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 1225 : char_update_prime(struct charact *S, GEN p)
37 : {
38 1225 : if (!S->isprime) { S->isprime = 1; S->q = p; }
39 1225 : if (!equalii(p, S->q)) pari_err_MODULUS("characteristic", S->q, p);
40 1218 : }
41 : static void
42 6594 : char_update_int(struct charact *S, GEN n)
43 : {
44 6594 : if (S->isprime)
45 : {
46 7 : if (dvdii(n, S->q)) return;
47 7 : pari_err_MODULUS("characteristic", S->q, n);
48 : }
49 6587 : S->q = gcdii(S->q, n);
50 : }
51 : static void
52 179046 : charact(struct charact *S, GEN x)
53 : {
54 179046 : const long tx = typ(x);
55 : long i, l;
56 179046 : switch(tx)
57 : {
58 5145 : case t_INTMOD:char_update_int(S, gel(x,1)); break;
59 1134 : case t_FFELT: char_update_prime(S, gel(x,4)); break;
60 26761 : 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 26761 : l = lg(x);
64 178171 : for (i=lontyp[tx]; i < l; i++) charact(S,gel(x,i));
65 26747 : break;
66 7 : case t_LIST:
67 7 : x = list_data(x);
68 7 : if (x) charact(S, x);
69 7 : break;
70 : }
71 179018 : }
72 : static void
73 4739 : charact_res(struct charact *S, GEN x)
74 : {
75 4739 : const long tx = typ(x);
76 : long i, l;
77 4739 : 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, padic_p(x)); break;
82 1722 : 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 1722 : l = lg(x);
86 6132 : for (i=lontyp[tx]; i < l; i++) charact_res(S,gel(x,i));
87 1722 : 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 4739 : }
94 : GEN
95 27622 : characteristic(GEN x)
96 : {
97 : struct charact S;
98 27622 : S.q = gen_0; S.isprime = 0;
99 27622 : 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 71020954 : Rg_is_Fp(GEN x, GEN *pp)
111 : {
112 : GEN mod;
113 71020954 : switch(typ(x))
114 : {
115 2482676 : case t_INTMOD:
116 2482676 : mod = gel(x,1);
117 2482676 : if (!*pp) *pp = mod;
118 2341976 : 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 2482676 : return 1;
124 57129481 : case t_INT:
125 57129481 : return 1;
126 11408797 : default: return 0;
127 : }
128 : }
129 :
130 : int
131 28180615 : RgX_is_FpX(GEN x, GEN *pp)
132 : {
133 28180615 : long i, lx = lg(x);
134 87766632 : for (i=2; i<lx; i++)
135 70994815 : if (!Rg_is_Fp(gel(x, i), pp))
136 11408800 : return 0;
137 16771817 : 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 60802 : Rg_is_FpXQ(GEN x, GEN *pT, GEN *pp)
160 : {
161 : GEN pol, mod, p;
162 60802 : switch(typ(x))
163 : {
164 26131 : case t_INTMOD:
165 26131 : return Rg_is_Fp(x, pp);
166 8561 : case t_INT:
167 8561 : 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 3381 : RgX_is_FpXQX(GEN x, GEN *pT, GEN *pp)
202 : {
203 3381 : long i, lx = lg(x);
204 63427 : for (i = 2; i < lx; i++)
205 60144 : if (!Rg_is_FpXQ(gel(x,i), pT, pp)) return 0;
206 3283 : 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 52284967 : Rg_to_Fp(GEN x, GEN p)
219 : {
220 52284967 : if (lgefint(p) == 3) return utoi(Rg_to_Fl(x, uel(p,2)));
221 4554870 : switch(typ(x))
222 : {
223 288436 : 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 gc_INT(av, remii(mulii(z, Fp_inv(gel(x,2), p)), p));
229 : }
230 70 : case t_PADIC: return padic_to_Fp(x, p);
231 4247577 : case t_INTMOD: {
232 4247577 : GEN q = gel(x,1), a = gel(x,2);
233 4247577 : 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 1291958 : Rg_to_FpXQ(GEN x, GEN T, GEN p)
244 : {
245 1291958 : long ta, tx = typ(x), v = get_FpX_var(T);
246 : GEN a, b;
247 1291958 : if (is_const_t(tx))
248 : {
249 59175 : if (tx == t_FFELT)
250 : {
251 17355 : GEN z = FF_to_FpXQ(x);
252 17355 : setvarn(z, v);
253 17355 : return z;
254 : }
255 41820 : return scalar_ZX(degpol(T)? Rg_to_Fp(x, p): gen_0, v);
256 : }
257 1232783 : switch(tx)
258 : {
259 1230676 : case t_POLMOD:
260 1230676 : b = gel(x,1);
261 1230676 : a = gel(x,2); ta = typ(a);
262 1230676 : if (is_const_t(ta))
263 3885 : 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 2107 : case t_POL:
270 2107 : if (varn(x) != v) break;
271 2100 : 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 7 : pari_err_TYPE("Rg_to_FpXQ",x);
278 : return NULL; /* LCOV_EXCL_LINE */
279 : }
280 : GEN
281 3334999 : RgX_to_FpX(GEN x, GEN p)
282 : {
283 : long i, l;
284 3334999 : GEN z = cgetg_copy(x, &l); z[1] = x[1];
285 14762908 : for (i = 2; i < l; i++) gel(z,i) = Rg_to_Fp(gel(x,i), p);
286 3334999 : 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 1751090 : RgC_to_FpC(GEN x, GEN p)
295 28499485 : { pari_APPLY_type(t_COL, Rg_to_Fp(gel(x,i), p)) }
296 :
297 : GEN
298 222349 : RgM_to_FpM(GEN x, GEN p)
299 1973397 : { pari_APPLY_same(RgC_to_FpC(gel(x,i), p)) }
300 :
301 : GEN
302 369001 : RgV_to_Flv(GEN x, ulong p)
303 1676894 : { pari_APPLY_ulong(Rg_to_Fl(gel(x,i), p)) }
304 :
305 : GEN
306 118296 : RgM_to_Flm(GEN x, ulong p)
307 422998 : { pari_APPLY_same(RgV_to_Flv(gel(x,i), p)) }
308 :
309 : GEN
310 5098 : RgX_to_FpXQX(GEN x, GEN T, GEN p)
311 : {
312 5098 : long i, l = lg(x);
313 5098 : GEN z = cgetg(l, t_POL); z[1] = x[1];
314 43366 : for (i = 2; i < l; i++) gel(z,i) = Rg_to_FpXQ(gel(x,i), T,p);
315 5098 : 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 791394 : 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 = gc_upto(av, z);
356 : }
357 851487 : return gc_upto(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 1059 : FqX_Fq_sub(GEN y, GEN x, GEN T, GEN p)
377 : {
378 1059 : long i, lz = lg(y);
379 : GEN z;
380 1059 : if (!T) return FpX_Fp_sub(y, x, p);
381 1059 : if (lz == 2) return scalarpol(x, varn(y));
382 1059 : z = cgetg(lz,t_POL); z[1] = y[1];
383 1059 : gel(z,2) = Fq_sub(gel(y,2), x, T, p);
384 1059 : if (lz == 3) z = FpXX_renormalize(z,lz);
385 : else
386 10278 : for(i=3;i<lz;i++) gel(z,i) = gcopy(gel(y,i));
387 1059 : return z;
388 : }
389 :
390 : GEN
391 149052 : FqX_Fq_mul_to_monic(GEN P, GEN U, GEN T, GEN p)
392 : {
393 : long i, lP;
394 149052 : GEN res = cgetg_copy(P, &lP); res[1] = P[1];
395 918799 : for(i=2; i<lP-1; i++) gel(res,i) = Fq_mul(U,gel(P,i), T,p);
396 149052 : gel(res,lP-1) = gen_1; return res;
397 : }
398 :
399 : GEN
400 38188 : FpXQX_normalize(GEN z, GEN T, GEN p)
401 : {
402 : GEN lc;
403 38188 : if (lg(z) == 2) return z;
404 38174 : lc = leading_coeff(z);
405 38174 : if (typ(lc) == t_POL)
406 : {
407 2167 : if (lg(lc) > 3) /* nonconstant */
408 1902 : return FqX_Fq_mul_to_monic(z, Fq_inv(lc,T,p), T,p);
409 : /* constant */
410 265 : lc = gel(lc,2);
411 265 : z = shallowcopy(z);
412 265 : gel(z, lg(z)-1) = lc;
413 : }
414 : /* lc a t_INT */
415 36272 : if (equali1(lc)) return z;
416 87 : return FqX_Fq_mul_to_monic(z, Fp_inv(lc,p), T,p);
417 : }
418 :
419 : GEN
