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 175238 : charact(struct charact *S, GEN x)
53 : {
54 175238 : const long tx = typ(x);
55 : long i, l;
56 175238 : switch(tx)
57 : {
58 5124 : case t_INTMOD:char_update_int(S, gel(x,1)); break;
59 1148 : case t_FFELT: char_update_prime(S, gel(x,4)); break;
60 26334 : case t_COMPLEX: case t_QUAD:
61 : case t_POLMOD: case t_POL: case t_SER: case t_RFRAC:
62 : case t_VEC: case t_COL: case t_MAT:
63 26334 : l = lg(x);
64 174223 : for (i=lontyp[tx]; i < l; i++) charact(S,gel(x,i));
65 26320 : break;
66 7 : case t_LIST:
67 7 : x = list_data(x);
68 7 : if (x) charact(S, x);
69 7 : break;
70 : }
71 175210 : }
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 27335 : characteristic(GEN x)
96 : {
97 : struct charact S;
98 27335 : S.q = gen_0; S.isprime = 0;
99 27335 : 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 69150534 : Rg_is_Fp(GEN x, GEN *pp)
111 : {
112 : GEN mod;
113 69150534 : switch(typ(x))
114 : {
115 3203536 : case t_INTMOD:
116 3203536 : mod = gel(x,1);
117 3203536 : if (!*pp) *pp = mod;
118 2954266 : 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 3203536 : return 1;
124 54630733 : case t_INT:
125 54630733 : return 1;
126 11316265 : default: return 0;
127 : }
128 : }
129 :
130 : int
131 27887708 : RgX_is_FpX(GEN x, GEN *pp)
132 : {
133 27887708 : long i, lx = lg(x);
134 85695878 : for (i=2; i<lx; i++)
135 69124443 : if (!Rg_is_Fp(gel(x, i), pp))
136 11316247 : return 0;
137 16571435 : return 1;
138 : }
139 :
140 : int
141 0 : RgV_is_FpV(GEN x, GEN *pp)
142 : {
143 0 : long i, lx = lg(x);
144 0 : for (i=1; i<lx; i++)
145 0 : if (!Rg_is_Fp(gel(x,i), pp)) return 0;
146 0 : return 1;
147 : }
148 :
149 : int
150 0 : RgM_is_FpM(GEN x, GEN *pp)
151 : {
152 0 : long i, lx = lg(x);
153 0 : for (i=1; i<lx; i++)
154 0 : if (!RgV_is_FpV(gel(x, i), pp)) return 0;
155 0 : return 1;
156 : }
157 :
158 : int
159 59304 : Rg_is_FpXQ(GEN x, GEN *pT, GEN *pp)
160 : {
161 : GEN pol, mod, p;
162 59304 : switch(typ(x))
163 : {
164 26089 : case t_INTMOD:
165 26089 : return Rg_is_Fp(x, pp);
166 7105 : case t_INT:
167 7105 : return 1;
168 21 : case t_POL:
169 21 : return RgX_is_FpX(x, pp);
170 21350 : case t_FFELT:
171 21350 : mod = x; p = FF_p_i(x);
172 21350 : if (!*pp) *pp = p;
173 21350 : if (!*pT) *pT = mod;
174 19824 : else if (typ(*pT)!=t_FFELT || !FF_samefield(*pT,mod))
175 : {
176 42 : if (DEBUGLEVEL) pari_warn(warner,"different moduli in Rg_is_FpXQ");
177 42 : return 0;
178 : }
179 21308 : return 1;
180 4585 : case t_POLMOD:
181 4585 : mod = gel(x,1); pol = gel(x, 2);
182 4585 : if (!RgX_is_FpX(mod, pp)) return 0;
183 4585 : if (typ(pol)==t_POL)
184 : {
185 4578 : if (!RgX_is_FpX(pol, pp)) return 0;
186 : }
187 7 : else if (!Rg_is_Fp(pol, pp)) return 0;
188 4585 : if (!*pT) *pT = mod;
189 4431 : else if (mod != *pT && !gequal(mod, *pT))
190 : {
191 0 : if (DEBUGLEVEL) pari_warn(warner,"different moduli in Rg_is_FpXQ");
192 0 : return 0;
193 : }
194 4585 : return 1;
195 :
196 154 : default: return 0;
197 : }
198 : }
199 :
200 : int
201 3206 : RgX_is_FpXQX(GEN x, GEN *pT, GEN *pp)
202 : {
203 3206 : long i, lx = lg(x);
204 61754 : for (i = 2; i < lx; i++)
205 58646 : if (!Rg_is_FpXQ(gel(x,i), pT, pp)) return 0;
206 3108 : return 1;
207 : }
208 :
209 : /************************************************************************
210 : ** **
211 : ** Ring conversion **
212 : ** **
213 : ************************************************************************/
214 :
215 : /* p > 0 a t_INT, return lift(x * Mod(1,p)).
216 : * If x is an INTMOD, assume modulus is a multiple of p. */
217 : GEN
218 36903668 : Rg_to_Fp(GEN x, GEN p)
219 : {
220 36903668 : if (lgefint(p) == 3) return utoi(Rg_to_Fl(x, uel(p,2)));
221 4556047 : switch(typ(x))
222 : {
223 288470 : 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 4248723 : case t_INTMOD: {
232 4248723 : GEN q = gel(x,1), a = gel(x,2);
233 4248723 : 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 1291510 : Rg_to_FpXQ(GEN x, GEN T, GEN p)
244 : {
245 1291510 : long ta, tx = typ(x), v = get_FpX_var(T);
246 : GEN a, b;
247 1291510 : if (is_const_t(tx))
248 : {
249 58538 : if (tx == t_FFELT)
250 : {
251 17355 : GEN z = FF_to_FpXQ(x);
252 17355 : setvarn(z, v);
253 17355 : return z;
254 : }
255 41183 : return scalar_ZX(degpol(T)? Rg_to_Fp(x, p): gen_0, v);
256 : }
257 1232972 : switch(tx)
258 : {
259 1230893 : case t_POLMOD:
260 1230893 : b = gel(x,1);
261 1230893 : a = gel(x,2); ta = typ(a);
262 1230893 : if (is_const_t(ta))
263 4102 : return scalar_ZX(degpol(T)? Rg_to_Fp(a, p): gen_0, v);
264 1226791 : b = RgX_to_FpX(b, p); if (varn(b) != v) break;
265 1226791 : a = RgX_to_FpX(a, p);
266 1226791 : if (ZX_equal(b,get_FpX_mod(T)) || signe(FpX_rem(b,T,p))==0)
267 1226791 : return FpX_rem(a, T, p);
268 0 : break;
269 2079 : case t_POL:
270 2079 : if (varn(x) != v) break;
271 2079 : return FpX_rem(RgX_to_FpX(x,p), T, p);
272 0 : case t_RFRAC:
273 0 : a = Rg_to_FpXQ(gel(x,1), T,p);
274 0 : b = Rg_to_FpXQ(gel(x,2), T,p);
275 0 : return FpXQ_div(a,b, T,p);
276 : }
277 0 : pari_err_TYPE("Rg_to_FpXQ",x);
278 : return NULL; /* LCOV_EXCL_LINE */
279 : }
280 : GEN
281 3552175 : RgX_to_FpX(GEN x, GEN p)
282 : {
283 : long i, l;
284 3552175 : GEN z = cgetg_copy(x, &l); z[1] = x[1];
285 15794634 : for (i = 2; i < l; i++) gel(z,i) = Rg_to_Fp(gel(x,i), p);
286 3552175 : 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 933038 : RgC_to_FpC(GEN x, GEN p)
295 11542048 : { pari_APPLY_type(t_COL, Rg_to_Fp(gel(x,i), p)) }
296 :
297 : GEN
298 133812 : RgM_to_FpM(GEN x, GEN p)
299 1066808 : { pari_APPLY_same(RgC_to_FpC(gel(x,i), p)) }
300 :
301 : GEN
302 342814 : RgV_to_Flv(GEN x, ulong p)
303 1343113 : { 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 5028 : RgX_to_FpXQX(GEN x, GEN T, GEN p)
311 : {
312 5028 : long i, l = lg(x);
313 5028 : GEN z = cgetg(l, t_POL); z[1] = x[1];
314 42939 : for (i = 2; i < l; i++) gel(z,i) = Rg_to_FpXQ(gel(x,i), T,p);
315 5028 : return FpXQX_renormalize(z, l);
316 : }
317 : GEN
318 49186 : RgX_to_FqX(GEN x, GEN T, GEN p)
319 : {
320 49186 : long i, l = lg(x);
321 49186 : GEN z = cgetg(l, t_POL); z[1] = x[1];
322 49186 : if (T)
323 10990 : for (i = 2; i < l; i++) gel(z,i) = Rg_to_FpXQ(gel(x,i), T, p);
324 : else
325 791282 : for (i = 2; i < l; i++) gel(z,i) = Rg_to_Fp(gel(x,i), p);
326 49186 : return FpXQX_renormalize(z, l);
327 : }
328 :
329 : GEN
330 219128 : RgC_to_FqC(GEN x, GEN T, GEN p)
331 : {
332 219128 : long i, l = lg(x);
333 219128 : GEN z = cgetg(l, t_COL);
334 219128 : if (T)
335 1430310 : for (i = 1; i < l; i++) gel(z,i) = Rg_to_FpXQ(gel(x,i), T, p);
336 : else
337 0 : for (i = 1; i < l; i++) gel(z,i) = Rg_to_Fp(gel(x,i), p);
338 219128 : return z;
339 : }
340 :
341 : GEN
342 52325 : RgM_to_FqM(GEN x, GEN T, GEN p)
343 271418 : { pari_APPLY_same(RgC_to_FqC(gel(x, i), T, p)) }
344 :
345 : /* lg(V) > 1 */
346 : GEN
347 851487 : FpXV_FpC_mul(GEN V, GEN W, GEN p)
348 : {
349 851487 : pari_sp av = avma;
350 851487 : long i, l = lg(V);
351 851487 : GEN z = ZX_Z_mul(gel(V,1),gel(W,1));
352 4201029 : for(i=2; i<l; i++)
353 : {
354 3349542 : z = ZX_add(z, ZX_Z_mul(gel(V,i),gel(W,i)));
355 3349542 : if ((i & 7) == 0) z = gerepileupto(av, z);
356 : }
357 851487 : return gerepileupto(av, FpX_red(z,p));
358 : }
359 :
360 : GEN
361 55832 : FqX_Fq_add(GEN y, GEN x, GEN T, GEN p)
362 : {
363 55832 : long i, lz = lg(y);
364 : GEN z;
365 55832 : if (!T) return FpX_Fp_add(y, x, p);
366 8666 : if (lz == 2) return scalarpol(x, varn(y));
367 7952 : z = cgetg(lz,t_POL); z[1] = y[1];
368 7952 : gel(z,2) = Fq_add(gel(y,2),x, T, p);
369 7952 : if (lz == 3) z = FpXX_renormalize(z,lz);
370 : else
371 33145 : for(i=3;i<lz;i++) gel(z,i) = gcopy(gel(y,i));
372 7952 : return z;
373 : }
374 :
375 : GEN
376 1094 : FqX_Fq_sub(GEN y, GEN x, GEN T, GEN p)
377 : {
378 1094 : long i, lz = lg(y);
379 : GEN z;
380 1094 : if (!T) return FpX_Fp_sub(y, x, p);
381 1094 : if (lz == 2) return scalarpol(x, varn(y));
382 1094 : z = cgetg(lz,t_POL); z[1] = y[1];
383 1094 : gel(z,2) = Fq_sub(gel(y,2), x, T, p);
384 1094 : if (lz == 3) z = FpXX_renormalize(z,lz);
385 : else
386 10303 : for(i=3;i<lz;i++) gel(z,i) = gcopy(gel(y,i));
387 1094 : return z;
388 : }
389 :
390 : GEN
391 149030 : FqX_Fq_mul_to_monic(GEN P, GEN U, GEN T, GEN p)
392 : {
393 : long i, lP;
394 149030 : GEN res = cgetg_copy(P, &lP); res[1] = P[1];
395 918551 : for(i=2; i<lP-1; i++) gel(res,i) = Fq_mul(U,gel(P,i), T,p);
396 149030 : 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 80 : return FqX_Fq_mul_to_monic(z, Fp_inv(lc,p), T,p);
417 : }
418 :
419 : GEN
420 398831 : FqX_eval(GEN x, GEN y, GEN T, GEN p)
421 : {
422 : pari_sp av;
423 : GEN p1, r;
424 398831 : long j, i=lg(x)-1;
425 398831 : if (i<=2)
426 45957 : return (i==2)? Fq_red(gel(x,2), T, p): gen_0;
427 352874 : 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 1173882 : for (i--; i>=2; i=j-1)
431 : {
432 821710 : 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 821010 : r = (i==j)? y: Fq_pow(y, utoipos(i-j+1), T, p);
439 821010 : p1 = Fq_add(Fq_mul(p1,r,T,p), gel(x,j), T, p);
440 : }
441 352382 : 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 10495354 : monomial(GEN a, long d, long v)
471 : {
472 : long i, n;
473 : GEN P;
474 10495354 : if (d < 0) {
475 14 : if (isrationalzero(a)) return pol_0(v);
476 14 : retmkrfrac(a, pol_xn(-d, v));
477 : }
478 10495340 : 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 10402065 : n = d+2; P = cgetg(n+1, t_POL);
487 10402068 : P[1] = evalsigne(1) | evalvarn(v);
488 : }
489 29528739 : for (i = 2; i < n; i++) gel(P,i) = gen_0;
490 10402068 : gel(P,i) = a; return P;
491 : }
492 : GEN
493 1863415 : monomialcopy(GEN a, long d, long v)
494 : {
495 : long i, n;
496 : GEN P;
497 1863415 : if (d < 0) {
498 14 : if (isrationalzero(a)) return pol_0(v);
499 14 : retmkrfrac(gcopy(a), pol_xn(-d, v));
500 : }
501 1863401 : 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 1863387 : n = d+2; P = cgetg(n+1, t_POL);
510 1863387 : P[1] = evalsigne(1) | evalvarn(v);
511 : }
512 3510717 : for (i = 2; i < n; i++) gel(P,i) = gen_0;
513 1863394 : gel(P,i) = gcopy(a); return P;
514 : }
515 : GEN
516 324430 : pol_x_powers(long N, long v)
517 : {
518 324430 : GEN L = cgetg(N+1,t_VEC);
519 : long i;
520 784260 : for (i=1; i<=N; i++) gel(L,i) = pol_xn(i-1, v);
521 324441 : 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 11608980 : Fq_add(GEN x, GEN y, GEN T/*unused*/, GEN p)
544 : {
545 : (void)T;
546 11608980 : switch((typ(x)==t_POL)|((typ(y)==t_POL)<<1))
547 : {
548 1143057 : case 0: return Fp_add(x,y,p);
549 764628 : case 1: return FpX_Fp_add(x,y,p);
550 92070 : case 2: return FpX_Fp_add(y,x,p);
551 9609225 : case 3: return FpX_add(x,y,p);
552 : }
553 : return NULL;/*LCOV_EXCL_LINE*/
554 : }
555 :
556 : GEN
557 8349602 : Fq_sub(GEN x, GEN y, GEN T/*unused*/, GEN p)
558 : {
559 : (void)T;
560 8349602 : switch((typ(x)==t_POL)|((typ(y)==t_POL)<<1))
561 : {
562 256056 : 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 8045158 : case 3: return FpX_sub(x,y,p);
566 : }
567 : return NULL;/*LCOV_EXCL_LINE*/
568 : }
569 :
570 : GEN
571 1078431 : Fq_neg(GEN x, GEN T/*unused*/, GEN p)
572 : {
573 : (void)T;
574 1078431 : 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 8377893 : Fq_mul(GEN x, GEN y, GEN T, GEN p)
587 : {
588 8377893 : switch((typ(x)==t_POL)|((typ(y)==t_POL)<<1))
589 : {
590 1037518 : case 0: return Fp_mul(x,y,p);
