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 162428 : charact(struct charact *S, GEN x)
53 : {
54 162428 : const long tx = typ(x);
55 : long i, l;
56 162428 : 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 25823 : 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 25823 : l = lg(x);
64 172753 : for (i=lontyp[tx]; i < l; i++) charact(S,gel(x,i));
65 25809 : break;
66 7 : case t_LIST:
67 7 : x = list_data(x);
68 7 : if (x) charact(S, x);
69 7 : break;
70 : }
71 162400 : }
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 15484 : characteristic(GEN x)
96 : {
97 : struct charact S;
98 15484 : S.q = gen_0; S.isprime = 0;
99 15484 : 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 66105070 : Rg_is_Fp(GEN x, GEN *pp)
111 : {
112 : GEN mod;
113 66105070 : switch(typ(x))
114 : {
115 3196935 : case t_INTMOD:
116 3196935 : mod = gel(x,1);
117 3196935 : if (!*pp) *pp = mod;
118 2948932 : 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 3196935 : return 1;
124 54651277 : case t_INT:
125 54651277 : return 1;
126 8256858 : default: return 0;
127 : }
128 : }
129 :
130 : int
131 24834536 : RgX_is_FpX(GEN x, GEN *pp)
132 : {
133 24834536 : long i, lx = lg(x);
134 82656651 : for (i=2; i<lx; i++)
135 66078975 : if (!Rg_is_Fp(gel(x, i), pp))
136 8256864 : return 0;
137 16577676 : 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 59269 : Rg_is_FpXQ(GEN x, GEN *pT, GEN *pp)
160 : {
161 : GEN pol, mod, p;
162 59269 : switch(typ(x))
163 : {
164 26089 : case t_INTMOD:
165 26089 : return Rg_is_Fp(x, pp);
166 7070 : case t_INT:
167 7070 : 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 3199 : RgX_is_FpXQX(GEN x, GEN *pT, GEN *pp)
202 : {
203 3199 : long i, lx = lg(x);
204 61712 : for (i = 2; i < lx; i++)
205 58611 : if (!Rg_is_FpXQ(gel(x,i), pT, pp)) return 0;
206 3101 : 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 34339417 : Rg_to_Fp(GEN x, GEN p)
219 : {
220 34339417 : if (lgefint(p) == 3) return utoi(Rg_to_Fl(x, uel(p,2)));
221 4543285 : switch(typ(x))
222 : {
223 274756 : 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 4249680 : case t_INTMOD: {
232 4249680 : GEN q = gel(x,1), a = gel(x,2);
233 4249680 : 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 1290320 : Rg_to_FpXQ(GEN x, GEN T, GEN p)
244 : {
245 1290320 : long ta, tx = typ(x), v = get_FpX_var(T);
246 : GEN a, b;
247 1290320 : 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 1231789 : switch(tx)
258 : {
259 1229710 : case t_POLMOD:
260 1229710 : b = gel(x,1);
261 1229710 : a = gel(x,2); ta = typ(a);
262 1229710 : if (is_const_t(ta))
263 4095 : return scalar_ZX(degpol(T)? Rg_to_Fp(a, p): gen_0, v);
264 1225615 : b = RgX_to_FpX(b, p); if (varn(b) != v) break;
265 1225615 : a = RgX_to_FpX(a, p);
266 1225615 : if (ZX_equal(b,get_FpX_mod(T)) || signe(FpX_rem(b,T,p))==0)
267 1225615 : 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 3547500 : RgX_to_FpX(GEN x, GEN p)
282 : {
283 : long i, l;
284 3547500 : GEN z = cgetg_copy(x, &l); z[1] = x[1];
285 15777839 : for (i = 2; i < l; i++) gel(z,i) = Rg_to_Fp(gel(x,i), p);
286 3547500 : 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 948696 : RgC_to_FpC(GEN x, GEN p)
295 11932542 : { pari_APPLY_type(t_COL, Rg_to_Fp(gel(x,i), p)) }
296 :
297 : GEN
298 134233 : RgM_to_FpM(GEN x, GEN p)
299 1082887 : { pari_APPLY_same(RgC_to_FpC(gel(x,i), p)) }
300 :
301 : GEN
302 285930 : RgV_to_Flv(GEN x, ulong p)
303 1172349 : { 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 45437 : RgX_to_FqX(GEN x, GEN T, GEN p)
319 : {
320 45437 : long i, l = lg(x);
321 45437 : GEN z = cgetg(l, t_POL); z[1] = x[1];
322 45437 : if (T)
323 10990 : for (i = 2; i < l; i++) gel(z,i) = Rg_to_FpXQ(gel(x,i), T, p);
324 : else
325 760088 : for (i = 2; i < l; i++) gel(z,i) = Rg_to_Fp(gel(x,i), p);
326 45437 : return FpXQX_renormalize(z, l);
327 : }
328 :
329 : GEN
330 219121 : RgC_to_FqC(GEN x, GEN T, GEN p)
331 : {
332 219121 : long i, l = lg(x);
333 219121 : GEN z = cgetg(l, t_COL);
334 219121 : if (T)
335 1429127 : 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 219121 : 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 149016 : FqX_Fq_mul_to_monic(GEN P, GEN U, GEN T, GEN p)
392 : {
393 : long i, lP;
394 149016 : GEN res = cgetg_copy(P, &lP); res[1] = P[1];
395 918537 : for(i=2; i<lP-1; i++) gel(res,i) = Fq_mul(U,gel(P,i), T,p);
396 149016 : gel(res,lP-1) = gen_1; return res;
397 : }
398 :
399 : GEN
400 38167 : FpXQX_normalize(GEN z, GEN T, GEN p)
401 : {
402 : GEN lc;
403 38167 : if (lg(z) == 2) return z;
404 38153 : lc = leading_coeff(z);
405 38153 : if (typ(lc) == t_POL)
406 : {
407 2159 : if (lg(lc) > 3) /* nonconstant */
408 1887 : 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 36266 : if (equali1(lc)) return z;
416 66 : return FqX_Fq_mul_to_monic(z, Fp_inv(lc,p), T,p);
417 : }
418 :
419 : GEN
420 398859 : FqX_eval(GEN x, GEN y, GEN T, GEN p)
421 : {
422 : pari_sp av;
423 : GEN p1, r;
424 398859 : long j, i=lg(x)-1;
425 398859 : if (i<=2)
426 45957 : return (i==2)? Fq_red(gel(x,2), T, p): gen_0;
427 352902 : 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 1173802 : for (i--; i>=2; i=j-1)
431 : {
432 821600 : 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 820900 : r = (i==j)? y: Fq_pow(y, utoipos(i-j+1), T, p);
439 820900 : p1 = Fq_add(Fq_mul(p1,r,T,p), gel(x,j), T, p);
440 : }
441 352412 : 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 10009376 : monomial(GEN a, long d, long v)
471 : {
472 : long i, n;
473 : GEN P;
474 10009376 : if (d < 0) {
475 14 : if (isrationalzero(a)) return pol_0(v);
476 14 : retmkrfrac(a, pol_xn(-d, v));
477 : }
478 10009362 : 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 9916089 : n = d+2; P = cgetg(n+1, t_POL);
487 9916091 : P[1] = evalsigne(1) | evalvarn(v);
488 : }
489 29040154 : for (i = 2; i < n; i++) gel(P,i) = gen_0;
490 9916091 : gel(P,i) = a; return P;
491 : }
492 : GEN
493 1863744 : monomialcopy(GEN a, long d, long v)
494 : {
495 : long i, n;
496 : GEN P;
497 1863744 : if (d < 0) {
498 14 : if (isrationalzero(a)) return pol_0(v);
499 14 : retmkrfrac(gcopy(a), pol_xn(-d, v));
500 : }
501 1863730 : 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 1863716 : n = d+2; P = cgetg(n+1, t_POL);
510 1863716 : P[1] = evalsigne(1) | evalvarn(v);
511 : }
512 3511270 : for (i = 2; i < n; i++) gel(P,i) = gen_0;
513 1863723 : gel(P,i) = gcopy(a); return P;
514 : }
515 : GEN
516 323515 : pol_x_powers(long N, long v)
517 : {
518 323515 : GEN L = cgetg(N+1,t_VEC);
519 : long i;
520 781248 : for (i=1; i<=N; i++) gel(L,i) = pol_xn(i-1, v);
521 323514 : 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 11010528 : Fq_add(GEN x, GEN y, GEN T/*unused*/, GEN p)
544 : {
545 : (void)T;
546 11010528 : 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 9608958 : case 3: return FpX_add(x,y,p);
552 : }
553 : return NULL;/*LCOV_EXCL_LINE*/
554 : }
555 :
556 : GEN
557 8336935 : Fq_sub(GEN x, GEN y, GEN T/*unused*/, GEN p)
558 : {
559 : (void)T;
560 8336935 : 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 8044548 : case 3: return FpX_sub(x,y,p);
566 : }
567 : return NULL;/*LCOV_EXCL_LINE*/
568 : }
569 :
570 : GEN
571 892325 : Fq_neg(GEN x, GEN T/*unused*/, GEN p)
572 : {
573 : (void)T;
574 892325 : 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 8008215 : Fq_mul(GEN x, GEN y, GEN T, GEN p)
587 : {
588 8008215 : 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 402242 : case 2: return FpX_Fp_mul(y,x,p);
593 6808067 : case 3: if (T) return FpXQ_mul(x,y,T,p);
594 4231198 : 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 492741 : Fq_mulu(GEN x, ulong y, /*unused*/GEN T, GEN p)
602 : {
603 : (void) T;
604 492741 : 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 369591 : Fq_sqr(GEN x, GEN T, GEN p)
618 : {
619 369591 : 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 296747 : 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 89374 : Fq_inv(GEN x, GEN pol, GEN p)
644 : {
645 89374 : if (typ(x) == t_INT) return Fp_inv(x,p);
646 81608 : 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 777262 : Fq_pow(GEN x, GEN n, GEN pol, GEN p)
664 : {
665 777262 : if (typ(x) == t_INT) return Fp_pow(x,n,p);
666 131061 : 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 710597 : if (!T || odd(get_FpX_degree(T))) return Fp_sqrt(x,p);
682 9610 : x = scalarpol_shallow(x, get_FpX_var(T));
683 : }
684 79743 : return FpXQ_sqrt(x,T,p);
685 : }
686 : GEN
687 60284 : Fq_sqrtn(GEN x, GEN n, GEN T, GEN p, GEN *zeta)
688 : {
689 60284 : if (typ(x) == t_INT)
690 : {
691 : long d;
692 59948 : 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 40551 : FqV_roots_to_pol(GEN V, GEN T, GEN p, long v)
831 : {
832 40551 : pari_sp ltop = avma;
833 : long k;
834 : GEN W;
835 40551 : if (lgefint(p) == 3)
836 : {
837 31754 : ulong pp = p[2];
838 31754 : GEN Tl = ZX_to_Flx(T, pp);
839 31753 : GEN Vl = ZXC_to_FlxC(V, pp, get_Flx_var(Tl));
840 31752 : Tl = FlxqV_roots_to_pol(Vl, Tl, pp, v);
841 31754 : return gerepileupto(ltop, FlxX_to_ZXX(Tl));
842 : }
843 8797 : W = cgetg(lg(V),t_VEC);
844 78691 : for(k=1; k < lg(V); k++)
845 69894 : gel(W,k) = deg1pol_shallow(gen_1,Fq_neg(gel(V,k),T,p),v);
846 8797 : return gerepileupto(ltop, FpXQXV_prod(W, T, p));
847 : }
848 :
849 : GEN
850 157171 : FqV_red(GEN x, GEN T, GEN p)
851 1249142 : { 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 6005435 : primelist(forprime_t *S, long n, GEN dB)
905 : {
906 6005435 : GEN P = cgetg(n+1, t_VECSMALL);
907 6005390 : long i = 1;
908 : ulong p;
909 14901784 : while (i <= n && (p = u_forprime_next(S)))
910 8896393 : if (!dB || umodiu(dB, p)) P[i++] = p;
911 6005388 : return P;
912 : }
913 :
914 : void
915 5573125 : 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 5573125 : long m = mmin? minss(mmin, n): usqrt(n);
920 : GEN H, P, mod;
921 : pari_timer ti;
922 5573115 : if (DEBUGLEVEL > 4)
923 : {
924 0 : timer_start(&ti);
925 0 : err_printf("%s: nb primes: %ld\n",str, n);
926 : }
927 5573110 : if (m == 1)
928 : {
929 5358414 : GEN P = primelist(S, n, dB);
930 5358376 : GEN done = closure_callgen1(worker, P);
