************************************************************************
2604 : * *
2605 : * ZXQ_minpoly *
2606 : * *
2607 : ************************************************************************/
2608 :
2609 : static GEN
2610 3523 : ZXQ_minpoly_slice(GEN A, GEN B, long d, GEN P, GEN *mod)
2611 : {
2612 3523 : pari_sp av = avma;
2613 3523 : long i, n = lg(P)-1, v = evalvarn(varn(B));
2614 : GEN H, T;
2615 3523 : if (n == 1)
2616 : {
2617 716 : ulong p = uel(P,1);
2618 716 : GEN a = ZX_to_Flx(A, p), b = ZX_to_Flx(B, p);
2619 716 : GEN Hp = Flxq_minpoly(a, b, p);
2620 716 : if (degpol(Hp) != d) { p = 1; Hp = pol0_Flx(v); }
2621 716 : H = gerepileupto(av, Flx_to_ZX(Hp));
2622 716 : *mod = utoipos(p); return H;
2623 : }
2624 2807 : T = ZV_producttree(P);
2625 2807 : A = ZX_nv_mod_tree(A, P, T);
2626 2807 : B = ZX_nv_mod_tree(B, P, T);
2627 2807 : H = cgetg(n+1, t_VEC);
2628 16838 : for(i=1; i <= n; i++)
2629 : {
2630 14031 : ulong p = P[i];
2631 14031 : GEN a = gel(A,i), b = gel(B,i);
2632 14031 : GEN m = Flxq_minpoly(a, b, p);
2633 14031 : if (degpol(m) != d) { P[i] = 1; m = pol0_Flx(v); }
2634 14031 : gel(H, i) = m;
2635 : }
2636 2807 : H = nxV_chinese_center_tree(H, P, T, ZV_chinesetree(P, T));
2637 2807 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2638 : }
2639 :
2640 : GEN
2641 3523 : ZXQ_minpoly_worker(GEN P, GEN A, GEN B, long d)
2642 : {
2643 3523 : GEN V = cgetg(3, t_VEC);
2644 3523 : gel(V,1) = ZXQ_minpoly_slice(A, B, d, P, &gel(V,2));
2645 3523 : return V;
2646 : }
2647 :
2648 : GEN
2649 1701 : ZXQ_minpoly(GEN A, GEN B, long d, ulong bound)
2650 : {
2651 1701 : pari_sp av = avma;
2652 : GEN worker, H, dB;
2653 : forprime_t S;
2654 1701 : B = Q_remove_denom(B, &dB);
2655 1701 : worker = strtoclosure("_ZXQ_minpoly_worker", 3, A, B, stoi(d));
2656 1701 : init_modular_big(&S);
2657 1701 : H = gen_crt("ZXQ_minpoly", worker, &S, dB, bound, 0, NULL,
2658 : nxV_chinese_center, FpX_center_i);
2659 1701 : return gerepilecopy(av, H);
2660 : }
2661 :
2662 : /************************************************************************
2663 : * *
2664 : * ZX_ZXY_resultant *
2665 : * *
2666 : ************************************************************************/
2667 :
2668 : static GEN
2669 173515 : ZX_ZXY_resultant_prime(GEN a, GEN b, ulong dp, ulong p,
2670 : long degA, long degB, long dres, long sX)
2671 : {
2672 173515 : long dropa = degA - degpol(a), dropb = degB - degpol(b);
2673 173513 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2674 173513 : GEN Hp = Flx_FlxY_resultant_polint(a, b, p, pi, dres, sX);
2675 173515 : if (dropa && dropb)
2676 0 : Hp = zero_Flx(sX);
2677 : else {
2678 173515 : if (dropa)
2679 : { /* multiply by ((-1)^deg B lc(B))^(deg A - deg a) */
2680 0 : GEN c = gel(b,degB+2); /* lc(B) */
2681 0 : if (odd(degB)) c = Flx_neg(c, p);
2682 0 : if (!Flx_equal1(c)) {
2683 0 : c = Flx_powu_pre(c, dropa, p, pi);
2684 0 : if (!Flx_equal1(c)) Hp = Flx_mul_pre(Hp, c, p, pi);
2685 : }
2686 : }
2687 173515 : else if (dropb)
2688 : { /* multiply by lc(A)^(deg B - deg b) */
2689 0 : ulong c = uel(a, degA+2); /* lc(A) */
2690 0 : c = Fl_powu(c, dropb, p);
2691 0 : if (c != 1) Hp = Flx_Fl_mul_pre(Hp, c, p, pi);
2692 : }
2693 : }
2694 173515 : if (dp != 1) Hp = Flx_Fl_mul_pre(Hp, Fl_powu_pre(Fl_inv(dp,p), degA, p, pi), p, pi);
2695 173515 : return Hp;
2696 : }
2697 :
2698 : static GEN
2699 69332 : ZX_ZXY_resultant_slice(GEN A, GEN B, GEN dB, long degA, long degB, long dres,
2700 : GEN P, GEN *mod, long sX, long vY)
2701 : {
2702 69332 : pari_sp av = avma;
2703 69332 : long i, n = lg(P)-1;