420 390934 : FqX_eval(GEN x, GEN y, GEN T, GEN p)
421 : {
422 : pari_sp av;
423 : GEN p1, r;
424 390934 : long j, i=lg(x)-1;
425 390934 : if (i<=2)
426 45107 : return (i==2)? Fq_red(gel(x,2), T, p): gen_0;
427 345827 : 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 1150443 : for (i--; i>=2; i=j-1)
431 : {
432 805301 : for (j=i; !signe(gel(x,j)); j--)
433 686 : if (j==2)
434 : {
435 490 : if (i!=j) y = Fq_pow(y,utoipos(i-j+1), T, p);
436 490 : return gc_upto(av, Fq_mul(p1,y, T, p));
437 : }
438 804615 : r = (i==j)? y: Fq_pow(y, utoipos(i-j+1), T, p);
439 804615 : p1 = Fq_add(Fq_mul(p1,r,T,p), gel(x,j), T, p);
440 : }
441 345338 : return gc_upto(av, p1);
442 : }
443 :
444 : GEN
445 97321 : FqXY_evalx(GEN Q, GEN x, GEN T, GEN p)
446 : {
447 97321 : long i, lb = lg(Q);
448 : GEN z;
449 97321 : if (!T) return FpXY_evalx(Q, x, p);
450 86961 : z = cgetg(lb, t_POL); z[1] = Q[1];
451 462945 : for (i=2; i<lb; i++)
452 : {
453 375984 : GEN q = gel(Q,i);
454 375984 : gel(z,i) = typ(q) == t_INT? modii(q,p): FqX_eval(q, x, T, p);
455 : }
456 86961 : 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 gc_upto(av, FqX_eval(FqXY_evalx(Q, x, T, p), y, T, p));
466 : }
467 :
468 : /* a X^d */
469 : GEN
470 13104941 : monomial(GEN a, long d, long v)
471 : {
472 : long i, n;
473 : GEN P;
474 13104941 : if (d < 0) {
475 14 : if (isrationalzero(a)) return pol_0(v);
476 14 : retmkrfrac(a, pol_xn(-d, v));
477 : }
478 13104927 : if (gequal0(a))
479 : {
480 93989 : 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 13010939 : n = d+2; P = cgetg(n+1, t_POL);
487 13010944 : P[1] = evalsigne(1) | evalvarn(v);
488 : }
489 32179405 : for (i = 2; i < n; i++) gel(P,i) = gen_0;
490 13010944 : gel(P,i) = a; return P;
491 : }
492 : GEN
493 157969 : monomialcopy(GEN a, long d, long v)
494 : {
495 : long i, n;
496 : GEN P;
497 157969 : if (d < 0) {
498 14 : if (isrationalzero(a)) return pol_0(v);
499 14 : retmkrfrac(gcopy(a), pol_xn(-d, v));
500 : }
501 157955 : 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 157941 : n = d+2; P = cgetg(n+1, t_POL);
510 157941 : P[1] = evalsigne(1) | evalvarn(v);
511 : }
512 314678 : for (i = 2; i < n; i++) gel(P,i) = gen_0;
513 157948 : gel(P,i) = gcopy(a); return P;
514 : }
515 : GEN
516 325814 : pol_x_powers(long N, long v)
517 : {
518 325814 : GEN L = cgetg(N+1,t_VEC);
519 : long i;
520 788688 : for (i=1; i<=N; i++) gel(L,i) = pol_xn(i-1, v);
521 325820 : 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 11592875 : Fq_add(GEN x, GEN y, GEN T/*unused*/, GEN p)
544 : {
545 : (void)T;
546 11592875 : switch((typ(x)==t_POL)|((typ(y)==t_POL)<<1))
547 : {
548 1143687 : case 0: return Fp_add(x,y,p);
549 748136 : case 1: return FpX_Fp_add(x,y,p);
550 91991 : case 2: return FpX_Fp_add(y,x,p);
551 9609061 : case 3: return FpX_add(x,y,p);
552 : }
553 : return NULL;/*LCOV_EXCL_LINE*/
554 : }
555 :
556 : GEN
557 8562717 : Fq_sub(GEN x, GEN y, GEN T/*unused*/, GEN p)
558 : {
559 : (void)T;
560 8562717 : switch((typ(x)==t_POL)|((typ(y)==t_POL)<<1))
561 : {
562 256024 : case 0: return Fp_sub(x,y,p);
563 24540 : case 1: return FpX_Fp_sub(x,y,p);
564 23896 : case 2: return Fp_FpX_sub(x,y,p);
565 8258257 : case 3: return FpX_sub(x,y,p);
566 : }
567 : return NULL;/*LCOV_EXCL_LINE*/
568 : }
569 :
570 : GEN
571 1079572 : Fq_neg(GEN x, GEN T/*unused*/, GEN p)
572 : {
573 : (void)T;
574 1079572 : return (typ(x)==t_POL)? FpX_neg(x,p): Fp_neg(x,p);
575 : }
576 :
577 : GEN
578 81354 : Fq_halve(GEN x, GEN T/*unused*/, GEN p)
579 : {
580 : (void)T;
581 81354 : 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 8608651 : Fq_mul(GEN x, GEN y, GEN T, GEN p)
587 : {
588 8608651 : switch((typ(x)==t_POL)|((typ(y)==t_POL)<<1))
589 : {
590 1037917 : case 0: return Fp_mul(x,y,p);
591 128565 : case 1: return FpX_Fp_mul(x,y,p);
592 394685 : case 2: return FpX_Fp_mul(y,x,p);
593 7047486 : case 3: if (T) return FpXQ_mul(x,y,T,p);
594 4478770 : 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 490623 : Fq_mulu(GEN x, ulong y, /*unused*/GEN T, GEN p)
602 : {
603 : (void) T;
604 490623 : 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 43929 : Fq_Fp_mul(GEN x, GEN y, GEN T/*unused*/, GEN p)
610 : {
611 : (void)T;
612 6822 : return (typ(x) == t_POL)? FpX_Fp_mul(x,y,p)
613 50751 : : Fp_mul(x,y,p);
614 : }
615 : /* If T==NULL do not reduce*/
616 : GEN
617 611138 : Fq_sqr(GEN x, GEN T, GEN p)
618 : {
619 611138 : if (typ(x) == t_POL)
620 : {
621 70585 : if (T) return FpXQ_sqr(x,T,p);
622 0 : else return FpX_sqr(x,p);
623 : }
624 : else
625 540553 : 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 89312 : Fq_inv(GEN x, GEN pol, GEN p)
644 : {
645 89312 : if (typ(x) == t_INT) return Fp_inv(x,p);
646 81546 : return FpXQ_inv(x,pol,p);
647 : }
648 :
649 : GEN
650 343791 : Fq_div(GEN x, GEN y, GEN pol, GEN p)
651 : {
652 343791 : switch((typ(x)==t_POL)|((typ(y)==t_POL)<<1))
653 : {
654 318402 : 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 8281 : case 3: return FpXQ_div(x,y,pol,p);
658 : }
659 : return NULL;/*LCOV_EXCL_LINE*/
660 : }
661 :
662 : GEN
663 795515 : Fq_pow(GEN x, GEN n, GEN pol, GEN p)
664 : {
665 795515 : if (typ(x) == t_INT) return Fp_pow(x,n,p);
666 136709 : 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 1892926 : Fq_sqrt(GEN x, GEN T, GEN p)
678 : {
679 1892926 : if (typ(x) == t_INT)
680 : {
681 1825064 : if (!T || odd(get_FpX_degree(T))) return Fp_sqrt(x,p);
682 9621 : x = scalarpol_shallow(x, get_FpX_var(T));
683 : }
684 77483 : return FpXQ_sqrt(x,T,p);
685 : }
686 : GEN
687 170786 : Fq_sqrtn(GEN x, GEN n, GEN T, GEN p, GEN *zeta)
688 : {
689 170786 : if (typ(x) == t_INT)
690 : {
691 : long d;
692 170415 : if (!T) return Fp_sqrtn(x,n,p,zeta);
693 126 : d = get_FpX_degree(T);
694 126 : 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 77 : x = scalarpol(x, get_FpX_var(T)); /* left on stack */
702 : }
703 448 : 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 303247 : _Fq_red(void *E, GEN x)
713 303247 : { struct _Fq_field *s = (struct _Fq_field *)E;
714 303247 : 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 243341 : _Fq_mul(void *E, GEN x, GEN y)
735 : {
736 : (void) E;
737 243341 : switch((typ(x)==t_POL)|((typ(y)==t_POL)<<1))
738 : {
739 679 : 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 4151 : _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 4725 : const struct bb_field *get_Fq_field(void **E, GEN T, GEN p)
762 : {
763 4725 : if (!T)
764 0 : return get_Fp_field(E, p);
765 : else
766 : {
767 4725 : GEN z = new_chunk(sizeof(struct _Fq_field));
768 4725 : struct _Fq_field *e = (struct _Fq_field *) z;