931 5358381 : H = gel(done,1);
932 5358381 : mod = gel(done,2);
933 5358381 : if (!*pH && center) H = center(H, mod, shifti(mod,-1));
934 5358333 : if (DEBUGLEVEL>4) timer_printf(&ti,"%s: modular", str);
935 : }
936 : else
937 : {
938 214696 : long i, s = (n+m-1)/m, r = m - (m*s-n), di = 0;
939 : struct pari_mt pt;
940 214696 : long pending = 0;
941 214696 : H = cgetg(m+1, t_VEC); P = cgetg(m+1, t_VEC);
942 214696 : mt_queue_start_lim(&pt, worker, m);
943 909943 : for (i=1; i<=m || pending; i++)
944 : {
945 : GEN done;
946 695247 : GEN pr = i <= m ? mkvec(primelist(S, i<=r ? s: s-1, dB)): NULL;
947 695248 : mt_queue_submit(&pt, i, pr);
948 695249 : done = mt_queue_get(&pt, NULL, &pending);
949 695249 : if (done)
950 : {
951 647027 : di++;
952 647027 : gel(H, di) = gel(done,1);
953 647027 : gel(P, di) = gel(done,2);
954 647027 : if (DEBUGLEVEL>5) err_printf("%ld%% ",100*di/m);
955 : }
956 : }
957 214696 : mt_queue_end(&pt);
958 214696 : if (DEBUGLEVEL>5) err_printf("\n");
959 214696 : if (DEBUGLEVEL>4) timer_printf(&ti,"%s: modular", str);
960 214696 : H = crt(H, P, &mod);
961 214696 : if (DEBUGLEVEL>4) timer_printf(&ti,"%s: chinese", str);
962 : }
963 5573029 : if (*pH) H = crt(mkvec2(*pH, H), mkvec2(*pmod, mod), &mod);
964 5573029 : *pH = H; *pmod = mod;
965 5573029 : }
966 : void
967 1990646 : 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 1990646 : pari_sp av = avma;
972 1990646 : gen_inccrt_i(str, worker, dB, n, mmin, S, pH, pmod, crt, center);
973 1990555 : gerepileall(av, 2, pH, pmod);
974 1990693 : }
975 :
976 : GEN
977 1390292 : 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 1390292 : GEN mod = gen_1, H = NULL;
981 : ulong e;
982 :
983 1390292 : bound++;
984 2780647 : while (bound > (e = expi(mod)))
985 : {
986 1390263 : long n = (bound - e) / expu(S->p) + 1;
987 1390284 : gen_inccrt(str, worker, dB, n, mmin, S, &H, &mod, crt, center);
988 : }
989 1390336 : if (pmod) *pmod = mod;
990 1390336 : 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 5109246 : Fl_chinese_coprime(GEN a, ulong b, GEN q, ulong p, ulong qinv, GEN pq, GEN pq2)
1001 : {
1002 5109246 : ulong d, amod = umodiu(a, p);
1003 5109245 : pari_sp av = avma;
1004 : GEN ax;
1005 :
1006 5109245 : if (b == amod) return NULL;
1007 2104484 : d = Fl_mul(Fl_sub(b, amod, p), qinv, p); /* != 0 */
1008 2104951 : if (d >= 1 + (p>>1))
1009 1026751 : ax = subii(a, mului(p-d, q));
1010 : else
1011 : {
1012 1078200 : ax = addii(a, mului(d, q)); /* in ]0, pq[ assuming a in ]-q,q[ */
1013 1077819 : if (cmpii(ax,pq2) > 0) ax = subii(ax,pq);
1014 : }
1015 2104390 : 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 794415 : for (i=2; i<l; i++)
1026 762873 : 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 147063 : ZX_incremental_CRT_raw(GEN *ptH, GEN Hp, GEN q, GEN qp, ulong p)
1060 : {
1061 147063 : GEN H = *ptH, h, qp2 = shifti(qp,-1);
1062 147059 : ulong qinv = Fl_inv(umodiu(q,p), p);
1063 147063 : long i, l = lg(H), lp = lg(Hp);
1064 147063 : int stable = 1;
1065 :
1066 147063 : 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 147063 : } 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 4928808 : for (i=2; i<lp; i++)
1081 : {
1082 4781818 : h = Fl_chinese_coprime(gel(H,i),Hp[i],q,p,qinv,qp,qp2);
1083 4781745 : if (h) { gel(H,i) = h; stable = 0; }
1084 : }
1085 146990 : (void)ZX_renormalize(H,lp);
1086 147065 : 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 5804 : ZM_incremental_CRT(GEN *pH, GEN Hp, GEN *ptq, ulong p)
1099 : {
1100 5804 : GEN h, H = *pH, q = *ptq, qp = muliu(q, p), qp2 = shifti(qp,-1);
1101 5804 : ulong qinv = Fl_inv(umodiu(q,p), p);
1102 5804 : long i,j, l = lg(H), m = lgcols(H);
1103 5804 : int stable = 1;
1104 20974 : for (j=1; j<l; j++)
1105 157241 : for (i=1; i<m; i++)
1106 : {
1107 142071 : h = Fl_chinese_coprime(gcoeff(H,i,j), coeff(Hp,i,j),q,p,qinv,qp,qp2);
1108 142071 : if (h) { gcoeff(H,i,j) = h; stable = 0; }
1109 : }
1110 5804 : *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 23261 : Flx_resultant_set_dglist(GEN a, GEN b, GEN dglist, ulong p)
1174 : {
1175 : long da,db,dc, ind;
1176 23261 : pari_sp av = avma;
1177 :
1178 23261 : if (lgpol(a)==0 || lgpol(b)==0) return;
1179 21994 : da = degpol(a);
1180 21994 : db = degpol(b);
1181 21994 : if (db > da)
1182 0 : { swapspec(a,b, da,db); }
1183 21994 : else if (!da) return;
1184 21994 : ind = 0;
1185 144333 : while (db)
1186 : {
1187 122340 : GEN c = Flx_rem(a,b, p);
1188 122339 : a = b; b = c; dc = degpol(c);
1189 122340 : if (dc < 0) break;
1190 :
1191 122340 : ind++;
1192 122340 : if (dc > dglist[ind]) dglist[ind] = dc;
1193 122340 : 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 122339 : db = dc; /* = degpol(b) */
1199 : }
1200 21993 : if (ind+1 > lg(dglist)) setlg(dglist,ind+1);
1201 21993 : 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 2082176 : Flx_resultant_all(GEN a, GEN b, long *C0, long *C1, GEN dglist, ulong p)
1208 : {
1209 : long da,db,dc, ind;
1210 2082176 : ulong lb, res, g = 1UL, h = 1UL, ca = 1UL, cb = 1UL;
1211 2082176 : int s = 1;
1212 2082176 : pari_sp av = avma;
1213 :
1214 2082176 : *C0 = 1; *C1 = 0;
1215 2082176 : if (lgpol(a)==0 || lgpol(b)==0) return 0;
1216 2072804 : da = degpol(a);
1217 2072805 : db = degpol(b);
1218 2072777 : if (db > da)
1219 : {
1220 0 : swapspec(a,b, da,db);
1221 0 : if (both_odd(da,db)) s = -s;
1222 : }
1223 2072777 : else if (!da) return 1; /* = a[2] ^ db, since 0 <= db <= da = 0 */
1224 2072777 : ind = 0;
1225 19789582 : while (db)
1226 : { /* sub-resultant algo., applied to ca * a and cb * b, ca,cb scalars,
1227 : * da = deg a, db = deg b */
1228 17721404 : GEN c = Flx_rem(a,b, p);
1229 17617082 : long delta = da - db;
1230 :
1231 17617082 : if (both_odd(da,db)) s = -s;
1232 17617477 : lb = Fl_mul(b[db+2], cb, p);
1233 17643608 : a = b; b = c; dc = degpol(c);
1234 17660283 : ind++;
1235 17660283 : if (dc != dglist[ind]) return gc_ulong(av,0); /* degenerates */
1236 17655392 : if (g == h)
1237 : { /* frequent */
1238 17595525 : ulong cc = Fl_mul(ca, Fl_powu(Fl_div(lb,g,p), delta+1, p), p);
1239 17660239 : ca = cb;
1240 17660239 : cb = cc;
1241 : }
1242 : else
1243 : {
1244 59867 : ulong cc = Fl_mul(ca, Fl_powu(lb, delta+1, p), p);
1245 59869 : ulong ghdelta = Fl_mul(g, Fl_powu(h, delta, p), p);
1246 59869 : ca = cb;
1247 59869 : cb = Fl_div(cc, ghdelta, p);
1248 : }
1249 17717192 : da = db; /* = degpol(a) */
1250 17717192 : db = dc; /* = degpol(b) */
1251 :
1252 17717192 : g = lb;
1253 17717192 : if (delta == 1)
1254 17617869 : h = g; /* frequent */
1255 : else
1256 99323 : h = Fl_mul(h, Fl_powu(Fl_div(g,h,p), delta, p), p);
1257 :
1258 17717500 : 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 2068178 : if (da > 1) return 0; /* Failure */
1265 : /* last nonconstant polynomial has degree 1 */
1266 2068178 : *C0 = Fl_mul(ca, a[2], p);
1267 2068161 : *C1 = Fl_mul(ca, a[3], p);
1268 2068160 : res = Fl_mul(cb, b[2], p);
1269 2068185 : if (s == -1) res = p - res;
1270 2068185 : 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 2105635 : FlxY_evalx_drop(GEN Q, ulong x, ulong p)
1277 : {
1278 : GEN z;
1279 2105635 : long i, lb = lg(Q);
1280 2105635 : ulong leadz = Flx_eval(leading_coeff(Q), x, p);
1281 2105226 : long vs=mael(Q,2,1);
1282 2105226 : if (!leadz) return zero_Flx(vs);
1283 :
1284 2094566 : z = cgetg(lb, t_VECSMALL); z[1] = vs;
1285 20053536 : for (i=2; i<lb-1; i++) z[i] = Flx_eval(gel(Q,i), x, p);
1286 2092561 : 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 272219 : ZX_norml1(GEN x)
1311 : {
1312 272219 : long i, l = lg(x);
1313 : GEN s;
1314 :
1315 272219 : if (l == 2) return gen_0;
1316 179665 : s = gel(x, l-1); /* != 0 */
1317 657529 : for (i = l-2; i > 1; i--) {
1318 477870 : GEN xi = gel(x,i);
1319 477870 : if (!signe(xi)) continue;
1320 239790 : s = addii_sign(s,1, xi,1);
1321 : }
1322 179659 : return s;
1323 : }
1324 : /* x >= 0, y != 0, return x + |y| */
1325 : static GEN
1326 25584 : addii_abs(GEN x, GEN y)
1327 : {
1328 25584 : if (!signe(x)) return absi_shallow(y);
1329 16048 : 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 31705 : ZX_norml1_1(GEN x, long k)
1335 : {
1336 31705 : long i, d = degpol(x);
1337 : GEN s, C; /* = binomial(i, k) */
1338 :
1339 31704 : if (!d || k > d) return gen_0;
1340 31706 : s = absi_shallow(gel(x, k+2)); /* may be 0 */
1341 31707 : C = gen_1;
1342 68175 : for (i = k+1; i <= d; i++) {
1343 36466 : GEN xi = gel(x,i+2);
1344 36466 : if (k) C = diviuexact(muliu(C, i), i-k);
1345 36471 : if (signe(xi)) s = addii_abs(s, mulii(C, xi));
1346 : }
1347 31709 : 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 7108 : sqrN2(GEN A, long prec)
1370 : {
1371 7108 : pari_sp av = avma;
1372 7108 : long i, l = lg(A);
1373 7108 : GEN a = gen_0;
1374 34823 : for (i = 2; i < l; i++)
1375 : {
1376 27715 : a = gadd(a, gabs(gnorm(gel(A,i)), prec));
1377 27715 : 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 7108 : 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 6261 : RgX_RgXY_ResBound(GEN A, GEN B, long prec)
1391 : {
1392 6261 : pari_sp av = avma;
1393 6261 : GEN b = gen_0, bnd;
1394 6261 : long i, lB = lg(B);
1395 24712 : for (i=2; i<lB; i++)
1396 : {
1397 18451 : GEN t = gel(B,i);
1398 18451 : if (typ(t) == t_POL) t = gnorml1(t, prec);
1399 18451 : b = gadd(b, gabs(gsqr(t), prec));
1400 18451 : 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 6261 : bnd = gsqrt(gmul(gpowgs(sqrN2(A,prec), degpol(B)),
1407 : gpowgs(b, degpol(A))), prec);
1408 6261 : 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 184100 : log2N2(GEN A)
1437 : {
1438 184100 : pari_sp av = avma;
1439 184100 : long i, l = lg(A);
1440 184100 : GEN a = gen_0;
1441 1114283 : for (i=2; i < l; i++)
1442 : {
1443 930200 : a = addii(a, sqri(gel(A,i)));
1444 930183 : 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 184083 : 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 174001 : ZX_ZXY_ResBound(GEN A, GEN B, GEN dB)
1458 : {
1459 174001 : pari_sp av = avma;
1460 174001 : GEN b = gen_0;
1461 174001 : long i, lB = lg(B);
1462 : double logb;
1463 1015953 : for (i=2; i<lB; i++)