591 128947 : case 1: return FpX_Fp_mul(x,y,p);
592 402234 : case 2: return FpX_Fp_mul(y,x,p);
593 6809198 : case 3: if (T) return FpXQ_mul(x,y,T,p);
594 4231894 : 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 492815 : Fq_mulu(GEN x, ulong y, /*unused*/GEN T, GEN p)
602 : {
603 : (void) T;
604 492815 : 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 43965 : Fq_Fp_mul(GEN x, GEN y, GEN T/*unused*/, GEN p)
610 : {
611 : (void)T;
612 6844 : return (typ(x) == t_POL)? FpX_Fp_mul(x,y,p)
613 50809 : : Fp_mul(x,y,p);
614 : }
615 : /* If T==NULL do not reduce*/
616 : GEN
617 613228 : Fq_sqr(GEN x, GEN T, GEN p)
618 : {
619 613228 : 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 540384 : 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 89360 : Fq_inv(GEN x, GEN pol, GEN p)
644 : {
645 89360 : if (typ(x) == t_INT) return Fp_inv(x,p);
646 81594 : return FpXQ_inv(x,pol,p);
647 : }
648 :
649 : GEN
650 343588 : Fq_div(GEN x, GEN y, GEN pol, GEN p)
651 : {
652 343588 : switch((typ(x)==t_POL)|((typ(y)==t_POL)<<1))
653 : {
654 318269 : 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 794194 : Fq_pow(GEN x, GEN n, GEN pol, GEN p)
664 : {
665 794194 : if (typ(x) == t_INT) return Fp_pow(x,n,p);
666 136192 : 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 1894003 : Fq_sqrt(GEN x, GEN T, GEN p)
678 : {
679 1894003 : if (typ(x) == t_INT)
680 : {
681 1823898 : if (!T || odd(get_FpX_degree(T))) return Fp_sqrt(x,p);
682 9596 : x = scalarpol_shallow(x, get_FpX_var(T));
683 : }
684 79701 : return FpXQ_sqrt(x,T,p);
685 : }
686 : GEN
687 170723 : Fq_sqrtn(GEN x, GEN n, GEN T, GEN p, GEN *zeta)
688 : {
689 170723 : if (typ(x) == t_INT)
690 : {
691 : long d;
692 170366 : 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 : }
703 427 : 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 40404 : FqV_roots_to_pol(GEN V, GEN T, GEN p, long v)
831 : {
832 40404 : pari_sp ltop = avma;
833 : long k;
834 : GEN W;
835 40404 : if (lgefint(p) == 3)
836 : {
837 31739 : ulong pp = p[2];
838 31739 : GEN Tl = ZX_to_Flx(T, pp);
839 31739 : GEN Vl = ZXC_to_FlxC(V, pp, get_Flx_var(Tl));
840 31739 : Tl = FlxqV_roots_to_pol(Vl, Tl, pp, v);
841 31739 : return gerepileupto(ltop, FlxX_to_ZXX(Tl));
842 : }
843 8665 : W = cgetg(lg(V),t_VEC);
844 77799 : for(k=1; k < lg(V); k++)
845 69134 : gel(W,k) = deg1pol_shallow(gen_1,Fq_neg(gel(V,k),T,p),v);
846 8665 : return gerepileupto(ltop, FpXQXV_prod(W, T, p));
847 : }
848 :
849 : GEN
850 191195 : FqV_red(GEN x, GEN T, GEN p)
851 1347338 : { 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 8169754 : primelist(forprime_t *S, long n, GEN dB)
905 : {
906 8169754 : GEN P = cgetg(n+1, t_VECSMALL);
907 8169737 : long i = 1;
908 : ulong p;
909 19386684 : while (i <= n && (p = u_forprime_next(S)))
910 11216948 : if (!dB || umodiu(dB, p)) P[i++] = p;
911 8169730 : return P;
912 : }
913 :
914 : void
915 7665083 : 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 7665083 : long m = mmin? minss(mmin, n): usqrt(n);
920 : GEN H, P, mod;
921 : pari_timer ti;
922 7665085 : if (DEBUGLEVEL > 4)
923 : {
924 0 : timer_start(&ti);
925 0 : err_printf("%s: nb primes: %ld\n",str, n);
926 : }
927 7665079 : if (m == 1)
928 : {
929 7400439 : GEN P = primelist(S, n, dB);
930 7400416 : GEN done = closure_callgen1(worker, P);
931 7400392 : H = gel(done,1);
932 7400392 : mod = gel(done,2);
933 7400392 : if (!*pH && center) H = center(H, mod, shifti(mod,-1));
934 7400327 : if (DEBUGLEVEL>4) timer_printf(&ti,"%s: modular", str);
935 : }
936 : else
937 : {
938 264640 : long i, s = (n+m-1)/m, r = m - (m*s-n), di = 0;
939 : struct pari_mt pt;
940 264640 : long pending = 0;
941 264640 : H = cgetg(m+1, t_VEC); P = cgetg(m+1, t_VEC);
942 264640 : mt_queue_start_lim(&pt, worker, m);
943 1089237 : for (i=1; i<=m || pending; i++)
944 : {
945 : GEN done;
946 824597 : GEN pr = i <= m ? mkvec(primelist(S, i<=r ? s: s-1, dB)): NULL;
947 824597 : mt_queue_submit(&pt, i, pr);
948 824596 : done = mt_queue_get(&pt, NULL, &pending);
949 824596 : if (done)
950 : {
951 769314 : di++;
952 769314 : gel(H, di) = gel(done,1);
953 769314 : gel(P, di) = gel(done,2);
954 769314 : if (DEBUGLEVEL>5) err_printf("%ld%% ",100*di/m);
955 : }
956 : }
957 264640 : mt_queue_end(&pt);
958 264640 : if (DEBUGLEVEL>5) err_printf("\n");
959 264640 : if (DEBUGLEVEL>4) timer_printf(&ti,"%s: modular", str);
960 264640 : H = crt(H, P, &mod);
961 264640 : if (DEBUGLEVEL>4) timer_printf(&ti,"%s: chinese", str);
962 : }
963 7664967 : if (*pH) H = crt(mkvec2(*pH, H), mkvec2(*pmod, mod), &mod);
964 7664967 : *pH = H; *pmod = mod;
965 7664967 : }
966 : void
967 2016542 : 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 2016542 : pari_sp av = avma;
972 2016542 : gen_inccrt_i(str, worker, dB, n, mmin, S, pH, pmod, crt, center);
973 2016466 : gerepileall(av, 2, pH, pmod);
974 2016600 : }
975 :
976 : GEN
977 1241441 : 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 1241441 : GEN mod = gen_1, H = NULL;
981 : ulong e;
982 :
983 1241441 : bound++;
984 2482932 : while (bound > (e = expi(mod)))
985 : {
986 1241392 : long n = (bound - e) / expu(S->p) + 1;
987 1241419 : gen_inccrt(str, worker, dB, n, mmin, S, &H, &mod, crt, center);
988 : }
989 1241469 : if (pmod) *pmod = mod;
990 1241469 : 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 5113144 : Fl_chinese_coprime(GEN a, ulong b, GEN q, ulong p, ulong qinv, GEN pq, GEN pq2)
1001 : {
1002 5113144 : ulong d, amod = umodiu(a, p);
1003 5113191 : pari_sp av = avma;
1004 : GEN ax;
1005 :
1006 5113191 : if (b == amod) return NULL;
1007 2104573 : d = Fl_mul(Fl_sub(b, amod, p), qinv, p); /* != 0 */
1008 2105338 : if (d >= 1 + (p>>1))
1009 1026999 : ax = subii(a, mului(p-d, q));
1010 : else
1011 : {
1012 1078339 : ax = addii(a, mului(d, q)); /* in ]0, pq[ assuming a in ]-q,q[ */
1013 1077770 : if (cmpii(ax,pq2) > 0) ax = subii(ax,pq);
1014 : }
1015 2104226 : 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 31646 : ZX_init_CRT(GEN Hp, ulong p, long v)
1021 : {
1022 31646 : long i, l = lg(Hp), lim = (long)(p>>1);
1023 31646 : GEN H = cgetg(l, t_POL);
1024 31646 : H[1] = evalsigne(1) | evalvarn(v);
1025 795731 : for (i=2; i<l; i++)
1026 764084 : gel(H,i) = stoi(Fl_center(Hp[i], p, lim));
1027 31647 : return ZX_renormalize(H,l);
1028 : }
1029 :
1030 : GEN
1031 5775 : ZM_init_CRT(GEN Hp, ulong p)
1032 : {
1033 5775 : long i,j, m, l = lg(Hp), lim = (long)(p>>1);
1034 5775 : GEN c, cp, H = cgetg(l, t_MAT);
1035 5775 : if (l==1) return H;
1036 5775 : m = lgcols(Hp);
1037 18970 : for (j=1; j<l; j++)
1038 : {
1039 13195 : cp = gel(Hp,j);
1040 13195 : c = cgetg(m, t_COL);
1041 13195 : gel(H,j) = c;
1042 166411 : for (i=1; i<m; i++) gel(c,i) = stoi(Fl_center(cp[i],p, lim));
1043 : }
1044 5775 : 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 147430 : ZX_incremental_CRT_raw(GEN *ptH, GEN Hp, GEN q, GEN qp, ulong p)
1060 : {
1061 147430 : GEN H = *ptH, h, qp2 = shifti(qp,-1);
1062 147420 : ulong qinv = Fl_inv(umodiu(q,p), p);
1063 147430 : long i, l = lg(H), lp = lg(Hp);
1064 147430 : int stable = 1;
1065 :
1066 147430 : 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 147430 : } 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 4932992 : for (i=2; i<lp; i++)
1081 : {
1082 4785703 : h = Fl_chinese_coprime(gel(H,i),Hp[i],q,p,qinv,qp,qp2);
1083 4785562 : if (h) { gel(H,i) = h; stable = 0; }
1084 : }
1085 147289 : (void)ZX_renormalize(H,lp);
1086 147432 : return stable;
1087 : }
1088 :
1089 : int
1090 0 : ZX_incremental_CRT(GEN *ptH, GEN Hp, GEN *ptq, ulong p)
1091 : {
1092 0 : GEN q = *ptq, qp = muliu(q,p);
1093 0 : int stable = ZX_incremental_CRT_raw(ptH, Hp, q, qp, p);
1094 0 : *ptq = qp; return stable;
1095 : }
1096 :
1097 : int
1098 5801 : ZM_incremental_CRT(GEN *pH, GEN Hp, GEN *ptq, ulong p)
1099 : {
1100 5801 : GEN h, H = *pH, q = *ptq, qp = muliu(q, p), qp2 = shifti(qp,-1);
1101 5801 : ulong qinv = Fl_inv(umodiu(q,p), p);
1102 5801 : long i,j, l = lg(H), m = lgcols(H);
1103 5801 : int stable = 1;
1104 20944 : for (j=1; j<l; j++)
1105 157160 : for (i=1; i<m; i++)
1106 : {
1107 142017 : h = Fl_chinese_coprime(gcoeff(H,i,j), coeff(Hp,i,j),q,p,qinv,qp,qp2);
1108 142017 : if (h) { gcoeff(H,i,j) = h; stable = 0; }
1109 : }
1110 5801 : *ptq = qp; return stable;
1111 : }
1112 :
1113 : GEN
1114 623 : ZXM_init_CRT(GEN Hp, long deg, ulong p)
1115 : {
1116 : long i, j, k;
1117 : GEN H;
1118 623 : long m, l = lg(Hp), lim = (long)(p>>1), n;
1119 623 : H = cgetg(l, t_MAT);
1120 623 : if (l==1) return H;
1121 623 : m = lgcols(Hp);
1122 623 : n = deg + 3;
1123 2114 : for (j=1; j<l; j++)
1124 : {
1125 1491 : GEN cp = gel(Hp,j);
1126 1491 : GEN c = cgetg(m, t_COL);
1127 1491 : gel(H,j) = c;
1128 23905 : for (i=1; i<m; i++)
1129 : {
1130 22414 : GEN dp = gel(cp, i);
1131 22414 : long l = lg(dp);
1132 22414 : GEN d = cgetg(n, t_POL);
1133 22414 : gel(c, i) = d;
1134 22414 : d[1] = dp[1] | evalsigne(1);
1135 45647 : for (k=2; k<l; k++)
1136 23233 : gel(d,k) = stoi(Fl_center(dp[k], p, lim));
1137 44457 : for ( ; k<n; k++)
1138 22043 : gel(d,k) = gen_0;
1139 : }
1140 : }
1141 623 : return H;
1142 : }
1143 :
1144 : int
1145 653 : ZXM_incremental_CRT(GEN *pH, GEN Hp, GEN *ptq, ulong p)
1146 : {
1147 653 : GEN v, H = *pH, q = *ptq, qp = muliu(q, p), qp2 = shifti(qp,-1);
1148 653 : ulong qinv = Fl_inv(umodiu(q,p), p);
1149 653 : long i,j,k, l = lg(H), m = lgcols(H), n = lg(gmael(H,1,1));
1150 653 : int stable = 1;
1151 2225 : for (j=1; j<l; j++)
1152 90418 : for (i=1; i<m; i++)
1153 : {
1154 88846 : GEN h = gmael(H,j,i), hp = gmael(Hp,j,i);
1155 88846 : long lh = lg(hp);
1156 246641 : for (k=2; k<lh; k++)
1157 : {
1158 157795 : v = Fl_chinese_coprime(gel(h,k),uel(hp,k),q,p,qinv,qp,qp2);
1159 157795 : if (v) { gel(h,k) = v; stable = 0; }
1160 : }
1161 108763 : for (; k<n; k++)
1162 : {
1163 19917 : v = Fl_chinese_coprime(gel(h,k),0,q,p,qinv,qp,qp2);
1164 19917 : if (v) { gel(h,k) = v; stable = 0; }
1165 : }
1166 : }
1167 653 : *ptq = qp; return stable;
1168 : }
1169 :
1170 : /* record the degrees of Euclidean remainders (make them as large as
1171 : * possible : smaller values correspond to a degenerate sequence) */
1172 : static void
1173 23195 : Flx_resultant_set_dglist(GEN a, GEN b, GEN dglist, ulong p)
1174 : {
1175 : long da,db,dc, ind;
1176 23195 : pari_sp av = avma;
1177 :
1178 23195 : if (lgpol(a)==0 || lgpol(b)==0) return;
1179 21928 : da = degpol(a);
1180 21928 : db = degpol(b);
1181 21928 : if (db > da)
1182 0 : { swapspec(a,b, da,db); }
1183 21928 : else if (!da) return;
1184 21928 : ind = 0;
1185 144150 : while (db)
1186 : {
1187 122225 : GEN c = Flx_rem(a,b, p);
1188 122222 : a = b; b = c; dc = degpol(c);
1189 122222 : if (dc < 0) break;
1190 :
1191 122222 : ind++;
1192 122222 : if (dc > dglist[ind]) dglist[ind] = dc;
1193 122222 : 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 122222 : db = dc; /* = degpol(b) */
1199 : }
1200 21925 : if (ind+1 > lg(dglist)) setlg(dglist,ind+1);
1201 21925 : 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 2084431 : Flx_resultant_all(GEN a, GEN b, long *C0, long *C1, GEN dglist, ulong p)
1208 : {
1209 : long da,db,dc, ind;
1210 2084431 : ulong lb, res, g = 1UL, h = 1UL, ca = 1UL, cb = 1UL;
1211 2084431 : int s = 1;
1212 2084431 : pari_sp av = avma;
1213 :
1214 2084431 : *C0 = 1; *C1 = 0;
1215 2084431 : if (lgpol(a)==0 || lgpol(b)==0) return 0;
1216 2074965 : da = degpol(a);
1217 2075007 : db = degpol(b);
1218 2074994 : if (db > da)
1219 : {