2704 : GEN H, T, D;
2705 69332 : if (n == 1)
2706 : {
2707 40101 : ulong p = uel(P,1);
2708 40101 : ulong dp = dB ? umodiu(dB, p): 1;
2709 40101 : GEN a = ZX_to_Flx(A, p), b = ZXX_to_FlxX(B, p, vY);
2710 40100 : GEN Hp = ZX_ZXY_resultant_prime(a, b, dp, p, degA, degB, dres, sX);
2711 40102 : H = gerepileupto(av, Flx_to_ZX(Hp));
2712 40102 : *mod = utoipos(p); return H;
2713 : }
2714 29231 : T = ZV_producttree(P);
2715 29231 : A = ZX_nv_mod_tree(A, P, T);
2716 29231 : B = ZXX_nv_mod_tree(B, P, T, vY);
2717 29231 : D = dB ? Z_ZV_mod_tree(dB, P, T): NULL;
2718 29231 : H = cgetg(n+1, t_VEC);
2719 117397 : for(i=1; i <= n; i++)
2720 : {
2721 88167 : ulong p = P[i];
2722 88167 : GEN a = gel(A,i), b = gel(B,i);
2723 88167 : ulong dp = D ? uel(D, i): 1;
2724 88167 : gel(H,i) = ZX_ZXY_resultant_prime(a, b, dp, p, degA, degB, dres, sX);
2725 : }
2726 29230 : H = nxV_chinese_center_tree(H, P, T, ZV_chinesetree(P, T));
2727 29231 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2728 : }
2729 :
2730 : GEN
2731 69332 : ZX_ZXY_resultant_worker(GEN P, GEN A, GEN B, GEN dB, GEN v)
2732 : {
2733 69332 : GEN V = cgetg(3, t_VEC);
2734 69333 : if (isintzero(dB)) dB = NULL;
2735 69332 : gel(V,1) = ZX_ZXY_resultant_slice(A, B, dB, v[1], v[2], v[3], P, &gel(V,2), v[4], v[5]);
2736 69332 : return V;
2737 : }
2738 :
2739 : GEN
2740 60043 : ZX_ZXY_resultant(GEN A, GEN B)
2741 : {
2742 60043 : pari_sp av = avma;
2743 : forprime_t S;
2744 : ulong bound;
2745 60043 : long v = fetch_var_higher();
2746 60043 : long degA = degpol(A), degB, dres = degA * degpol(B);
2747 60043 : long vX = varn(B), vY = varn(A); /* assume vY has lower priority */
2748 60043 : long sX = evalvarn(vX);
2749 : GEN worker, H, dB;
2750 60043 : B = Q_remove_denom(B, &dB);
2751 60043 : if (!dB) B = leafcopy(B);
2752 60043 : A = leafcopy(A); setvarn(A,v);
2753 60043 : B = swap_vars(B, vY); setvarn(B,v); degB = degpol(B);
2754 60043 : bound = ZX_ZXY_ResBound(A, B, dB);
2755 60041 : if (DEBUGLEVEL>4) err_printf("bound for resultant coeffs: 2^%ld\n",bound);
2756 120082 : worker = snm_closure(is_entry("_ZX_ZXY_resultant_worker"),
2757 60041 : mkvec4(A, B, dB? dB: gen_0,
2758 : mkvecsmall5(degA, degB, dres, sX, vY)));
2759 60043 : init_modular_big(&S);
2760 60042 : H = gen_crt("ZX_ZXY_resultant_all", worker, &S, dB, bound, 0, NULL,
2761 : nxV_chinese_center, FpX_center_i);
2762 60043 : setvarn(H, vX); (void)delete_var();
2763 60043 : return gerepilecopy(av, H);
2764 : }
2765 :
2766 : static long
2767 40488 : ZX_ZXY_rnfequation_lambda(GEN A, GEN B0, long lambda)
2768 : {
2769 40488 : pari_sp av = avma;
2770 40488 : long degA = degpol(A), degB, dres = degA*degpol(B0);
2771 40488 : long v = fetch_var_higher();
2772 40488 : long vX = varn(B0), vY = varn(A); /* assume vY has lower priority */
2773 40488 : long sX = evalvarn(vX);
2774 : GEN dB, B, a, b, Hp;
2775 : forprime_t S;
2776 :
2777 40488 : B0 = Q_remove_denom(B0, &dB);
2778 40487 : if (!dB) B0 = leafcopy(B0);
2779 40488 : A = leafcopy(A);
2780 40488 : B = B0;
2781 40488 : setvarn(A,v);
2782 45248 : INIT:
2783 45248 : if (lambda) B = RgX_translate(B0, monomial(stoi(lambda), 1, vY));
2784 45248 : B = swap_vars(B, vY); setvarn(B,v);
2785 : /* B0(lambda v + x, v) */
2786 45248 : if (DEBUGLEVEL>4) err_printf("Trying lambda = %ld\n", lambda);
2787 :
2788 45248 : degB = degpol(B);
2789 45248 : init_modular_big(&S);
2790 : while (1)
2791 0 : {
2792 45248 : ulong p = u_forprime_next(&S);
2793 45248 : ulong dp = dB ? umodiu(dB, p): 1;
2794 45248 : if (!dp) continue;
2795 45248 : a = ZX_to_Flx(A, p);