1464 : {
1465 841960 : GEN t = gel(B,i);
1466 841960 : if (typ(t) == t_POL) t = ZX_norml1(t);
1467 841961 : b = addii(b, sqri(t));
1468 841951 : 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 173993 : logb = dbllog2(b); if (dB) logb -= 2 * dbllog2(dB);
1475 173999 : i = (long)((degpol(B) * log2N2(A) + degpol(A) * logb) / 2);
1476 173998 : 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 10106 : ZX_ZXY_ResBound_1(GEN A, GEN B)
1481 : {
1482 10106 : pari_sp av = avma;
1483 10106 : GEN b = gen_0;
1484 10106 : long i, lB = lg(B);
1485 41806 : for (i=2; i<lB; i++)
1486 : {
1487 31704 : b = addii(b, sqri(ZX_norml1_1(B, i-2)));
1488 31699 : 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 10102 : i = (long)((degpol(B) * log2N2(A) + degpol(A) * dbllog2(b)) / 2);
1495 10104 : return gc_ulong(av, (i <= 0)? 1: 1 + (ulong)i);
1496 : }
1497 : /* special case B = A' */
1498 : static ulong
1499 1130682 : ZX_discbound(GEN A)
1500 : {
1501 1130682 : pari_sp av = avma;
1502 1130682 : GEN a = gen_0, b = gen_0;
1503 1130682 : long i , lA = lg(A), dA = degpol(A);
1504 : double loga, logb;
1505 6744136 : for (i = 2; i < lA; i++)
1506 : {
1507 5613840 : GEN c = sqri(gel(A,i));
1508 5613240 : a = addii(a, c);
1509 5613348 : if (i > 2) b = addii(b, mulii(c, sqru(i-2)));
1510 5613391 : 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 1130296 : loga = dbllog2(a);
1517 1130568 : logb = dbllog2(b); set_avma(av);
1518 1130629 : i = (long)(((dA-1) * loga + dA * logb) / 2);
1519 1130629 : 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 2254545 : Flx_FlxY_eval_resultant(GEN a, GEN b, ulong n, ulong p, ulong pi, ulong la)
1525 : {
1526 2254545 : GEN ev = FlxY_evalx_pre(b, n, p, pi);
1527 2254530 : long drop = lg(b) - lg(ev);
1528 2254530 : ulong r = Flx_resultant_pre(a, ev, p, pi);
1529 2254506 : if (drop && la != 1) r = Fl_mul(r, Fl_powu_pre(la, drop, p, pi), p);
1530 2254511 : 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 176868 : Flx_FlxY_resultant_polint(GEN a, GEN b, ulong p, ulong pi, long dres, long sx)
1546 : {
1547 : long i;
1548 176868 : ulong n, la = Flx_lead(a);
1549 176867 : GEN x = cgetg(dres+2, t_VECSMALL);
1550 176867 : 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 1218730 : for (i=0,n = 1; i < dres; n++)
1554 : {
1555 1041865 : x[++i] = n; y[i] = Flx_FlxY_eval_resultant(a,b, x[i], p,pi,la);
1556 1041805 : x[++i] = p-n; y[i] = Flx_FlxY_eval_resultant(a,b, x[i], p,pi,la);
1557 : }
1558 176865 : if (i == dres)
1559 : {
1560 171439 : x[++i] = 0; y[i] = Flx_FlxY_eval_resultant(a,b, x[i], p,pi,la);
1561 : }
1562 176868 : return Flv_polint(x,y, p, sx);
1563 : }
1564 :
1565 : static GEN
1566 8007 : FlxX_pseudorem(GEN x, GEN y, ulong p, ulong pi)
1567 : {
1568 8007 : long vx = varn(x), dx, dy, dz, i, lx, dp;
1569 8007 : pari_sp av = avma, av2;
1570 :
1571 8007 : if (!signe(y)) pari_err_INV("FlxX_pseudorem",y);
1572 8007 : (void)new_chunk(2);
1573 8009 : dx=degpol(x); x = RgX_recip_i(x)+2;
1574 8010 : dy=degpol(y); y = RgX_recip_i(y)+2; dz=dx-dy; dp = dz+1;
1575 8011 : av2 = avma;
1576 : for (;;)
1577 : {
1578 65389 : gel(x,0) = Flx_neg(gel(x,0), p); dp--;
1579 244147 : for (i=1; i<=dy; i++)
1580 177356 : gel(x,i) = Flx_add( Flx_mul_pre(gel(y,0), gel(x,i), p, pi),
1581 178721 : Flx_mul_pre(gel(x,0), gel(y,i), p, pi), p );
1582 1137195 : for ( ; i<=dx; i++)
1583 1072776 : gel(x,i) = Flx_mul_pre(gel(y,0), gel(x,i), p, pi);
1584 69458 : do { x++; dx--; } while (dx >= 0 && lg(gel(x,0))==2);
1585 64419 : if (dx < dy) break;
1586 56415 : 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 8004 : if (dx < 0) return zero_Flx(0);
1593 8004 : lx = dx+3; x -= 2;
1594 8004 : x[0]=evaltyp(t_POL) | evallg(lx);
1595 8004 : x[1]=evalsigne(1) | evalvarn(vx);
1596 8004 : x = RgX_recip_i(x);
1597 8007 : if (dp)
1598 : { /* multiply by y[0]^dp [beware dummy vars from FpX_FpXY_resultant] */
1599 2095 : GEN t = Flx_powu_pre(gel(y,0), dp, p, pi);
1600 8383 : for (i=2; i<lx; i++) gel(x,i) = Flx_mul_pre(gel(x,i), t, p, pi);
1601 : }
1602 8011 : return gerepilecopy(av, x);
1603 : }
1604 :
1605 : /* return a Flx */
1606 : GEN
1607 2679 : FlxX_resultant(GEN u, GEN v, ulong p, long sx)
1608 : {
1609 2679 : 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 2679 : dx = degpol(u); dy = degpol(v); signh = 1;
1615 2679 : if (dx < dy)
1616 : {
1617 7 : swap(u,v); lswap(dx,dy);
1618 7 : if (both_odd(dx, dy)) signh = -signh;
1619 : }
1620 2679 : if (dy < 0) return zero_Flx(sx);
1621 2679 : pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
1622 2679 : if (dy==0) return gerepileupto(av, Flx_powu_pre(gel(v,2),dx,p,pi));
1623 :
1624 2679 : g = h = pol1_Flx(sx); av2 = avma;
1625 : for(;;)
1626 : {
1627 8007 : r = FlxX_pseudorem(u,v,p,pi); dr = lg(r);
1628 8009 : if (dr == 2) { set_avma(av); return zero_Flx(sx); }
1629 8009 : du = degpol(u); dv = degpol(v); degq = du-dv;
1630 8009 : u = v; p1 = g; g = leading_coeff(u);
1631 8010 : switch(degq)
1632 : {
1633 0 : case 0: break;
1634 5901 : case 1:
1635 5901 : p1 = Flx_mul_pre(h,p1, p, pi); h = g; break;
1636 2109 : default:
1637 2109 : p1 = Flx_mul_pre(Flx_powu_pre(h,degq,p,pi), p1, p, pi);
1638 2109 : h = Flx_div_pre(Flx_powu_pre(g,degq,p,pi),
1639 2107 : Flx_powu_pre(h,degq-1,p,pi), p, pi);
1640 : }
1641 8004 : if (both_odd(du,dv)) signh = -signh;
1642 8003 : v = FlxY_Flx_div(r, p1, p);
1643 8005 : if (dr==3) break;
1644 5325 : 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 2680 : z = gel(v,2);
1651 2680 : 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 2680 : if (signh < 0) z = Flx_neg(z,p);
1654 2680 : return gerepileupto(av, z);
1655 : }
1656 :
1657 : /* Warning:
1658 : * This function switches between valid and invalid variable ordering*/
1659 :
1660 : static GEN
1661 6296 : FlxY_to_FlyX(GEN b, long sv)
1662 : {
1663 6296 : long i, n=-1;
1664 6296 : long sw = b[1]&VARNBITS;
1665 21574 : for(i=2;i<lg(b);i++) n = maxss(n,lgpol(gel(b,i)));
1666 6296 : return Flm_to_FlxX(Flm_transpose(FlxX_to_Flm(b,n)),sv,sw);
1667 : }
1668 :
1669 : /* Return a Fly*/
1670 : GEN
1671 6297 : Flx_FlxY_resultant(GEN a, GEN b, ulong p)
1672 : {
1673 6297 : pari_sp ltop=avma;
1674 6297 : long dres = degpol(a)*degpol(b);
1675 6296 : long sx=a[1], sy=b[1]&VARNBITS;
1676 : GEN z;
1677 6296 : b = FlxY_to_FlyX(b,sx);
1678 6293 : if ((ulong)dres >= p)
1679 2676 : z = FlxX_resultant(Fly_to_FlxY(a, sy), b, p, sx);
1680 : else
1681 : {
1682 3617 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
1683 3617 : z = Flx_FlxY_resultant_polint(a, b, p, pi, (ulong)dres, sy);
1684 : }
1685 6297 : 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 125740 : swap_vars(GEN b0, long v)
1696 : {
1697 125740 : long i, n = RgX_degree(b0, v);
1698 : GEN b, x;
1699 125742 : if (n < 0) return pol_0(v);
1700 125742 : b = cgetg(n+3, t_POL); x = b + 2;
1701 125742 : b[1] = evalsigne(1) | evalvarn(v);
1702 639503 : for (i=0; i<=n; i++) gel(x,i) = polcoef_i(b0, i, v);
1703 125738 : 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 20902 : ZX_ZXY_resultant_LERS(GEN A, GEN B0, long *plambda, GEN *LERS)
1810 : {
1811 : ulong bound, dp;
1812 20902 : pari_sp av = avma, av2 = 0;
1813 20902 : long lambda = *plambda, degA = degpol(A), dres = degA*degpol(B0);
1814 : long stable, checksqfree, i,n, cnt, degB;
1815 20902 : 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 20902 : if (degA == 1)
1820 : {
1821 987 : GEN a1 = gel(A,3), a0 = gel(A,2);
1822 987 : B = lambda? RgX_translate(B0, monomial(stoi(lambda), 1, vY)): B0;
1823 987 : H = gsubst(B, vY, gdiv(gneg(a0),a1));
1824 987 : if (!equali1(a1)) H = RgX_Rg_mul(H, powiu(a1, poldegree(B,vY)));
1825 987 : *LERS = mkvec2(scalarpol_shallow(a0,vX), scalarpol_shallow(a1,vX));
1826 987 : return gc_all(av, 2, &H, LERS);
1827 : }
1828 :
1829 19915 : dglist = Hp = H0p = H1p = C0 = C1 = NULL; /* gcc -Wall */
1830 19915 : C0 = cgetg(dres+2, t_VECSMALL);
1831 19915 : C1 = cgetg(dres+2, t_VECSMALL);
1832 19915 : dglist = cgetg(dres+1, t_VECSMALL);
1833 19915 : x = cgetg(dres+2, t_VECSMALL);
1834 19915 : y = cgetg(dres+2, t_VECSMALL);
1835 19915 : B0 = leafcopy(B0);
1836 19915 : A = leafcopy(A);
1837 19914 : B = B0;
1838 19914 : v = fetch_var_higher(); setvarn(A,v);
1839 : /* make sure p large enough */
1840 20558 : INIT:
1841 : /* always except the first time */
1842 20558 : if (av2) { set_avma(av2); lambda = next_lambda(lambda); }
1843 20558 : if (lambda) B = RgX_translate(B0, monomial(stoi(lambda), 1, vY));
1844 20558 : B = swap_vars(B, vY); setvarn(B,v);
1845 : /* B0(lambda v + x, v) */
1846 20559 : if (DEBUGLEVEL>4) err_printf("Trying lambda = %ld\n", lambda);
1847 20559 : av2 = avma;
1848 :
1849 20559 : if (degA <= 3)
1850 : { /* sub-resultant faster for small degrees */
1851 9926 : H = RgX_resultant_all(A,B,&q);
1852 9926 : if (typ(q) != t_POL || degpol(q)!=1) goto INIT;
1853 9443 : H0 = gel(q,2);
1854 9443 : if (typ(H0) == t_POL) setvarn(H0,vX); else H0 = scalarpol(H0,vX);
1855 9443 : H1 = gel(q,3);
1856 9443 : if (typ(H1) == t_POL) setvarn(H1,vX); else H1 = scalarpol(H1,vX);
1857 9443 : if (!ZX_is_squarefree(H)) goto INIT;
1858 9401 : goto END;
1859 : }
1860 :
1861 10633 : H = H0 = H1 = NULL;
1862 10633 : degB = degpol(B);
1863 10633 : bound = ZX_ZXY_ResBound(A, B, NULL);
1864 10633 : if (DEBUGLEVEL>4) err_printf("bound for resultant coeffs: 2^%ld\n",bound);
1865 10633 : dp = 1;
1866 10633 : init_modular_big(&S);
1867 10633 : for(cnt = 0, checksqfree = 1;;)
1868 49021 : {
1869 59654 : ulong p = u_forprime_next(&S);
1870 : GEN Hi;
1871 59654 : a = ZX_to_Flx(A, p);
1872 59655 : b = ZXX_to_FlxX(B, p, varn(A));
1873 59655 : if (degpol(a) < degA || degpol(b) < degB) continue; /* p | lc(A)lc(B) */
1874 59655 : if (checksqfree)
1875 : { /* find degree list for generic Euclidean Remainder Sequence */
1876 10633 : long goal = minss(degpol(a), degpol(b)); /* longest possible */
1877 72970 : for (n=1; n <= goal; n++) dglist[n] = 0;
1878 10633 : setlg(dglist, 1);
1879 23662 : for (n=0; n <= dres; n++)
1880 : {
1881 23262 : ev = FlxY_evalx_drop(b, n, p);
1882 23261 : Flx_resultant_set_dglist(a, ev, dglist, p);