1220 0 : swapspec(a,b, da,db);
1221 0 : if (both_odd(da,db)) s = -s;
1222 : }
1223 2074994 : else if (!da) return 1; /* = a[2] ^ db, since 0 <= db <= da = 0 */
1224 2074994 : ind = 0;
1225 19786436 : while (db)
1226 : { /* sub-resultant algo., applied to ca * a and cb * b, ca,cb scalars,
1227 : * da = deg a, db = deg b */
1228 17715983 : GEN c = Flx_rem(a,b, p);
1229 17587097 : long delta = da - db;
1230 :
1231 17587097 : if (both_odd(da,db)) s = -s;
1232 17583976 : lb = Fl_mul(b[db+2], cb, p);
1233 17604901 : a = b; b = c; dc = degpol(c);
1234 17603186 : ind++;
1235 17603186 : if (dc != dglist[ind]) return gc_ulong(av,0); /* degenerates */
1236 17598295 : if (g == h)
1237 : { /* frequent */
1238 17538453 : ulong cc = Fl_mul(ca, Fl_powu(Fl_div(lb,g,p), delta+1, p), p);
1239 17652973 : ca = cb;
1240 17652973 : cb = cc;
1241 : }
1242 : else
1243 : {
1244 59842 : ulong cc = Fl_mul(ca, Fl_powu(lb, delta+1, p), p);
1245 59842 : ulong ghdelta = Fl_mul(g, Fl_powu(h, delta, p), p);
1246 59843 : ca = cb;
1247 59843 : cb = Fl_div(cc, ghdelta, p);
1248 : }
1249 17713107 : da = db; /* = degpol(a) */
1250 17713107 : db = dc; /* = degpol(b) */
1251 :
1252 17713107 : g = lb;
1253 17713107 : if (delta == 1)
1254 17613836 : h = g; /* frequent */
1255 : else
1256 99271 : h = Fl_mul(h, Fl_powu(Fl_div(g,h,p), delta, p), p);
1257 :
1258 17712967 : 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 2070453 : if (da > 1) return 0; /* Failure */
1265 : /* last nonconstant polynomial has degree 1 */
1266 2070453 : *C0 = Fl_mul(ca, a[2], p);
1267 2070390 : *C1 = Fl_mul(ca, a[3], p);
1268 2070390 : res = Fl_mul(cb, b[2], p);
1269 2070356 : if (s == -1) res = p - res;
1270 2070356 : 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 2107816 : FlxY_evalx_drop(GEN Q, ulong x, ulong p)
1277 : {
1278 : GEN z;
1279 2107816 : long i, lb = lg(Q);
1280 2107816 : ulong leadz = Flx_eval(leading_coeff(Q), x, p);
1281 2107399 : long vs=mael(Q,2,1);
1282 2107399 : if (!leadz) return zero_Flx(vs);
1283 :
1284 2096739 : z = cgetg(lb, t_VECSMALL); z[1] = vs;
1285 20046399 : for (i=2; i<lb-1; i++) z[i] = Flx_eval(gel(Q,i), x, p);
1286 2094545 : 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 272558 : ZX_norml1(GEN x)
1311 : {
1312 272558 : long i, l = lg(x);
1313 : GEN s;
1314 :
1315 272558 : if (l == 2) return gen_0;
1316 180004 : s = gel(x, l-1); /* != 0 */
1317 658861 : for (i = l-2; i > 1; i--) {
1318 478863 : GEN xi = gel(x,i);
1319 478863 : if (!signe(xi)) continue;
1320 240190 : s = addii_sign(s,1, xi,1);
1321 : }
1322 179998 : return s;
1323 : }
1324 : /* x >= 0, y != 0, return x + |y| */
1325 : static GEN
1326 25555 : addii_abs(GEN x, GEN y)
1327 : {
1328 25555 : if (!signe(x)) return absi_shallow(y);
1329 16045 : 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 31650 : ZX_norml1_1(GEN x, long k)
1335 : {
1336 31650 : long i, d = degpol(x);
1337 : GEN s, C; /* = binomial(i, k) */
1338 :
1339 31649 : if (!d || k > d) return gen_0;
1340 31649 : s = absi_shallow(gel(x, k+2)); /* may be 0 */
1341 31652 : C = gen_1;
1342 68058 : for (i = k+1; i <= d; i++) {
1343 36410 : GEN xi = gel(x,i+2);
1344 36410 : if (k) C = diviuexact(muliu(C, i), i-k);
1345 36410 : if (signe(xi)) s = addii_abs(s, mulii(C, xi));
1346 : }
1347 31648 : 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 7409 : sqrN2(GEN A, long prec)
1370 : {
1371 7409 : pari_sp av = avma;
1372 7409 : long i, l = lg(A);
1373 7409 : GEN a = gen_0;
1374 36671 : for (i = 2; i < l; i++)
1375 : {
1376 29262 : a = gadd(a, gabs(gnorm(gel(A,i)), prec));
1377 29262 : 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 7409 : 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 6562 : RgX_RgXY_ResBound(GEN A, GEN B, long prec)
1391 : {
1392 6562 : pari_sp av = avma;
1393 6562 : GEN b = gen_0, bnd;
1394 6562 : long i, lB = lg(B);
1395 26322 : for (i=2; i<lB; i++)
1396 : {
1397 19760 : GEN t = gel(B,i);
1398 19760 : if (typ(t) == t_POL) t = gnorml1(t, prec);
1399 19760 : b = gadd(b, gabs(gsqr(t), prec));
1400 19760 : 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 6562 : bnd = gsqrt(gmul(gpowgs(sqrN2(A,prec), degpol(B)),
1407 : gpowgs(b, degpol(A))), prec);
1408 6562 : 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 150630 : log2N2(GEN A)
1437 : {
1438 150630 : pari_sp av = avma;
1439 150630 : long i, l = lg(A);
1440 150630 : GEN a = gen_0;
1441 954727 : for (i=2; i < l; i++)
1442 : {
1443 804099 : a = addii(a, sqri(gel(A,i)));
1444 804095 : 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 150628 : 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 140546 : ZX_ZXY_ResBound(GEN A, GEN B, GEN dB)
1458 : {
1459 140546 : pari_sp av = avma;
1460 140546 : GEN b = gen_0;
1461 140546 : long i, lB = lg(B);
1462 : double logb;
1463 908704 : for (i=2; i<lB; i++)
1464 : {
1465 768158 : GEN t = gel(B,i);
1466 768158 : if (typ(t) == t_POL) t = ZX_norml1(t);
1467 768158 : b = addii(b, sqri(t));
1468 768158 : 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 140546 : logb = dbllog2(b); if (dB) logb -= 2 * dbllog2(dB);
1475 140545 : i = (long)((degpol(B) * log2N2(A) + degpol(A) * logb) / 2);
1476 140545 : 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 10083 : ZX_ZXY_ResBound_1(GEN A, GEN B)
1481 : {
1482 10083 : pari_sp av = avma;
1483 10083 : GEN b = gen_0;
1484 10083 : long i, lB = lg(B);
1485 41738 : for (i=2; i<lB; i++)
1486 : {
1487 31650 : b = addii(b, sqri(ZX_norml1_1(B, i-2)));
1488 31655 : 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 10088 : i = (long)((degpol(B) * log2N2(A) + degpol(A) * dbllog2(b)) / 2);
1495 10087 : return gc_ulong(av, (i <= 0)? 1: 1 + (ulong)i);
1496 : }
1497 : /* special case B = A' */
1498 : static ulong
1499 1128679 : ZX_discbound(GEN A)
1500 : {
1501 1128679 : pari_sp av = avma;
1502 1128679 : GEN a = gen_0, b = gen_0;
1503 1128679 : long i , lA = lg(A), dA = degpol(A);
1504 : double loga, logb;
1505 6727655 : for (i = 2; i < lA; i++)
1506 : {
1507 5599211 : GEN c = sqri(gel(A,i));
1508 5598847 : a = addii(a, c);
1509 5598948 : if (i > 2) b = addii(b, mulii(c, sqru(i-2)));
1510 5599025 : 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 1128444 : loga = dbllog2(a);
1517 1128594 : logb = dbllog2(b); set_avma(av);
1518 1128617 : i = (long)(((dA-1) * loga + dA * logb) / 2);
1519 1128617 : 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 2270706 : Flx_FlxY_eval_resultant(GEN a, GEN b, ulong n, ulong p, ulong pi, ulong la)
1525 : {
1526 2270706 : GEN ev = FlxY_evalx_pre(b, n, p, pi);
1527 2270809 : long drop = lg(b) - lg(ev);
1528 2270809 : ulong r = Flx_resultant_pre(a, ev, p, pi);
1529 2270620 : if (drop && la != 1) r = Fl_mul(r, Fl_powu_pre(la, drop, p, pi), p);
1530 2270631 : 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 177360 : Flx_FlxY_resultant_polint(GEN a, GEN b, ulong p, ulong pi, long dres, long sx)
1546 : {
1547 : long i;
1548 177360 : ulong n, la = Flx_lead(a);
1549 177360 : GEN x = cgetg(dres+2, t_VECSMALL);
1550 177358 : 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 1227055 : for (i=0,n = 1; i < dres; n++)
1554 : {
1555 1049700 : x[++i] = n; y[i] = Flx_FlxY_eval_resultant(a,b, x[i], p,pi,la);
1556 1049672 : x[++i] = p-n; y[i] = Flx_FlxY_eval_resultant(a,b, x[i], p,pi,la);
1557 : }
1558 177355 : if (i == dres)
1559 : {
1560 171929 : x[++i] = 0; y[i] = Flx_FlxY_eval_resultant(a,b, x[i], p,pi,la);
1561 : }
1562 177356 : return Flv_polint(x,y, p, sx);
1563 : }
1564 :
1565 : static GEN
1566 7574 : FlxX_pseudorem(GEN x, GEN y, ulong p, ulong pi)
1567 : {
1568 7574 : long vx = varn(x), dx, dy, dz, i, lx, dp;
1569 7574 : pari_sp av = avma, av2;
1570 :
1571 7574 : if (!signe(y)) pari_err_INV("FlxX_pseudorem",y);
1572 7574 : (void)new_chunk(2);
1573 7575 : dx=degpol(x); x = RgX_recip_i(x)+2;
1574 7575 : dy=degpol(y); y = RgX_recip_i(y)+2; dz=dx-dy; dp = dz+1;
1575 7575 : av2 = avma;
1576 : for (;;)
1577 : {
1578 62000 : gel(x,0) = Flx_neg(gel(x,0), p); dp--;
1579 232290 : for (i=1; i<=dy; i++)
1580 169869 : gel(x,i) = Flx_add( Flx_mul_pre(gel(y,0), gel(x,i), p, pi),
1581 170186 : Flx_mul_pre(gel(x,0), gel(y,i), p, pi), p );
1582 1119015 : for ( ; i<=dx; i++)
1583 1052455 : gel(x,i) = Flx_mul_pre(gel(y,0), gel(x,i), p, pi);
1584 69038 : do { x++; dx--; } while (dx >= 0 && lg(gel(x,0))==2);
1585 66560 : if (dx < dy) break;
1586 58987 : 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 7573 : if (dx < 0) return zero_Flx(0);
1593 7573 : lx = dx+3; x -= 2;
1594 7573 : x[0]=evaltyp(t_POL) | _evallg(lx);
1595 7573 : x[1]=evalsigne(1) | evalvarn(vx);
1596 7573 : x = RgX_recip_i(x);
1597 7575 : if (dp)
1598 : { /* multiply by y[0]^dp [beware dummy vars from FpX_FpXY_resultant] */
1599 1978 : GEN t = Flx_powu_pre(gel(y,0), dp, p, pi);
1600 7921 : for (i=2; i<lx; i++) gel(x,i) = Flx_mul_pre(gel(x,i), t, p, pi);
1601 : }
1602 7575 : return gerepilecopy(av, x);
1603 : }
1604 :
1605 : /* return a Flx */
1606 : GEN
1607 2533 : FlxX_resultant(GEN u, GEN v, ulong p, long sx)
1608 : {
1609 2533 : 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 2533 : dx = degpol(u); dy = degpol(v); signh = 1;
1615 2533 : if (dx < dy)
1616 : {
1617 7 : swap(u,v); lswap(dx,dy);
1618 7 : if (both_odd(dx, dy)) signh = -signh;
1619 : }
1620 2533 : if (dy < 0) return zero_Flx(sx);
1621 2533 : pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
1622 2534 : if (dy==0) return gerepileupto(av, Flx_powu_pre(gel(v,2),dx,p,pi));
1623 :
1624 2534 : g = h = pol1_Flx(sx); av2 = avma;
1625 : for(;;)
1626 : {
1627 7574 : r = FlxX_pseudorem(u,v,p,pi); dr = lg(r);
1628 7576 : if (dr == 2) { set_avma(av); return zero_Flx(sx); }
1629 7576 : du = degpol(u); dv = degpol(v); degq = du-dv;
1630 7576 : u = v; p1 = g; g = leading_coeff(u);
1631 7576 : switch(degq)
1632 : {
1633 0 : case 0: break;
1634 5584 : case 1:
1635 5584 : p1 = Flx_mul_pre(h,p1, p, pi); h = g; break;
1636 1992 : default:
1637 1992 : p1 = Flx_mul_pre(Flx_powu_pre(h,degq,p,pi), p1, p, pi);
1638 1992 : h = Flx_div_pre(Flx_powu_pre(g,degq,p,pi),
1639 1992 : Flx_powu_pre(h,degq-1,p,pi), p, pi);
1640 : }
1641 7576 : if (both_odd(du,dv)) signh = -signh;
1642 7575 : v = FlxY_Flx_div(r, p1, p);
1643 7575 : if (dr==3) break;
1644 5040 : 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 2535 : z = gel(v,2);
1651 2535 : 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 2535 : if (signh < 0) z = Flx_neg(z,p);
1654 2535 : return gerepileupto(av, z);
1655 : }
1656 :
1657 : /* Warning:
1658 : * This function switches between valid and invalid variable ordering*/
1659 :
1660 : static GEN
1661 6146 : FlxY_to_FlyX(GEN b, long sv)
1662 : {
1663 6146 : long i, n=-1;
1664 6146 : long sw = b[1]&VARNBITS;
1665 20977 : for(i=2;i<lg(b);i++) n = maxss(n,lgpol(gel(b,i)));
1666 6145 : return Flm_to_FlxX(Flm_transpose(FlxX_to_Flm(b,n)),sv,sw);
1667 : }
1668 :
1669 : /* Return a Fly*/
1670 : GEN
1671 6145 : Flx_FlxY_resultant(GEN a, GEN b, ulong p)
1672 : {
1673 6145 : pari_sp ltop=avma;
1674 6145 : long dres = degpol(a)*degpol(b);
1675 6145 : long sx=a[1], sy=b[1]&VARNBITS;
1676 : GEN z;
1677 6145 : b = FlxY_to_FlyX(b,sx);
1678 6143 : if ((ulong)dres >= p)
1679 2532 : z = FlxX_resultant(Fly_to_FlxY(a, sy), b, p, sx);
1680 : else
1681 : {
1682 3611 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
1683 3611 : z = Flx_FlxY_resultant_polint(a, b, p, pi, (ulong)dres, sy);
1684 : }
1685 6146 : 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 126145 : swap_vars(GEN b0, long v)
1696 : {