769 4725 : e->T = T; e->p = p; *E = (void*)e;
770 4725 : return &Fq_field;
771 : }
772 : }
773 :
774 : /*******************************************************************/
775 : /* */
776 : /* Fq[X] */
777 : /* */
778 : /*******************************************************************/
779 : /* P(X + c) */
780 : static GEN
781 434 : Fp_XpN_powu(GEN u, long n, GEN p, long v)
782 : {
783 : pari_sp av;
784 : long k;
785 : GEN B, C, V;
786 434 : if (!n) return pol_1(v);
787 434 : if (is_pm1(u))
788 434 : return Xpm1_powu(n, signe(u), v);
789 0 : av = avma;
790 0 : V = Fp_powers(u, n, p);
791 0 : B = FpV_red(vecbinomial(n), p);
792 0 : C = cgetg(n+3, t_POL);
793 0 : C[1] = evalsigne(1)| evalvarn(v);
794 0 : for (k=1; k <= n+1; k++)
795 0 : gel(C,k+1) = Fp_mul(gel(V,n+2-k), gel(B,k), p);
796 0 : return gc_upto(av, C);
797 : }
798 :
799 : static GEN
800 700 : FpX_translate_basecase(GEN P, GEN c, GEN p)
801 : {
802 700 : pari_sp av = avma;
803 : GEN Q, *R;
804 : long i, k, n;
805 :
806 700 : if (!signe(P) || !signe(c)) return ZX_copy(P);
807 560 : Q = leafcopy(P);
808 560 : R = (GEN*)(Q+2); n = degpol(P);
809 1316 : for (i=1; i<=n; i++)
810 : {
811 2016 : for (k=n-i; k<n; k++)
812 1260 : R[k] = Fp_add(R[k], Fp_mul(c, R[k+1], p), p);
813 :
814 756 : if (gc_needed(av,2))
815 : {
816 0 : if(DEBUGMEM>1) pari_warn(warnmem,"FpX_translate, i = %ld/%ld", i,n);
817 0 : Q = gc_GEN(av, Q); R = (GEN*)Q+2;
818 : }
819 : }
820 560 : return gc_GEN(av, FpX_renormalize(Q, lg(Q)));
821 : }
822 :
823 : GEN
824 1134 : FpX_translate(GEN P, GEN c, GEN p)
825 : {
826 1134 : pari_sp av = avma;
827 1134 : long n = degpol(P);
828 1134 : if (n<=3 || 25*(n-3) < expi(p))
829 700 : return FpX_translate_basecase(P, c, p);
830 : else
831 : {
832 434 : long d = n >> 1;
833 434 : GEN Q = FpX_translate(RgX_shift_shallow(P, -d), c, p);
834 434 : GEN R = FpX_translate(RgXn_red_shallow(P, d), c, p);
835 434 : GEN S = Fp_XpN_powu(c, d, p, varn(P));
836 434 : return gc_upto(av, FpX_add(FpX_mul(Q, S, p), R, p));
837 : }
838 : }
839 :
840 : /* P(X + c), c an Fq */
841 : GEN
842 33880 : FqX_translate(GEN P, GEN c, GEN T, GEN p)
843 : {
844 33880 : pari_sp av = avma;
845 : GEN Q, *R;
846 : long i, k, n;
847 :
848 : /* signe works for t_(INT|POL) */
849 33880 : if (!signe(P) || !signe(c)) return RgX_copy(P);
850 33880 : Q = leafcopy(P);
851 33880 : R = (GEN*)(Q+2); n = degpol(P);
852 150059 : for (i=1; i<=n; i++)
853 : {
854 433559 : for (k=n-i; k<n; k++)
855 317380 : R[k] = Fq_add(R[k], Fq_mul(c, R[k+1], T, p), T, p);
856 :
857 116179 : if (gc_needed(av,2))
858 : {
859 0 : if(DEBUGMEM>1) pari_warn(warnmem,"FqX_translate, i = %ld/%ld", i,n);
860 0 : Q = gc_GEN(av, Q); R = (GEN*)Q+2;
861 : }
862 : }
863 33880 : return gc_GEN(av, FpXQX_renormalize(Q, lg(Q)));
864 : }
865 :
866 : GEN
867 40457 : FqV_roots_to_pol(GEN V, GEN T, GEN p, long v)
868 : {
869 40457 : pari_sp ltop = avma;
870 : long k;
871 : GEN W;
872 40457 : if (lgefint(p) == 3)
873 : {
874 31746 : ulong pp = p[2];
875 31746 : GEN Tl = ZX_to_Flx(T, pp);
876 31746 : GEN Vl = ZXC_to_FlxC(V, pp, get_Flx_var(Tl));
877 31746 : Tl = FlxqV_roots_to_pol(Vl, Tl, pp, v);
878 31746 : return gc_upto(ltop, FlxX_to_ZXX(Tl));
879 : }
880 8711 : W = cgetg(lg(V),t_VEC);
881 78117 : for(k=1; k < lg(V); k++)
882 69406 : gel(W,k) = deg1pol_shallow(gen_1,Fq_neg(gel(V,k),T,p),v);
883 8711 : return gc_upto(ltop, FpXQXV_prod(W, T, p));
884 : }
885 :
886 : GEN
887 109466 : FqV_red(GEN x, GEN T, GEN p)
888 778210 : { pari_APPLY_type(t_VEC, Fq_red(gel(x,i), T, p)) }
889 :
890 : GEN
891 24058 : FqC_red(GEN x, GEN T, GEN p)
892 163384 : { pari_APPLY_type(t_COL, Fq_red(gel(x,i), T, p)) }
893 :
894 : GEN
895 1701 : FqM_red(GEN x, GEN T, GEN p)
896 5411 : { pari_APPLY_same(FqC_red(gel(x,i), T, p)) }
897 :
898 : GEN
899 0 : FqC_add(GEN x, GEN y, GEN T, GEN p)
900 : {
901 0 : if (!T) return FpC_add(x, y, p);
902 0 : pari_APPLY_type(t_COL, Fq_add(gel(x,i), gel(y,i), T, p))
903 : }
904 :
905 : GEN
906 0 : FqC_sub(GEN x, GEN y, GEN T, GEN p)
907 : {
908 0 : if (!T) return FpC_sub(x, y, p);
909 0 : pari_APPLY_type(t_COL, Fq_sub(gel(x,i), gel(y,i), T, p))
910 : }
911 :
912 : GEN
913 0 : FqC_Fq_mul(GEN x, GEN y, GEN T, GEN p)
914 : {
915 0 : if (!T) return FpC_Fp_mul(x, y, p);
916 0 : pari_APPLY_type(t_COL, Fq_mul(gel(x,i),y,T,p))
917 : }
918 :
919 : GEN
920 105 : FqC_FqV_mul(GEN x, GEN y, GEN T, GEN p)
921 : {
922 105 : long i,j, lx=lg(x), ly=lg(y);
923 : GEN z;
924 105 : if (ly==1) return cgetg(1,t_MAT);
925 105 : z = cgetg(ly,t_MAT);
926 819 : for (j=1; j < ly; j++)
927 : {
928 714 : GEN zj = cgetg(lx,t_COL);
929 4200 : for (i=1; i<lx; i++) gel(zj,i) = Fq_mul(gel(x,i),gel(y,j), T, p);
930 714 : gel(z, j) = zj;
931 : }
932 105 : return z;
933 : }
934 :
935 : GEN
936 5467 : FpXC_center(GEN x, GEN p, GEN pov2)
937 41476 : { pari_APPLY_type(t_COL, FpX_center(gel(x,i), p, pov2)) }
938 :
939 : GEN
940 1800 : FpXM_center(GEN x, GEN p, GEN pov2)
941 7267 : { pari_APPLY_same(FpXC_center(gel(x,i), p, pov2)) }
942 :
943 : /*******************************************************************/
944 : /* */
945 : /* GENERIC CRT */
946 : /* */
947 : /*******************************************************************/
948 : static GEN
949 8309535 : primelist(forprime_t *S, long n, GEN dB)
950 : {
951 8309535 : GEN P = cgetg(n+1, t_VECSMALL);
952 8309523 : long i = 1;
953 : ulong p;
954 20071888 : while (i <= n && (p = u_forprime_next(S)))
955 11762366 : if (!dB || umodiu(dB, p)) P[i++] = p;
956 8309532 : return P;
957 : }
958 :
959 : void
960 7726953 : gen_inccrt_i(const char *str, GEN worker, GEN dB, long n, long mmin,
961 : forprime_t *S, GEN *pH, GEN *pmod, GEN crt(GEN, GEN, GEN*),
962 : GEN center(GEN, GEN, GEN))
963 : {
964 7726953 : long m = mmin? minss(mmin, n): usqrt(n);
965 : GEN H, P, mod;
966 : pari_timer ti;
967 7726950 : if (DEBUGLEVEL > 4)
968 : {
969 0 : timer_start(&ti);
970 0 : err_printf("%s: nb primes: %ld\n",str, n);
971 : }
972 7726948 : if (m == 1)
973 : {
974 7415886 : GEN P = primelist(S, n, dB);
975 7415868 : GEN done = closure_callgen1(worker, P);
976 7415852 : H = gel(done,1);
977 7415852 : mod = gel(done,2);
978 7415852 : if (!*pH && center) H = center(H, mod, shifti(mod,-1));
979 7415795 : if (DEBUGLEVEL>4) timer_printf(&ti,"%s: modular", str);
980 : }
981 : else
982 : {
983 311062 : long i, s = (n+m-1)/m, r = m - (m*s-n), di = 0;
984 : struct pari_mt pt;
985 311062 : long pending = 0;
986 311062 : H = cgetg(m+1, t_VEC); P = cgetg(m+1, t_VEC);
987 311062 : mt_queue_start_lim(&pt, worker, m);
988 1270314 : for (i=1; i<=m || pending; i++)
989 : {
990 : GEN done;