1883 23262 : if (lg(dglist)-1 == goal) break;
1884 : }
1885 : /* last pol in ERS has degree > 1 ? */
1886 10633 : goal = lg(dglist)-1;
1887 10633 : if (degpol(B) == 1) { if (!goal) goto INIT; }
1888 : else
1889 : {
1890 10577 : if (goal <= 1) goto INIT;
1891 10520 : if (dglist[goal] != 0 || dglist[goal-1] != 1) goto INIT;
1892 : }
1893 10576 : if (DEBUGLEVEL>4)
1894 0 : err_printf("Degree list for ERS (trials: %ld) = %Ps\n",n+1,dglist);
1895 : }
1896 :
1897 2141992 : for (i=0,n = 0; i <= dres; n++)
1898 : {
1899 2082407 : ev = FlxY_evalx_drop(b, n, p);
1900 2082150 : x[++i] = n; y[i] = Flx_resultant_all(a, ev, C0+i, C1+i, dglist, p);
1901 2082394 : if (!C1[i]) i--; /* C1(i) = 0. No way to recover C0(i) */
1902 : }
1903 59585 : Hi = Flv_Flm_polint(x, mkvec3(y,C0,C1), p, 0);
1904 59598 : Hp = gel(Hi,1); H0p = gel(Hi,2); H1p = gel(Hi,3);
1905 59598 : if (!H && degpol(Hp) != dres) continue;
1906 59598 : if (dp != 1) Hp = Flx_Fl_mul(Hp, Fl_powu(Fl_inv(dp,p), degA, p), p);
1907 59598 : if (checksqfree) {
1908 10576 : 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 59536 : 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 49022 : GEN qp = muliu(q,p);
1923 49020 : stable = ZX_incremental_CRT_raw(&H, Hp, q,qp, p)
1924 49022 : & ZX_incremental_CRT_raw(&H0,H0p, q,qp, p)
1925 49022 : & ZX_incremental_CRT_raw(&H1,H1p, q,qp, p);
1926 49021 : q = qp;
1927 : }
1928 : /* could make it probabilistic for H ? [e.g if stable twice, etc]
1929 : * Probabilistic anyway for H0, H1 */
1930 59535 : if (DEBUGLEVEL>5 && (stable || ++cnt==100))
1931 0 : { cnt=0; err_printf("%ld%%%s ",100*expi(q)/bound,stable?"s":""); }
1932 59535 : if (stable && (ulong)expi(q) >= bound) break; /* DONE */
1933 49021 : 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 19915 : END:
1940 19915 : if (DEBUGLEVEL>5) err_printf(" done\n");
1941 19915 : setvarn(H, vX); (void)delete_var();
1942 19915 : *LERS = mkvec2(H0,H1);
1943 19915 : *plambda = lambda; return gc_all(av, 2, &H, LERS);
1944 : }
1945 :
1946 : GEN
1947 58583 : ZX_ZXY_resultant_all(GEN A, GEN B, long *plambda, GEN *LERS)
1948 : {
1949 58583 : if (LERS)
1950 : {
1951 20902 : if (!plambda)
1952 0 : pari_err_BUG("ZX_ZXY_resultant_all [LERS != NULL needs lambda]");
1953 20902 : return ZX_ZXY_resultant_LERS(A, B, plambda, LERS);
1954 : }
1955 37681 : 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 3525 : ZXQ_charpoly_sqf(GEN A, GEN T, long *lambda, long v)
1963 : {
1964 3525 : pari_sp av = avma;
1965 : GEN R, a;
1966 : long dA;
1967 : int delvar;
1968 :
1969 3525 : if (v < 0) v = 0;
1970 3525 : switch (typ(A))
1971 : {
1972 3525 : 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 3525 : delvar = 0;
1979 3525 : if (varn(T) == 0)
1980 : {
1981 3319 : long v0 = fetch_var(); delvar = 1;
1982 3319 : T = leafcopy(T); setvarn(T,v0);
1983 3319 : A = leafcopy(A); setvarn(A,v0);
1984 : }
1985 3525 : R = ZX_ZXY_rnfequation(T, deg1pol_shallow(gen_1, gneg_i(A), 0), lambda);
1986 3525 : if (delvar) (void)delete_var();
1987 3525 : setvarn(R, v); a = leading_coeff(T);
1988 3525 : if (!gequal1(a)) R = gdiv(R, powiu(a, dA));
1989 3525 : 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 119224 : ZXQ_charpoly(GEN A, GEN T, long v)
1995 : {
1996 119224 : 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 3962775 : ZX_resultant_prime(GEN a, GEN b, GEN dB, long degA, long degB, ulong p)
2010 : {
2011 3962775 : long dropa = degA - degpol(a), dropb = degB - degpol(b);
2012 : ulong H, dp;
2013 3962657 : if (dropa && dropb) return 0; /* p | lc(A), p | lc(B) */
2014 3962657 : H = Flx_resultant(a, b, p);
2015 3962531 : 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 3962531 : 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 3962545 : dp = dB ? umodiu(dB, p): 1;
2029 3962545 : if (dp != 1) H = Fl_mul(H, Fl_powu(Fl_inv(dp,p), degA, p), p);
2030 3962543 : return H;
2031 : }
2032 :
2033 : /* If B=NULL, assume B=A' */
2034 : static GEN
2035 1628120 : ZX_resultant_slice(GEN A, GEN B, GEN dB, GEN P, GEN *mod)
2036 : {
2037 1628120 : pari_sp av = avma, av2;
2038 1628120 : long degA, degB, i, n = lg(P)-1;
2039 : GEN H, T;
2040 :
2041 1628120 : degA = degpol(A);
2042 1628118 : degB = B? degpol(B): degA - 1;
2043 1628116 : if (n == 1)
2044 : {
2045 943273 : ulong Hp, p = uel(P,1);
2046 943273 : GEN a = ZX_to_Flx(A, p), b = B? ZX_to_Flx(B, p): Flx_deriv(a, p);
2047 943233 : Hp = ZX_resultant_prime(a, b, dB, degA, degB, p);
2048 943288 : set_avma(av); *mod = utoipos(p); return utoi(Hp);
2049 : }
2050 684843 : T = ZV_producttree(P);
2051 684839 : A = ZX_nv_mod_tree(A, P, T);
2052 684842 : if (B) B = ZX_nv_mod_tree(B, P, T);
2053 684842 : H = cgetg(n+1, t_VECSMALL); av2 = avma;
2054 3704159 : for(i=1; i <= n; i++, set_avma(av2))
2055 : {
2056 3019324 : ulong p = P[i];
2057 3019324 : GEN a = gel(A,i), b = B? gel(B,i): Flx_deriv(a, p);
2058 3019561 : H[i] = ZX_resultant_prime(a, b, dB, degA, degB, p);
2059 : }
2060 684835 : H = ZV_chinese_tree(H, P, T, ZV_chinesetree(P,T));
2061 684842 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2062 : }
2063 :
2064 : GEN
2065 1628161 : ZX_resultant_worker(GEN P, GEN A, GEN B, GEN dB)
2066 : {
2067 1628161 : GEN V = cgetg(3, t_VEC);
2068 1628120 : if (typ(B) == t_INT) B = NULL;
2069 1628120 : if (!signe(dB)) dB = NULL;
2070 1628120 : gel(V,1) = ZX_resultant_slice(A, B, dB, P, &gel(V,2));
2071 1628149 : 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 1297734 : ZX_resultant_all(GEN A, GEN B, GEN dB, ulong bound)
2078 : {
2079 1297734 : pari_sp av = avma;
2080 : forprime_t S;
2081 : GEN H, worker;
2082 1297734 : if (B)
2083 : {
2084 105109 : long a = degpol(A), b = degpol(B);
2085 105109 : if (a < 0 || b < 0) return gen_0;
2086 105079 : if (!a) return powiu(gel(A,2), b);
2087 105079 : if (!b) return powiu(gel(B,2), a);
2088 103362 : if (!bound) bound = ZX_ZXY_ResBound(A, B, dB);
2089 : }
2090 1295987 : worker = snm_closure(is_entry("_ZX_resultant_worker"),
2091 : mkvec3(A, B? B: gen_0, dB? dB: gen_0));
2092 1296062 : init_modular_big(&S);
2093 1296016 : H = gen_crt("ZX_resultant_all", worker, &S, dB, bound, 0, NULL,
2094 : ZV_chinese_center, Fp_center);
2095 1296046 : 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 24353 : 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 18732 : QXQ_norm(GEN A, GEN B)
2140 : {
2141 : GEN c, R, lB;
2142 18732 : long dA = degpol(A), dB = degpol(B);
2143 18732 : pari_sp av = avma;
2144 18732 : if (dA < 0) return gen_0;
2145 18732 : A = Q_primitive_part(A, &c);
2146 18732 : R = ZX_resultant(B, A);
2147 18732 : if (c) R = gmul(R, gpowgs(c, dB));
2148 18732 : lB = leading_coeff(B);
2149 18732 : if (!equali1(lB)) R = gdiv(R, gpowgs(lB, dA));
2150 18732 : return gerepileupto(av, R);
2151 : }
2152 :
2153 : /* assume x has integral coefficients */
2154 : GEN
2155 1195689 : ZX_disc_all(GEN x, ulong bound)
2156 : {
2157 1195689 : pari_sp av = avma;
2158 1195689 : long s, d = degpol(x);
2159 : GEN l, R;
2160 :
2161 1195692 : if (d <= 1) return d == 1? gen_1: gen_0;
2162 1192672 : s = (d & 2) ? -1: 1;
2163 1192672 : l = leading_coeff(x);
2164 1192679 : if (!bound) bound = ZX_discbound(x);
2165 1192594 : R = ZX_resultant_all(x, NULL, NULL, bound);
2166 1192655 : if (is_pm1(l))
2167 1017406 : { if (signe(l) < 0) s = -s; }
2168 : else
2169 175245 : R = diviiexact(R,l);
2170 1192651 : if (s == -1) togglesign_safe(&R);
2171 1192654 : return gerepileuptoint(av,R);
2172 : }
2173 :
2174 : GEN
2175 1133641 : ZX_disc(GEN x) { return ZX_disc_all(x,0); }
2176 :
2177 : static GEN
2178 7273 : ZXQX_resultant_prime(GEN a, GEN b, GEN dB, long degA, long degB, GEN T, ulong p)
2179 : {
2180 7273 : long dropa = degA - degpol(a), dropb = degB - degpol(b);
2181 : GEN H, dp;
2182 7272 : if (dropa && dropb) return pol0_Flx(T[1]); /* p | lc(A), p | lc(B) */
2183 7272 : H = FlxqX_saferesultant(a, b, T, p);
2184 7271 : if (!H) return NULL;
2185 7271 : 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 7271 : 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 7271 : dp = dB ? ZX_to_Flx(dB, p): pol1_Flx(T[1]);
2199 7272 : 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 7272 : return H;
2206 : }
2207 :
2208 : /* If B=NULL, assume B=A' */
2209 : static GEN
2210 3547 : ZXQX_resultant_slice(GEN A, GEN B, GEN U, GEN dB, GEN P, GEN *mod)
2211 : {
2212 3547 : pari_sp av = avma;
2213 3547 : long degA, degB, i, n = lg(P)-1;
2214 : GEN H, T;
2215 3547 : long v = varn(U), redo = 0;
2216 :
2217 3547 : degA = degpol(A);
2218 3547 : degB = B? degpol(B): degA - 1;
2219 3547 : if (n == 1)
2220 : {
2221 2307 : ulong p = uel(P,1);
2222 2307 : GEN a = ZXX_to_FlxX(A, p, v), b = B? ZXX_to_FlxX(B, p, v): FlxX_deriv(a, p);
2223 2307 : GEN u = ZX_to_Flx(U, p);
2224 2307 : GEN Hp = ZXQX_resultant_prime(a, b, dB, degA, degB, u, p);
2225 2307 : if (!Hp) { set_avma(av); *mod = gen_1; return pol_0(v); }
2226 2307 : Hp = gerepileupto(av, Flx_to_ZX(Hp)); *mod = utoipos(p); return Hp;
2227 : }
2228 1240 : T = ZV_producttree(P);
2229 1240 : A = ZXX_nv_mod_tree(A, P, T, v);
2230 1240 : if (B) B = ZXX_nv_mod_tree(B, P, T, v);
2231 1240 : U = ZX_nv_mod_tree(U, P, T);
2232 1240 : H = cgetg(n+1, t_VEC);
2233 6204 : for(i=1; i <= n; i++)
2234 : {
2235 4964 : ulong p = P[i];
2236 4964 : GEN a = gel(A,i), b = B? gel(B,i): FlxX_deriv(a, p), u = gel(U, i);
2237 4966 : GEN h = ZXQX_resultant_prime(a, b, dB, degA, degB, u, p);
2238 4964 : if (!h)
2239 : {
2240 0 : gel(H,i) = pol_0(v);
2241 0 : P[i] = 1; redo = 1;
2242 : }
2243 : else
2244 4964 : gel(H,i) = h;
2245 : }
2246 1240 : if (redo) T = ZV_producttree(P);
2247 1240 : H = nxV_chinese_center_tree(H, P, T, ZV_chinesetree(P, T));
2248 1239 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2249 : }
2250 :
2251 : GEN
2252 3547 : ZXQX_resultant_worker(GEN P, GEN A, GEN B, GEN T, GEN dB)
2253 : {
2254 3547 : GEN V = cgetg(3, t_VEC);
2255 3547 : if (isintzero(B)) B = NULL;
2256 3547 : if (!signe(dB)) dB = NULL;
2257 3547 : gel(V,1) = ZXQX_resultant_slice(A, B, T, dB, P, &gel(V,2));
2258 3547 : return V;
2259 : }
2260 :
2261 : static ulong
2262 3265 : ZXQX_resultant_bound_i(GEN nf, GEN A, GEN B, GEN (*f)(GEN,GEN,long))