1697 126145 : long i, n = RgX_degree(b0, v);
1698 : GEN b, x;
1699 126145 : if (n < 0) return pol_0(v);
1700 126145 : b = cgetg(n+3, t_POL); x = b + 2;
1701 126145 : b[1] = evalsigne(1) | evalvarn(v);
1702 641200 : for (i=0; i<=n; i++) gel(x,i) = polcoef_i(b0, i, v);
1703 126142 : 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 21048 : ZX_ZXY_resultant_LERS(GEN A, GEN B0, long *plambda, GEN *LERS)
1810 : {
1811 : ulong bound, dp;
1812 21048 : pari_sp av = avma, av2 = 0;
1813 21048 : long lambda = *plambda, degA = degpol(A), dres = degA*degpol(B0);
1814 : long stable, checksqfree, i,n, cnt, degB;
1815 21048 : 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 21048 : if (degA == 1)
1820 : {
1821 1043 : GEN a1 = gel(A,3), a0 = gel(A,2);
1822 1043 : B = lambda? RgX_translate(B0, monomial(stoi(lambda), 1, vY)): B0;
1823 1043 : H = gsubst(B, vY, gdiv(gneg(a0),a1));
1824 1043 : if (!equali1(a1)) H = RgX_Rg_mul(H, powiu(a1, poldegree(B,vY)));
1825 1043 : *LERS = mkvec2(scalarpol_shallow(a0,vX), scalarpol_shallow(a1,vX));
1826 1043 : return gc_all(av, 2, &H, LERS);
1827 : }
1828 :
1829 20005 : dglist = Hp = H0p = H1p = C0 = C1 = NULL; /* gcc -Wall */
1830 20005 : C0 = cgetg(dres+2, t_VECSMALL);
1831 20006 : C1 = cgetg(dres+2, t_VECSMALL);
1832 20006 : dglist = cgetg(dres+1, t_VECSMALL);
1833 20006 : x = cgetg(dres+2, t_VECSMALL);
1834 20006 : y = cgetg(dres+2, t_VECSMALL);
1835 20006 : B0 = leafcopy(B0);
1836 20006 : A = leafcopy(A);
1837 20006 : B = B0;
1838 20006 : v = fetch_var_higher(); setvarn(A,v);
1839 : /* make sure p large enough */
1840 20668 : INIT:
1841 : /* always except the first time */
1842 20668 : if (av2) { set_avma(av2); lambda = next_lambda(lambda); }
1843 20668 : if (lambda) B = RgX_translate(B0, monomial(stoi(lambda), 1, vY));
1844 20668 : B = swap_vars(B, vY); setvarn(B,v);
1845 : /* B0(lambda v + x, v) */
1846 20668 : if (DEBUGLEVEL>4) err_printf("Trying lambda = %ld\n", lambda);
1847 20668 : av2 = avma;
1848 :
1849 20668 : if (degA <= 3)
1850 : { /* sub-resultant faster for small degrees */
1851 10010 : H = RgX_resultant_all(A,B,&q);
1852 10010 : if (typ(q) != t_POL || degpol(q)!=1) goto INIT;
1853 9499 : H0 = gel(q,2);
1854 9499 : if (typ(H0) == t_POL) setvarn(H0,vX); else H0 = scalarpol(H0,vX);
1855 9499 : H1 = gel(q,3);
1856 9499 : if (typ(H1) == t_POL) setvarn(H1,vX); else H1 = scalarpol(H1,vX);
1857 9499 : if (!ZX_is_squarefree(H)) goto INIT;
1858 9457 : goto END;
1859 : }
1860 :
1861 10658 : H = H0 = H1 = NULL;
1862 10658 : degB = degpol(B);
1863 10658 : bound = ZX_ZXY_ResBound(A, B, NULL);
1864 10658 : if (DEBUGLEVEL>4) err_printf("bound for resultant coeffs: 2^%ld\n",bound);
1865 10658 : dp = 1;
1866 10658 : init_modular_big(&S);
1867 10658 : for(cnt = 0, checksqfree = 1;;)
1868 49144 : {
1869 59802 : ulong p = u_forprime_next(&S);
1870 : GEN Hi;
1871 59803 : a = ZX_to_Flx(A, p);
1872 59801 : b = ZXX_to_FlxX(B, p, varn(A));
1873 59800 : if (degpol(a) < degA || degpol(b) < degB) continue; /* p | lc(A)lc(B) */
1874 59799 : if (checksqfree)
1875 : { /* find degree list for generic Euclidean Remainder Sequence */
1876 10658 : long goal = minss(degpol(a), degpol(b)); /* longest possible */
1877 73035 : for (n=1; n <= goal; n++) dglist[n] = 0;
1878 10658 : setlg(dglist, 1);
1879 23587 : for (n=0; n <= dres; n++)
1880 : {
1881 23195 : ev = FlxY_evalx_drop(b, n, p);
1882 23195 : Flx_resultant_set_dglist(a, ev, dglist, p);
1883 23195 : if (lg(dglist)-1 == goal) break;
1884 : }
1885 : /* last pol in ERS has degree > 1 ? */
1886 10658 : goal = lg(dglist)-1;
1887 10658 : if (degpol(B) == 1) { if (!goal) goto INIT; }
1888 : else
1889 : {
1890 10602 : if (goal <= 1) goto INIT;
1891 10546 : if (dglist[goal] != 0 || dglist[goal-1] != 1) goto INIT;
1892 : }
1893 10602 : if (DEBUGLEVEL>4)
1894 0 : err_printf("Degree list for ERS (trials: %ld) = %Ps\n",n+1,dglist);
1895 : }
1896 :
1897 2144375 : for (i=0,n = 0; i <= dres; n++)
1898 : {
1899 2084639 : ev = FlxY_evalx_drop(b, n, p);
1900 2084383 : x[++i] = n; y[i] = Flx_resultant_all(a, ev, C0+i, C1+i, dglist, p);
1901 2084632 : if (!C1[i]) i--; /* C1(i) = 0. No way to recover C0(i) */
1902 : }
1903 59736 : Hi = Flv_Flm_polint(x, mkvec3(y,C0,C1), p, 0);
1904 59747 : Hp = gel(Hi,1); H0p = gel(Hi,2); H1p = gel(Hi,3);
1905 59747 : if (!H && degpol(Hp) != dres) continue;
1906 59747 : if (dp != 1) Hp = Flx_Fl_mul(Hp, Fl_powu(Fl_inv(dp,p), degA, p), p);
1907 59747 : if (checksqfree) {
1908 10602 : if (!Flx_is_squarefree(Hp, p)) goto INIT;
1909 10548 : if (DEBUGLEVEL>4) err_printf("Final lambda = %ld\n", lambda);
1910 10548 : checksqfree = 0;
1911 : }
1912 :
1913 59693 : if (!H)
1914 : { /* initialize */
1915 10548 : q = utoipos(p); stable = 0;
1916 10548 : H = ZX_init_CRT(Hp, p,vX);
1917 10549 : H0= ZX_init_CRT(H0p, p,vX);
1918 10549 : H1= ZX_init_CRT(H1p, p,vX);
1919 : }
1920 : else
1921 : {
1922 49145 : GEN qp = muliu(q,p);
1923 49145 : stable = ZX_incremental_CRT_raw(&H, Hp, q,qp, p)
1924 49144 : & ZX_incremental_CRT_raw(&H0,H0p, q,qp, p)
1925 49145 : & ZX_incremental_CRT_raw(&H1,H1p, q,qp, p);
1926 49144 : q = qp;
1927 : }
1928 : /* could make it probabilistic for H ? [e.g if stable twice, etc]
1929 : * Probabilistic anyway for H0, H1 */
1930 59693 : if (DEBUGLEVEL>5 && (stable || ++cnt==100))
1931 0 : { cnt=0; err_printf("%ld%%%s ",100*expi(q)/bound,stable?"s":""); }
1932 59693 : if (stable && (ulong)expi(q) >= bound) break; /* DONE */
1933 49144 : 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 20006 : END:
1940 20006 : if (DEBUGLEVEL>5) err_printf(" done\n");
1941 20006 : setvarn(H, vX); (void)delete_var();
1942 20006 : *LERS = mkvec2(H0,H1);
1943 20006 : *plambda = lambda; return gc_all(av, 2, &H, LERS);
1944 : }
1945 :
1946 : GEN
1947 58848 : ZX_ZXY_resultant_all(GEN A, GEN B, long *plambda, GEN *LERS)
1948 : {
1949 58848 : if (LERS)
1950 : {
1951 21048 : if (!plambda)
1952 0 : pari_err_BUG("ZX_ZXY_resultant_all [LERS != NULL needs lambda]");
1953 21048 : return ZX_ZXY_resultant_LERS(A, B, plambda, LERS);
1954 : }
1955 37800 : 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 3493 : ZXQ_charpoly_sqf(GEN A, GEN T, long *lambda, long v)
1963 : {
1964 3493 : pari_sp av = avma;
1965 : GEN R, a;
1966 : long dA;
1967 : int delvar;
1968 :
1969 3493 : if (v < 0) v = 0;
1970 3493 : switch (typ(A))
1971 : {
1972 3493 : 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 3493 : delvar = 0;
1979 3493 : if (varn(T) == 0)
1980 : {
1981 3296 : long v0 = fetch_var(); delvar = 1;
1982 3296 : T = leafcopy(T); setvarn(T,v0);
1983 3296 : A = leafcopy(A); setvarn(A,v0);
1984 : }
1985 3493 : R = ZX_ZXY_rnfequation(T, deg1pol_shallow(gen_1, gneg_i(A), 0), lambda);
1986 3493 : if (delvar) (void)delete_var();
1987 3493 : setvarn(R, v); a = leading_coeff(T);
1988 3493 : if (!gequal1(a)) R = gdiv(R, powiu(a, dA));
1989 3493 : 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 120729 : ZXQ_charpoly(GEN A, GEN T, long v)
1995 : {
1996 120729 : 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 3756565 : ZX_resultant_prime(GEN a, GEN b, GEN dB, long degA, long degB, ulong p)
2010 : {
2011 3756565 : long dropa = degA - degpol(a), dropb = degB - degpol(b);
2012 : ulong H, dp;
2013 3756445 : if (dropa && dropb) return 0; /* p | lc(A), p | lc(B) */
2014 3756445 : H = Flx_resultant(a, b, p);
2015 3756324 : 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 3756324 : 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 3756320 : dp = dB ? umodiu(dB, p): 1;
2029 3756320 : if (dp != 1) H = Fl_mul(H, Fl_powu(Fl_inv(dp,p), degA, p), p);
2030 3756321 : return H;
2031 : }
2032 :
2033 : /* If B=NULL, assume B=A' */
2034 : static GEN
2035 1469425 : ZX_resultant_slice(GEN A, GEN B, GEN dB, GEN P, GEN *mod)
2036 : {
2037 1469425 : pari_sp av = avma, av2;
2038 1469425 : long degA, degB, i, n = lg(P)-1;
2039 : GEN H, T;
2040 :
2041 1469425 : degA = degpol(A);
2042 1469421 : degB = B? degpol(B): degA - 1;
2043 1469422 : if (n == 1)
2044 : {
2045 804393 : ulong Hp, p = uel(P,1);
2046 804393 : GEN a = ZX_to_Flx(A, p), b = B? ZX_to_Flx(B, p): Flx_deriv(a, p);
2047 804368 : Hp = ZX_resultant_prime(a, b, dB, degA, degB, p);
2048 804381 : set_avma(av); *mod = utoipos(p); return utoi(Hp);
2049 : }
2050 665029 : T = ZV_producttree(P);
2051 665031 : A = ZX_nv_mod_tree(A, P, T);
2052 665029 : if (B) B = ZX_nv_mod_tree(B, P, T);
2053 665029 : H = cgetg(n+1, t_VECSMALL); av2 = avma;
2054 3617007 : for(i=1; i <= n; i++, set_avma(av2))
2055 : {
2056 2951974 : ulong p = P[i];
2057 2951974 : GEN a = gel(A,i), b = B? gel(B,i): Flx_deriv(a, p);
2058 2952198 : H[i] = ZX_resultant_prime(a, b, dB, degA, degB, p);
2059 : }
2060 665033 : H = ZV_chinese_tree(H, P, T, ZV_chinesetree(P,T));
2061 665031 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2062 : }
2063 :
2064 : GEN
2065 1469428 : ZX_resultant_worker(GEN P, GEN A, GEN B, GEN dB)
2066 : {
2067 1469428 : GEN V = cgetg(3, t_VEC);
2068 1469423 : if (typ(B) == t_INT) B = NULL;
2069 1469423 : if (!signe(dB)) dB = NULL;
2070 1469423 : gel(V,1) = ZX_resultant_slice(A, B, dB, P, &gel(V,2));
2071 1469398 : 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 1313063 : ZX_resultant_all(GEN A, GEN B, GEN dB, ulong bound)
2078 : {
2079 1313063 : pari_sp av = avma;
2080 : forprime_t S;
2081 : GEN H, worker;
2082 1313063 : if (!B && degpol(A)==2)
2083 : {
2084 113412 : GEN a = gel(A,4), b = gel(A,3), c = gel(A,2);
2085 113412 : H = mulii(a, subii(shifti(mulii(a, c), 2), sqri(b)));
2086 113404 : if (dB) H = diviiexact(H, sqri(dB));
2087 113404 : return gerepileuptoint(av, H);
2088 : }
2089 1199653 : if (B)
2090 : {
2091 122755 : long a = degpol(A), b = degpol(B);
2092 122755 : if (a < 0 || b < 0) return gen_0;
2093 122725 : if (!a) return powiu(gel(A,2), b);
2094 122725 : if (!b) return powiu(gel(B,2), a);
2095 120980 : if (minss(a, b) <= 1)
2096 : {
2097 51187 : H = RgX_resultant_all(A, B, NULL);
2098 51187 : if (dB) H = diviiexact(H, powiu(dB, a));
2099 51187 : return gerepileuptoint(av, H);
2100 : }
2101 69793 : if (!bound) bound = ZX_ZXY_ResBound(A, B, dB);
2102 : }
2103 1146700 : worker = snm_closure(is_entry("_ZX_resultant_worker"),
2104 : mkvec3(A, B? B: gen_0, dB? dB: gen_0));
2105 1146779 : init_modular_big(&S);
2106 1146749 : H = gen_crt("ZX_resultant_all", worker, &S, dB, bound, 0, NULL,
2107 : ZV_chinese_center, Fp_center);
2108 1146767 : return gerepileuptoint(av, H);
2109 : }
2110 :
2111 : /* A0 and B0 in Q[X] */
2112 : GEN
2113 56 : QX_resultant(GEN A0, GEN B0)
2114 : {
2115 : GEN s, a, b, A, B;
2116 56 : pari_sp av = avma;
2117 :
2118 56 : A = Q_primitive_part(A0, &a);
2119 56 : B = Q_primitive_part(B0, &b);
2120 56 : s = ZX_resultant(A, B);
2121 56 : if (!signe(s)) { set_avma(av); return gen_0; }
2122 56 : if (a) s = gmul(s, gpowgs(a,degpol(B)));
2123 56 : if (b) s = gmul(s, gpowgs(b,degpol(A)));
2124 56 : return gerepileupto(av, s);
2125 : }
2126 :
2127 : GEN
2128 25319 : ZX_resultant(GEN A, GEN B) { return ZX_resultant_all(A,B,NULL,0); }
2129 :
2130 : GEN
2131 0 : QXQ_intnorm(GEN A, GEN B)
2132 : {
2133 : GEN c, n, R, lB;
2134 0 : long dA = degpol(A), dB = degpol(B);
2135 0 : pari_sp av = avma;
2136 0 : if (dA < 0) return gen_0;
2137 0 : A = Q_primitive_part(A, &c);
2138 0 : if (!c || typ(c) == t_INT) {
2139 0 : n = c;
2140 0 : R = ZX_resultant(B, A);
2141 : } else {
2142 0 : n = gel(c,1);
2143 0 : R = ZX_resultant_all(B, A, gel(c,2), 0);
2144 : }
2145 0 : if (n && !equali1(n)) R = mulii(R, powiu(n, dB));