2796 45248 : b = ZXX_to_FlxX(B, p, v);
2797 45248 : Hp = ZX_ZXY_resultant_prime(a, b, dp, p, degA, degB, dres, sX);
2798 45247 : if (degpol(Hp) != dres) continue;
2799 45247 : if (dp != 1) Hp = Flx_Fl_mul(Hp, Fl_powu(Fl_inv(dp,p), degA, p), p);
2800 45247 : if (!Flx_is_squarefree(Hp, p)) { lambda = next_lambda(lambda); goto INIT; }
2801 40488 : if (DEBUGLEVEL>4) err_printf("Final lambda = %ld\n", lambda);
2802 40488 : (void)delete_var(); return gc_long(av,lambda);
2803 : }
2804 : }
2805 :
2806 : GEN
2807 41409 : ZX_ZXY_rnfequation(GEN A, GEN B, long *lambda)
2808 : {
2809 41409 : if (lambda)
2810 : {
2811 40488 : *lambda = ZX_ZXY_rnfequation_lambda(A, B, *lambda);
2812 40488 : if (*lambda) B = RgX_translate(B, monomial(stoi(*lambda), 1, varn(A)));
2813 : }
2814 41409 : return ZX_ZXY_resultant(A,B);
2815 : }
2816 :
2817 : static GEN
2818 10359 : ZX_composedsum_slice(GEN A, GEN B, GEN P, GEN *mod)
2819 : {
2820 10359 : pari_sp av = avma;
2821 10359 : long i, n = lg(P)-1;
2822 : GEN H, T;
2823 10359 : if (n == 1)
2824 : {
2825 9860 : ulong p = uel(P,1);
2826 9860 : GEN a = ZX_to_Flx(A, p), b = ZX_to_Flx(B, p);
2827 9858 : GEN Hp = Flx_composedsum(a, b, p);
2828 9866 : H = gerepileupto(av, Flx_to_ZX(Hp));
2829 9870 : *mod = utoipos(p); return H;
2830 : }
2831 499 : T = ZV_producttree(P);
2832 502 : A = ZX_nv_mod_tree(A, P, T);
2833 502 : B = ZX_nv_mod_tree(B, P, T);
2834 502 : H = cgetg(n+1, t_VEC);
2835 4526 : for(i=1; i <= n; i++)
2836 : {
2837 4024 : ulong p = P[i];
2838 4024 : GEN a = gel(A,i), b = gel(B,i);
2839 4024 : gel(H,i) = Flx_composedsum(a, b, p);
2840 : }
2841 502 : H = nxV_chinese_center_tree(H, P, T, ZV_chinesetree(P, T));
2842 502 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2843 : }
2844 :
2845 : GEN
2846 10357 : ZX_composedsum_worker(GEN P, GEN A, GEN B)
2847 : {
2848 10357 : GEN V = cgetg(3, t_VEC);
2849 10358 : gel(V,1) = ZX_composedsum_slice(A, B, P, &gel(V,2));
2850 10366 : return V;
2851 : }
2852 :
2853 : static GEN
2854 10103 : ZX_composedsum_i(GEN A, GEN B, GEN lead)
2855 : {
2856 10103 : pari_sp av = avma;
2857 : forprime_t S;
2858 : ulong bound;
2859 : GEN H, worker, mod;
2860 10103 : if (degpol(A) < degpol(B)) swap(A, B);
2861 10102 : if (!lead) lead = mulii(leading_coeff(A),leading_coeff(B));
2862 10102 : bound = ZX_ZXY_ResBound_1(A, B);
2863 10102 : worker = snm_closure(is_entry("_ZX_composedsum_worker"), mkvec2(A,B));
2864 10101 : init_modular_big(&S);
2865 10097 : H = gen_crt("ZX_composedsum", worker, &S, lead, bound, 0, &mod,
2866 : nxV_chinese_center, FpX_center);
2867 10104 : return gerepileupto(av, H);
2868 : }
2869 :
2870 : static long
2871 9718 : ZX_compositum_lambda(GEN A, GEN B, GEN lead, long lambda)
2872 : {
2873 9718 : pari_sp av = avma;
2874 : forprime_t S;
2875 : ulong p;
2876 9718 : init_modular_big(&S);
2877 9719 : p = u_forprime_next(&S);
2878 : while (1)
2879 112 : {
2880 : GEN Hp, a;
2881 9831 : if (DEBUGLEVEL>4) err_printf("Trying lambda = %ld\n", lambda);
2882 9831 : if (lead && dvdiu(lead,p)) { p = u_forprime_next(&S); continue; }
2883 9824 : a = ZX_to_Flx(ZX_rescale(A, stoi(-lambda)), p);
2884 9816 : Hp = Flx_composedsum(a, ZX_to_Flx(B, p), p);
2885 9819 : if (!Flx_is_squarefree(Hp, p)) { lambda = next_lambda(lambda); continue; }
2886 9716 : if (DEBUGLEVEL>4) err_printf("Final lambda = %ld\n", lambda);
2887 9716 : return gc_long(av, lambda);
2888 : }
2889 : }
2890 :
2891 : GEN
2892 9718 : ZX_compositum(GEN A, GEN B, long *lambda)
2893 : {
2894 9718 : GEN lead = mulii(leading_coeff(A),leading_coeff(B));
2895 9718 : if (lambda)
2896 : {