2263 : {
2264 3265 : pari_sp av = avma;
2265 3265 : GEN r, M = nf_L2_bound(nf, NULL, &r);
2266 3265 : long v = nf_get_varn(nf), i, l = lg(r);
2267 3265 : GEN a = cgetg(l, t_COL);
2268 10373 : for (i = 1; i < l; i++)
2269 7108 : gel(a, i) = f(gsubst(A, v, gel(r,i)), gsubst(B, v, gel(r,i)), DEFAULTPREC);
2270 3265 : return gc_ulong(av, (ulong) dbllog2(gmul(M,RgC_fpnorml2(a, DEFAULTPREC))));
2271 : }
2272 : static ulong
2273 2957 : ZXQX_resultant_bound(GEN nf, GEN A, GEN B)
2274 2957 : { return ZXQX_resultant_bound_i(nf, A, B, &RgX_RgXY_ResBound); }
2275 :
2276 : /* Compute Res(A, B/dB) in Z[X]/T, assuming A,B in Z[X,Y], dB in Z or NULL (= 1)
2277 : * If B=NULL, take B = A' and assume deg A > 1 */
2278 : static GEN
2279 2954 : ZXQX_resultant_all(GEN A, GEN B, GEN T, GEN dB, ulong bound)
2280 : {
2281 2954 : pari_sp av = avma;
2282 : forprime_t S;
2283 : GEN H, worker;
2284 2954 : if (B)
2285 : {
2286 70 : long a = degpol(A), b = degpol(B);
2287 70 : if (a < 0 || b < 0) return gen_0;
2288 70 : if (!a) return gpowgs(gel(A,2), b);
2289 70 : if (!b) return gpowgs(gel(B,2), a);
2290 : } else
2291 2884 : if (!bound) B = RgX_deriv(A);
2292 2933 : if (!bound) bound = ZXQX_resultant_bound(nfinit(T, DEFAULTPREC), A, B);
2293 2933 : worker = snm_closure(is_entry("_ZXQX_resultant_worker"),
2294 : mkvec4(A, B? B: gen_0, T, dB? dB: gen_0));
2295 2933 : init_modular_big(&S);
2296 2933 : H = gen_crt("ZXQX_resultant_all", worker, &S, dB, bound, 0, NULL,
2297 : nxV_chinese_center, FpX_center);
2298 2933 : if (DEBUGLEVEL)
2299 0 : err_printf("ZXQX_resultant_all: a priori bound: %lu, a posteriori: %lu\n",
2300 : bound, expi(gsupnorm(H, DEFAULTPREC)));
2301 2933 : return gerepileupto(av, H);
2302 : }
2303 :
2304 : GEN
2305 70 : nfX_resultant(GEN nf, GEN x, GEN y)
2306 : {
2307 70 : pari_sp av = avma;
2308 70 : GEN cx, cy, D, T = nf_get_pol(nf);
2309 : ulong bound;
2310 70 : long d = degpol(x), v = varn(T);
2311 70 : if (d <= 1) return d == 1? pol_1(v): pol_0(v);
2312 70 : x = Q_primitive_part(x, &cx);
2313 70 : y = Q_primitive_part(y, &cy);
2314 70 : bound = ZXQX_resultant_bound(nf, x, y);
2315 70 : D = ZXQX_resultant_all(x, y, T, NULL, bound);
2316 70 : if (cx) D = gmul(D, gpowgs(cx, degpol(y)));
2317 70 : if (cy) D = gmul(D, gpowgs(cy, degpol(x)));
2318 70 : return gerepileupto(av, D);
2319 : }
2320 :
2321 : static GEN
2322 217 : to_ZX(GEN a, long v) { return typ(a)==t_INT? scalarpol(a,v): a; }
2323 :
2324 : static GEN
2325 2884 : ZXQX_disc_all(GEN x, GEN T, ulong bound)
2326 : {
2327 2884 : pari_sp av = avma;
2328 2884 : long s, d = degpol(x), v = varn(T);
2329 : GEN l, R;
2330 :
2331 2884 : if (d <= 1) return d == 1? pol_1(v): pol_0(v);
2332 2884 : s = (d & 2) ? -1: 1;
2333 2884 : l = leading_coeff(x);
2334 2884 : R = ZXQX_resultant_all(x, NULL, T, NULL, bound);
2335 2884 : if (!gequal1(l)) R = QXQ_div(R, to_ZX(l,v), T);
2336 2884 : if (s == -1) R = RgX_neg(R);
2337 2884 : return gerepileupto(av, R);
2338 : }
2339 :
2340 : GEN
2341 7 : QX_disc(GEN x)
2342 : {
2343 7 : pari_sp av = avma;
2344 7 : GEN c, d = ZX_disc( Q_primitive_part(x, &c) );
2345 7 : if (c) d = gmul(d, gpowgs(c, 2*degpol(x) - 2));
2346 7 : return gerepileupto(av, d);
2347 : }
2348 :
2349 : GEN
2350 3038 : nfX_disc(GEN nf, GEN x)
2351 : {
2352 3038 : pari_sp av = avma;
2353 3038 : GEN c, D, T = nf_get_pol(nf);
2354 : ulong bound;
2355 3038 : long d = degpol(x), v = varn(T);
2356 3038 : if (d <= 1) return d == 1? pol_1(v): pol_0(v);
2357 2884 : x = Q_primitive_part(x, &c);
2358 2884 : bound = ZXQX_resultant_bound(nf, x, RgX_deriv(x));
2359 2884 : D = ZXQX_disc_all(x, T, bound);
2360 2884 : if (c) D = gmul(D, gpowgs(c, 2*d - 2));
2361 2884 : return gerepileupto(av, D);
2362 : }
2363 :
2364 : GEN
2365 559658 : QXQ_mul(GEN x, GEN y, GEN T)
2366 : {
2367 559658 : GEN dx, nx = Q_primitive_part(x, &dx);
2368 559657 : GEN dy, ny = Q_primitive_part(y, &dy);
2369 559659 : GEN z = ZXQ_mul(nx, ny, T);
2370 559661 : if (dx || dy)
2371 : {
2372 556670 : GEN d = dx ? dy ? gmul(dx, dy): dx : dy;
2373 556670 : if (!gequal1(d)) z = ZX_Q_mul(z, d);
2374 : }
2375 559659 : return z;
2376 : }
2377 :
2378 : GEN
2379 92939 : QXQ_sqr(GEN x, GEN T)
2380 : {
2381 92939 : GEN dx, nx = Q_primitive_part(x, &dx);
2382 92939 : GEN z = ZXQ_sqr(nx, T);
2383 92939 : if (dx)
2384 91147 : z = ZX_Q_mul(z, gsqr(dx));
2385 92939 : return z;
2386 : }
2387 :
2388 : static GEN
2389 101844 : QXQ_inv_slice(GEN A, GEN B, GEN P, GEN *mod)
2390 : {
2391 101844 : pari_sp av = avma;
2392 101844 : long i, n = lg(P)-1, v = varn(A), redo = 0;
2393 : GEN H, T;
2394 101844 : if (n == 1)
2395 : {
2396 56637 : ulong p = uel(P,1);
2397 56637 : GEN a = ZX_to_Flx(A, p), b = ZX_to_Flx(B, p);
2398 56638 : GEN U = Flxq_invsafe(a, b, p);
2399 56637 : if (!U)
2400 : {
2401 24 : set_avma(av);
2402 24 : *mod = gen_1; return pol_0(v);
2403 : }
2404 56613 : H = gerepilecopy(av, Flx_to_ZX(U));
2405 56614 : *mod = utoipos(p); return H;
2406 : }
2407 45207 : T = ZV_producttree(P);
2408 45207 : A = ZX_nv_mod_tree(A, P, T);
2409 45207 : B = ZX_nv_mod_tree(B, P, T);
2410 45207 : H = cgetg(n+1, t_VEC);
2411 224519 : for(i=1; i <= n; i++)
2412 : {
2413 179313 : ulong p = P[i];
2414 179313 : GEN a = gel(A,i), b = gel(B,i);
2415 179313 : GEN U = Flxq_invsafe(a, b, p);
2416 179309 : if (!U)
2417 : {
2418 601 : gel(H,i) = pol_0(v);
2419 601 : P[i] = 1; redo = 1;
2420 : }
2421 : else
2422 178708 : gel(H,i) = U;
2423 : }
2424 45206 : if (redo) T = ZV_producttree(P);
2425 45206 : H = nxV_chinese_center_tree(H, P, T, ZV_chinesetree(P, T));
2426 45207 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2427 : }
2428 :
2429 : GEN
2430 101844 : QXQ_inv_worker(GEN P, GEN A, GEN B)
2431 : {
2432 101844 : GEN V = cgetg(3, t_VEC);
2433 101844 : gel(V,1) = QXQ_inv_slice(A, B, P, &gel(V,2));
2434 101845 : return V;
2435 : }
2436 :
2437 : /* lift(1 / Mod(A,B)). B a ZX, A a scalar or a QX */
2438 : GEN
2439 37557 : QXQ_inv(GEN A, GEN B)
2440 : {
2441 : GEN D, Ap, Bp;
2442 : ulong pp;
2443 37557 : pari_sp av2, av = avma;
2444 : forprime_t S;
2445 37557 : GEN worker, U, H = NULL, mod = gen_1;
2446 : pari_timer ti;
2447 : long k, dA, dB;
2448 37557 : if (is_scalar_t(typ(A))) return scalarpol(ginv(A), varn(B));
2449 : /* A a QX, B a ZX */
2450 37557 : A = Q_primitive_part(A, &D);
2451 37558 : dA = degpol(A); dB= degpol(B);
2452 : /* A, B in Z[X] */
2453 37558 : init_modular_small(&S);
2454 : do {
2455 37557 : pp = u_forprime_next(&S);
2456 37557 : Ap = ZX_to_Flx(A, pp);
2457 37556 : Bp = ZX_to_Flx(B, pp);
2458 37556 : } while (degpol(Ap) != dA || degpol(Bp) != dB);
2459 37556 : if (degpol(Flx_gcd(Ap, Bp, pp)) != 0 && degpol(ZX_gcd(A,B))!=0)
2460 14 : pari_err_INV("QXQ_inv",mkpolmod(A,B));
2461 37544 : worker = snm_closure(is_entry("_QXQ_inv_worker"), mkvec2(A, B));
2462 37544 : av2 = avma;
2463 37544 : for (k = 1; ;k *= 2)
2464 41111 : {
2465 : GEN res, b, N, den;
2466 78655 : gen_inccrt_i("QXQ_inv", worker, NULL, (k+1)>>1, 0, &S, &H, &mod,
2467 : nxV_chinese_center, FpX_center);
2468 78655 : gerepileall(av2, 2, &H, &mod);
2469 78655 : b = sqrti(shifti(mod,-1));
2470 78653 : if (DEBUGLEVEL>5) timer_start(&ti);
2471 78653 : U = FpX_ratlift(H, mod, b, b, NULL);
2472 78654 : if (DEBUGLEVEL>5) timer_printf(&ti,"QXQ_inv: ratlift");
2473 84279 : if (!U) continue;
2474 43168 : N = Q_remove_denom(U, &den); if (!den) den = gen_1;
2475 43168 : res = Flx_rem(Flx_Fl_sub(Flx_mul(Ap, ZX_to_Flx(N,pp), pp),
2476 : umodiu(den, pp), pp), Bp, pp);
2477 43168 : if (degpol(res) >= 0) continue;
2478 37544 : res = ZX_Z_sub(ZX_mul(A, N), den);
2479 37544 : res = ZX_is_monic(B) ? ZX_rem(res, B): RgX_pseudorem(res, B);
2480 37544 : if (DEBUGLEVEL>5) timer_printf(&ti,"QXQ_inv: final check");
2481 37544 : if (degpol(res)<0)
2482 : {
2483 37544 : if (D) U = RgX_Rg_div(U, D);
2484 37544 : return gerepilecopy(av, U);
2485 : }
2486 : }
2487 : }
2488 :
2489 : static GEN
2490 116909 : QXQ_div_slice(GEN A, GEN B, GEN C, GEN P, GEN *mod)
2491 : {
2492 116909 : pari_sp av = avma;
2493 116909 : long i, n = lg(P)-1, v = varn(A), redo = 0;
2494 : GEN H, T;
2495 116909 : if (n == 1)
2496 : {
2497 42725 : ulong p = uel(P,1);
2498 42725 : GEN a = ZX_to_Flx(A, p), b = ZX_to_Flx(B, p), c = ZX_to_Flx(C, p);
2499 42725 : GEN bi = Flxq_invsafe(b, c, p), U;
2500 42725 : if (!bi)
2501 : {
2502 0 : set_avma(av);
2503 0 : *mod = gen_1; return pol_0(v);
2504 : }
2505 42725 : U = Flxq_mul(a, bi, c, p);
2506 42725 : H = gerepilecopy(av, Flx_to_ZX(U));
2507 42725 : *mod = utoipos(p); return H;
2508 : }
2509 74184 : T = ZV_producttree(P);
2510 74183 : A = ZX_nv_mod_tree(A, P, T);
2511 74184 : B = ZX_nv_mod_tree(B, P, T);
2512 74184 : C = ZX_nv_mod_tree(C, P, T);
2513 74184 : H = cgetg(n+1, t_VEC);
2514 326480 : for(i=1; i <= n; i++)
2515 : {
2516 252296 : ulong p = P[i];
2517 252296 : GEN a = gel(A,i), b = gel(B,i), c = gel(C, i);
2518 252296 : GEN bi = Flxq_invsafe(b, c, p);
2519 252296 : if (!bi)
2520 : {
2521 0 : gel(H,i) = pol_0(v);
2522 0 : P[i] = 1; redo = 1;
2523 : }
2524 : else
2525 252296 : gel(H,i) = Flxq_mul(a, bi, c, p);
2526 : }
2527 74184 : if (redo) T = ZV_producttree(P);
2528 74184 : H = nxV_chinese_center_tree(H, P, T, ZV_chinesetree(P, T));
2529 74184 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2530 : }
2531 :
2532 : GEN
2533 116909 : QXQ_div_worker(GEN P, GEN A, GEN B, GEN C)
2534 : {
2535 116909 : GEN V = cgetg(3, t_VEC);
2536 116909 : gel(V,1) = QXQ_div_slice(A, B, C, P, &gel(V,2));
2537 116909 : return V;
2538 : }
2539 :
2540 : /* lift(Mod(A/B, C)). C a ZX, A, B a scalar or a QX */
2541 : GEN
2542 31661 : QXQ_div(GEN A, GEN B, GEN C)
2543 : {
2544 : GEN DA, DB, Ap, Bp, Cp;
2545 : ulong pp;
2546 31661 : pari_sp av2, av = avma;
2547 : forprime_t S;
2548 31661 : GEN worker, U, H = NULL, mod = gen_1;
2549 : pari_timer ti;
2550 : long k, dA, dB, dC;
2551 31661 : if (is_scalar_t(typ(A))) return scalarpol(ginv(A), varn(B));
2552 : /* A a QX, B a ZX */
2553 31661 : A = Q_primitive_part(A, &DA);
2554 31661 : B = Q_primitive_part(B, &DB);
2555 31661 : dA = degpol(A); dB = degpol(B); dC = degpol(C);
2556 : /* A, B in Z[X] */