991 959251 : GEN pr = i <= m ? mkvec(primelist(S, i<=r ? s: s-1, dB)): NULL;
992 959254 : mt_queue_submit(&pt, i, pr);
993 959252 : done = mt_queue_get(&pt, NULL, &pending);
994 959252 : if (done)
995 : {
996 893652 : di++;
997 893652 : gel(H, di) = gel(done,1);
998 893652 : gel(P, di) = gel(done,2);
999 893652 : if (DEBUGLEVEL>5) err_printf("%ld%% ",100*di/m);
1000 : }
1001 : }
1002 311063 : mt_queue_end(&pt);
1003 311063 : if (DEBUGLEVEL>5) err_printf("\n");
1004 311063 : if (DEBUGLEVEL>4) timer_printf(&ti,"%s: modular", str);
1005 311063 : H = crt(H, P, &mod);
1006 311063 : if (DEBUGLEVEL>4) timer_printf(&ti,"%s: chinese", str);
1007 : }
1008 7726858 : if (*pH) H = crt(mkvec2(*pH, H), mkvec2(*pmod, mod), &mod);
1009 7726858 : *pH = H; *pmod = mod;
1010 7726858 : }
1011 : void
1012 2060092 : gen_inccrt(const char *str, GEN worker, GEN dB, long n, long mmin,
1013 : forprime_t *S, GEN *pH, GEN *pmod, GEN crt(GEN, GEN, GEN*),
1014 : GEN center(GEN, GEN, GEN))
1015 : {
1016 2060092 : pari_sp av = avma;
1017 2060092 : gen_inccrt_i(str, worker, dB, n, mmin, S, pH, pmod, crt, center);
1018 2060034 : (void)gc_all(av, 2, pH, pmod);
1019 2060152 : }
1020 :
1021 : GEN
1022 1273458 : gen_crt(const char *str, GEN worker, forprime_t *S, GEN dB, ulong bound, long mmin, GEN *pmod,
1023 : GEN crt(GEN, GEN, GEN*), GEN center(GEN, GEN, GEN))
1024 : {
1025 1273458 : GEN mod = gen_1, H = NULL;
1026 : ulong e;
1027 :
1028 1273458 : bound++;
1029 2546962 : while (bound > (e = expi(mod)))
1030 : {
1031 1273414 : long n = (bound - e) / expu(S->p) + 1;
1032 1273442 : gen_inccrt(str, worker, dB, n, mmin, S, &H, &mod, crt, center);
1033 : }
1034 1273491 : if (pmod) *pmod = mod;
1035 1273491 : return H;
1036 : }
1037 :
1038 : /*******************************************************************/
1039 : /* */
1040 : /* MODULAR GCD */
1041 : /* */
1042 : /*******************************************************************/
1043 : /* return z = a mod q, b mod p (p,q) = 1; qinv = 1/q mod p; a in ]-q,q] */
1044 : static GEN
1045 5154724 : Fl_chinese_coprime(GEN a, ulong b, GEN q, ulong p, ulong qinv, GEN pq, GEN pq2)
1046 : {
1047 5154724 : ulong d, amod = umodiu(a, p);
1048 5154732 : pari_sp av = avma;
1049 : GEN ax;
1050 :
1051 5154732 : if (b == amod) return NULL;
1052 2125754 : d = Fl_mul(Fl_sub(b, amod, p), qinv, p); /* != 0 */
1053 2126862 : if (d >= 1 + (p>>1))
1054 1037768 : ax = subii(a, mului(p-d, q));
1055 : else
1056 : {
1057 1089094 : ax = addii(a, mului(d, q)); /* in ]0, pq[ assuming a in ]-q,q[ */
1058 1088495 : if (cmpii(ax,pq2) > 0) ax = subii(ax,pq);
1059 : }
1060 2125707 : return gc_INT(av, ax);
1061 : }
1062 : GEN
1063 378 : Z_init_CRT(ulong Hp, ulong p) { return stoi(Fl_center(Hp, p, p>>1)); }
1064 : GEN
1065 31773 : ZX_init_CRT(GEN Hp, ulong p, long v)
1066 : {
1067 31773 : long i, l = lg(Hp), lim = (long)(p>>1);
1068 31773 : GEN H = cgetg(l, t_POL);
1069 31773 : H[1] = evalsigne(1) | evalvarn(v);
1070 795984 : for (i=2; i<l; i++)
1071 764211 : gel(H,i) = stoi(Fl_center(Hp[i], p, lim));
1072 31773 : return ZX_renormalize(H,l);
1073 : }
1074 :
1075 : GEN
1076 5796 : ZM_init_CRT(GEN Hp, ulong p)
1077 : {
1078 5796 : long i,j, m, l = lg(Hp), lim = (long)(p>>1);
1079 5796 : GEN c, cp, H = cgetg(l, t_MAT);
1080 5796 : if (l==1) return H;
1081 5796 : m = lgcols(Hp);
1082 19026 : for (j=1; j<l; j++)
1083 : {
1084 13230 : cp = gel(Hp,j);
1085 13230 : c = cgetg(m, t_COL);
1086 13230 : gel(H,j) = c;
1087 166376 : for (i=1; i<m; i++) gel(c,i) = stoi(Fl_center(cp[i],p, lim));
1088 : }
1089 5796 : return H;
1090 : }
1091 :
1092 : int
1093 7616 : Z_incremental_CRT(GEN *H, ulong Hp, GEN *ptq, ulong p)
1094 : {
1095 7616 : GEN h, q = *ptq, qp = muliu(q,p);
1096 7616 : ulong qinv = Fl_inv(umodiu(q,p), p);
1097 7616 : int stable = 1;
1098 7616 : h = Fl_chinese_coprime(*H,Hp,q,p,qinv,qp,shifti(qp,-1));
1099 7616 : if (h) { *H = h; stable = 0; }
1100 7616 : *ptq = qp; return stable;
1101 : }
1102 :
1103 : static int
1104 147586 : ZX_incremental_CRT_raw(GEN *ptH, GEN Hp, GEN q, GEN qp, ulong p)
1105 : {
1106 147586 : GEN H = *ptH, h, qp2 = shifti(qp,-1);
1107 147576 : ulong qinv = Fl_inv(umodiu(q,p), p);
1108 147587 : long i, l = lg(H), lp = lg(Hp);
1109 147587 : int stable = 1;
1110 :
1111 147587 : if (l < lp)
1112 : { /* degree increases */
1113 0 : GEN x = cgetg(lp, t_POL);
1114 0 : for (i=1; i<l; i++) x[i] = H[i];
1115 0 : for ( ; i<lp; i++) gel(x,i) = gen_0;
1116 0 : *ptH = H = x;
1117 0 : stable = 0;
1118 147587 : } else if (l > lp)
1119 : { /* degree decreases */
1120 0 : GEN x = cgetg(l, t_VECSMALL);
1121 0 : for (i=1; i<lp; i++) x[i] = Hp[i];
1122 0 : for ( ; i<l; i++) x[i] = 0;
1123 0 : Hp = x; lp = l;
1124 : }
1125 4932705 : for (i=2; i<lp; i++)
1126 : {
1127 4785243 : h = Fl_chinese_coprime(gel(H,i),Hp[i],q,p,qinv,qp,qp2);
1128 4785118 : if (h) { gel(H,i) = h; stable = 0; }
1129 : }
1130 147462 : (void)ZX_renormalize(H,lp);
1131 147584 : return stable;
1132 : }
1133 :
1134 : int
1135 0 : ZX_incremental_CRT(GEN *ptH, GEN Hp, GEN *ptq, ulong p)
1136 : {
1137 0 : GEN q = *ptq, qp = muliu(q,p);
1138 0 : int stable = ZX_incremental_CRT_raw(ptH, Hp, q, qp, p);
1139 0 : *ptq = qp; return stable;
1140 : }
1141 :
1142 : int
1143 7611 : ZM_incremental_CRT(GEN *pH, GEN Hp, GEN *ptq, ulong p)
1144 : {
1145 7611 : GEN h, H = *pH, q = *ptq, qp = muliu(q, p), qp2 = shifti(qp,-1);
1146 7611 : ulong qinv = Fl_inv(umodiu(q,p), p);
1147 7611 : long i,j, l = lg(H), m = lgcols(H);
1148 7611 : int stable = 1;
1149 26290 : for (j=1; j<l; j++)
1150 202771 : for (i=1; i<m; i++)
1151 : {
1152 184092 : h = Fl_chinese_coprime(gcoeff(H,i,j), coeff(Hp,i,j),q,p,qinv,qp,qp2);
1153 184092 : if (h) { gcoeff(H,i,j) = h; stable = 0; }
1154 : }
1155 7611 : *ptq = qp; return stable;
1156 : }
1157 :
1158 : GEN
1159 679 : ZXM_init_CRT(GEN Hp, long deg, ulong p)
1160 : {
1161 : long i, j, k;
1162 : GEN H;
1163 679 : long m, l = lg(Hp), lim = (long)(p>>1), n;
1164 679 : H = cgetg(l, t_MAT);
1165 679 : if (l==1) return H;
1166 679 : m = lgcols(Hp);
1167 679 : n = deg + 3;
1168 2268 : for (j=1; j<l; j++)
1169 : {
1170 1589 : GEN cp = gel(Hp,j);
1171 1589 : GEN c = cgetg(m, t_COL);
1172 1589 : gel(H,j) = c;
1173 24465 : for (i=1; i<m; i++)
1174 : {
1175 22876 : GEN dp = gel(cp, i);
1176 22876 : long l = lg(dp);
1177 22876 : GEN d = cgetg(n, t_POL);
1178 22876 : gel(c, i) = d;
1179 22876 : d[1] = dp[1] | evalsigne(1);
1180 46459 : for (k=2; k<l; k++)
1181 23583 : gel(d,k) = stoi(Fl_center(dp[k], p, lim));
1182 45493 : for ( ; k<n; k++)
1183 22617 : gel(d,k) = gen_0;
1184 : }
1185 : }
1186 679 : return H;
1187 : }
1188 :
1189 : int