2897 9718 : *lambda = ZX_compositum_lambda(A, B, lead, *lambda);
2898 9715 : A = ZX_rescale(A, stoi(-*lambda));
2899 : }
2900 9718 : return ZX_composedsum_i(A, B, lead);
2901 : }
2902 :
2903 : GEN
2904 385 : ZX_composedsum(GEN A, GEN B)
2905 385 : { return ZX_composedsum_i(A, B, NULL); }
2906 :
2907 : static GEN
2908 352 : ZXQX_composedsum_slice(GEN A, GEN B, GEN C, GEN P, GEN *mod)
2909 : {
2910 352 : pari_sp av = avma;
2911 352 : long i, n = lg(P)-1, dC = degpol(C), v = varn(C);
2912 : GEN H, T;
2913 352 : if (n == 1)
2914 : {
2915 174 : ulong p = uel(P,1);
2916 174 : GEN a = ZXX_to_FlxX(A, p, v), b = ZXX_to_FlxX(B, p, v);
2917 174 : GEN c = ZX_to_Flx(C, p);
2918 174 : GEN Hp = FlxX_to_Flm(FlxqX_composedsum(a, b, c, p), dC);
2919 174 : H = gerepileupto(av, Flm_to_ZM(Hp));
2920 174 : *mod = utoipos(p); return H;
2921 : }
2922 178 : T = ZV_producttree(P);
2923 178 : A = ZXX_nv_mod_tree(A, P, T, v);
2924 178 : B = ZXX_nv_mod_tree(B, P, T, v);
2925 178 : C = ZX_nv_mod_tree(C, P, T);
2926 178 : H = cgetg(n+1, t_VEC);
2927 660 : for(i=1; i <= n; i++)
2928 : {
2929 482 : ulong p = P[i];
2930 482 : GEN a = gel(A,i), b = gel(B,i), c = gel(C,i);
2931 482 : gel(H,i) = FlxX_to_Flm(FlxqX_composedsum(a, b, c, p), dC);
2932 : }
2933 178 : H = nmV_chinese_center_tree_seq(H, P, T, ZV_chinesetree(P, T));
2934 178 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2935 : }
2936 :
2937 : GEN
2938 352 : ZXQX_composedsum_worker(GEN P, GEN A, GEN B, GEN C)
2939 : {
2940 352 : GEN V = cgetg(3, t_VEC);
2941 352 : gel(V,1) = ZXQX_composedsum_slice(A, B, C, P, &gel(V,2));
2942 352 : return V;
2943 : }
2944 :
2945 : static GEN
2946 308 : ZXQX_composedsum(GEN A, GEN B, GEN T, ulong bound)
2947 : {
2948 308 : pari_sp av = avma;
2949 : forprime_t S;
2950 : GEN H, worker, mod;
2951 308 : GEN lead = mulii(Q_content(leading_coeff(A)), Q_content(leading_coeff(B)));
2952 308 : worker = snm_closure(is_entry("_ZXQX_composedsum_worker")
2953 : , mkvec3(A,B,T));
2954 308 : init_modular_big(&S);
2955 308 : H = gen_crt("ZXQX_composedsum", worker, &S, lead, bound, 0, &mod,
2956 : nmV_chinese_center, FpM_center);
2957 308 : if (DEBUGLEVEL > 4)
2958 0 : err_printf("nfcompositum: a priori bound: %lu, a posteriori: %lu\n",
2959 : bound, expi(gsupnorm(H, DEFAULTPREC)));
2960 308 : return gerepilecopy(av, RgM_to_RgXX(H, varn(A), varn(T)));
2961 : }
2962 :
2963 : static long
2964 308 : ZXQX_composedsum_bound(GEN nf, GEN A, GEN B)
2965 308 : { return ZXQX_resultant_bound_i(nf, A, B, &RgX_RgXY_ResBound_1); }
2966 :
2967 : GEN
2968 308 : nf_direct_compositum(GEN nf, GEN A, GEN B)
2969 : {
2970 308 : ulong bnd = ZXQX_composedsum_bound(nf, A, B);
2971 308 : return ZXQX_composedsum(A, B, nf_get_pol(nf), bnd);
2972 : }
2973 :
2974 : /************************************************************************
2975 : * *
2976 : * IRREDUCIBLE POLYNOMIAL / Fp *
2977 : * *
2978 : ************************************************************************/
2979 :
2980 : /* irreducible (unitary) polynomial of degree n over Fp */
2981 : GEN
2982 0 : ffinit_rand(GEN p,long n)
2983 : {
2984 0 : for(;;) {
2985 0 : pari_sp av = avma;
2986 0 : GEN pol = ZX_add(pol_xn(n, 0), random_FpX(n-1,0, p));
2987 0 : if (FpX_is_irred(pol, p)) return pol;
2988 0 : set_avma(av);
2989 : }
2990 : }
2991 :
2992 : /* return an extension of degree 2^l of F_2, assume l > 0
2993 : * Not stack clean. */
2994 : static GEN
2995 637 : ffinit_Artin_Schreier_2(long l)
2996 : {
2997 : GEN Q, T, S;
2998 : long i, v;
2999 :
3000 637 : if (l == 1) return mkvecsmall4(0,1,1,1); /*x^2 + x + 1*/