2557 31661 : init_modular_small(&S);
2558 : do {
2559 31661 : pp = u_forprime_next(&S);
2560 31661 : Ap = ZX_to_Flx(A, pp);
2561 31661 : Bp = ZX_to_Flx(B, pp);
2562 31661 : Cp = ZX_to_Flx(C, pp);
2563 31661 : } while (degpol(Ap) != dA || degpol(Bp) != dB || degpol(Cp) != dC);
2564 31661 : if (degpol(Flx_gcd(Bp, Cp, pp)) != 0 && degpol(ZX_gcd(B,C))!=0)
2565 0 : pari_err_INV("QXQ_div",mkpolmod(B,C));
2566 31661 : worker = snm_closure(is_entry("_QXQ_div_worker"), mkvec3(A, B, C));
2567 31661 : av2 = avma;
2568 31661 : for (k = 1; ;k *= 2)
2569 45307 : {
2570 : GEN res, b, N, den;
2571 76968 : gen_inccrt_i("QXQ_div", worker, NULL, (k+1)>>1, 0, &S, &H, &mod,
2572 : nxV_chinese_center, FpX_center);
2573 76968 : gerepileall(av2, 2, &H, &mod);
2574 76968 : b = sqrti(shifti(mod,-1));
2575 76966 : if (DEBUGLEVEL>5) timer_start(&ti);
2576 76966 : U = FpX_ratlift(H, mod, b, b, NULL);
2577 76968 : if (DEBUGLEVEL>5) timer_printf(&ti,"QXQ_div: ratlift");
2578 87578 : if (!U) continue;
2579 42271 : N = Q_remove_denom(U, &den); if (!den) den = gen_1;
2580 42271 : res = Flx_rem(Flx_sub(Flx_mul(Bp, ZX_to_Flx(N,pp), pp),
2581 : Flx_Fl_mul(Ap, umodiu(den, pp), pp), pp), Cp, pp);
2582 42271 : if (degpol(res) >= 0) continue;
2583 31661 : res = ZX_sub(ZX_mul(B, N), ZX_Z_mul(A,den));
2584 31661 : res = ZX_is_monic(C) ? ZX_rem(res, C): RgX_pseudorem(res, C);
2585 31661 : if (DEBUGLEVEL>5) timer_printf(&ti,"QXQ_div: final check");
2586 31661 : if (degpol(res)<0)
2587 : {
2588 31661 : if (DA && DB) U = RgX_Rg_mul(U, gdiv(DA,DB));
2589 26929 : else if (DA) U = RgX_Rg_mul(U, DA);
2590 15141 : else if (DB) U = RgX_Rg_div(U, DB);
2591 31661 : return gerepilecopy(av, U);
2592 : }
2593 : }
2594 : }
2595 :
2596 : /************************************************************************
2597 : * *
2598 : * ZXQ_minpoly *
2599 : * *
2600 : ************************************************************************/
2601 :
2602 : static GEN
2603 3523 : ZXQ_minpoly_slice(GEN A, GEN B, long d, GEN P, GEN *mod)
2604 : {
2605 3523 : pari_sp av = avma;
2606 3523 : long i, n = lg(P)-1, v = evalvarn(varn(B));
2607 : GEN H, T;
2608 3523 : if (n == 1)
2609 : {
2610 716 : ulong p = uel(P,1);
2611 716 : GEN a = ZX_to_Flx(A, p), b = ZX_to_Flx(B, p);
2612 716 : GEN Hp = Flxq_minpoly(a, b, p);
2613 716 : if (degpol(Hp) != d) { p = 1; Hp = pol0_Flx(v); }
2614 716 : H = gerepileupto(av, Flx_to_ZX(Hp));
2615 716 : *mod = utoipos(p); return H;
2616 : }
2617 2807 : T = ZV_producttree(P);
2618 2807 : A = ZX_nv_mod_tree(A, P, T);
2619 2807 : B = ZX_nv_mod_tree(B, P, T);
2620 2807 : H = cgetg(n+1, t_VEC);
2621 16838 : for(i=1; i <= n; i++)
2622 : {
2623 14031 : ulong p = P[i];
2624 14031 : GEN a = gel(A,i), b = gel(B,i);
2625 14031 : GEN m = Flxq_minpoly(a, b, p);
2626 14031 : if (degpol(m) != d) { P[i] = 1; m = pol0_Flx(v); }
2627 14031 : gel(H, i) = m;
2628 : }
2629 2807 : H = nxV_chinese_center_tree(H, P, T, ZV_chinesetree(P, T));
2630 2807 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2631 : }
2632 :
2633 : GEN
2634 3523 : ZXQ_minpoly_worker(GEN P, GEN A, GEN B, long d)
2635 : {
2636 3523 : GEN V = cgetg(3, t_VEC);
2637 3523 : gel(V,1) = ZXQ_minpoly_slice(A, B, d, P, &gel(V,2));
2638 3523 : return V;
2639 : }
2640 :
2641 : GEN
2642 1701 : ZXQ_minpoly(GEN A, GEN B, long d, ulong bound)
2643 : {
2644 1701 : pari_sp av = avma;
2645 : GEN worker, H, dB;
2646 : forprime_t S;
2647 1701 : B = Q_remove_denom(B, &dB);
2648 1701 : worker = strtoclosure("_ZXQ_minpoly_worker", 3, A, B, stoi(d));
2649 1701 : init_modular_big(&S);
2650 1701 : H = gen_crt("ZXQ_minpoly", worker, &S, dB, bound, 0, NULL,
2651 : nxV_chinese_center, FpX_center_i);
2652 1701 : return gerepilecopy(av, H);
2653 : }
2654 :
2655 : /************************************************************************
2656 : * *
2657 : * ZX_ZXY_resultant *
2658 : * *
2659 : ************************************************************************/
2660 :
2661 : static GEN
2662 173253 : ZX_ZXY_resultant_prime(GEN a, GEN b, ulong dp, ulong p,
2663 : long degA, long degB, long dres, long sX)
2664 : {
2665 173253 : long dropa = degA - degpol(a), dropb = degB - degpol(b);
2666 173252 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2667 173251 : GEN Hp = Flx_FlxY_resultant_polint(a, b, p, pi, dres, sX);
2668 173256 : if (dropa && dropb)
2669 0 : Hp = zero_Flx(sX);
2670 : else {
2671 173256 : if (dropa)
2672 : { /* multiply by ((-1)^deg B lc(B))^(deg A - deg a) */
2673 0 : GEN c = gel(b,degB+2); /* lc(B) */
2674 0 : if (odd(degB)) c = Flx_neg(c, p);
2675 0 : if (!Flx_equal1(c)) {
2676 0 : c = Flx_powu_pre(c, dropa, p, pi);
2677 0 : if (!Flx_equal1(c)) Hp = Flx_mul_pre(Hp, c, p, pi);
2678 : }
2679 : }
2680 173256 : else if (dropb)
2681 : { /* multiply by lc(A)^(deg B - deg b) */
2682 0 : ulong c = uel(a, degA+2); /* lc(A) */
2683 0 : c = Fl_powu(c, dropb, p);
2684 0 : if (c != 1) Hp = Flx_Fl_mul_pre(Hp, c, p, pi);
2685 : }
2686 : }
2687 173256 : if (dp != 1) Hp = Flx_Fl_mul_pre(Hp, Fl_powu_pre(Fl_inv(dp,p), degA, p, pi), p, pi);
2688 173256 : return Hp;
2689 : }
2690 :
2691 : static GEN
2692 69272 : ZX_ZXY_resultant_slice(GEN A, GEN B, GEN dB, long degA, long degB, long dres,
2693 : GEN P, GEN *mod, long sX, long vY)
2694 : {
2695 69272 : pari_sp av = avma;
2696 69272 : long i, n = lg(P)-1;
2697 : GEN H, T, D;
2698 69272 : if (n == 1)
2699 : {
2700 39962 : ulong p = uel(P,1);
2701 39962 : ulong dp = dB ? umodiu(dB, p): 1;
2702 39962 : GEN a = ZX_to_Flx(A, p), b = ZXX_to_FlxX(B, p, vY);
2703 39962 : GEN Hp = ZX_ZXY_resultant_prime(a, b, dp, p, degA, degB, dres, sX);
2704 39962 : H = gerepileupto(av, Flx_to_ZX(Hp));
2705 39961 : *mod = utoipos(p); return H;
2706 : }
2707 29310 : T = ZV_producttree(P);
2708 29310 : A = ZX_nv_mod_tree(A, P, T);
2709 29310 : B = ZXX_nv_mod_tree(B, P, T, vY);
2710 29310 : D = dB ? Z_ZV_mod_tree(dB, P, T): NULL;
2711 29310 : H = cgetg(n+1, t_VEC);
2712 117461 : for(i=1; i <= n; i++)
2713 : {
2714 88150 : ulong p = P[i];
2715 88150 : GEN a = gel(A,i), b = gel(B,i);
2716 88150 : ulong dp = D ? uel(D, i): 1;
2717 88150 : gel(H,i) = ZX_ZXY_resultant_prime(a, b, dp, p, degA, degB, dres, sX);
2718 : }
2719 29311 : H = nxV_chinese_center_tree(H, P, T, ZV_chinesetree(P, T));
2720 29310 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2721 : }
2722 :
2723 : GEN
2724 69272 : ZX_ZXY_resultant_worker(GEN P, GEN A, GEN B, GEN dB, GEN v)
2725 : {
2726 69272 : GEN V = cgetg(3, t_VEC);
2727 69272 : if (isintzero(dB)) dB = NULL;
2728 69272 : gel(V,1) = ZX_ZXY_resultant_slice(A, B, dB, v[1], v[2], v[3], P, &gel(V,2), v[4], v[5]);
2729 69271 : return V;
2730 : }
2731 :
2732 : GEN
2733 60006 : ZX_ZXY_resultant(GEN A, GEN B)
2734 : {
2735 60006 : pari_sp av = avma;
2736 : forprime_t S;
2737 : ulong bound;
2738 60006 : long v = fetch_var_higher();
2739 60006 : long degA = degpol(A), degB, dres = degA * degpol(B);
2740 60006 : long vX = varn(B), vY = varn(A); /* assume vY has lower priority */
2741 60006 : long sX = evalvarn(vX);
2742 : GEN worker, H, dB;
2743 60006 : B = Q_remove_denom(B, &dB);
2744 60005 : if (!dB) B = leafcopy(B);
2745 60005 : A = leafcopy(A); setvarn(A,v);
2746 60005 : B = swap_vars(B, vY); setvarn(B,v); degB = degpol(B);
2747 60006 : bound = ZX_ZXY_ResBound(A, B, dB);
2748 60004 : if (DEBUGLEVEL>4) err_printf("bound for resultant coeffs: 2^%ld\n",bound);
2749 120009 : worker = snm_closure(is_entry("_ZX_ZXY_resultant_worker"),
2750 60005 : mkvec4(A, B, dB? dB: gen_0,
2751 : mkvecsmall5(degA, degB, dres, sX, vY)));
2752 60006 : init_modular_big(&S);
2753 60006 : H = gen_crt("ZX_ZXY_resultant_all", worker, &S, dB, bound, 0, NULL,
2754 : nxV_chinese_center, FpX_center_i);
2755 60005 : setvarn(H, vX); (void)delete_var();
2756 60005 : return gerepilecopy(av, H);
2757 : }
2758 :
2759 : static long
2760 40402 : ZX_ZXY_rnfequation_lambda(GEN A, GEN B0, long lambda)
2761 : {
2762 40402 : pari_sp av = avma;
2763 40402 : long degA = degpol(A), degB, dres = degA*degpol(B0);
2764 40401 : long v = fetch_var_higher();
2765 40401 : long vX = varn(B0), vY = varn(A); /* assume vY has lower priority */
2766 40401 : long sX = evalvarn(vX);
2767 : GEN dB, B, a, b, Hp;
2768 : forprime_t S;
2769 :
2770 40401 : B0 = Q_remove_denom(B0, &dB);
2771 40401 : if (!dB) B0 = leafcopy(B0);
2772 40401 : A = leafcopy(A);
2773 40401 : B = B0;
2774 40401 : setvarn(A,v);
2775 45143 : INIT:
2776 45143 : if (lambda) B = RgX_translate(B0, monomial(stoi(lambda), 1, vY));
2777 45143 : B = swap_vars(B, vY); setvarn(B,v);
2778 : /* B0(lambda v + x, v) */
2779 45143 : if (DEBUGLEVEL>4) err_printf("Trying lambda = %ld\n", lambda);
2780 :
2781 45143 : degB = degpol(B);
2782 45143 : init_modular_big(&S);
2783 : while (1)
2784 0 : {
2785 45143 : ulong p = u_forprime_next(&S);
2786 45143 : ulong dp = dB ? umodiu(dB, p): 1;
2787 45143 : if (!dp) continue;
2788 45143 : a = ZX_to_Flx(A, p);
2789 45143 : b = ZXX_to_FlxX(B, p, v);
2790 45143 : Hp = ZX_ZXY_resultant_prime(a, b, dp, p, degA, degB, dres, sX);
2791 45144 : if (degpol(Hp) != dres) continue;
2792 45144 : if (dp != 1) Hp = Flx_Fl_mul(Hp, Fl_powu(Fl_inv(dp,p), degA, p), p);
2793 45144 : if (!Flx_is_squarefree(Hp, p)) { lambda = next_lambda(lambda); goto INIT; }
2794 40402 : if (DEBUGLEVEL>4) err_printf("Final lambda = %ld\n", lambda);
2795 40402 : (void)delete_var(); return gc_long(av,lambda);
2796 : }
2797 : }
2798 :
2799 : GEN
2800 41372 : ZX_ZXY_rnfequation(GEN A, GEN B, long *lambda)
2801 : {
2802 41372 : if (lambda)
2803 : {
2804 40402 : *lambda = ZX_ZXY_rnfequation_lambda(A, B, *lambda);
2805 40402 : if (*lambda) B = RgX_translate(B, monomial(stoi(*lambda), 1, varn(A)));
2806 : }
2807 41372 : return ZX_ZXY_resultant(A,B);
2808 : }
2809 :
2810 : static GEN
2811 10371 : ZX_composedsum_slice(GEN A, GEN B, GEN P, GEN *mod)
2812 : {
2813 10371 : pari_sp av = avma;
2814 10371 : long i, n = lg(P)-1;
2815 : GEN H, T;
2816 10371 : if (n == 1)
2817 : {
2818 9869 : ulong p = uel(P,1);
2819 9869 : GEN a = ZX_to_Flx(A, p), b = ZX_to_Flx(B, p);
2820 9868 : GEN Hp = Flx_composedsum(a, b, p);
2821 9866 : H = gerepileupto(av, Flx_to_ZX(Hp));
2822 9870 : *mod = utoipos(p); return H;