2146 0 : lB = leading_coeff(B);
2147 0 : if (!equali1(lB)) R = diviiexact(R, powiu(lB, dA));
2148 0 : return gerepileuptoint(av, R);
2149 : }
2150 :
2151 : GEN
2152 18858 : QXQ_norm(GEN A, GEN B)
2153 : {
2154 : GEN c, R, lB;
2155 18858 : long dA = degpol(A), dB = degpol(B);
2156 18858 : pari_sp av = avma;
2157 18858 : if (dA < 0) return gen_0;
2158 18858 : A = Q_primitive_part(A, &c);
2159 18858 : R = ZX_resultant(B, A);
2160 18858 : if (c) R = gmul(R, gpowgs(c, dB));
2161 18858 : lB = leading_coeff(B);
2162 18858 : if (!equali1(lB)) R = gdiv(R, gpowgs(lB, dA));
2163 18858 : return gerepileupto(av, R);
2164 : }
2165 :
2166 : /* assume x has integral coefficients */
2167 : GEN
2168 1193432 : ZX_disc_all(GEN x, ulong bound)
2169 : {
2170 1193432 : pari_sp av = avma;
2171 1193432 : long s, d = degpol(x);
2172 : GEN l, R;
2173 :
2174 1193433 : if (d <= 1) return d == 1? gen_1: gen_0;
2175 1190343 : s = (d & 2) ? -1: 1;
2176 1190343 : l = leading_coeff(x);
2177 1190336 : if (!bound) bound = ZX_discbound(x);
2178 1190277 : R = ZX_resultant_all(x, NULL, NULL, bound);
2179 1190351 : if (is_pm1(l))
2180 1015064 : { if (signe(l) < 0) s = -s; }
2181 : else
2182 175285 : R = diviiexact(R,l);
2183 1190349 : if (s == -1) togglesign_safe(&R);
2184 1190348 : return gerepileuptoint(av,R);
2185 : }
2186 :
2187 : GEN
2188 1131722 : ZX_disc(GEN x) { return ZX_disc_all(x,0); }
2189 :
2190 : static GEN
2191 8616 : ZXQX_resultant_prime(GEN a, GEN b, GEN dB, long degA, long degB, GEN T, ulong p)
2192 : {
2193 8616 : long dropa = degA - degpol(a), dropb = degB - degpol(b);
2194 : GEN H, dp;
2195 8616 : if (dropa && dropb) return pol0_Flx(T[1]); /* p | lc(A), p | lc(B) */
2196 8616 : H = FlxqX_saferesultant(a, b, T, p);
2197 8616 : if (!H) return NULL;
2198 8616 : if (dropa)
2199 : { /* multiply by ((-1)^deg B lc(B))^(deg A - deg a) */
2200 0 : GEN c = gel(b,degB+2); /* lc(B) */
2201 0 : if (odd(degB)) c = Flx_neg(c, p);
2202 0 : c = Flxq_powu(c, dropa, T, p);
2203 0 : if (!Flx_equal1(c)) H = Flxq_mul(H, c, T, p);
2204 : }
2205 8616 : else if (dropb)
2206 : { /* multiply by lc(A)^(deg B - deg b) */
2207 0 : GEN c = gel(a,degA+2); /* lc(A) */
2208 0 : c = Flxq_powu(c, dropb, T, p);
2209 0 : if (!Flx_equal1(c)) H = Flxq_mul(H, c, T, p);
2210 : }
2211 8616 : dp = dB ? ZX_to_Flx(dB, p): pol1_Flx(T[1]);
2212 8616 : if (!Flx_equal1(dp))
2213 : {
2214 0 : GEN idp = Flxq_invsafe(dp, T, p);
2215 0 : if (!idp) return NULL;
2216 0 : H = Flxq_mul(H, Flxq_powu(idp, degA, T, p), T, p);
2217 : }
2218 8616 : return H;
2219 : }
2220 :
2221 : /* If B=NULL, assume B=A' */
2222 : static GEN
2223 3894 : ZXQX_resultant_slice(GEN A, GEN B, GEN U, GEN dB, GEN P, GEN *mod)
2224 : {
2225 3894 : pari_sp av = avma;
2226 3894 : long degA, degB, i, n = lg(P)-1;
2227 : GEN H, T;
2228 3894 : long v = varn(U), redo = 0;
2229 :
2230 3894 : degA = degpol(A);
2231 3894 : degB = B? degpol(B): degA - 1;
2232 3894 : if (n == 1)
2233 : {
2234 2370 : ulong p = uel(P,1);
2235 2370 : GEN a = ZXX_to_FlxX(A, p, v), b = B? ZXX_to_FlxX(B, p, v): FlxX_deriv(a, p);
2236 2370 : GEN u = ZX_to_Flx(U, p);
2237 2370 : GEN Hp = ZXQX_resultant_prime(a, b, dB, degA, degB, u, p);
2238 2370 : if (!Hp) { set_avma(av); *mod = gen_1; return pol_0(v); }
2239 2370 : Hp = gerepileupto(av, Flx_to_ZX(Hp)); *mod = utoipos(p); return Hp;
2240 : }
2241 1524 : T = ZV_producttree(P);
2242 1524 : A = ZXX_nv_mod_tree(A, P, T, v);
2243 1524 : if (B) B = ZXX_nv_mod_tree(B, P, T, v);
2244 1524 : U = ZX_nv_mod_tree(U, P, T);
2245 1524 : H = cgetg(n+1, t_VEC);
2246 7770 : for(i=1; i <= n; i++)
2247 : {
2248 6246 : ulong p = P[i];
2249 6246 : GEN a = gel(A,i), b = B? gel(B,i): FlxX_deriv(a, p), u = gel(U, i);
2250 6246 : GEN h = ZXQX_resultant_prime(a, b, dB, degA, degB, u, p);
2251 6246 : if (!h)
2252 : {
2253 0 : gel(H,i) = pol_0(v);
2254 0 : P[i] = 1; redo = 1;
2255 : }
2256 : else
2257 6246 : gel(H,i) = h;
2258 : }
2259 1524 : if (redo) T = ZV_producttree(P);
2260 1524 : H = nxV_chinese_center_tree(H, P, T, ZV_chinesetree(P, T));
2261 1524 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2262 : }
2263 :
2264 : GEN
2265 3894 : ZXQX_resultant_worker(GEN P, GEN A, GEN B, GEN T, GEN dB)
2266 : {
2267 3894 : GEN V = cgetg(3, t_VEC);
2268 3894 : if (isintzero(B)) B = NULL;
2269 3894 : if (!signe(dB)) dB = NULL;
2270 3894 : gel(V,1) = ZXQX_resultant_slice(A, B, T, dB, P, &gel(V,2));
2271 3894 : return V;
2272 : }
2273 :
2274 : static ulong
2275 3433 : ZXQX_resultant_bound_i(GEN nf, GEN A, GEN B, GEN (*f)(GEN,GEN,long))
2276 : {
2277 3433 : pari_sp av = avma;
2278 3433 : GEN r, M = nf_L2_bound(nf, NULL, &r);
2279 3433 : long v = nf_get_varn(nf), i, l = lg(r);
2280 3433 : GEN a = cgetg(l, t_COL);
2281 10842 : for (i = 1; i < l; i++)
2282 7409 : gel(a, i) = f(gsubst(A, v, gel(r,i)), gsubst(B, v, gel(r,i)), DEFAULTPREC);
2283 3433 : return gc_ulong(av, (ulong) dbllog2(gmul(M,RgC_fpnorml2(a, DEFAULTPREC))));
2284 : }
2285 : static ulong
2286 3125 : ZXQX_resultant_bound(GEN nf, GEN A, GEN B)
2287 3125 : { return ZXQX_resultant_bound_i(nf, A, B, &RgX_RgXY_ResBound); }
2288 :
2289 : static GEN
2290 56 : _ZXQ_powu(GEN x, ulong u, GEN T)
2291 56 : { return typ(x) == t_INT? powiu(x, u): ZXQ_powu(x, u, T); }
2292 :
2293 : /* Compute Res(A, B/dB) in Z[X]/T, assuming A,B in Z[X,Y], dB in Z or NULL (= 1)
2294 : * If B=NULL, take B = A' and assume deg A > 1 */
2295 : static GEN
2296 3122 : ZXQX_resultant_all(GEN A, GEN B, GEN T, GEN dB, ulong bound)
2297 : {
2298 3122 : pari_sp av = avma;
2299 : forprime_t S;
2300 : GEN H, worker;
2301 3122 : if (B)
2302 : {
2303 63 : long a = degpol(A), b = degpol(B);
2304 63 : if (a < 0 || b < 0) return gen_0;
2305 63 : if (!a) return _ZXQ_powu(gel(A,2), b, T);
2306 63 : if (!b) return _ZXQ_powu(gel(B,2), a, T);
2307 : } else
2308 3059 : if (!bound) B = RgX_deriv(A);
2309 3122 : if (!bound) bound = ZXQX_resultant_bound(nfinit(T, DEFAULTPREC), A, B);
2310 3122 : worker = snm_closure(is_entry("_ZXQX_resultant_worker"),
2311 : mkvec4(A, B? B: gen_0, T, dB? dB: gen_0));
2312 3122 : init_modular_big(&S);
2313 3122 : H = gen_crt("ZXQX_resultant_all", worker, &S, dB, bound, 0, NULL,
2314 : nxV_chinese_center, FpX_center);
2315 3122 : if (DEBUGLEVEL)
2316 0 : err_printf("ZXQX_resultant_all: a priori bound: %lu, a posteriori: %lu\n",
2317 : bound, expi(gsupnorm(H, DEFAULTPREC)));
2318 3122 : return gerepileupto(av, H);
2319 : }
2320 :
2321 : GEN
2322 119 : nfX_resultant(GEN nf, GEN x, GEN y)
2323 : {
2324 119 : pari_sp av = avma;
2325 119 : GEN cx, cy, D, T = nf_get_pol(nf);
2326 119 : long dx = degpol(x), dy = degpol(y);
2327 119 : if (dx < 0 || dy < 0) return gen_0;
2328 119 : x = Q_primitive_part(x, &cx); if (cx) cx = gpowgs(cx, dy);
2329 119 : y = Q_primitive_part(y, &cy); if (cy) cy = gpowgs(cy, dx);
2330 119 : if (!dx) D = _ZXQ_powu(gel(x,2), dy, T);
2331 119 : else if (!dy) D = _ZXQ_powu(gel(y,2), dx, T);
2332 : else
2333 : {
2334 63 : ulong bound = ZXQX_resultant_bound(nf, x, y);
2335 63 : D = ZXQX_resultant_all(x, y, T, NULL, bound);
2336 : }
2337 119 : cx = mul_content(cx, cy); if (cx) D = gmul(D, cx);
2338 119 : return gerepileupto(av, D);
2339 : }
2340 :
2341 : static GEN
2342 217 : to_ZX(GEN a, long v) { return typ(a)==t_INT? scalarpol(a,v): a; }
2343 :
2344 : static GEN
2345 3059 : ZXQX_disc_all(GEN x, GEN T, ulong bound)
2346 : {
2347 3059 : pari_sp av = avma;
2348 3059 : long s, d = degpol(x), v = varn(T);
2349 : GEN l, R;
2350 :
2351 3059 : if (d <= 1) return d == 1? pol_1(v): pol_0(v);
2352 3059 : s = (d & 2) ? -1: 1;
2353 3059 : l = leading_coeff(x);
2354 3059 : R = ZXQX_resultant_all(x, NULL, T, NULL, bound);
2355 3059 : if (!gequal1(l)) R = QXQ_div(R, to_ZX(l,v), T);
2356 3059 : if (s == -1) R = RgX_neg(R);
2357 3059 : return gerepileupto(av, R);
2358 : }
2359 :
2360 : GEN
2361 7 : QX_disc(GEN x)
2362 : {
2363 7 : pari_sp av = avma;
2364 7 : GEN c, d = ZX_disc( Q_primitive_part(x, &c) );
2365 7 : if (c) d = gmul(d, gpowgs(c, 2*degpol(x) - 2));
2366 7 : return gerepileupto(av, d);
2367 : }
2368 :
2369 : GEN
2370 3220 : nfX_disc(GEN nf, GEN x)
2371 : {
2372 3220 : pari_sp av = avma;
2373 3220 : GEN c, D, T = nf_get_pol(nf);
2374 : ulong bound;
2375 3220 : long d = degpol(x), v = varn(T);
2376 3220 : if (d <= 1) return d == 1? pol_1(v): pol_0(v);
2377 3059 : x = Q_primitive_part(x, &c);
2378 3059 : bound = ZXQX_resultant_bound(nf, x, RgX_deriv(x));
2379 3059 : D = ZXQX_disc_all(x, T, bound);
2380 3059 : if (c) D = gmul(D, gpowgs(c, 2*d - 2));
2381 3059 : return gerepileupto(av, D);
2382 : }
2383 :
2384 : GEN
2385 828624 : QXQ_mul(GEN x, GEN y, GEN T)
2386 : {
2387 828624 : GEN dx, nx = Q_primitive_part(x, &dx);
2388 828619 : GEN dy, ny = Q_primitive_part(y, &dy);
2389 828622 : GEN z = ZXQ_mul(nx, ny, T);
2390 828624 : if (dx || dy)
2391 : {
2392 825824 : GEN d = dx ? dy ? gmul(dx, dy): dx : dy;
2393 825824 : if (!gequal1(d)) z = ZX_Q_mul(z, d);
2394 : }
2395 828623 : return z;
2396 : }
2397 :
2398 : GEN
2399 398516 : QXQ_sqr(GEN x, GEN T)
2400 : {
2401 398516 : GEN dx, nx = Q_primitive_part(x, &dx);
2402 398516 : GEN z = ZXQ_sqr(nx, T);
2403 398516 : if (dx)
2404 396780 : z = ZX_Q_mul(z, gsqr(dx));
2405 398516 : return z;
2406 : }
2407 :
2408 : static GEN
2409 210338 : QXQ_inv_slice(GEN A, GEN B, GEN P, GEN *mod)
2410 : {
2411 210338 : pari_sp av = avma;
2412 210338 : long i, n = lg(P)-1, v = varn(A), redo = 0;
2413 : GEN H, T;
2414 210338 : if (n == 1)
2415 : {
2416 164625 : ulong p = uel(P,1);
2417 164625 : GEN a = ZX_to_Flx(A, p), b = ZX_to_Flx(B, p);
2418 164625 : GEN U = Flxq_invsafe(a, b, p);
2419 164625 : if (!U)
2420 : {
2421 24 : set_avma(av);
2422 24 : *mod = gen_1; return pol_0(v);
2423 : }
2424 164601 : H = gerepilecopy(av, Flx_to_ZX(U));
2425 164600 : *mod = utoipos(p); return H;
2426 : }
2427 45713 : T = ZV_producttree(P);
2428 45713 : A = ZX_nv_mod_tree(A, P, T);
2429 45712 : B = ZX_nv_mod_tree(B, P, T);
2430 45710 : H = cgetg(n+1, t_VEC);
2431 226333 : for(i=1; i <= n; i++)
2432 : {
2433 180621 : ulong p = P[i];
2434 180621 : GEN a = gel(A,i), b = gel(B,i);
2435 180621 : GEN U = Flxq_invsafe(a, b, p);
2436 180623 : if (!U)
2437 : {
2438 601 : gel(H,i) = pol_0(v);
2439 601 : P[i] = 1; redo = 1;
2440 : }
2441 : else
2442 180022 : gel(H,i) = U;
2443 : }
2444 45712 : if (redo) T = ZV_producttree(P);
2445 45712 : H = nxV_chinese_center_tree(H, P, T, ZV_chinesetree(P, T));
2446 45713 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2447 : }
2448 :
2449 : GEN
2450 210338 : QXQ_inv_worker(GEN P, GEN A, GEN B)
2451 : {
2452 210338 : GEN V = cgetg(3, t_VEC);
2453 210338 : gel(V,1) = QXQ_inv_slice(A, B, P, &gel(V,2));
2454 210336 : return V;
2455 : }
2456 :
2457 : /* lift(1 / Mod(A,B)). B a ZX, A a scalar or a QX */
2458 : GEN
2459 145297 : QXQ_inv(GEN A, GEN B)
2460 : {
2461 : GEN D, Ap, Bp;
2462 : ulong pp;
2463 145297 : pari_sp av2, av = avma;
2464 : forprime_t S;
2465 145297 : GEN worker, U, H = NULL, mod = gen_1;
2466 : pari_timer ti;
2467 : long k, dA, dB;
2468 145297 : if (is_scalar_t(typ(A))) return scalarpol(ginv(A), varn(B));
2469 : /* A a QX, B a ZX */
2470 145297 : A = Q_primitive_part(A, &D);
2471 145296 : dA = degpol(A); dB= degpol(B);
2472 : /* A, B in Z[X] */
2473 145296 : init_modular_small(&S);
2474 : do {
2475 145296 : pp = u_forprime_next(&S);
2476 145297 : Ap = ZX_to_Flx(A, pp);
2477 145297 : Bp = ZX_to_Flx(B, pp);
2478 145297 : } while (degpol(Ap) != dA || degpol(Bp) != dB);
2479 145297 : if (degpol(Flx_gcd(Ap, Bp, pp)) != 0 && degpol(ZX_gcd(A,B))!=0)