3001 588 : v = fetch_var_higher();
3002 588 : S = mkvecsmall5(0, 0, 0, 1, 1); /* y(y^2 + y) */
3003 588 : Q = mkpoln(3, pol1_Flx(0), pol1_Flx(0), S); /* x^2 + x + y(y^2+y) */
3004 587 : setvarn(Q, v);
3005 :
3006 : /* x^4+x+1, irred over F_2, minimal polynomial of a root of Q */
3007 587 : T = mkvecsmalln(6,evalvarn(v),1UL,1UL,0UL,0UL,1UL);
3008 : /* Q = x^2 + x + a(y) irred. over K = F2[y] / (T(y))
3009 : * ==> x^2 + x + a(y) b irred. over K for any root b of Q
3010 : * ==> x^2 + x + (b^2+b)b */
3011 3262 : for (i=2; i<l; i++) T = Flx_FlxY_resultant(T, Q, 2); /* minpoly(b) / F2*/
3012 588 : (void)delete_var(); T[1] = 0; return T;
3013 : }
3014 :
3015 : /* return an extension of degree p^l of F_p, assume l > 0
3016 : * Not stack clean. */
3017 : GEN
3018 994 : ffinit_Artin_Schreier(ulong p, long l)
3019 : {
3020 : long i, v;
3021 : GEN Q, R, S, T, xp;
3022 994 : if (p==2) return ffinit_Artin_Schreier_2(l);
3023 357 : xp = polxn_Flx(p,0); /* x^p */
3024 357 : T = Flx_sub(xp, mkvecsmall3(0,1,1),p); /* x^p - x - 1 */
3025 357 : if (l == 1) return T;
3026 :
3027 7 : v = evalvarn(fetch_var_higher());
3028 7 : xp[1] = v;
3029 7 : R = Flx_sub(polxn_Flx(2*p-1,0), polxn_Flx(p,0),p);
3030 7 : S = Flx_sub(xp, polx_Flx(0), p);
3031 7 : Q = FlxX_Flx_sub(Flx_to_FlxX(S, v), R, p); /* x^p - x - (y^(2p-1)-y^p) */
3032 14 : for (i = 2; i <= l; ++i) T = Flx_FlxY_resultant(T, Q, p);
3033 7 : (void)delete_var(); T[1] = 0; return T;
3034 : }
3035 :
3036 : static long
3037 147949 : flinit_check(ulong p, long n, long l)
3038 : {
3039 : ulong q;
3040 147949 : if (!uisprime(n)) return 0;
3041 101208 : q = p % n; if (!q) return 0;
3042 98730 : return ugcd((n-1)/Fl_order(q, n-1, n), l) == 1;
3043 : }
3044 :
3045 : static GEN
3046 31727 : flinit(ulong p, long l)
3047 : {
3048 31727 : ulong n = 1+l;
3049 95809 : while (!flinit_check(p,n,l)) n += l;
3050 31727 : if (DEBUGLEVEL>=4) err_printf("FFInit: using polsubcyclo(%ld, %ld)\n",n,l);
3051 31727 : return ZX_to_Flx(polsubcyclo(n,l,0), p);
3052 : }
3053 :
3054 : static GEN
3055 28895 : ffinit_fact_Flx(ulong p, long n)
3056 : {
3057 28895 : GEN P, F = factoru_pow(n), Fp = gel(F,1), Fe = gel(F,2), Fm = gel(F,3);
3058 28895 : long i, l = lg(Fm);
3059 28895 : P = cgetg(l, t_VEC);
3060 61616 : for (i = 1; i < l; ++i)
3061 32721 : gel(P,i) = p==uel(Fp,i) ?
3062 994 : ffinit_Artin_Schreier(uel(Fp,i), Fe[i])
3063 32721 : : flinit(p, uel(Fm,i));
3064 28895 : return FlxV_composedsum(P, p);
3065 : }
3066 :
3067 : static GEN
3068 52140 : init_Flxq_i(ulong p, long n, long sv)
3069 : {
3070 : GEN P;
3071 52140 : if (n == 1) return polx_Flx(sv);
3072 52140 : if (flinit_check(p, n+1, n))
3073 : {
3074 23245 : P = const_vecsmall(n+2,1);
3075 23245 : P[1] = sv; return P;
3076 : }
3077 28895 : P = ffinit_fact_Flx(p,n);
3078 28895 : P[1] = sv; return P;
3079 : }
3080 :
3081 : GEN
3082 0 : init_Flxq(ulong p, long n, long v)
3083 : {
3084 0 : pari_sp av = avma;
3085 0 : return gerepileupto(av, init_Flxq_i(p, n, v));
3086 : }
3087 :
3088 : /* check if polsubcyclo(n,l,0) is irreducible modulo p */
3089 : static long
3090 7185 : fpinit_check(GEN p, long n, long l)
3091 : {
3092 : ulong q;
3093 7185 : if (!uisprime(n)) return 0;
3094 4450 : q = umodiu(p,n); if (!q) return 0;
3095 4450 : return ugcd((n-1)/Fl_order(q, n-1, n), l) == 1;
3096 : }
3097 :
3098 : /* let k=2 if p%4==1, and k=4 else and assume k*p does not divide l.
3099 : * Return an irreducible polynomial of degree l over F_p.
3100 : * Variant of Adleman and Lenstra "Finding irreducible polynomials over
3101 : * finite fields", ACM, 1986 (5) 350--355.