2823 : }
2824 502 : T = ZV_producttree(P);
2825 502 : A = ZX_nv_mod_tree(A, P, T);
2826 502 : B = ZX_nv_mod_tree(B, P, T);
2827 502 : H = cgetg(n+1, t_VEC);
2828 4526 : for(i=1; i <= n; i++)
2829 : {
2830 4024 : ulong p = P[i];
2831 4024 : GEN a = gel(A,i), b = gel(B,i);
2832 4024 : gel(H,i) = Flx_composedsum(a, b, p);
2833 : }
2834 502 : H = nxV_chinese_center_tree(H, P, T, ZV_chinesetree(P, T));
2835 502 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2836 : }
2837 :
2838 : GEN
2839 10370 : ZX_composedsum_worker(GEN P, GEN A, GEN B)
2840 : {
2841 10370 : GEN V = cgetg(3, t_VEC);
2842 10371 : gel(V,1) = ZX_composedsum_slice(A, B, P, &gel(V,2));
2843 10371 : return V;
2844 : }
2845 :
2846 : static GEN
2847 10105 : ZX_composedsum_i(GEN A, GEN B, GEN lead)
2848 : {
2849 10105 : pari_sp av = avma;
2850 : forprime_t S;
2851 : ulong bound;
2852 : GEN H, worker, mod;
2853 10105 : if (degpol(A) < degpol(B)) swap(A, B);
2854 10106 : if (!lead) lead = mulii(leading_coeff(A),leading_coeff(B));
2855 10106 : bound = ZX_ZXY_ResBound_1(A, B);
2856 10103 : worker = snm_closure(is_entry("_ZX_composedsum_worker"), mkvec2(A,B));
2857 10107 : init_modular_big(&S);
2858 10101 : H = gen_crt("ZX_composedsum", worker, &S, lead, bound, 0, &mod,
2859 : nxV_chinese_center, FpX_center);
2860 10107 : return gerepileupto(av, H);
2861 : }
2862 :
2863 : static long
2864 9716 : ZX_compositum_lambda(GEN A, GEN B, GEN lead, long lambda)
2865 : {
2866 9716 : pari_sp av = avma;
2867 : forprime_t S;
2868 : ulong p;
2869 9716 : init_modular_big(&S);
2870 9719 : p = u_forprime_next(&S);
2871 : while (1)
2872 112 : {
2873 : GEN Hp, a;
2874 9831 : if (DEBUGLEVEL>4) err_printf("Trying lambda = %ld\n", lambda);
2875 9831 : if (lead && dvdiu(lead,p)) { p = u_forprime_next(&S); continue; }
2876 9825 : a = ZX_to_Flx(ZX_rescale(A, stoi(-lambda)), p);
2877 9820 : Hp = Flx_composedsum(a, ZX_to_Flx(B, p), p);
2878 9824 : if (!Flx_is_squarefree(Hp, p)) { lambda = next_lambda(lambda); continue; }
2879 9716 : if (DEBUGLEVEL>4) err_printf("Final lambda = %ld\n", lambda);
2880 9716 : return gc_long(av, lambda);
2881 : }
2882 : }
2883 :
2884 : GEN
2885 9720 : ZX_compositum(GEN A, GEN B, long *lambda)
2886 : {
2887 9720 : GEN lead = mulii(leading_coeff(A),leading_coeff(B));
2888 9717 : if (lambda)
2889 : {
2890 9718 : *lambda = ZX_compositum_lambda(A, B, lead, *lambda);
2891 9716 : A = ZX_rescale(A, stoi(-*lambda));
2892 : }
2893 9720 : return ZX_composedsum_i(A, B, lead);
2894 : }
2895 :
2896 : GEN
2897 385 : ZX_composedsum(GEN A, GEN B)
2898 385 : { return ZX_composedsum_i(A, B, NULL); }
2899 :
2900 : static GEN
2901 351 : ZXQX_composedsum_slice(GEN A, GEN B, GEN C, GEN P, GEN *mod)
2902 : {
2903 351 : pari_sp av = avma;
2904 351 : long i, n = lg(P)-1, dC = degpol(C), v = varn(C);
2905 : GEN H, T;
2906 351 : if (n == 1)
2907 : {
2908 174 : ulong p = uel(P,1);
2909 174 : GEN a = ZXX_to_FlxX(A, p, v), b = ZXX_to_FlxX(B, p, v);
2910 174 : GEN c = ZX_to_Flx(C, p);
2911 174 : GEN Hp = FlxX_to_Flm(FlxqX_composedsum(a, b, c, p), dC);
2912 174 : H = gerepileupto(av, Flm_to_ZM(Hp));
2913 174 : *mod = utoipos(p); return H;
2914 : }
2915 177 : T = ZV_producttree(P);
2916 177 : A = ZXX_nv_mod_tree(A, P, T, v);
2917 177 : B = ZXX_nv_mod_tree(B, P, T, v);
2918 177 : C = ZX_nv_mod_tree(C, P, T);
2919 177 : H = cgetg(n+1, t_VEC);
2920 651 : for(i=1; i <= n; i++)
2921 : {
2922 474 : ulong p = P[i];
2923 474 : GEN a = gel(A,i), b = gel(B,i), c = gel(C,i);
2924 474 : gel(H,i) = FlxX_to_Flm(FlxqX_composedsum(a, b, c, p), dC);
2925 : }
2926 177 : H = nmV_chinese_center_tree_seq(H, P, T, ZV_chinesetree(P, T));
2927 177 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2928 : }
2929 :
2930 : GEN
2931 351 : ZXQX_composedsum_worker(GEN P, GEN A, GEN B, GEN C)
2932 : {
2933 351 : GEN V = cgetg(3, t_VEC);
2934 351 : gel(V,1) = ZXQX_composedsum_slice(A, B, C, P, &gel(V,2));
2935 351 : return V;
2936 : }
2937 :
2938 : static GEN
2939 308 : ZXQX_composedsum(GEN A, GEN B, GEN T, ulong bound)
2940 : {
2941 308 : pari_sp av = avma;
2942 : forprime_t S;
2943 : GEN H, worker, mod;
2944 308 : GEN lead = mulii(Q_content(leading_coeff(A)), Q_content(leading_coeff(B)));
2945 308 : worker = snm_closure(is_entry("_ZXQX_composedsum_worker")
2946 : , mkvec3(A,B,T));
2947 308 : init_modular_big(&S);
2948 308 : H = gen_crt("ZXQX_composedsum", worker, &S, lead, bound, 0, &mod,
2949 : nmV_chinese_center, FpM_center);
2950 308 : if (DEBUGLEVEL > 4)
2951 0 : err_printf("nfcompositum: a priori bound: %lu, a posteriori: %lu\n",
2952 : bound, expi(gsupnorm(H, DEFAULTPREC)));
2953 308 : return gerepilecopy(av, RgM_to_RgXX(H, varn(A), varn(T)));
2954 : }
2955 :
2956 : static long
2957 308 : ZXQX_composedsum_bound(GEN nf, GEN A, GEN B)
2958 308 : { return ZXQX_resultant_bound_i(nf, A, B, &RgX_RgXY_ResBound_1); }
2959 :
2960 : GEN
2961 308 : nf_direct_compositum(GEN nf, GEN A, GEN B)
2962 : {
2963 308 : ulong bnd = ZXQX_composedsum_bound(nf, A, B);
2964 308 : return ZXQX_composedsum(A, B, nf_get_pol(nf), bnd);
2965 : }
2966 :
2967 : /************************************************************************
2968 : * *
2969 : * IRREDUCIBLE POLYNOMIAL / Fp *
2970 : * *
2971 : ************************************************************************/
2972 :
2973 : /* irreducible (unitary) polynomial of degree n over Fp */
2974 : GEN
2975 0 : ffinit_rand(GEN p,long n)
2976 : {
2977 0 : for(;;) {
2978 0 : pari_sp av = avma;
2979 0 : GEN pol = ZX_add(pol_xn(n, 0), random_FpX(n-1,0, p));
2980 0 : if (FpX_is_irred(pol, p)) return pol;
2981 0 : set_avma(av);
2982 : }
2983 : }
2984 :
2985 : /* return an extension of degree 2^l of F_2, assume l > 0
2986 : * Not stack clean. */
2987 : static GEN
2988 634 : ffinit_Artin_Schreier_2(long l)
2989 : {
2990 : GEN Q, T, S;
2991 : long i, v;
2992 :
2993 634 : if (l == 1) return mkvecsmall4(0,1,1,1); /*x^2 + x + 1*/
2994 585 : v = fetch_var_higher();
2995 585 : S = mkvecsmall5(0, 0, 0, 1, 1); /* y(y^2 + y) */
2996 585 : Q = mkpoln(3, pol1_Flx(0), pol1_Flx(0), S); /* x^2 + x + y(y^2+y) */
2997 585 : setvarn(Q, v);
2998 :
2999 : /* x^4+x+1, irred over F_2, minimal polynomial of a root of Q */
3000 585 : T = mkvecsmalln(6,evalvarn(v),1UL,1UL,0UL,0UL,1UL);
3001 : /* Q = x^2 + x + a(y) irred. over K = F2[y] / (T(y))
3002 : * ==> x^2 + x + a(y) b irred. over K for any root b of Q
3003 : * ==> x^2 + x + (b^2+b)b */
3004 3244 : for (i=2; i<l; i++) T = Flx_FlxY_resultant(T, Q, 2); /* minpoly(b) / F2*/
3005 585 : (void)delete_var(); T[1] = 0; return T;
3006 : }
3007 :
3008 : /* return an extension of degree p^l of F_p, assume l > 0
3009 : * Not stack clean. */
3010 : GEN
3011 991 : ffinit_Artin_Schreier(ulong p, long l)
3012 : {
3013 : long i, v;
3014 : GEN Q, R, S, T, xp;
3015 991 : if (p==2) return ffinit_Artin_Schreier_2(l);
3016 357 : xp = polxn_Flx(p,0); /* x^p */
3017 357 : T = Flx_sub(xp, mkvecsmall3(0,1,1),p); /* x^p - x - 1 */
3018 357 : if (l == 1) return T;
3019 :
3020 7 : v = evalvarn(fetch_var_higher());
3021 7 : xp[1] = v;
3022 7 : R = Flx_sub(polxn_Flx(2*p-1,0), polxn_Flx(p,0),p);
3023 7 : S = Flx_sub(xp, polx_Flx(0), p);
3024 7 : Q = FlxX_Flx_sub(Flx_to_FlxX(S, v), R, p); /* x^p - x - (y^(2p-1)-y^p) */
3025 14 : for (i = 2; i <= l; ++i) T = Flx_FlxY_resultant(T, Q, p);
3026 7 : (void)delete_var(); T[1] = 0; return T;
3027 : }
3028 :
3029 : static long
3030 147890 : flinit_check(ulong p, long n, long l)
3031 : {
3032 : ulong q;
3033 147890 : if (!uisprime(n)) return 0;
3034 101180 : q = p % n; if (!q) return 0;
3035 98702 : return ugcd((n-1)/Fl_order(q, n-1, n), l) == 1;
3036 : }
3037 :
3038 : static GEN
3039 31713 : flinit(ulong p, long l)
3040 : {
3041 31713 : ulong n = 1+l;
3042 95767 : while (!flinit_check(p,n,l)) n += l;
3043 31713 : if (DEBUGLEVEL>=4) err_printf("FFInit: using polsubcyclo(%ld, %ld)\n",n,l);
3044 31713 : return ZX_to_Flx(polsubcyclo(n,l,0), p);
3045 : }
3046 :
3047 : static GEN
3048 28878 : ffinit_fact_Flx(ulong p, long n)
3049 : {
3050 28878 : GEN P, F = factoru_pow(n), Fp = gel(F,1), Fe = gel(F,2), Fm = gel(F,3);
3051 28878 : long i, l = lg(Fm);
3052 28878 : P = cgetg(l, t_VEC);
3053 61582 : for (i = 1; i < l; ++i)
3054 32704 : gel(P,i) = p==uel(Fp,i) ?
3055 991 : ffinit_Artin_Schreier(uel(Fp,i), Fe[i])
3056 32704 : : flinit(p, uel(Fm,i));
3057 28878 : return FlxV_composedsum(P, p);
3058 : }
3059 :
3060 : static GEN
3061 52123 : init_Flxq_i(ulong p, long n, long sv)
3062 : {
3063 : GEN P;
3064 52123 : if (n == 1) return polx_Flx(sv);
3065 52123 : if (flinit_check(p, n+1, n))
3066 : {
3067 23245 : P = const_vecsmall(n+2,1);
3068 23245 : P[1] = sv; return P;
3069 : }
3070 28878 : P = ffinit_fact_Flx(p,n);
3071 28878 : P[1] = sv; return P;
3072 : }
3073 :
3074 : GEN
3075 0 : init_Flxq(ulong p, long n, long v)
3076 : {
3077 0 : pari_sp av = avma;
3078 0 : return gerepileupto(av, init_Flxq_i(p, n, v));
3079 : }
3080 :
3081 : /* check if polsubcyclo(n,l,0) is irreducible modulo p */
3082 : static long
3083 7185 : fpinit_check(GEN p, long n, long l)
3084 : {
3085 : ulong q;
3086 7185 : if (!uisprime(n)) return 0;
3087 4450 : q = umodiu(p,n); if (!q) return 0;
3088 4450 : return ugcd((n-1)/Fl_order(q, n-1, n), l) == 1;
3089 : }
3090 :
3091 : /* let k=2 if p%4==1, and k=4 else and assume k*p does not divide l.
3092 : * Return an irreducible polynomial of degree l over F_p.
3093 : * Variant of Adleman and Lenstra "Finding irreducible polynomials over
3094 : * finite fields", ACM, 1986 (5) 350--355.
3095 : * Not stack clean */
3096 : static GEN
3097 1653 : fpinit(GEN p, long l)
3098 : {
3099 1653 : ulong n = 1+l;
3100 5202 : while (!fpinit_check(p,n,l)) n += l;
3101 1653 : if (DEBUGLEVEL>=4) err_printf("FFInit: using polsubcyclo(%ld, %ld)\n",n,l);
3102 1653 : return FpX_red(polsubcyclo(n,l,0),p);
3103 : }
3104 :
3105 : static GEN
3106 1574 : ffinit_fact(GEN p, long n)
3107 : {
3108 1574 : GEN P, F = factoru_pow(n), Fp = gel(F,1), Fe = gel(F,2), Fm = gel(F,3);
3109 1574 : long i, l = lg(Fm);
3110 1574 : P = cgetg(l, t_VEC);
3111 3227 : for (i = 1; i < l; ++i)
3112 3306 : gel(P,i) = absequaliu(p, Fp[i]) ?