2480 14 : pari_err_INV("QXQ_inv",mkpolmod(A,B));
2481 145282 : worker = snm_closure(is_entry("_QXQ_inv_worker"), mkvec2(A, B));
2482 145283 : av2 = avma;
2483 145283 : for (k = 1; ;k *= 2)
2484 41741 : {
2485 : GEN res, b, N, den;
2486 187024 : gen_inccrt_i("QXQ_inv", worker, NULL, (k+1)>>1, 0, &S, &H, &mod,
2487 : nxV_chinese_center, FpX_center);
2488 187023 : gerepileall(av2, 2, &H, &mod);
2489 187024 : b = sqrti(shifti(mod,-1));
2490 187022 : if (DEBUGLEVEL>5) timer_start(&ti);
2491 187022 : U = FpX_ratlift(H, mod, b, b, NULL);
2492 187024 : if (DEBUGLEVEL>5) timer_printf(&ti,"QXQ_inv: ratlift");
2493 192640 : if (!U) continue;
2494 150899 : N = Q_remove_denom(U, &den); if (!den) den = gen_1;
2495 150898 : res = Flx_rem(Flx_Fl_sub(Flx_mul(Ap, ZX_to_Flx(N,pp), pp),
2496 : umodiu(den, pp), pp), Bp, pp);
2497 150898 : if (degpol(res) >= 0) continue;
2498 145282 : res = ZX_Z_sub(ZX_mul(A, N), den);
2499 145283 : res = ZX_is_monic(B) ? ZX_rem(res, B): RgX_pseudorem(res, B);
2500 145283 : if (DEBUGLEVEL>5) timer_printf(&ti,"QXQ_inv: final check");
2501 145283 : if (degpol(res)<0)
2502 : {
2503 145283 : if (D) U = RgX_Rg_div(U, D);
2504 145283 : return gerepilecopy(av, U);
2505 : }
2506 : }
2507 : }
2508 :
2509 : static GEN
2510 117339 : QXQ_div_slice(GEN A, GEN B, GEN C, GEN P, GEN *mod)
2511 : {
2512 117339 : pari_sp av = avma;
2513 117339 : long i, n = lg(P)-1, v = varn(A), redo = 0;
2514 : GEN H, T;
2515 117339 : if (n == 1)
2516 : {
2517 42963 : ulong p = uel(P,1);
2518 42963 : GEN a = ZX_to_Flx(A, p), b = ZX_to_Flx(B, p), c = ZX_to_Flx(C, p);
2519 42963 : GEN bi = Flxq_invsafe(b, c, p), U;
2520 42963 : if (!bi)
2521 : {
2522 0 : set_avma(av);
2523 0 : *mod = gen_1; return pol_0(v);
2524 : }
2525 42963 : U = Flxq_mul(a, bi, c, p);
2526 42963 : H = gerepilecopy(av, Flx_to_ZX(U));
2527 42963 : *mod = utoipos(p); return H;
2528 : }
2529 74376 : T = ZV_producttree(P);
2530 74376 : A = ZX_nv_mod_tree(A, P, T);
2531 74376 : B = ZX_nv_mod_tree(B, P, T);
2532 74376 : C = ZX_nv_mod_tree(C, P, T);
2533 74376 : H = cgetg(n+1, t_VEC);
2534 327242 : for(i=1; i <= n; i++)
2535 : {
2536 252866 : ulong p = P[i];
2537 252866 : GEN a = gel(A,i), b = gel(B,i), c = gel(C, i);
2538 252866 : GEN bi = Flxq_invsafe(b, c, p);
2539 252866 : if (!bi)
2540 : {
2541 0 : gel(H,i) = pol_0(v);
2542 0 : P[i] = 1; redo = 1;
2543 : }
2544 : else
2545 252866 : gel(H,i) = Flxq_mul(a, bi, c, p);
2546 : }
2547 74376 : if (redo) T = ZV_producttree(P);
2548 74376 : H = nxV_chinese_center_tree(H, P, T, ZV_chinesetree(P, T));
2549 74376 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2550 : }
2551 :
2552 : GEN
2553 117339 : QXQ_div_worker(GEN P, GEN A, GEN B, GEN C)
2554 : {
2555 117339 : GEN V = cgetg(3, t_VEC);
2556 117339 : gel(V,1) = QXQ_div_slice(A, B, C, P, &gel(V,2));
2557 117338 : return V;
2558 : }
2559 :
2560 : /* lift(Mod(A/B, C)). C a ZX, A, B a scalar or a QX */
2561 : GEN
2562 31807 : QXQ_div(GEN A, GEN B, GEN C)
2563 : {
2564 : GEN DA, DB, Ap, Bp, Cp;
2565 : ulong pp;
2566 31807 : pari_sp av2, av = avma;
2567 : forprime_t S;
2568 31807 : GEN worker, U, H = NULL, mod = gen_1;
2569 : pari_timer ti;
2570 : long k, dA, dB, dC;
2571 31807 : if (is_scalar_t(typ(A))) return scalarpol(ginv(A), varn(B));
2572 : /* A a QX, B a ZX */
2573 31807 : A = Q_primitive_part(A, &DA);
2574 31807 : B = Q_primitive_part(B, &DB);
2575 31807 : dA = degpol(A); dB = degpol(B); dC = degpol(C);
2576 : /* A, B in Z[X] */
2577 31807 : init_modular_small(&S);
2578 : do {
2579 31807 : pp = u_forprime_next(&S);
2580 31807 : Ap = ZX_to_Flx(A, pp);
2581 31807 : Bp = ZX_to_Flx(B, pp);
2582 31807 : Cp = ZX_to_Flx(C, pp);
2583 31807 : } while (degpol(Ap) != dA || degpol(Bp) != dB || degpol(Cp) != dC);
2584 31807 : if (degpol(Flx_gcd(Bp, Cp, pp)) != 0 && degpol(ZX_gcd(B,C))!=0)
2585 0 : pari_err_INV("QXQ_div",mkpolmod(B,C));
2586 31807 : worker = snm_closure(is_entry("_QXQ_div_worker"), mkvec3(A, B, C));
2587 31807 : av2 = avma;
2588 31807 : for (k = 1; ;k *= 2)
2589 45518 : {
2590 : GEN res, b, N, den;
2591 77325 : gen_inccrt_i("QXQ_div", worker, NULL, (k+1)>>1, 0, &S, &H, &mod,
2592 : nxV_chinese_center, FpX_center);
2593 77325 : gerepileall(av2, 2, &H, &mod);
2594 77325 : b = sqrti(shifti(mod,-1));
2595 77325 : if (DEBUGLEVEL>5) timer_start(&ti);
2596 77325 : U = FpX_ratlift(H, mod, b, b, NULL);
2597 77325 : if (DEBUGLEVEL>5) timer_printf(&ti,"QXQ_div: ratlift");
2598 87736 : if (!U) continue;
2599 42218 : N = Q_remove_denom(U, &den); if (!den) den = gen_1;
2600 42218 : res = Flx_rem(Flx_sub(Flx_mul(Bp, ZX_to_Flx(N,pp), pp),
2601 : Flx_Fl_mul(Ap, umodiu(den, pp), pp), pp), Cp, pp);
2602 42218 : if (degpol(res) >= 0) continue;
2603 31807 : res = ZX_sub(ZX_mul(B, N), ZX_Z_mul(A,den));
2604 31807 : res = ZX_is_monic(C) ? ZX_rem(res, C): RgX_pseudorem(res, C);
2605 31807 : if (DEBUGLEVEL>5) timer_printf(&ti,"QXQ_div: final check");
2606 31807 : if (degpol(res)<0)
2607 : {
2608 31807 : if (DA && DB) U = RgX_Rg_mul(U, gdiv(DA,DB));
2609 27005 : else if (DA) U = RgX_Rg_mul(U, DA);
2610 15211 : else if (DB) U = RgX_Rg_div(U, DB);
2611 31807 : return gerepilecopy(av, U);
2612 : }
2613 : }
2614 : }
2615 :
2616 : /************************************************************************
2617 : * *
2618 : * ZXQ_minpoly *
2619 : * *
2620 : ************************************************************************/
2621 :
2622 : static GEN
2623 3523 : ZXQ_minpoly_slice(GEN A, GEN B, long d, GEN P, GEN *mod)
2624 : {
2625 3523 : pari_sp av = avma;
2626 3523 : long i, n = lg(P)-1, v = evalvarn(varn(B));
2627 : GEN H, T;
2628 3523 : if (n == 1)
2629 : {
2630 716 : ulong p = uel(P,1);
2631 716 : GEN a = ZX_to_Flx(A, p), b = ZX_to_Flx(B, p);
2632 716 : GEN Hp = Flxq_minpoly(a, b, p);
2633 716 : if (degpol(Hp) != d) { p = 1; Hp = pol0_Flx(v); }
2634 716 : H = gerepileupto(av, Flx_to_ZX(Hp));
2635 716 : *mod = utoipos(p); return H;
2636 : }
2637 2807 : T = ZV_producttree(P);
2638 2807 : A = ZX_nv_mod_tree(A, P, T);
2639 2807 : B = ZX_nv_mod_tree(B, P, T);
2640 2807 : H = cgetg(n+1, t_VEC);
2641 16838 : for(i=1; i <= n; i++)
2642 : {
2643 14031 : ulong p = P[i];
2644 14031 : GEN a = gel(A,i), b = gel(B,i);
2645 14031 : GEN m = Flxq_minpoly(a, b, p);
2646 14031 : if (degpol(m) != d) { P[i] = 1; m = pol0_Flx(v); }
2647 14031 : gel(H, i) = m;
2648 : }
2649 2807 : H = nxV_chinese_center_tree(H, P, T, ZV_chinesetree(P, T));
2650 2807 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2651 : }
2652 :
2653 : GEN
2654 3523 : ZXQ_minpoly_worker(GEN P, GEN A, GEN B, long d)
2655 : {
2656 3523 : GEN V = cgetg(3, t_VEC);
2657 3523 : gel(V,1) = ZXQ_minpoly_slice(A, B, d, P, &gel(V,2));
2658 3523 : return V;
2659 : }
2660 :
2661 : GEN
2662 1701 : ZXQ_minpoly(GEN A, GEN B, long d, ulong bound)
2663 : {
2664 1701 : pari_sp av = avma;
2665 : GEN worker, H, dB;
2666 : forprime_t S;
2667 1701 : B = Q_remove_denom(B, &dB);
2668 1701 : worker = strtoclosure("_ZXQ_minpoly_worker", 3, A, B, stoi(d));
2669 1701 : init_modular_big(&S);
2670 1701 : H = gen_crt("ZXQ_minpoly", worker, &S, dB, bound, 0, NULL,
2671 : nxV_chinese_center, FpX_center_i);
2672 1701 : return gerepilecopy(av, H);
2673 : }
2674 :
2675 : /************************************************************************
2676 : * *
2677 : * ZX_ZXY_resultant *
2678 : * *
2679 : ************************************************************************/
2680 :
2681 : static GEN
2682 173749 : ZX_ZXY_resultant_prime(GEN a, GEN b, ulong dp, ulong p,
2683 : long degA, long degB, long dres, long sX)
2684 : {
2685 173749 : long dropa = degA - degpol(a), dropb = degB - degpol(b);
2686 173749 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2687 173749 : GEN Hp = Flx_FlxY_resultant_polint(a, b, p, pi, dres, sX);
2688 173749 : if (dropa && dropb)
2689 0 : Hp = zero_Flx(sX);
2690 : else {
2691 173749 : if (dropa)
2692 : { /* multiply by ((-1)^deg B lc(B))^(deg A - deg a) */
2693 0 : GEN c = gel(b,degB+2); /* lc(B) */
2694 0 : if (odd(degB)) c = Flx_neg(c, p);
2695 0 : if (!Flx_equal1(c)) {
2696 0 : c = Flx_powu_pre(c, dropa, p, pi);
2697 0 : if (!Flx_equal1(c)) Hp = Flx_mul_pre(Hp, c, p, pi);
2698 : }
2699 : }
2700 173749 : else if (dropb)
2701 : { /* multiply by lc(A)^(deg B - deg b) */
2702 0 : ulong c = uel(a, degA+2); /* lc(A) */
2703 0 : c = Fl_powu(c, dropb, p);
2704 0 : if (c != 1) Hp = Flx_Fl_mul_pre(Hp, c, p, pi);
2705 : }
2706 : }
2707 173749 : if (dp != 1) Hp = Flx_Fl_mul_pre(Hp, Fl_powu_pre(Fl_inv(dp,p), degA, p, pi), p, pi);
2708 173749 : return Hp;
2709 : }
2710 :
2711 : static GEN
2712 69401 : ZX_ZXY_resultant_slice(GEN A, GEN B, GEN dB, long degA, long degB, long dres,
2713 : GEN P, GEN *mod, long sX, long vY)
2714 : {
2715 69401 : pari_sp av = avma;
2716 69401 : long i, n = lg(P)-1;
2717 : GEN H, T, D;
2718 69401 : if (n == 1)
2719 : {
2720 40137 : ulong p = uel(P,1);
2721 40137 : ulong dp = dB ? umodiu(dB, p): 1;
2722 40137 : GEN a = ZX_to_Flx(A, p), b = ZXX_to_FlxX(B, p, vY);
2723 40137 : GEN Hp = ZX_ZXY_resultant_prime(a, b, dp, p, degA, degB, dres, sX);
2724 40137 : H = gerepileupto(av, Flx_to_ZX(Hp));
2725 40137 : *mod = utoipos(p); return H;
2726 : }
2727 29264 : T = ZV_producttree(P);
2728 29264 : A = ZX_nv_mod_tree(A, P, T);
2729 29264 : B = ZXX_nv_mod_tree(B, P, T, vY);
2730 29264 : D = dB ? Z_ZV_mod_tree(dB, P, T): NULL;
2731 29264 : H = cgetg(n+1, t_VEC);
2732 117529 : for(i=1; i <= n; i++)
2733 : {
2734 88265 : ulong p = P[i];
2735 88265 : GEN a = gel(A,i), b = gel(B,i);
2736 88265 : ulong dp = D ? uel(D, i): 1;
2737 88265 : gel(H,i) = ZX_ZXY_resultant_prime(a, b, dp, p, degA, degB, dres, sX);
2738 : }
2739 29264 : H = nxV_chinese_center_tree(H, P, T, ZV_chinesetree(P, T));
2740 29264 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2741 : }
2742 :
2743 : GEN
2744 69401 : ZX_ZXY_resultant_worker(GEN P, GEN A, GEN B, GEN dB, GEN v)
2745 : {
2746 69401 : GEN V = cgetg(3, t_VEC);
2747 69401 : if (isintzero(dB)) dB = NULL;
2748 69401 : gel(V,1) = ZX_ZXY_resultant_slice(A, B, dB, v[1], v[2], v[3], P, &gel(V,2), v[4], v[5]);
2749 69401 : return V;
2750 : }
2751 :
2752 : GEN
2753 60095 : ZX_ZXY_resultant(GEN A, GEN B)
2754 : {
2755 60095 : pari_sp av = avma;
2756 : forprime_t S;
2757 : ulong bound;
2758 60095 : long v = fetch_var_higher();
2759 60095 : long degA = degpol(A), degB, dres = degA * degpol(B);
2760 60095 : long vX = varn(B), vY = varn(A); /* assume vY has lower priority */
2761 60095 : long sX = evalvarn(vX);
2762 : GEN worker, H, dB;
2763 60095 : B = Q_remove_denom(B, &dB);
2764 60095 : if (!dB) B = leafcopy(B);
2765 60095 : A = leafcopy(A); setvarn(A,v);
2766 60095 : B = swap_vars(B, vY); setvarn(B,v); degB = degpol(B);
2767 60095 : bound = ZX_ZXY_ResBound(A, B, dB);
2768 60095 : if (DEBUGLEVEL>4) err_printf("bound for resultant coeffs: 2^%ld\n",bound);
2769 120190 : worker = snm_closure(is_entry("_ZX_ZXY_resultant_worker"),
2770 60095 : mkvec4(A, B, dB? dB: gen_0,
2771 : mkvecsmall5(degA, degB, dres, sX, vY)));
2772 60095 : init_modular_big(&S);
2773 60095 : H = gen_crt("ZX_ZXY_resultant_all", worker, &S, dB, bound, 0, NULL,
2774 : nxV_chinese_center, FpX_center_i);
2775 60095 : setvarn(H, vX); (void)delete_var();
2776 60095 : return gerepilecopy(av, H);
2777 : }
2778 :
2779 : static long
2780 40523 : ZX_ZXY_rnfequation_lambda(GEN A, GEN B0, long lambda)
2781 : {
2782 40523 : pari_sp av = avma;
2783 40523 : long degA = degpol(A), degB, dres = degA*degpol(B0);
2784 40523 : long v = fetch_var_higher();