1190 653 : ZXM_incremental_CRT(GEN *pH, GEN Hp, GEN *ptq, ulong p)
1191 : {
1192 653 : GEN v, H = *pH, q = *ptq, qp = muliu(q, p), qp2 = shifti(qp,-1);
1193 653 : ulong qinv = Fl_inv(umodiu(q,p), p);
1194 653 : long i,j,k, l = lg(H), m = lgcols(H), n = lg(gmael(H,1,1));
1195 653 : int stable = 1;
1196 2225 : for (j=1; j<l; j++)
1197 90418 : for (i=1; i<m; i++)
1198 : {
1199 88846 : GEN h = gmael(H,j,i), hp = gmael(Hp,j,i);
1200 88846 : long lh = lg(hp);
1201 246641 : for (k=2; k<lh; k++)
1202 : {
1203 157795 : v = Fl_chinese_coprime(gel(h,k),uel(hp,k),q,p,qinv,qp,qp2);
1204 157795 : if (v) { gel(h,k) = v; stable = 0; }
1205 : }
1206 108763 : for (; k<n; k++)
1207 : {
1208 19917 : v = Fl_chinese_coprime(gel(h,k),0,q,p,qinv,qp,qp2);
1209 19917 : if (v) { gel(h,k) = v; stable = 0; }
1210 : }
1211 : }
1212 653 : *ptq = qp; return stable;
1213 : }
1214 :
1215 : /* record the degrees of Euclidean remainders (make them as large as
1216 : * possible : smaller values correspond to a degenerate sequence) */
1217 : static void
1218 23272 : Flx_resultant_set_dglist(GEN a, GEN b, GEN dglist, ulong p)
1219 : {
1220 : long da,db,dc, ind;
1221 23272 : pari_sp av = avma;
1222 :
1223 23272 : if (lgpol(a)==0 || lgpol(b)==0) return;
1224 22005 : da = degpol(a);
1225 22005 : db = degpol(b);
1226 22005 : if (db > da)
1227 0 : { swapspec(a,b, da,db); }
1228 22005 : else if (!da) return;
1229 22005 : ind = 0;
1230 144341 : while (db)
1231 : {
1232 122338 : GEN c = Flx_rem(a,b, p);
1233 122336 : a = b; b = c; dc = degpol(c);
1234 122336 : if (dc < 0) break;
1235 :
1236 122336 : ind++;
1237 122336 : if (dc > dglist[ind]) dglist[ind] = dc;
1238 122336 : if (gc_needed(av,2))
1239 : {
1240 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flx_resultant_all");
1241 0 : (void)gc_all(av, 2, &a,&b);
1242 : }
1243 122336 : db = dc; /* = degpol(b) */
1244 : }
1245 22003 : if (ind+1 > lg(dglist)) setlg(dglist,ind+1);
1246 22003 : set_avma(av);
1247 : }
1248 : /* assuming the PRS finishes on a degree 1 polynomial C0 + C1X, with
1249 : * "generic" degree sequence as given by dglist, set *Ci and return
1250 : * resultant(a,b). Modular version of Collins's subresultant */
1251 : static ulong
1252 2084725 : Flx_resultant_all(GEN a, GEN b, long *C0, long *C1, GEN dglist, ulong p)
1253 : {
1254 : long da,db,dc, ind;
1255 2084725 : ulong lb, res, g = 1UL, h = 1UL, ca = 1UL, cb = 1UL;
1256 2084725 : int s = 1;
1257 2084725 : pari_sp av = avma;
1258 :
1259 2084725 : *C0 = 1; *C1 = 0;
1260 2084725 : if (lgpol(a)==0 || lgpol(b)==0) return 0;
1261 2075181 : da = degpol(a);
1262 2075177 : db = degpol(b);
1263 2075163 : if (db > da)
1264 : {
1265 0 : swapspec(a,b, da,db);
1266 0 : if (both_odd(da,db)) s = -s;
1267 : }
1268 2075163 : else if (!da) return 1; /* = a[2] ^ db, since 0 <= db <= da = 0 */
1269 2075163 : ind = 0;
1270 19793007 : while (db)
1271 : { /* sub-resultant algo., applied to ca * a and cb * b, ca,cb scalars,
1272 : * da = deg a, db = deg b */
1273 17722305 : GEN c = Flx_rem(a,b, p);
1274 17639427 : long delta = da - db;
1275 :
1276 17639427 : if (both_odd(da,db)) s = -s;
1277 17638203 : lb = Fl_mul(b[db+2], cb, p);
1278 17665014 : a = b; b = c; dc = degpol(c);
1279 17664707 : ind++;
1280 17664707 : if (dc != dglist[ind]) return gc_ulong(av,0); /* degenerates */
1281 17659774 : if (g == h)
1282 : { /* frequent */
1283 17599931 : ulong cc = Fl_mul(ca, Fl_powu(Fl_div(lb,g,p), delta+1, p), p);
1284 17659026 : ca = cb;
1285 17659026 : cb = cc;
1286 : }
1287 : else
1288 : {
1289 59843 : ulong cc = Fl_mul(ca, Fl_powu(lb, delta+1, p), p);
1290 59843 : ulong ghdelta = Fl_mul(g, Fl_powu(h, delta, p), p);
1291 59843 : ca = cb;
1292 59843 : cb = Fl_div(cc, ghdelta, p);
1293 : }
1294 17719930 : da = db; /* = degpol(a) */
1295 17719930 : db = dc; /* = degpol(b) */
1296 :
1297 17719930 : g = lb;
1298 17719930 : if (delta == 1)
1299 17619894 : h = g; /* frequent */
1300 : else
1301 100036 : h = Fl_mul(h, Fl_powu(Fl_div(g,h,p), delta, p), p);
1302 :
1303 17719256 : if (gc_needed(av,2))
1304 : {
1305 0 : if (DEBUGMEM>1) pari_warn(warnmem,"Flx_resultant_all");
1306 0 : (void)gc_all(av, 2, &a,&b);
1307 : }
1308 : }
1309 2070702 : if (da > 1) return 0; /* Failure */
1310 : /* last nonconstant polynomial has degree 1 */
1311 2070702 : *C0 = Fl_mul(ca, a[2], p);
1312 2070653 : *C1 = Fl_mul(ca, a[3], p);
1313 2070642 : res = Fl_mul(cb, b[2], p);
1314 2070620 : if (s == -1) res = p - res;
1315 2070620 : return gc_ulong(av,res);
1316 : }
1317 :
1318 : /* Q a vector of polynomials representing B in Fp[X][Y], evaluate at X = x,
1319 : * Return 0 in case of degree drop. */
1320 : static GEN
1321 2108174 : FlxY_evalx_drop(GEN Q, ulong x, ulong p)
1322 : {
1323 : GEN z;
1324 2108174 : long i, lb = lg(Q);
1325 2108174 : ulong leadz = Flx_eval(leading_coeff(Q), x, p);
1326 2107842 : long vs=mael(Q,2,1);
1327 2107842 : if (!leadz) return zero_Flx(vs);
1328 :
1329 2097182 : z = cgetg(lb, t_VECSMALL); z[1] = vs;
1330 20055004 : for (i=2; i<lb-1; i++) z[i] = Flx_eval(gel(Q,i), x, p);
1331 2097441 : z[i] = leadz; return z;
1332 : }
1333 :
1334 : GEN
1335 2072 : FpXY_FpXQ_evaly(GEN Q, GEN y, GEN T, GEN p, long vx)
1336 : {
1337 2072 : pari_sp av = avma;
1338 2072 : long i, lb = lg(Q);
1339 : GEN z;
1340 2072 : if (lb == 2) return pol_0(vx);
1341 2072 : z = gel(Q, lb-1);
1342 2072 : if (lb == 3 || !signe(y)) return typ(z)==t_INT? scalar_ZX(z, vx): ZX_copy(z);
1343 :
1344 2072 : if (typ(z) == t_INT) z = scalar_ZX_shallow(z, vx);
1345 48636 : for (i=lb-2; i>=2; i--)
1346 : {
1347 46564 : GEN c = gel(Q,i);
1348 46564 : z = FqX_Fq_mul(z, y, T, p);
1349 46564 : z = typ(c) == t_INT? FqX_Fq_add(z,c,T,p): FqX_add(z,c,T,p);
1350 : }
1351 2072 : return gc_upto(av, z);
1352 : }
1353 :
1354 : static GEN
1355 291749 : ZX_norml1(GEN x)
1356 : {
1357 291749 : long i, l = lg(x);
1358 : GEN s;
1359 :
1360 291749 : if (l == 2) return gen_0;
1361 199195 : s = gel(x, l-1); /* != 0 */
1362 697224 : for (i = l-2; i > 1; i--) {
1363 498035 : GEN xi = gel(x,i);
1364 498035 : if (!signe(xi)) continue;
1365 259420 : s = addii_sign(s,1, xi,1);
1366 : }
1367 199189 : return s;
1368 : }
1369 : /* x >= 0, y != 0, return x + |y| */
1370 : static GEN
1371 25552 : addii_abs(GEN x, GEN y)
1372 : {
1373 25552 : if (!signe(x)) return absi_shallow(y);
1374 16045 : return addii_sign(x,1, y,1);
1375 : }
1376 :
1377 : /* x a ZX, return sum_{i >= k} |x[i]| binomial(i, k) */
1378 : static GEN
1379 31640 : ZX_norml1_1(GEN x, long k)
1380 : {
1381 31640 : long i, d = degpol(x);
1382 : GEN s, C; /* = binomial(i, k) */
1383 :
1384 31640 : if (!d || k > d) return gen_0;