3113 0 : Flx_to_ZX(ffinit_Artin_Schreier(Fp[i], Fe[i]))
3114 1653 : : fpinit(p, Fm[i]);
3115 1574 : return FpXV_composedsum(P, p);
3116 : }
3117 :
3118 : static GEN
3119 54365 : init_Fq_i(GEN p, long n, long v)
3120 : {
3121 : GEN P;
3122 54365 : if (n <= 0) pari_err_DOMAIN("ffinit", "degree", "<=", gen_0, stoi(n));
3123 54365 : if (typ(p) != t_INT) pari_err_TYPE("ffinit",p);
3124 54365 : if (cmpiu(p, 2) < 0) pari_err_PRIME("ffinit",p);
3125 54358 : if (v < 0) v = 0;
3126 54358 : if (n == 1) return pol_x(v);
3127 54106 : if (lgefint(p) == 3)
3128 52123 : return Flx_to_ZX(init_Flxq_i(p[2], n, evalvarn(v)));
3129 1983 : if (fpinit_check(p, n+1, n)) return polcyclo(n+1, v);
3130 1574 : P = ffinit_fact(p,n);
3131 1574 : setvarn(P, v); return P;
3132 : }
3133 : GEN
3134 53812 : init_Fq(GEN p, long n, long v)
3135 : {
3136 53812 : pari_sp av = avma;
3137 53812 : return gerepileupto(av, init_Fq_i(p, n, v));
3138 : }
3139 : GEN
3140 553 : ffinit(GEN p, long n, long v)
3141 : {
3142 553 : pari_sp av = avma;
3143 553 : return gerepileupto(av, FpX_to_mod(init_Fq_i(p, n, v), p));
3144 : }
3145 :
3146 : GEN
3147 3178 : ffnbirred(GEN p, long n)
3148 : {
3149 3178 : pari_sp av = avma;
3150 3178 : GEN s = powiu(p,n), F = factoru(n), D = divisorsu_moebius(gel(F, 1));
3151 3178 : long j, l = lg(D);
3152 6797 : for (j = 2; j < l; j++) /* skip d = 1 */
3153 : {
3154 3619 : long md = D[j]; /* mu(d) * d, d squarefree */
3155 3619 : GEN pd = powiu(p, n / labs(md)); /* p^{n/d} */
3156 3619 : s = md > 0? addii(s, pd): subii(s,pd);
3157 : }
3158 3178 : return gerepileuptoint(av, diviuexact(s, n));
3159 : }
3160 :
3161 : GEN
3162 616 : ffsumnbirred(GEN p, long n)
3163 : {
3164 616 : pari_sp av = avma, av2;
3165 616 : GEN q, t = p, v = vecfactoru_i(1, n);
3166 : long i;
3167 616 : q = cgetg(n+1,t_VEC); gel(q,1) = p;
3168 1764 : for (i=2; i<=n; i++) gel(q,i) = mulii(gel(q,i-1), p);
3169 616 : av2 = avma;
3170 1764 : for (i=2; i<=n; i++)
3171 : {
3172 1148 : GEN s = gel(q,i), F = gel(v,i), D = divisorsu_moebius(gel(F,1));
3173 1148 : long j, l = lg(D);
3174 2534 : for (j = 2; j < l; j++) /* skip 1 */
3175 : {
3176 1386 : long md = D[j];
3177 1386 : GEN pd = gel(q, i / labs(md)); /* p^{i/d} */
3178 1386 : s = md > 0? addii(s, pd): subii(s, pd);
3179 : }
3180 1148 : t = gerepileuptoint(av2, addii(t, diviuexact(s, i)));
3181 : }
3182 616 : return gerepileuptoint(av, t);
3183 : }
3184 :
3185 : GEN
3186 140 : ffnbirred0(GEN p, long n, long flag)
3187 : {
3188 140 : if (typ(p) != t_INT) pari_err_TYPE("ffnbirred", p);
3189 140 : if (n <= 0) pari_err_DOMAIN("ffnbirred", "degree", "<=", gen_0, stoi(n));
3190 140 : switch(flag)
3191 : {
3192 70 : case 0: return ffnbirred(p, n);
3193 70 : case 1: return ffsumnbirred(p, n);
3194 : }
3195 0 : pari_err_FLAG("ffnbirred");
3196 : return NULL; /* LCOV_EXCL_LINE */
3197 : }
3198 :
3199 : static void
3200 2261 : checkmap(GEN m, const char *s)
3201 : {
3202 2261 : if (typ(m)!=t_VEC || lg(m)!=3 || typ(gel(m,1))!=t_FFELT)
3203 0 : pari_err_TYPE(s,m);
3204 2261 : }
3205 :
3206 : GEN
3207 189 : ffembed(GEN a, GEN b)
3208 : {
3209 189 : pari_sp av = avma;
3210 189 : GEN p, Ta, Tb, g, r = NULL;
3211 189 : if (typ(a)!=t_FFELT) pari_err_TYPE("ffembed",a);
3212 189 : if (typ(b)!=t_FFELT) pari_err_TYPE("ffembed",b);
3213 189 : p = FF_p_i(a); g = FF_gen(a);
3214 189 : if (!equalii(p, FF_p_i(b))) pari_err_MODULUS("ffembed",a,b);
3215 189 : Ta = FF_mod(a);
3216 189 : Tb = FF_mod(b);
3217 189 : if (degpol(Tb)%degpol(Ta)!=0)
3218 7 : pari_err_DOMAIN("ffembed",GENtostr_raw(a),"is not a subfield of",b,a);
3219 182 : r = gel(FFX_roots(Ta, b), 1);
3220 182 : return gerepilecopy(av, mkvec2(g,r));
3221 : }
3222 :
3223 : GEN
3224 91 : ffextend(GEN a, GEN P, long v)
3225 : {
3226 91 : pari_sp av = avma;
3227 : long n;
3228 : GEN p, T, R, g, m;
3229 91 : if (typ(a)!=t_FFELT) pari_err_TYPE("ffextend",a);
3230 91 : T = a; p = FF_p_i(a);
3231 91 : if (typ(P)!=t_POL || !RgX_is_FpXQX(P,&T,&p)) pari_err_TYPE("ffextend", P);
3232 49 : if (!FF_samefield(a, T)) pari_err_MODULUS("ffextend",a,T);
3233 49 : if (v < 0) v = varn(P);
3234 49 : n = FF_f(T) * degpol(P); R = ffinit(p, n, v); g = ffgen(R, v);
3235 49 : m = ffembed(a, g);
3236 49 : R = FFX_roots(ffmap(m, P),g);
3237 49 : return gerepilecopy(av, mkvec2(gel(R,1), m));
3238 : }
3239 :
3240 : GEN
3241 42 : fffrobenius(GEN a, long n)
3242 : {
3243 42 : if (typ(a)!=t_FFELT) pari_err_TYPE("fffrobenius",a);
3244 42 : retmkvec2(FF_gen(a), FF_Frobenius(a, n));
3245 : }
3246 :
3247 : GEN
3248 133 : ffinvmap(GEN m)
3249 : {
3250 133 : pari_sp av = avma;
3251 : long i, l;
3252 133 : GEN T, F, a, g, r, f = NULL;
3253 133 : checkmap(m, "ffinvmap");
3254 133 : a = gel(m,1); r = gel(m,2);
3255 133 : if (typ(r) != t_FFELT)
3256 7 : pari_err_TYPE("ffinvmap", m);
3257 126 : g = FF_gen(a);
3258 126 : T = FF_mod(r);
3259 126 : F = gel(FFX_factor(T, a), 1);
3260 126 : l = lg(F);
3261 490 : for(i=1; i<l; i++)
3262 : {
3263 490 : GEN s = FFX_rem(FF_to_FpXQ_i(r), gel(F, i), a);
3264 490 : if (degpol(s)==0 && gequal(constant_coeff(s),g)) { f = gel(F, i); break; }
3265 : }
3266 126 : if (f==NULL) pari_err_TYPE("ffinvmap", m);
3267 126 : if (degpol(f)==1) f = FF_neg_i(gel(f,2));
3268 126 : return gerepilecopy(av, mkvec2(FF_gen(r),f));
3269 : }
3270 :
3271 : static GEN
3272 1260 : ffpartmapimage(const char *s, GEN r)
3273 : {
3274 1260 : GEN a = NULL, p = NULL;
3275 1260 : if (typ(r)==t_POL && degpol(r) >= 1
3276 1260 : && RgX_is_FpXQX(r,&a,&p) && a && typ(a)==t_FFELT) return a;
3277 0 : pari_err_TYPE(s, r);
3278 : return NULL; /* LCOV_EXCL_LINE */
3279 : }
3280 :
3281 : static GEN
3282 2709 : ffeltmap_i(GEN m, GEN x)
3283 : {
3284 2709 : GEN r = gel(m,2);
3285 2709 : if (!FF_samefield(x, gel(m,1)))
3286 84 : pari_err_DOMAIN("ffmap","m","domain does not contain", x, r);
3287 2625 : if (typ(r)==t_FFELT)
3288 1659 : return FF_map(r, x);
3289 : else
3290 966 : return FFX_preimage(x, r, ffpartmapimage("ffmap", r));
3291 : }
3292 :
3293 : static GEN
3294 4459 : ffmap_i(GEN m, GEN x)
3295 : {
3296 : GEN y;
3297 4459 : long i, lx, tx = typ(x);
3298 4459 : switch(tx)
3299 : {
3300 2541 : case t_FFELT:
3301 2541 : return ffeltmap_i(m, x);
3302 1267 : case t_POL: case t_RFRAC: case t_SER:
3303 : case t_VEC: case t_COL: case t_MAT:
3304 1267 : y = cgetg_copy(x, &lx);
3305 1988 : for (i=1; i<lontyp[tx]; i++) y[i] = x[1];
3306 4564 : for (i=lontyp[tx]; i<lx; i++)
3307 : {
3308 3339 : GEN yi = ffmap_i(m, gel(x,i));
3309 3297 : if (!yi) return NULL;
3310 3297 : gel(y,i) = yi;
3311 : }
3312 1225 : return y;
3313 : }
3314 651 : return gcopy(x);
3315 : }
3316 :
3317 : GEN
3318 1036 : ffmap(GEN m, GEN x)
3319 : {
3320 1036 : pari_sp ltop = avma;
3321 : GEN y;
3322 1036 : checkmap(m, "ffmap");
3323 1036 : y = ffmap_i(m, x);
3324 1036 : if (y) return y;
3325 42 : set_avma(ltop); return cgetg(1,t_VEC);
3326 : }
3327 :
3328 : static GEN
3329 252 : ffeltmaprel_i(GEN m, GEN x)
3330 : {
3331 252 : GEN g = gel(m,1), r = gel(m,2);
3332 252 : if (!FF_samefield(x, g))
3333 0 : pari_err_DOMAIN("ffmap","m","domain does not contain", x, r);
3334 252 : if (typ(r)==t_FFELT)
3335 84 : retmkpolmod(FF_map(r, x), pol_x(FF_var(g)));
3336 : else
3337 168 : retmkpolmod(FFX_preimagerel(x, r, ffpartmapimage("ffmap", r)), gcopy(r));
3338 : }
3339 :
3340 : static GEN
3341 252 : ffmaprel_i(GEN m, GEN x)
3342 : {
3343 : GEN y;
3344 252 : long i, lx, tx = typ(x);
3345 252 : switch(tx)
3346 : {
3347 252 : case t_FFELT:
3348 252 : return ffeltmaprel_i(m, x);
3349 0 : case t_POL: case t_RFRAC: case t_SER:
3350 : case t_VEC: case t_COL: case t_MAT:
3351 0 : y = cgetg_copy(x, &lx);
3352 0 : for (i=1; i<lontyp[tx]; i++) y[i] = x[1];
3353 0 : for (i=lontyp[tx]; i<lx; i++)
3354 0 : gel(y,i) = ffmaprel_i(m, gel(x,i));
3355 0 : return y;
3356 : }
3357 0 : return gcopy(x);
3358 : }
3359 :
3360 : GEN
3361 252 : ffmaprel(GEN m, GEN x)
3362 : {
3363 252 : checkmap(m, "ffmaprel");
3364 252 : return ffmaprel_i(m, x);
3365 : }
3366 :
3367 : static void
3368 84 : err_compo(GEN m, GEN n)
3369 84 : { pari_err_DOMAIN("ffcompomap","m","domain does not contain codomain of",n,m); }
3370 :
3371 : GEN
3372 420 : ffcompomap(GEN m, GEN n)
3373 : {
3374 420 : pari_sp av = avma;
3375 420 : GEN g = gel(n,1), r, m2, n2;
3376 420 : checkmap(m, "ffcompomap");
3377 420 : checkmap(n, "ffcompomap");
3378 420 : m2 = gel(m,2); n2 = gel(n,2);
3379 420 : switch((typ(m2)==t_POL)|((typ(n2)==t_POL)<<1))
3380 : {
3381 84 : case 0:
3382 84 : if (!FF_samefield(gel(m,1),n2)) err_compo(m,n);
3383 42 : r = FF_map(gel(m,2), n2);
3384 42 : break;
3385 84 : case 2:
3386 84 : r = ffmap_i(m, n2);
3387 42 : if (lg(r) == 1) err_compo(m,n);
3388 42 : break;
3389 168 : case 1:
3390 168 : r = ffeltmap_i(m, n2);
3391 126 : if (!r)
3392 : {
3393 : GEN a, A, R, M;
3394 : long dm, dn;
3395 42 : a = ffpartmapimage("ffcompomap",m2);
3396 42 : A = FF_to_FpXQ_i(FF_neg(n2));
3397 42 : setvarn(A, 1);
3398 42 : R = deg1pol(gen_1, A, 0);
3399 42 : setvarn(R, 0);
3400 42 : M = gcopy(m2);
3401 42 : setvarn(M, 1);
3402 42 : r = polresultant0(R, M, 1, 0);
3403 42 : dm = FF_f(gel(m,1)); dn = FF_f(gel(n,1));
3404 42 : if (dm % dn || !FFX_ispower(r, dm/dn, a, &r)) err_compo(m,n);
3405 42 : setvarn(r, varn(FF_mod(g)));
3406 : }
3407 126 : break;
3408 84 : case 3:
3409 : {
3410 : GEN M, R, T, p, a;
3411 84 : a = ffpartmapimage("ffcompomap",n2);
3412 84 : if (!FF_samefield(a, gel(m,1))) err_compo(m,n);
3413 42 : p = FF_p_i(gel(n,1));
3414 42 : T = FF_mod(gel(n,1));
3415 42 : setvarn(T, 1);
3416 42 : R = RgX_to_FpXQX(n2,T,p);
3417 42 : setvarn(R, 0);
3418 42 : M = gcopy(m2);
3419 42 : setvarn(M, 1);
3420 42 : r = polresultant0(R, M, 1, 0);
3421 42 : setvarn(r, varn(n2));
3422 : }
3423 : }
3424 252 : return gerepilecopy(av, mkvec2(g,r));
3425 : }
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