3102 : * Not stack clean */
3103 : static GEN
3104 1653 : fpinit(GEN p, long l)
3105 : {
3106 1653 : ulong n = 1+l;
3107 5202 : while (!fpinit_check(p,n,l)) n += l;
3108 1653 : if (DEBUGLEVEL>=4) err_printf("FFInit: using polsubcyclo(%ld, %ld)\n",n,l);
3109 1653 : return FpX_red(polsubcyclo(n,l,0),p);
3110 : }
3111 :
3112 : static GEN
3113 1574 : ffinit_fact(GEN p, long n)
3114 : {
3115 1574 : GEN P, F = factoru_pow(n), Fp = gel(F,1), Fe = gel(F,2), Fm = gel(F,3);
3116 1574 : long i, l = lg(Fm);
3117 1574 : P = cgetg(l, t_VEC);
3118 3227 : for (i = 1; i < l; ++i)
3119 3306 : gel(P,i) = absequaliu(p, Fp[i]) ?
3120 0 : Flx_to_ZX(ffinit_Artin_Schreier(Fp[i], Fe[i]))
3121 1653 : : fpinit(p, Fm[i]);
3122 1574 : return FpXV_composedsum(P, p);
3123 : }
3124 :
3125 : static GEN
3126 54382 : init_Fq_i(GEN p, long n, long v)
3127 : {
3128 : GEN P;
3129 54382 : if (n <= 0) pari_err_DOMAIN("ffinit", "degree", "<=", gen_0, stoi(n));
3130 54382 : if (typ(p) != t_INT) pari_err_TYPE("ffinit",p);
3131 54382 : if (cmpiu(p, 2) < 0) pari_err_PRIME("ffinit",p);
3132 54375 : if (v < 0) v = 0;
3133 54375 : if (n == 1) return pol_x(v);
3134 54123 : if (lgefint(p) == 3)
3135 52140 : return Flx_to_ZX(init_Flxq_i(p[2], n, evalvarn(v)));
3136 1983 : if (fpinit_check(p, n+1, n)) return polcyclo(n+1, v);
3137 1574 : P = ffinit_fact(p,n);
3138 1574 : setvarn(P, v); return P;
3139 : }
3140 : GEN
3141 53829 : init_Fq(GEN p, long n, long v)
3142 : {
3143 53829 : pari_sp av = avma;
3144 53829 : return gerepileupto(av, init_Fq_i(p, n, v));
3145 : }
3146 : GEN
3147 553 : ffinit(GEN p, long n, long v)
3148 : {
3149 553 : pari_sp av = avma;
3150 553 : return gerepileupto(av, FpX_to_mod(init_Fq_i(p, n, v), p));
3151 : }
3152 :
3153 : GEN
3154 3178 : ffnbirred(GEN p, long n)
3155 : {
3156 3178 : pari_sp av = avma;
3157 3178 : GEN s = powiu(p,n), F = factoru(n), D = divisorsu_moebius(gel(F, 1));
3158 3178 : long j, l = lg(D);
3159 6797 : for (j = 2; j < l; j++) /* skip d = 1 */
3160 : {
3161 3619 : long md = D[j]; /* mu(d) * d, d squarefree */
3162 3619 : GEN pd = powiu(p, n / labs(md)); /* p^{n/d} */
3163 3619 : s = md > 0? addii(s, pd): subii(s,pd);
3164 : }
3165 3178 : return gerepileuptoint(av, diviuexact(s, n));
3166 : }
3167 :
3168 : GEN
3169 616 : ffsumnbirred(GEN p, long n)
3170 : {
3171 616 : pari_sp av = avma, av2;
3172 616 : GEN q, t = p, v = vecfactoru_i(1, n);
3173 : long i;
3174 616 : q = cgetg(n+1,t_VEC); gel(q,1) = p;
3175 1764 : for (i=2; i<=n; i++) gel(q,i) = mulii(gel(q,i-1), p);
3176 616 : av2 = avma;
3177 1764 : for (i=2; i<=n; i++)
3178 : {
3179 1148 : GEN s = gel(q,i), F = gel(v,i), D = divisorsu_moebius(gel(F,1));
3180 1148 : long j, l = lg(D);
3181 2534 : for (j = 2; j < l; j++) /* skip 1 */
3182 : {
3183 1386 : long md = D[j];
3184 1386 : GEN pd = gel(q, i / labs(md)); /* p^{i/d} */
3185 1386 : s = md > 0? addii(s, pd): subii(s, pd);
3186 : }
3187 1148 : t = gerepileuptoint(av2, addii(t, diviuexact(s, i)));
3188 : }
3189 616 : return gerepileuptoint(av, t);
3190 : }
3191 :
3192 : GEN
3193 140 : ffnbirred0(GEN p, long n, long flag)
3194 : {
3195 140 : if (typ(p) != t_INT) pari_err_TYPE("ffnbirred", p);
3196 140 : if (n <= 0) pari_err_DOMAIN("ffnbirred", "degree", "<=", gen_0, stoi(n));
3197 140 : switch(flag)
3198 : {
3199 70 : case 0: return ffnbirred(p, n);
3200 70 : case 1: return ffsumnbirred(p, n);
3201 : }
3202 0 : pari_err_FLAG("ffnbirred");
3203 : return NULL; /* LCOV_EXCL_LINE */
3204 : }
3205 :