1385 31641 : s = absi_shallow(gel(x, k+2)); /* may be 0 */
1386 31643 : C = gen_1;
1387 68039 : for (i = k+1; i <= d; i++) {
1388 36407 : GEN xi = gel(x,i+2);
1389 36407 : if (k) C = diviuexact(muliu(C, i), i-k);
1390 36405 : if (signe(xi)) s = addii_abs(s, mulii(C, xi));
1391 : }
1392 31632 : return s;
1393 : }
1394 : /* x has non-negative real coefficients */
1395 : static GEN
1396 3283 : RgX_norml1_1(GEN x, long k)
1397 : {
1398 3283 : long i, d = degpol(x);
1399 : GEN s, C; /* = binomial(i, k) */
1400 :
1401 3283 : if (!d || k > d) return gen_0;
1402 3283 : s = gel(x, k+2); /* may be 0 */
1403 3283 : C = gen_1;
1404 9198 : for (i = k+1; i <= d; i++) {
1405 5915 : GEN xi = gel(x,i+2);
1406 5915 : if (k) C = diviuexact(muliu(C, i), i-k);
1407 5915 : if (!gequal0(xi)) s = gadd(s, gmul(C, xi));
1408 : }
1409 3283 : return s;
1410 : }
1411 :
1412 : /* N_2(A)^2 */
1413 : static GEN
1414 8571 : sqrN2(GEN A, long prec)
1415 : {
1416 8571 : pari_sp av = avma;
1417 8571 : long i, l = lg(A);
1418 8571 : GEN a = gen_0;
1419 41935 : for (i = 2; i < l; i++)
1420 : {
1421 33364 : a = gadd(a, gabs(gnorm(gel(A,i)), prec));
1422 33364 : if (gc_needed(av,1))
1423 : {
1424 0 : if(DEBUGMEM>1) pari_warn(warnmem,"RgX_RgXY_ResBound i = %ld",i);
1425 0 : a = gc_upto(av, a);
1426 : }
1427 : }
1428 8571 : return a;
1429 : }
1430 : /* Interpolate at roots of 1 and use Hadamard bound for univariate resultant:
1431 : * bound = N_2(A)^degpol B N_2(B)^degpol(A), where
1432 : * N_2(A) = sqrt(sum (N_1(Ai))^2)
1433 : * Return e such that Res(A, B) < 2^e */
1434 : static GEN
1435 7717 : RgX_RgXY_ResBound(GEN A, GEN B, long prec)
1436 : {
1437 7717 : pari_sp av = avma;
1438 7717 : GEN b = gen_0, bnd;
1439 7717 : long i, lB = lg(B);
1440 30403 : for (i=2; i<lB; i++)
1441 : {
1442 22686 : GEN t = gel(B,i);
1443 22686 : if (typ(t) == t_POL) t = gnorml1(t, prec);
1444 22686 : b = gadd(b, gabs(gsqr(t), prec));
1445 22686 : if (gc_needed(av,1))
1446 : {
1447 0 : if(DEBUGMEM>1) pari_warn(warnmem,"RgX_RgXY_ResBound i = %ld",i);
1448 0 : b = gc_upto(av, b);
1449 : }
1450 : }
1451 7717 : bnd = gsqrt(gmul(gpowgs(sqrN2(A,prec), degpol(B)),
1452 : gpowgs(b, degpol(A))), prec);
1453 7717 : return gc_upto(av, bnd);
1454 : }
1455 : /* A,B in C[X] return RgX_RgXY_ResBound(A, B(x+y)) */
1456 : static GEN
1457 854 : RgX_RgXY_ResBound_1(GEN A, GEN B, long prec)
1458 : {
1459 854 : pari_sp av = avma, av2;
1460 854 : GEN b = gen_0, bnd;
1461 854 : long i, lB = lg(B);
1462 854 : B = shallowcopy(B);
1463 4137 : for (i=2; i<lB; i++) gel(B,i) = gabs(gel(B,i), prec);
1464 854 : av2 = avma;
1465 4137 : for (i=2; i<lB; i++)
1466 : {
1467 3283 : b = gadd(b, gsqr(RgX_norml1_1(B, i-2)));
1468 3283 : if (gc_needed(av2,1))
1469 : {
1470 0 : if(DEBUGMEM>1) pari_warn(warnmem,"RgX_RgXY_ResBound i = %ld",i);
1471 0 : b = gc_upto(av2, b);
1472 : }
1473 : }
1474 854 : bnd = gsqrt(gmul(gpowgs(sqrN2(A,prec), degpol(B)),
1475 : gpowgs(b, degpol(A))), prec);
1476 854 : return gc_upto(av, bnd);
1477 : }
1478 :
1479 : /* log2 N_2(A)^2 */
1480 : static double
1481 176674 : log2N2(GEN A)
1482 : {
1483 176674 : pari_sp av = avma;
1484 176674 : long i, l = lg(A);
1485 176674 : GEN a = gen_0;
1486 1335119 : for (i=2; i < l; i++)
1487 : {
1488 1158444 : a = addii(a, sqri(gel(A,i)));
1489 1158445 : if (gc_needed(av,1))
1490 : {
1491 0 : if(DEBUGMEM>1) pari_warn(warnmem,"ZX_ZXY_ResBound i = %ld",i);
1492 0 : a = gc_upto(av, a);
1493 : }
1494 : }
1495 176675 : return gc_double(av, dbllog2(a));
1496 : }
1497 : /* Interpolate at roots of 1 and use Hadamard bound for univariate resultant:
1498 : * bound = N_2(A)^degpol B N_2(B)^degpol(A), where
1499 : * N_2(A) = sqrt(sum (N_1(Ai))^2)
1500 : * Return e such that Res(A, B) < 2^e */
1501 : ulong
1502 166596 : ZX_ZXY_ResBound(GEN A, GEN B, GEN dB)
1503 : {
1504 166596 : pari_sp av = avma;
1505 166596 : GEN b = gen_0;
1506 166596 : long i, lB = lg(B);
1507 : double logb;
1508 1260603 : for (i=2; i<lB; i++)
1509 : {
1510 1094010 : GEN t = gel(B,i);
1511 1094010 : if (typ(t) == t_POL) t = ZX_norml1(t);
1512 1094009 : b = addii(b, sqri(t));
1513 1094007 : if (gc_needed(av,1))
1514 : {
1515 0 : if(DEBUGMEM>1) pari_warn(warnmem,"ZX_ZXY_ResBound i = %ld",i);
1516 0 : b = gc_upto(av, b);
1517 : }
1518 : }
1519 166593 : logb = dbllog2(b); if (dB) logb -= 2 * dbllog2(dB);
1520 166593 : i = (long)((degpol(B) * log2N2(A) + degpol(A) * logb) / 2);
1521 166596 : return gc_ulong(av, (i <= 0)? 1: 1 + (ulong)i);
1522 : }
1523 : /* A,B ZX. Return ZX_ZXY_ResBound(A(x), B(x+y)) */
1524 : static ulong
1525 10082 : ZX_ZXY_ResBound_1(GEN A, GEN B)
1526 : {
1527 10082 : pari_sp av = avma;
1528 10082 : GEN b = gen_0;
1529 10082 : long i, lB = lg(B);
1530 41727 : for (i=2; i<lB; i++)
1531 : {
1532 31639 : b = addii(b, sqri(ZX_norml1_1(B, i-2)));
1533 31645 : if (gc_needed(av,1))
1534 : {
1535 0 : if(DEBUGMEM>1) pari_warn(warnmem,"ZX_ZXY_ResBound i = %ld",i);
1536 0 : b = gc_upto(av, b);
1537 : }
1538 : }
1539 10088 : i = (long)((degpol(B) * log2N2(A) + degpol(A) * dbllog2(b)) / 2);
1540 10085 : return gc_ulong(av, (i <= 0)? 1: 1 + (ulong)i);
1541 : }
1542 : /* special case B = A' */
1543 : static ulong
1544 1134132 : ZX_discbound(GEN A)
1545 : {
1546 1134132 : pari_sp av = avma;
1547 1134132 : GEN a = gen_0, b = gen_0;
1548 1134132 : long i , lA = lg(A), dA = degpol(A);
1549 : double loga, logb;
1550 6767811 : for (i = 2; i < lA; i++)
1551 : {
1552 5633847 : GEN c = sqri(gel(A,i));
1553 5633728 : a = addii(a, c);
1554 5633804 : if (i > 2) b = addii(b, mulii(c, sqru(i-2)));
1555 5633752 : if (gc_needed(av,1))
1556 : {
1557 0 : if(DEBUGMEM>1) pari_warn(warnmem,"ZX_discbound i = %ld",i);
1558 0 : (void)gc_all(av, 2, &a, &b);
1559 : }
1560 : }
1561 1133964 : loga = dbllog2(a);
1562 1134073 : logb = dbllog2(b); set_avma(av);
1563 1134076 : i = (long)(((dA-1) * loga + dA * logb) / 2);
1564 1134076 : return (i <= 0)? 1: 1 + (ulong)i;
1565 : }
1566 :
1567 : /* return Res(a(Y), b(n,Y)) over Fp. la = leading_coeff(a) [for efficiency] */
1568 : static ulong
1569 5538321 : Flx_FlxY_eval_resultant(GEN a, GEN b, ulong n, ulong p, ulong pi, ulong la)
1570 : {
1571 5538321 : GEN ev = FlxY_evalx_pre(b, n, p, pi);
1572 5538839 : long drop = lg(b) - lg(ev);
1573 5538839 : ulong r = Flx_resultant_pre(a, ev, p, pi);
1574 5538171 : if (drop && la != 1) r = Fl_mul(r, Fl_powu_pre(la, drop, p, pi), p);
1575 5538177 : return r;
1576 : }
1577 : static GEN
1578 284 : FpX_FpXY_eval_resultant(GEN a, GEN b, GEN n, GEN p, GEN la, long db, long vX)
1579 : {
1580 284 : GEN ev = FpXY_evaly(b, n, p, vX);
1581 284 : long drop = db-degpol(ev);
1582 284 : GEN r = FpX_resultant(a, ev, p);