2785 40523 : long vX = varn(B0), vY = varn(A); /* assume vY has lower priority */
2786 40523 : long sX = evalvarn(vX);
2787 : GEN dB, B, a, b, Hp;
2788 : forprime_t S;
2789 :
2790 40523 : B0 = Q_remove_denom(B0, &dB);
2791 40523 : if (!dB) B0 = leafcopy(B0);
2792 40523 : A = leafcopy(A);
2793 40523 : B = B0;
2794 40523 : setvarn(A,v);
2795 45347 : INIT:
2796 45347 : if (lambda) B = RgX_translate(B0, monomial(stoi(lambda), 1, vY));
2797 45347 : B = swap_vars(B, vY); setvarn(B,v);
2798 : /* B0(lambda v + x, v) */
2799 45347 : if (DEBUGLEVEL>4) err_printf("Trying lambda = %ld\n", lambda);
2800 :
2801 45347 : degB = degpol(B);
2802 45347 : init_modular_big(&S);
2803 : while (1)
2804 0 : {
2805 45347 : ulong p = u_forprime_next(&S);
2806 45347 : ulong dp = dB ? umodiu(dB, p): 1;
2807 45347 : if (!dp) continue;
2808 45347 : a = ZX_to_Flx(A, p);
2809 45347 : b = ZXX_to_FlxX(B, p, v);
2810 45347 : Hp = ZX_ZXY_resultant_prime(a, b, dp, p, degA, degB, dres, sX);
2811 45347 : if (degpol(Hp) != dres) continue;
2812 45347 : if (dp != 1) Hp = Flx_Fl_mul(Hp, Fl_powu(Fl_inv(dp,p), degA, p), p);
2813 45347 : if (!Flx_is_squarefree(Hp, p)) { lambda = next_lambda(lambda); goto INIT; }
2814 40523 : if (DEBUGLEVEL>4) err_printf("Final lambda = %ld\n", lambda);
2815 40523 : (void)delete_var(); return gc_long(av,lambda);
2816 : }
2817 : }
2818 :
2819 : GEN
2820 41461 : ZX_ZXY_rnfequation(GEN A, GEN B, long *lambda)
2821 : {
2822 41461 : if (lambda)
2823 : {
2824 40523 : *lambda = ZX_ZXY_rnfequation_lambda(A, B, *lambda);
2825 40523 : if (*lambda) B = RgX_translate(B, monomial(stoi(*lambda), 1, varn(A)));
2826 : }
2827 41461 : return ZX_ZXY_resultant(A,B);
2828 : }
2829 :
2830 : static GEN
2831 10346 : ZX_composedsum_slice(GEN A, GEN B, GEN P, GEN *mod)
2832 : {
2833 10346 : pari_sp av = avma;
2834 10346 : long i, n = lg(P)-1;
2835 : GEN H, T;
2836 10346 : if (n == 1)
2837 : {
2838 9844 : ulong p = uel(P,1);
2839 9844 : GEN a = ZX_to_Flx(A, p), b = ZX_to_Flx(B, p);
2840 9845 : GEN Hp = Flx_composedsum(a, b, p);
2841 9846 : H = gerepileupto(av, Flx_to_ZX(Hp));
2842 9849 : *mod = utoipos(p); return H;
2843 : }
2844 502 : T = ZV_producttree(P);
2845 502 : A = ZX_nv_mod_tree(A, P, T);
2846 502 : B = ZX_nv_mod_tree(B, P, T);
2847 502 : H = cgetg(n+1, t_VEC);
2848 4526 : for(i=1; i <= n; i++)
2849 : {
2850 4024 : ulong p = P[i];
2851 4024 : GEN a = gel(A,i), b = gel(B,i);
2852 4024 : gel(H,i) = Flx_composedsum(a, b, p);
2853 : }
2854 502 : H = nxV_chinese_center_tree(H, P, T, ZV_chinesetree(P, T));
2855 502 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2856 : }
2857 :
2858 : GEN
2859 10347 : ZX_composedsum_worker(GEN P, GEN A, GEN B)
2860 : {
2861 10347 : GEN V = cgetg(3, t_VEC);
2862 10346 : gel(V,1) = ZX_composedsum_slice(A, B, P, &gel(V,2));
2863 10350 : return V;
2864 : }
2865 :
2866 : static GEN
2867 10084 : ZX_composedsum_i(GEN A, GEN B, GEN lead)
2868 : {
2869 10084 : pari_sp av = avma;
2870 : forprime_t S;
2871 : ulong bound;
2872 : GEN H, worker, mod;
2873 10084 : if (degpol(A) < degpol(B)) swap(A, B);
2874 10083 : if (!lead) lead = mulii(leading_coeff(A),leading_coeff(B));
2875 10083 : bound = ZX_ZXY_ResBound_1(A, B);
2876 10087 : worker = snm_closure(is_entry("_ZX_composedsum_worker"), mkvec2(A,B));
2877 10084 : init_modular_big(&S);
2878 10083 : H = gen_crt("ZX_composedsum", worker, &S, lead, bound, 0, &mod,
2879 : nxV_chinese_center, FpX_center);
2880 10085 : return gerepileupto(av, H);
2881 : }
2882 :
2883 : static long
2884 9698 : ZX_compositum_lambda(GEN A, GEN B, GEN lead, long lambda)
2885 : {
2886 9698 : pari_sp av = avma;
2887 : forprime_t S;
2888 : ulong p;
2889 9698 : init_modular_big(&S);
2890 9699 : p = u_forprime_next(&S);
2891 : while (1)
2892 112 : {
2893 : GEN Hp, a;
2894 9811 : if (DEBUGLEVEL>4) err_printf("Trying lambda = %ld\n", lambda);
2895 9811 : if (lead && dvdiu(lead,p)) { p = u_forprime_next(&S); continue; }
2896 9804 : a = ZX_to_Flx(ZX_rescale(A, stoi(-lambda)), p);
2897 9804 : Hp = Flx_composedsum(a, ZX_to_Flx(B, p), p);
2898 9802 : if (!Flx_is_squarefree(Hp, p)) { lambda = next_lambda(lambda); continue; }
2899 9695 : if (DEBUGLEVEL>4) err_printf("Final lambda = %ld\n", lambda);
2900 9695 : return gc_long(av, lambda);
2901 : }
2902 : }
2903 :
2904 : GEN
2905 9698 : ZX_compositum(GEN A, GEN B, long *lambda)
2906 : {
2907 9698 : GEN lead = mulii(leading_coeff(A),leading_coeff(B));
2908 9698 : if (lambda)
2909 : {
2910 9698 : *lambda = ZX_compositum_lambda(A, B, lead, *lambda);
2911 9695 : A = ZX_rescale(A, stoi(-*lambda));
2912 : }
2913 9699 : return ZX_composedsum_i(A, B, lead);
2914 : }
2915 :
2916 : GEN
2917 385 : ZX_composedsum(GEN A, GEN B)
2918 385 : { return ZX_composedsum_i(A, B, NULL); }
2919 :
2920 : static GEN
2921 352 : ZXQX_composedsum_slice(GEN A, GEN B, GEN C, GEN P, GEN *mod)
2922 : {
2923 352 : pari_sp av = avma;
2924 352 : long i, n = lg(P)-1, dC = degpol(C), v = varn(C);
2925 : GEN H, T;
2926 352 : if (n == 1)
2927 : {
2928 174 : ulong p = uel(P,1);
2929 174 : GEN a = ZXX_to_FlxX(A, p, v), b = ZXX_to_FlxX(B, p, v);
2930 174 : GEN c = ZX_to_Flx(C, p);
2931 174 : GEN Hp = FlxX_to_Flm(FlxqX_composedsum(a, b, c, p), dC);
2932 174 : H = gerepileupto(av, Flm_to_ZM(Hp));
2933 174 : *mod = utoipos(p); return H;
2934 : }
2935 178 : T = ZV_producttree(P);
2936 178 : A = ZXX_nv_mod_tree(A, P, T, v);
2937 178 : B = ZXX_nv_mod_tree(B, P, T, v);
2938 178 : C = ZX_nv_mod_tree(C, P, T);
2939 178 : H = cgetg(n+1, t_VEC);
2940 660 : for(i=1; i <= n; i++)
2941 : {
2942 482 : ulong p = P[i];
2943 482 : GEN a = gel(A,i), b = gel(B,i), c = gel(C,i);
2944 482 : gel(H,i) = FlxX_to_Flm(FlxqX_composedsum(a, b, c, p), dC);
2945 : }
2946 178 : H = nmV_chinese_center_tree_seq(H, P, T, ZV_chinesetree(P, T));
2947 178 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2948 : }
2949 :
2950 : GEN
2951 352 : ZXQX_composedsum_worker(GEN P, GEN A, GEN B, GEN C)
2952 : {
2953 352 : GEN V = cgetg(3, t_VEC);
2954 352 : gel(V,1) = ZXQX_composedsum_slice(A, B, C, P, &gel(V,2));
2955 352 : return V;
2956 : }
2957 :
2958 : static GEN
2959 308 : ZXQX_composedsum(GEN A, GEN B, GEN T, ulong bound)
2960 : {
2961 308 : pari_sp av = avma;
2962 : forprime_t S;
2963 : GEN H, worker, mod;
2964 308 : GEN lead = mulii(Q_content(leading_coeff(A)), Q_content(leading_coeff(B)));
2965 308 : worker = snm_closure(is_entry("_ZXQX_composedsum_worker")
2966 : , mkvec3(A,B,T));
2967 308 : init_modular_big(&S);
2968 308 : H = gen_crt("ZXQX_composedsum", worker, &S, lead, bound, 0, &mod,
2969 : nmV_chinese_center, FpM_center);
2970 308 : if (DEBUGLEVEL > 4)
2971 0 : err_printf("nfcompositum: a priori bound: %lu, a posteriori: %lu\n",
2972 : bound, expi(gsupnorm(H, DEFAULTPREC)));
2973 308 : return gerepilecopy(av, RgM_to_RgXX(H, varn(A), varn(T)));
2974 : }
2975 :
2976 : static long
2977 308 : ZXQX_composedsum_bound(GEN nf, GEN A, GEN B)
2978 308 : { return ZXQX_resultant_bound_i(nf, A, B, &RgX_RgXY_ResBound_1); }
2979 :
2980 : GEN
2981 308 : nf_direct_compositum(GEN nf, GEN A, GEN B)
2982 : {
2983 308 : ulong bnd = ZXQX_composedsum_bound(nf, A, B);
2984 308 : return ZXQX_composedsum(A, B, nf_get_pol(nf), bnd);
2985 : }
2986 :
2987 : /************************************************************************
2988 : * *
2989 : * IRREDUCIBLE POLYNOMIAL / Fp *
2990 : * *
2991 : ************************************************************************/
2992 :
2993 : /* irreducible (unitary) polynomial of degree n over Fp */
2994 : GEN
2995 0 : ffinit_rand(GEN p,long n)
2996 : {
2997 0 : for(;;) {
2998 0 : pari_sp av = avma;
2999 0 : GEN pol = ZX_add(pol_xn(n, 0), random_FpX(n-1,0, p));
3000 0 : if (FpX_is_irred(pol, p)) return pol;
3001 0 : set_avma(av);
3002 : }
3003 : }
3004 :
3005 : /* return an extension of degree 2^l of F_2, assume l > 0
3006 : * Not stack clean. */
3007 : static GEN
3008 604 : ffinit_Artin_Schreier_2(long l)
3009 : {
3010 : GEN Q, T, S;
3011 : long i, v;
3012 :
3013 604 : if (l == 1) return mkvecsmall4(0,1,1,1); /*x^2 + x + 1*/
3014 555 : v = fetch_var_higher();
3015 555 : S = mkvecsmall5(0, 0, 0, 1, 1); /* y(y^2 + y) */
3016 555 : Q = mkpoln(3, pol1_Flx(0), pol1_Flx(0), S); /* x^2 + x + y(y^2+y) */
3017 555 : setvarn(Q, v);
3018 :
3019 : /* x^4+x+1, irred over F_2, minimal polynomial of a root of Q */
3020 555 : T = mkvecsmalln(6,evalvarn(v),1UL,1UL,0UL,0UL,1UL);
3021 : /* Q = x^2 + x + a(y) irred. over K = F2[y] / (T(y))
3022 : * ==> x^2 + x + a(y) b irred. over K for any root b of Q
3023 : * ==> x^2 + x + (b^2+b)b */
3024 3068 : for (i=2; i<l; i++) T = Flx_FlxY_resultant(T, Q, 2); /* minpoly(b) / F2*/
3025 556 : (void)delete_var(); T[1] = 0; return T;
3026 : }
3027 :
3028 : /* return an extension of degree p^l of F_p, assume l > 0
3029 : * Not stack clean. */
3030 : GEN
3031 961 : ffinit_Artin_Schreier(ulong p, long l)
3032 : {
3033 : long i, v;
3034 : GEN Q, R, S, T, xp;
3035 961 : if (p==2) return ffinit_Artin_Schreier_2(l);
3036 357 : xp = polxn_Flx(p,0); /* x^p */
3037 357 : T = Flx_sub(xp, mkvecsmall3(0,1,1),p); /* x^p - x - 1 */
3038 357 : if (l == 1) return T;
3039 :
3040 7 : v = evalvarn(fetch_var_higher());
3041 7 : xp[1] = v;
3042 7 : R = Flx_sub(polxn_Flx(2*p-1,0), polxn_Flx(p,0),p);
3043 7 : S = Flx_sub(xp, polx_Flx(0), p);
3044 7 : Q = FlxX_Flx_sub(Flx_to_FlxX(S, v), R, p); /* x^p - x - (y^(2p-1)-y^p) */
3045 14 : for (i = 2; i <= l; ++i) T = Flx_FlxY_resultant(T, Q, p);
3046 7 : (void)delete_var(); T[1] = 0; return T;
3047 : }
3048 :
3049 : static long
3050 147966 : flinit_check(ulong p, long n, long l)
3051 : {
3052 : ulong q;
3053 147966 : if (!uisprime(n)) return 0;
3054 101222 : q = p % n; if (!q) return 0;
3055 98737 : return ugcd((n-1)/Fl_order(q, n-1, n), l) == 1;
3056 : }
3057 :
3058 : static GEN
3059 31734 : flinit(ulong p, long l)
3060 : {
3061 31734 : ulong n = 1+l;
3062 95851 : while (!flinit_check(p,n,l)) n += l;
3063 31734 : if (DEBUGLEVEL>=4) err_printf("FFInit: using polsubcyclo(%ld, %ld)\n",n,l);
3064 31734 : return ZX_to_Flx(polsubcyclo(n,l,0), p);
3065 : }
3066 :
3067 : static GEN
3068 28870 : ffinit_fact_Flx(ulong p, long n)
3069 : {
3070 28870 : GEN P, F = factoru_pow(n), Fp = gel(F,1), Fe = gel(F,2), Fm = gel(F,3);
3071 28869 : long i, l = lg(Fm);
3072 28869 : P = cgetg(l, t_VEC);
3073 61565 : for (i = 1; i < l; i++)
3074 32695 : gel(P,i) = p==uel(Fp,i) ? ffinit_Artin_Schreier(p, Fe[i])
3075 32695 : : flinit(p, uel(Fm,i));
3076 28870 : return FlxV_composedsum(P, p);
3077 : }
3078 :
3079 : static GEN
3080 52122 : init_Flxq_i(ulong p, long n, long sv)
3081 : {
3082 : GEN P;
3083 52122 : if (!odd(p) && p != 2) pari_err_PRIME("ffinit", utoi(p));
3084 52115 : if (n == 1) return polx_Flx(sv);
3085 52115 : if (flinit_check(p, n+1, n))
3086 : {
3087 23245 : P = const_vecsmall(n+2,1);
3088 23245 : P[1] = sv; return P;
3089 : }
3090 28870 : P = ffinit_fact_Flx(p,n);
3091 28870 : P[1] = sv; return P;
3092 : }
3093 :
3094 : GEN
3095 0 : init_Flxq(ulong p, long n, long v)
3096 : {
3097 0 : pari_sp av = avma;
3098 0 : return gerepileupto(av, init_Flxq_i(p, n, v));
3099 : }
3100 :
3101 : /* check if polsubcyclo(n,l,0) is irreducible modulo p */
3102 : static long
3103 7185 : fpinit_check(GEN p, long n, long l)
3104 : {
3105 : ulong q;
3106 7185 : if (!uisprime(n)) return 0;
3107 4450 : q = umodiu(p,n); if (!q) return 0;
3108 4450 : return ugcd((n-1)/Fl_order(q, n-1, n), l) == 1;
3109 : }
3110 :
3111 : /* let k=2 if p%4==1, and k=4 else and assume k*p does not divide l.