3206 : static void
3207 2261 : checkmap(GEN m, const char *s)
3208 : {
3209 2261 : if (typ(m)!=t_VEC || lg(m)!=3 || typ(gel(m,1))!=t_FFELT)
3210 0 : pari_err_TYPE(s,m);
3211 2261 : }
3212 :
3213 : GEN
3214 189 : ffembed(GEN a, GEN b)
3215 : {
3216 189 : pari_sp av = avma;
3217 189 : GEN p, Ta, Tb, g, r = NULL;
3218 189 : if (typ(a)!=t_FFELT) pari_err_TYPE("ffembed",a);
3219 189 : if (typ(b)!=t_FFELT) pari_err_TYPE("ffembed",b);
3220 189 : p = FF_p_i(a); g = FF_gen(a);
3221 189 : if (!equalii(p, FF_p_i(b))) pari_err_MODULUS("ffembed",a,b);
3222 189 : Ta = FF_mod(a);
3223 189 : Tb = FF_mod(b);
3224 189 : if (degpol(Tb)%degpol(Ta)!=0)
3225 7 : pari_err_DOMAIN("ffembed",GENtostr_raw(a),"is not a subfield of",b,a);
3226 182 : r = gel(FFX_roots(Ta, b), 1);
3227 182 : return gerepilecopy(av, mkvec2(g,r));
3228 : }
3229 :
3230 : GEN
3231 91 : ffextend(GEN a, GEN P, long v)
3232 : {
3233 91 : pari_sp av = avma;
3234 : long n;
3235 : GEN p, T, R, g, m;
3236 91 : if (typ(a)!=t_FFELT) pari_err_TYPE("ffextend",a);
3237 91 : T = a; p = FF_p_i(a);
3238 91 : if (typ(P)!=t_POL || !RgX_is_FpXQX(P,&T,&p)) pari_err_TYPE("ffextend", P);
3239 49 : if (!FF_samefield(a, T)) pari_err_MODULUS("ffextend",a,T);
3240 49 : if (v < 0) v = varn(P);
3241 49 : n = FF_f(T) * degpol(P); R = ffinit(p, n, v); g = ffgen(R, v);
3242 49 : m = ffembed(a, g);
3243 49 : R = FFX_roots(ffmap(m, P),g);
3244 49 : return gerepilecopy(av, mkvec2(gel(R,1), m));
3245 : }
3246 :
3247 : GEN
3248 42 : fffrobenius(GEN a, long n)
3249 : {
3250 42 : if (typ(a)!=t_FFELT) pari_err_TYPE("fffrobenius",a);
3251 42 : retmkvec2(FF_gen(a), FF_Frobenius(a, n));
3252 : }
3253 :
3254 : GEN
3255 133 : ffinvmap(GEN m)
3256 : {
3257 133 : pari_sp av = avma;
3258 : long i, l;
3259 133 : GEN T, F, a, g, r, f = NULL;
3260 133 : checkmap(m, "ffinvmap");
3261 133 : a = gel(m,1); r = gel(m,2);
3262 133 : if (typ(r) != t_FFELT)
3263 7 : pari_err_TYPE("ffinvmap", m);
3264 126 : g = FF_gen(a);
3265 126 : T = FF_mod(r);
3266 126 : F = gel(FFX_factor(T, a), 1);
3267 126 : l = lg(F);
3268 490 : for(i=1; i<l; i++)
3269 : {
3270 490 : GEN s = FFX_rem(FF_to_FpXQ_i(r), gel(F, i), a);
3271 490 : if (degpol(s)==0 && gequal(constant_coeff(s),g)) { f = gel(F, i); break; }
3272 : }
3273 126 : if (f==NULL) pari_err_TYPE("ffinvmap", m);
3274 126 : if (degpol(f)==1) f = FF_neg_i(gel(f,2));
3275 126 : return gerepilecopy(av, mkvec2(FF_gen(r),f));
3276 : }
3277 :
3278 : static GEN
3279 1260 : ffpartmapimage(const char *s, GEN r)
3280 : {
3281 1260 : GEN a = NULL, p = NULL;
3282 1260 : if (typ(r)==t_POL && degpol(r) >= 1
3283 1260 : && RgX_is_FpXQX(r,&a,&p) && a && typ(a)==t_FFELT) return a;
3284 0 : pari_err_TYPE(s, r);
3285 : return NULL; /* LCOV_EXCL_LINE */
3286 : }
3287 :
3288 : static GEN
3289 2709 : ffeltmap_i(GEN m, GEN x)
3290 : {
3291 2709 : GEN r = gel(m,2);
3292 2709 : if (!FF_samefield(x, gel(m,1)))
3293 84 : pari_err_DOMAIN("ffmap","m","domain does not contain", x, r);
3294 2625 : if (typ(r)==t_FFELT)
3295 1659 : return FF_map(r, x);
3296 : else
3297 966 : return FFX_preimage(x, r, ffpartmapimage("ffmap", r));
3298 : }
3299 :
3300 : static GEN
3301 4459 : ffmap_i(GEN m, GEN x)
3302 : {
3303 : GEN y;
3304 4459 : long i, lx, tx = typ(x);
3305 4459 : switch(tx)
3306 : {
3307 2541 : case t_FFELT:
3308 2541 : return ffeltmap_i(m, x);
3309 1267 : case t_POL: case t_RFRAC: case t_SER:
3310 : case t_VEC: case t_COL: case t_MAT:
3311 1267 : y = cgetg_copy(x, &lx);