1583 284 : if (drop && !gequal1(la)) r = Fp_mul(r, Fp_powu(la, drop,p),p);
1584 284 : return r;
1585 : }
1586 :
1587 : /* assume dres := deg(Res_X(a,b), Y) <= deg(a,X) * deg(b,Y) < p */
1588 : /* Return a Fly */
1589 : static GEN
1590 368426 : Flx_FlxY_resultant_polint(GEN a, GEN b, ulong p, ulong pi, long dres, long sx)
1591 : {
1592 : long i;
1593 368426 : ulong n, la = Flx_lead(a);
1594 368425 : GEN x = cgetg(dres+2, t_VECSMALL);
1595 368426 : GEN y = cgetg(dres+2, t_VECSMALL);
1596 : /* Evaluate at dres+ 1 points: 0 (if dres even) and +/- n, so that P_n(X) =
1597 : * P_{-n}(-X), where P_i is Lagrange polynomial: P_i(j) = delta_{i,j} */
1598 2956781 : for (i=0,n = 1; i < dres; n++)
1599 : {
1600 2588362 : x[++i] = n; y[i] = Flx_FlxY_eval_resultant(a,b, x[i], p,pi,la);
1601 2588319 : x[++i] = p-n; y[i] = Flx_FlxY_eval_resultant(a,b, x[i], p,pi,la);
1602 : }
1603 368419 : if (i == dres)
1604 : {
1605 362913 : x[++i] = 0; y[i] = Flx_FlxY_eval_resultant(a,b, x[i], p,pi,la);
1606 : }
1607 368426 : return Flv_polint(x,y, p, sx);
1608 : }
1609 :
1610 : static GEN
1611 7572 : FlxX_pseudorem(GEN x, GEN y, ulong p, ulong pi)
1612 : {
1613 7572 : long vx = varn(x), dx, dy, dz, i, lx, dp;
1614 7572 : pari_sp av = avma, av2;
1615 :
1616 7572 : if (!signe(y)) pari_err_INV("FlxX_pseudorem",y);
1617 7572 : (void)new_chunk(2);
1618 7573 : dx=degpol(x); x = RgX_recip_i(x)+2;
1619 7574 : dy=degpol(y); y = RgX_recip_i(y)+2; dz=dx-dy; dp = dz+1;
1620 7573 : av2 = avma;
1621 : for (;;)
1622 : {
1623 62023 : gel(x,0) = Flx_neg(gel(x,0), p); dp--;
1624 232338 : for (i=1; i<=dy; i++)
1625 169950 : gel(x,i) = Flx_add( Flx_mul_pre(gel(y,0), gel(x,i), p, pi),
1626 170228 : Flx_mul_pre(gel(x,0), gel(y,i), p, pi), p );
1627 1121425 : for ( ; i<=dx; i++)
1628 1060219 : gel(x,i) = Flx_mul_pre(gel(y,0), gel(x,i), p, pi);
1629 65954 : do { x++; dx--; } while (dx >= 0 && lg(gel(x,0))==2);
1630 61206 : if (dx < dy) break;
1631 53634 : if (gc_needed(av2,1))
1632 : {
1633 0 : if(DEBUGMEM>1) pari_warn(warnmem,"FlxX_pseudorem dx = %ld >= %ld",dx,dy);
1634 0 : gc_slice(av2,x,dx+1);
1635 : }
1636 : }
1637 7572 : if (dx < 0) return zero_Flx(0);
1638 7572 : lx = dx+3; x -= 2;
1639 7572 : x[0]=evaltyp(t_POL) | _evallg(lx);
1640 7572 : x[1]=evalsigne(1) | evalvarn(vx);
1641 7572 : x = RgX_recip_i(x);
1642 7573 : if (dp)
1643 : { /* multiply by y[0]^dp [beware dummy vars from FpX_FpXY_resultant] */
1644 1977 : GEN t = Flx_powu_pre(gel(y,0), dp, p, pi);
1645 7912 : for (i=2; i<lx; i++) gel(x,i) = Flx_mul_pre(gel(x,i), t, p, pi);
1646 : }
1647 7575 : return gc_GEN(av, x);
1648 : }
1649 :
1650 : /* return a Flx */
1651 : GEN
1652 2533 : FlxX_resultant(GEN u, GEN v, ulong p, long sx)
1653 : {
1654 2533 : pari_sp av = avma, av2;
1655 : long degq, dx, dy, du, dv, dr, signh;
1656 : ulong pi;
1657 : GEN z, g, h, r, p1;
1658 :
1659 2533 : dx = degpol(u); dy = degpol(v); signh = 1;
1660 2534 : if (dx < dy)
1661 : {
1662 7 : swap(u,v); lswap(dx,dy);
1663 7 : if (both_odd(dx, dy)) signh = -signh;
1664 : }
1665 2534 : if (dy < 0) return zero_Flx(sx);
1666 2534 : pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
1667 2534 : if (dy==0) return gc_upto(av, Flx_powu_pre(gel(v,2),dx,p,pi));
1668 :
1669 2534 : g = h = pol1_Flx(sx); av2 = avma;
1670 : for(;;)
1671 : {
1672 7573 : r = FlxX_pseudorem(u,v,p,pi); dr = lg(r);
1673 7576 : if (dr == 2) { set_avma(av); return zero_Flx(sx); }
1674 7576 : du = degpol(u); dv = degpol(v); degq = du-dv;
1675 7576 : u = v; p1 = g; g = leading_coeff(u);
1676 7576 : switch(degq)
1677 : {
1678 0 : case 0: break;
1679 5583 : case 1:
1680 5583 : p1 = Flx_mul_pre(h,p1, p, pi); h = g; break;
1681 1993 : default:
1682 1993 : p1 = Flx_mul_pre(Flx_powu_pre(h,degq,p,pi), p1, p, pi);
1683 1992 : h = Flx_div_pre(Flx_powu_pre(g,degq,p,pi),
1684 1991 : Flx_powu_pre(h,degq-1,p,pi), p, pi);
1685 : }
1686 7570 : if (both_odd(du,dv)) signh = -signh;
1687 7570 : v = FlxY_Flx_div(r, p1, p);
1688 7572 : if (dr==3) break;
1689 5039 : if (gc_needed(av2,1))
1690 : {
1691 0 : if(DEBUGMEM>1) pari_warn(warnmem,"FlxX_resultant, dr = %ld",dr);
1692 0 : (void)gc_all(av2,4, &u, &v, &g, &h);
1693 : }
1694 : }
1695 2533 : z = gel(v,2);
1696 2533 : if (dv > 1) z = Flx_div_pre(Flx_powu_pre(z,dv,p,pi),
1697 0 : Flx_powu_pre(h,dv-1,p,pi), p, pi);
1698 2533 : if (signh < 0) z = Flx_neg(z,p);
1699 2533 : return gc_upto(av, z);
1700 : }
1701 :
1702 : /* Warning:
1703 : * This function switches between valid and invalid variable ordering*/
1704 :
1705 : static GEN
1706 6144 : FlxY_to_FlyX(GEN b, long sv)
1707 : {
1708 6144 : long i, n=-1;
1709 6144 : long sw = b[1]&VARNBITS;
1710 20973 : for(i=2;i<lg(b);i++) n = maxss(n,lgpol(gel(b,i)));
1711 6144 : return Flm_to_FlxX(Flm_transpose(FlxX_to_Flm(b,n)),sv,sw);
1712 : }
1713 :
1714 : /* Return a Fly*/
1715 : GEN
1716 6144 : Flx_FlxY_resultant(GEN a, GEN b, ulong p)
1717 : {
1718 6144 : pari_sp ltop=avma;
1719 6144 : long dres = degpol(a)*degpol(b);
1720 6144 : long sx=a[1], sy=b[1]&VARNBITS;
1721 : GEN z;
1722 6144 : b = FlxY_to_FlyX(b,sx);
1723 6145 : if ((ulong)dres >= p)
1724 2535 : z = FlxX_resultant(Fly_to_FlxY(a, sy), b, p, sx);
1725 : else
1726 : {
1727 3610 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
1728 3610 : z = Flx_FlxY_resultant_polint(a, b, p, pi, (ulong)dres, sy);
1729 : }
1730 6145 : return gc_upto(ltop,z);
1731 : }
1732 :
1733 : /* Return a t_POL in variable vc whose coeffs are the coeffs of b in
1734 : * variable v; vc must have higher priority than all variables occuring in b. */
1735 : GEN
1736 146017 : swap_vars(GEN b, long v, long vc)
1737 : {
1738 146017 : long i, n = RgX_degree(b, v);
1739 : GEN c, x;
1740 146015 : if (n < 0) return pol_0(vc);
1741 146015 : c = cgetg(n+3, t_POL); x = c + 2;
1742 146016 : c[1] = evalsigne(1) | evalvarn(vc);
1743 967876 : for (i = 0; i <= n; i++) gel(x,i) = polcoef_i(b, i, v);
1744 146013 : return c;
1745 : }
1746 :
1747 : /* assume varn(b) << varn(a) */
1748 : /* return a FpY*/
1749 : GEN
1750 15 : FpX_FpXY_resultant(GEN a, GEN b, GEN p)
1751 : {
1752 15 : long i,n,dres, db, vY = varn(b), vX = varn(a);
1753 : GEN la,x,y;
1754 :
1755 15 : if (lgefint(p) == 3)
1756 : {
1757 0 : ulong pp = uel(p,2);
1758 0 : b = ZXX_to_FlxX(b, pp, vX);
1759 0 : a = ZX_to_Flx(a, pp);
1760 0 : x = Flx_FlxY_resultant(a, b, pp);
1761 0 : return Flx_to_ZX(x);
1762 : }
1763 15 : db = RgXY_degreex(b);
1764 15 : dres = degpol(a)*degpol(b);
1765 15 : la = leading_coeff(a);
1766 15 : x = cgetg(dres+2, t_VEC);
1767 15 : y = cgetg(dres+2, t_VEC);
1768 : /* Evaluate at dres+ 1 points: 0 (if dres even) and +/- n, so that P_n(X) =