3112 : * Return an irreducible polynomial of degree l over F_p.
3113 : * Variant of Adleman and Lenstra "Finding irreducible polynomials over
3114 : * finite fields", ACM, 1986 (5) 350--355.
3115 : * Not stack clean */
3116 : static GEN
3117 1653 : fpinit(GEN p, long l)
3118 : {
3119 1653 : ulong n = 1+l;
3120 5202 : while (!fpinit_check(p,n,l)) n += l;
3121 1653 : if (DEBUGLEVEL>=4) err_printf("FFInit: using polsubcyclo(%ld, %ld)\n",n,l);
3122 1653 : return FpX_red(polsubcyclo(n,l,0),p);
3123 : }
3124 :
3125 : static GEN
3126 1574 : ffinit_fact(GEN p, long n)
3127 : {
3128 1574 : GEN P, F = factoru_pow(n), Fp = gel(F,1), Fe = gel(F,2), Fm = gel(F,3);
3129 1574 : long i, l = lg(Fm);
3130 1574 : P = cgetg(l, t_VEC);
3131 3227 : for (i = 1; i < l; ++i)
3132 3306 : gel(P,i) = absequaliu(p, Fp[i]) ?
3133 0 : Flx_to_ZX(ffinit_Artin_Schreier(Fp[i], Fe[i]))
3134 1653 : : fpinit(p, Fm[i]);
3135 1574 : return FpXV_composedsum(P, p);
3136 : }
3137 :
3138 : static GEN
3139 54371 : init_Fq_i(GEN p, long n, long v)
3140 : {
3141 : GEN P;
3142 54371 : if (n <= 0) pari_err_DOMAIN("ffinit", "degree", "<=", gen_0, stoi(n));
3143 54371 : if (typ(p) != t_INT) pari_err_TYPE("ffinit",p);
3144 54371 : if (cmpiu(p, 2) < 0) pari_err_PRIME("ffinit",p);
3145 54364 : if (v < 0) v = 0;
3146 54364 : if (n == 1) return pol_x(v);
3147 54112 : if (lgefint(p) == 3)
3148 52122 : return Flx_to_ZX(init_Flxq_i(p[2], n, evalvarn(v)));
3149 1990 : if (!mpodd(p)) pari_err_PRIME("ffinit", p);
3150 1983 : if (fpinit_check(p, n+1, n)) return polcyclo(n+1, v);
3151 1574 : P = ffinit_fact(p,n);
3152 1574 : setvarn(P, v); return P;
3153 : }
3154 : GEN
3155 53804 : init_Fq(GEN p, long n, long v)
3156 : {
3157 53804 : pari_sp av = avma;
3158 53804 : return gerepileupto(av, init_Fq_i(p, n, v));
3159 : }
3160 : GEN
3161 567 : ffinit(GEN p, long n, long v)
3162 : {
3163 567 : pari_sp av = avma;
3164 567 : return gerepileupto(av, FpX_to_mod(init_Fq_i(p, n, v), p));
3165 : }
3166 :
3167 : GEN
3168 3178 : ffnbirred(GEN p, long n)
3169 : {
3170 3178 : pari_sp av = avma;
3171 3178 : GEN s = powiu(p,n), F = factoru(n), D = divisorsu_moebius(gel(F, 1));
3172 3178 : long j, l = lg(D);
3173 6797 : for (j = 2; j < l; j++) /* skip d = 1 */
3174 : {
3175 3619 : long md = D[j]; /* mu(d) * d, d squarefree */
3176 3619 : GEN pd = powiu(p, n / labs(md)); /* p^{n/d} */
3177 3619 : s = md > 0? addii(s, pd): subii(s,pd);
3178 : }
3179 3178 : return gerepileuptoint(av, diviuexact(s, n));
3180 : }
3181 :
3182 : GEN
3183 616 : ffsumnbirred(GEN p, long n)
3184 : {
3185 616 : pari_sp av = avma, av2;
3186 616 : GEN q, t = p, v = vecfactoru_i(1, n);
3187 : long i;
3188 616 : q = cgetg(n+1,t_VEC); gel(q,1) = p;
3189 1764 : for (i=2; i<=n; i++) gel(q,i) = mulii(gel(q,i-1), p);
3190 616 : av2 = avma;
3191 1764 : for (i=2; i<=n; i++)
3192 : {
3193 1148 : GEN s = gel(q,i), F = gel(v,i), D = divisorsu_moebius(gel(F,1));
3194 1148 : long j, l = lg(D);
3195 2534 : for (j = 2; j < l; j++) /* skip 1 */
3196 : {
3197 1386 : long md = D[j];
3198 1386 : GEN pd = gel(q, i / labs(md)); /* p^{i/d} */
3199 1386 : s = md > 0? addii(s, pd): subii(s, pd);
3200 : }
3201 1148 : t = gerepileuptoint(av2, addii(t, diviuexact(s, i)));
3202 : }
3203 616 : return gerepileuptoint(av, t);
3204 : }
3205 :
3206 : GEN
3207 140 : ffnbirred0(GEN p, long n, long flag)
3208 : {
3209 140 : if (typ(p) != t_INT) pari_err_TYPE("ffnbirred", p);
3210 140 : if (n <= 0) pari_err_DOMAIN("ffnbirred", "degree", "<=", gen_0, stoi(n));
3211 140 : switch(flag)
3212 : {
3213 70 : case 0: return ffnbirred(p, n);
3214 70 : case 1: return ffsumnbirred(p, n);
3215 : }
3216 0 : pari_err_FLAG("ffnbirred");
3217 : return NULL; /* LCOV_EXCL_LINE */
3218 : }
3219 :
3220 : static void
3221 2261 : checkmap(GEN m, const char *s)
3222 : {
3223 2261 : if (typ(m)!=t_VEC || lg(m)!=3 || typ(gel(m,1))!=t_FFELT)
3224 0 : pari_err_TYPE(s,m);
3225 2261 : }
3226 :
3227 : GEN
3228 189 : ffembed(GEN a, GEN b)
3229 : {
3230 189 : pari_sp av = avma;
3231 189 : GEN p, Ta, Tb, g, r = NULL;
3232 189 : if (typ(a)!=t_FFELT) pari_err_TYPE("ffembed",a);
3233 189 : if (typ(b)!=t_FFELT) pari_err_TYPE("ffembed",b);
3234 189 : p = FF_p_i(a); g = FF_gen(a);
3235 189 : if (!equalii(p, FF_p_i(b))) pari_err_MODULUS("ffembed",a,b);
3236 189 : Ta = FF_mod(a);
3237 189 : Tb = FF_mod(b);
3238 189 : if (degpol(Tb)%degpol(Ta)!=0)
3239 7 : pari_err_DOMAIN("ffembed",GENtostr_raw(a),"is not a subfield of",b,a);
3240 182 : r = gel(FFX_roots(Ta, b), 1);
3241 182 : return gerepilecopy(av, mkvec2(g,r));
3242 : }
3243 :
3244 : GEN
3245 91 : ffextend(GEN a, GEN P, long v)
3246 : {
3247 91 : pari_sp av = avma;
3248 : long n;
3249 : GEN p, T, R, g, m;
3250 91 : if (typ(a)!=t_FFELT) pari_err_TYPE("ffextend",a);
3251 91 : T = a; p = FF_p_i(a);
3252 91 : if (typ(P)!=t_POL || !RgX_is_FpXQX(P,&T,&p)) pari_err_TYPE("ffextend", P);
3253 49 : if (!FF_samefield(a, T)) pari_err_MODULUS("ffextend",a,T);
3254 49 : if (v < 0) v = varn(P);
3255 49 : n = FF_f(T) * degpol(P); R = ffinit(p, n, v); g = ffgen(R, v);
3256 49 : m = ffembed(a, g);
3257 49 : R = FFX_roots(ffmap(m, P),g);
3258 49 : return gerepilecopy(av, mkvec2(gel(R,1), m));
3259 : }
3260 :
3261 : GEN
3262 42 : fffrobenius(GEN a, long n)
3263 : {
3264 42 : if (typ(a)!=t_FFELT) pari_err_TYPE("fffrobenius",a);
3265 42 : retmkvec2(FF_gen(a), FF_Frobenius(a, n));
3266 : }
3267 :
3268 : GEN
3269 133 : ffinvmap(GEN m)
3270 : {
3271 133 : pari_sp av = avma;
3272 : long i, l;
3273 133 : GEN T, F, a, g, r, f = NULL;
3274 133 : checkmap(m, "ffinvmap");
3275 133 : a = gel(m,1); r = gel(m,2);
3276 133 : if (typ(r) != t_FFELT)
3277 7 : pari_err_TYPE("ffinvmap", m);
3278 126 : g = FF_gen(a);
3279 126 : T = FF_mod(r);
3280 126 : F = gel(FFX_factor(T, a), 1);
3281 126 : l = lg(F);
3282 490 : for(i=1; i<l; i++)
3283 : {
3284 490 : GEN s = FFX_rem(FF_to_FpXQ_i(r), gel(F, i), a);
3285 490 : if (degpol(s)==0 && gequal(constant_coeff(s),g)) { f = gel(F, i); break; }
3286 : }
3287 126 : if (f==NULL) pari_err_TYPE("ffinvmap", m);
3288 126 : if (degpol(f)==1) f = FF_neg_i(gel(f,2));
3289 126 : return gerepilecopy(av, mkvec2(FF_gen(r),f));
3290 : }
3291 :
3292 : static GEN
3293 1260 : ffpartmapimage(const char *s, GEN r)
3294 : {
3295 1260 : GEN a = NULL, p = NULL;
3296 1260 : if (typ(r)==t_POL && degpol(r) >= 1
3297 1260 : && RgX_is_FpXQX(r,&a,&p) && a && typ(a)==t_FFELT) return a;
3298 0 : pari_err_TYPE(s, r);
3299 : return NULL; /* LCOV_EXCL_LINE */
3300 : }
3301 :
3302 : static GEN
3303 2709 : ffeltmap_i(GEN m, GEN x)
3304 : {
3305 2709 : GEN r = gel(m,2);
3306 2709 : if (!FF_samefield(x, gel(m,1)))
3307 84 : pari_err_DOMAIN("ffmap","m","domain does not contain", x, r);
3308 2625 : if (typ(r)==t_FFELT)
3309 1659 : return FF_map(r, x);
3310 : else
3311 966 : return FFX_preimage(x, r, ffpartmapimage("ffmap", r));
3312 : }
3313 :
3314 : static GEN
3315 4459 : ffmap_i(GEN m, GEN x)
3316 : {
3317 : GEN y;
3318 4459 : long i, lx, tx = typ(x);
3319 4459 : switch(tx)
3320 : {
3321 2541 : case t_FFELT:
3322 2541 : return ffeltmap_i(m, x);
3323 1267 : case t_POL: case t_RFRAC: case t_SER:
3324 : case t_VEC: case t_COL: case t_MAT:
3325 1267 : y = cgetg_copy(x, &lx);
3326 1988 : for (i=1; i<lontyp[tx]; i++) y[i] = x[1];
3327 4564 : for (i=lontyp[tx]; i<lx; i++)
3328 : {
3329 3339 : GEN yi = ffmap_i(m, gel(x,i));
3330 3297 : if (!yi) return NULL;
3331 3297 : gel(y,i) = yi;
3332 : }
3333 1225 : return y;
3334 : }
3335 651 : return gcopy(x);
3336 : }
3337 :
3338 : GEN
3339 1036 : ffmap(GEN m, GEN x)
3340 : {
3341 1036 : pari_sp ltop = avma;
3342 : GEN y;
3343 1036 : checkmap(m, "ffmap");
3344 1036 : y = ffmap_i(m, x);
3345 1036 : if (y) return y;
3346 42 : set_avma(ltop); return cgetg(1,t_VEC);
3347 : }
3348 :
3349 : static GEN
3350 252 : ffeltmaprel_i(GEN m, GEN x)
3351 : {
3352 252 : GEN g = gel(m,1), r = gel(m,2);
3353 252 : if (!FF_samefield(x, g))
3354 0 : pari_err_DOMAIN("ffmap","m","domain does not contain", x, r);
3355 252 : if (typ(r)==t_FFELT)
3356 84 : retmkpolmod(FF_map(r, x), pol_x(FF_var(g)));
3357 : else
3358 168 : retmkpolmod(FFX_preimagerel(x, r, ffpartmapimage("ffmap", r)), gcopy(r));
3359 : }
3360 :
3361 : static GEN
3362 252 : ffmaprel_i(GEN m, GEN x)
3363 : {
3364 : GEN y;
3365 252 : long i, lx, tx = typ(x);
3366 252 : switch(tx)
3367 : {
3368 252 : case t_FFELT:
3369 252 : return ffeltmaprel_i(m, x);
3370 0 : case t_POL: case t_RFRAC: case t_SER:
3371 : case t_VEC: case t_COL: case t_MAT:
3372 0 : y = cgetg_copy(x, &lx);
3373 0 : for (i=1; i<lontyp[tx]; i++) y[i] = x[1];
3374 0 : for (i=lontyp[tx]; i<lx; i++)
3375 0 : gel(y,i) = ffmaprel_i(m, gel(x,i));
3376 0 : return y;
3377 : }
3378 0 : return gcopy(x);
3379 : }
3380 :
3381 : GEN
3382 252 : ffmaprel(GEN m, GEN x)
3383 : {
3384 252 : checkmap(m, "ffmaprel");
3385 252 : return ffmaprel_i(m, x);
3386 : }
3387 :
3388 : static void
3389 84 : err_compo(GEN m, GEN n)
3390 84 : { pari_err_DOMAIN("ffcompomap","m","domain does not contain codomain of",n,m); }
3391 :
3392 : GEN
3393 420 : ffcompomap(GEN m, GEN n)
3394 : {
3395 420 : pari_sp av = avma;
3396 420 : GEN g = gel(n,1), r, m2, n2;
3397 420 : checkmap(m, "ffcompomap");
3398 420 : checkmap(n, "ffcompomap");
3399 420 : m2 = gel(m,2); n2 = gel(n,2);
3400 420 : switch((typ(m2)==t_POL)|((typ(n2)==t_POL)<<1))
3401 : {
3402 84 : case 0:
3403 84 : if (!FF_samefield(gel(m,1),n2)) err_compo(m,n);
3404 42 : r = FF_map(gel(m,2), n2);
3405 42 : break;
3406 84 : case 2:
3407 84 : r = ffmap_i(m, n2);
3408 42 : if (lg(r) == 1) err_compo(m,n);
3409 42 : break;
3410 168 : case 1:
3411 168 : r = ffeltmap_i(m, n2);
3412 126 : if (!r)
3413 : {
3414 : GEN a, A, R, M;
3415 : long dm, dn;
3416 42 : a = ffpartmapimage("ffcompomap",m2);
3417 42 : A = FF_to_FpXQ_i(FF_neg(n2));
3418 42 : setvarn(A, 1);
3419 42 : R = deg1pol(gen_1, A, 0);
3420 42 : setvarn(R, 0);
3421 42 : M = gcopy(m2);
3422 42 : setvarn(M, 1);
3423 42 : r = polresultant0(R, M, 1, 0);
3424 42 : dm = FF_f(gel(m,1)); dn = FF_f(gel(n,1));
3425 42 : if (dm % dn || !FFX_ispower(r, dm/dn, a, &r)) err_compo(m,n);
3426 42 : setvarn(r, varn(FF_mod(g)));
3427 : }
3428 126 : break;
3429 84 : case 3:
3430 : {
3431 : GEN M, R, T, p, a;
3432 84 : a = ffpartmapimage("ffcompomap",n2);
3433 84 : if (!FF_samefield(a, gel(m,1))) err_compo(m,n);
3434 42 : p = FF_p_i(gel(n,1));
3435 42 : T = FF_mod(gel(n,1));
3436 42 : setvarn(T, 1);
3437 42 : R = RgX_to_FpXQX(n2,T,p);
3438 42 : setvarn(R, 0);
3439 42 : M = gcopy(m2);
3440 42 : setvarn(M, 1);
3441 42 : r = polresultant0(R, M, 1, 0);
3442 42 : setvarn(r, varn(n2));
3443 : }
3444 : }
3445 252 : return gerepilecopy(av, mkvec2(g,r));
3446 : }
|