3312 1988 : for (i=1; i<lontyp[tx]; i++) y[i] = x[1];
3313 4564 : for (i=lontyp[tx]; i<lx; i++)
3314 : {
3315 3339 : GEN yi = ffmap_i(m, gel(x,i));
3316 3297 : if (!yi) return NULL;
3317 3297 : gel(y,i) = yi;
3318 : }
3319 1225 : return y;
3320 : }
3321 651 : return gcopy(x);
3322 : }
3323 :
3324 : GEN
3325 1036 : ffmap(GEN m, GEN x)
3326 : {
3327 1036 : pari_sp ltop = avma;
3328 : GEN y;
3329 1036 : checkmap(m, "ffmap");
3330 1036 : y = ffmap_i(m, x);
3331 1036 : if (y) return y;
3332 42 : set_avma(ltop); return cgetg(1,t_VEC);
3333 : }
3334 :
3335 : static GEN
3336 252 : ffeltmaprel_i(GEN m, GEN x)
3337 : {
3338 252 : GEN g = gel(m,1), r = gel(m,2);
3339 252 : if (!FF_samefield(x, g))
3340 0 : pari_err_DOMAIN("ffmap","m","domain does not contain", x, r);
3341 252 : if (typ(r)==t_FFELT)
3342 84 : retmkpolmod(FF_map(r, x), pol_x(FF_var(g)));
3343 : else
3344 168 : retmkpolmod(FFX_preimagerel(x, r, ffpartmapimage("ffmap", r)), gcopy(r));
3345 : }
3346 :
3347 : static GEN
3348 252 : ffmaprel_i(GEN m, GEN x)
3349 : {
3350 : GEN y;
3351 252 : long i, lx, tx = typ(x);
3352 252 : switch(tx)
3353 : {
3354 252 : case t_FFELT:
3355 252 : return ffeltmaprel_i(m, x);
3356 0 : case t_POL: case t_RFRAC: case t_SER:
3357 : case t_VEC: case t_COL: case t_MAT:
3358 0 : y = cgetg_copy(x, &lx);
3359 0 : for (i=1; i<lontyp[tx]; i++) y[i] = x[1];
3360 0 : for (i=lontyp[tx]; i<lx; i++)
3361 0 : gel(y,i) = ffmaprel_i(m, gel(x,i));
3362 0 : return y;
3363 : }
3364 0 : return gcopy(x);
3365 : }
3366 :
3367 : GEN
3368 252 : ffmaprel(GEN m, GEN x)
3369 : {
3370 252 : checkmap(m, "ffmaprel");
3371 252 : return ffmaprel_i(m, x);
3372 : }
3373 :
3374 : static void
3375 84 : err_compo(GEN m, GEN n)
3376 84 : { pari_err_DOMAIN("ffcompomap","m","domain does not contain codomain of",n,m); }
3377 :
3378 : GEN
3379 420 : ffcompomap(GEN m, GEN n)
3380 : {
3381 420 : pari_sp av = avma;
3382 420 : GEN g = gel(n,1), r, m2, n2;
3383 420 : checkmap(m, "ffcompomap");
3384 420 : checkmap(n, "ffcompomap");
3385 420 : m2 = gel(m,2); n2 = gel(n,2);
3386 420 : switch((typ(m2)==t_POL)|((typ(n2)==t_POL)<<1))
3387 : {
3388 84 : case 0:
3389 84 : if (!FF_samefield(gel(m,1),n2)) err_compo(m,n);
3390 42 : r = FF_map(gel(m,2), n2);
3391 42 : break;
3392 84 : case 2:
3393 84 : r = ffmap_i(m, n2);
3394 42 : if (lg(r) == 1) err_compo(m,n);
3395 42 : break;
3396 168 : case 1:
3397 168 : r = ffeltmap_i(m, n2);
3398 126 : if (!r)
3399 : {
3400 : GEN a, A, R, M;
3401 : long dm, dn;
3402 42 : a = ffpartmapimage("ffcompomap",m2);
3403 42 : A = FF_to_FpXQ_i(FF_neg(n2));
3404 42 : setvarn(A, 1);
3405 42 : R = deg1pol(gen_1, A, 0);
3406 42 : setvarn(R, 0);
3407 42 : M = gcopy(m2);
3408 42 : setvarn(M, 1);
3409 42 : r = polresultant0(R, M, 1, 0);
3410 42 : dm = FF_f(gel(m,1)); dn = FF_f(gel(n,1));
3411 42 : if (dm % dn || !FFX_ispower(r, dm/dn, a, &r)) err_compo(m,n);
3412 42 : setvarn(r, varn(FF_mod(g)));
3413 : }
3414 126 : break;
3415 84 : case 3:
3416 : {
3417 : GEN M, R, T, p, a;
3418 84 : a = ffpartmapimage("ffcompomap",n2);
3419 84 : if (!FF_samefield(a, gel(m,1))) err_compo(m,n);
3420 42 : p = FF_p_i(gel(n,1));
3421 42 : T = FF_mod(gel(n,1));
3422 42 : setvarn(T, 1);
3423 42 : R = RgX_to_FpXQX(n2,T,p);
3424 42 : setvarn(R, 0);
3425 42 : M = gcopy(m2);
3426 42 : setvarn(M, 1);
3427 42 : r = polresultant0(R, M, 1, 0);
3428 42 : setvarn(r, varn(n2));
3429 : }
3430 : }
3431 252 : return gerepilecopy(av, mkvec2(g,r));
3432 : }
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