1769 : * P_{-n}(-X), where P_i is Lagrange polynomial: P_i(j) = delta_{i,j} */
1770 157 : for (i=0,n = 1; i < dres; n++)
1771 : {
1772 142 : gel(x,++i) = utoipos(n);
1773 142 : gel(y,i) = FpX_FpXY_eval_resultant(a,b,gel(x,i),p,la,db,vY);
1774 142 : gel(x,++i) = subiu(p,n);
1775 142 : gel(y,i) = FpX_FpXY_eval_resultant(a,b,gel(x,i),p,la,db,vY);
1776 : }
1777 15 : if (i == dres)
1778 : {
1779 0 : gel(x,++i) = gen_0;
1780 0 : gel(y,i) = FpX_FpXY_eval_resultant(a,b, gel(x,i), p,la,db,vY);
1781 : }
1782 15 : return FpV_polint(x,y, p, vY);
1783 : }
1784 :
1785 : GEN
1786 191 : FpX_composedsum(GEN P, GEN Q, GEN p)
1787 : {
1788 191 : pari_sp av = avma;
1789 191 : if (lgefint(p)==3)
1790 : {
1791 0 : ulong pp = p[2];
1792 0 : GEN z = Flx_composedsum(ZX_to_Flx(P, pp), ZX_to_Flx(Q, pp), pp);
1793 0 : return gc_upto(av, Flx_to_ZX(z));
1794 : }
1795 : else
1796 : {
1797 191 : long n = 1+ degpol(P)*degpol(Q);
1798 191 : GEN Pl = FpX_invLaplace(FpX_Newton(P,n,p), p);
1799 191 : GEN Ql = FpX_invLaplace(FpX_Newton(Q,n,p), p);
1800 191 : GEN L = FpX_Laplace(FpXn_mul(Pl, Ql, n, p), p);
1801 191 : GEN lead = Fp_mul(Fp_powu(leading_coeff(P),degpol(Q), p),
1802 191 : Fp_powu(leading_coeff(Q),degpol(P), p), p);
1803 191 : GEN R = FpX_fromNewton(L, p);
1804 191 : return gc_upto(av, FpX_Fp_mul(R, lead, p));
1805 : }
1806 : }
1807 :
1808 : GEN
1809 0 : FpX_composedprod(GEN P, GEN Q, GEN p)
1810 : {
1811 0 : pari_sp av = avma;
1812 0 : if (lgefint(p)==3)
1813 : {
1814 0 : ulong pp = p[2];
1815 0 : GEN z = Flx_composedprod(ZX_to_Flx(P, pp), ZX_to_Flx(Q, pp), pp);
1816 0 : return gc_upto(av, Flx_to_ZX(z));
1817 : }
1818 : else
1819 : {
1820 0 : long n = 1+ degpol(P)*degpol(Q);
1821 0 : GEN L = FpX_convol(FpX_Newton(P,n,p), FpX_Newton(Q,n,p), p);
1822 0 : return gc_upto(av,FpX_fromNewton(L, p));
1823 : }
1824 : }
1825 :
1826 : static GEN
1827 191 : _FpX_composedsum(void *E, GEN a, GEN b)
1828 191 : { return FpX_composedsum(a,b, (GEN)E); }
1829 :
1830 : GEN
1831 1637 : FpXV_composedsum(GEN V, GEN p)
1832 : {
1833 1637 : if (lgefint(p)==3)
1834 : {
1835 0 : ulong pp = p[2];
1836 0 : return Flx_to_ZX(FlxV_composedsum(ZXV_to_FlxV(V, pp), pp));
1837 : }
1838 1637 : return gen_product(V, (void *)p, &_FpX_composedsum);
1839 : }
1840 :
1841 : /* 0, 1, -1, 2, -2, ... */
1842 : #define next_lambda(a) (a>0 ? -a : 1-a)
1843 :
1844 : /* Assume A in Z[Y], B in Q[Y][X], B squarefree in (Q[Y]/(A))[X] and
1845 : * Res_Y(A, B) in Z[X]. Find a small lambda (start from *lambda, use
1846 : * next_lambda successively) such that C(X) = Res_Y(A(Y), B(X + lambda Y))
1847 : * is squarefree, reset *lambda to the chosen value and return C. Set LERS to
1848 : * the Last nonconstant polynomial in the Euclidean Remainder Sequence */
1849 : static GEN
1850 21840 : ZX_ZXY_resultant_LERS(GEN A, GEN B0, long *plambda, GEN *LERS)
1851 : {
1852 : ulong bound, dp;
1853 21840 : pari_sp av = avma, av2 = 0;
1854 21840 : long lambda = *plambda, degA = degpol(A), dres = degA*degpol(B0);
1855 : long stable, checksqfree, i,n, cnt, degB;
1856 21840 : long v, vX = varn(B0), vY = varn(A); /* vY < vX */
1857 : GEN x, y, dglist, B, q, a, b, ev, H, H0, H1, Hp, H0p, H1p, C0, C1;
1858 : forprime_t S;
1859 :
1860 21840 : if (degA == 1)
1861 : {
1862 1260 : GEN a1 = gel(A,3), a0 = gel(A,2);
1863 1260 : B = lambda? RgX_translate(B0, monomial(stoi(lambda), 1, vY)): B0;
1864 1260 : H = gsubst(B, vY, gdiv(gneg(a0),a1));
1865 1260 : if (!equali1(a1)) H = RgX_Rg_mul(H, powiu(a1, poldegree(B,vY)));
1866 1260 : *LERS = mkvec2(scalarpol_shallow(a0,vX), scalarpol_shallow(a1,vX));
1867 1260 : return gc_all(av, 2, &H, LERS);
1868 : }
1869 :
1870 20580 : dglist = Hp = H0p = H1p = C0 = C1 = NULL; /* gcc -Wall */
1871 20580 : C0 = cgetg(dres+2, t_VECSMALL);
1872 20580 : C1 = cgetg(dres+2, t_VECSMALL);
1873 20580 : dglist = cgetg(dres+1, t_VECSMALL);
1874 20580 : x = cgetg(dres+2, t_VECSMALL);
1875 20580 : y = cgetg(dres+2, t_VECSMALL);
1876 20580 : B0 = leafcopy(B0);
1877 20580 : A = leafcopy(A);
1878 20580 : B = B0;
1879 20580 : v = fetch_var_higher(); setvarn(A,v);
1880 : /* make sure p large enough */
1881 21466 : INIT:
1882 : /* always except the first time */
1883 21466 : if (av2) { set_avma(av2); lambda = next_lambda(lambda); }
1884 21466 : if (lambda) B = RgX_translate(B0, monomial(stoi(lambda), 1, vY));
1885 21466 : B = swap_vars(B, vY, v);
1886 : /* B0(lambda v + x, v) */
1887 21466 : if (DEBUGLEVEL>4) err_printf("Trying lambda = %ld\n", lambda);
1888 21466 : av2 = avma;
1889 :
1890 21466 : if (degA <= 3)
1891 : { /* sub-resultant faster for small degrees */
1892 10752 : H = RgX_resultant_all(A,B,&q);
1893 10752 : if (typ(q) != t_POL || degpol(q)!=1) goto INIT;
1894 10031 : H0 = gel(q,2);
1895 10031 : if (typ(H0) == t_POL) setvarn(H0,vX); else H0 = scalarpol(H0,vX);
1896 10031 : H1 = gel(q,3);
1897 10031 : if (typ(H1) == t_POL) setvarn(H1,vX); else H1 = scalarpol(H1,vX);
1898 10031 : if (!ZX_is_squarefree(H)) goto INIT;
1899 9989 : goto END;
1900 : }
1901 :
1902 10714 : H = H0 = H1 = NULL;
1903 10714 : degB = degpol(B);
1904 10714 : bound = ZX_ZXY_ResBound(A, B, NULL);
1905 10714 : if (DEBUGLEVEL>4) err_printf("bound for resultant coeffs: 2^%ld\n",bound);
1906 10714 : dp = 1;
1907 10714 : init_modular_big(&S);
1908 10714 : for(cnt = 0, checksqfree = 1;;)
1909 49195 : {
1910 59909 : ulong p = u_forprime_next(&S);
1911 : GEN Hi;
1912 59909 : a = ZX_to_Flx(A, p);
1913 59910 : b = ZXX_to_FlxX(B, p, varn(A));
1914 59910 : if (degpol(a) < degA || degpol(b) < degB) continue; /* p | lc(A)lc(B) */
1915 59910 : if (checksqfree)
1916 : { /* find degree list for generic Euclidean Remainder Sequence */
1917 10714 : long goal = minss(degpol(a), degpol(b)); /* longest possible */
1918 73182 : for (n=1; n <= goal; n++) dglist[n] = 0;
1919 10714 : setlg(dglist, 1);
1920 23664 : for (n=0; n <= dres; n++)
1921 : {
1922 23272 : ev = FlxY_evalx_drop(b, n, p);
1923 23272 : Flx_resultant_set_dglist(a, ev, dglist, p);
1924 23272 : if (lg(dglist)-1 == goal) break;
1925 : }
1926 : /* last pol in ERS has degree > 1 ? */
1927 10714 : goal = lg(dglist)-1;
1928 10714 : if (degpol(B) == 1) { if (!goal) goto INIT; }
1929 : else
1930 : {
1931 10658 : if (goal <= 1) goto INIT;
1932 10595 : if (dglist[goal] != 0 || dglist[goal-1] != 1) goto INIT;
1933 : }
1934 10651 : if (DEBUGLEVEL>4)
1935 0 : err_printf("Degree list for ERS (trials: %ld) = %Ps\n",n+1,dglist);
1936 : }
1937 :
1938 2144761 : for (i=0,n = 0; i <= dres; n++)
1939 : {
1940 2084923 : ev = FlxY_evalx_drop(b, n, p);
1941 2084696 : x[++i] = n; y[i] = Flx_resultant_all(a, ev, C0+i, C1+i, dglist, p);
1942 2084914 : if (!C1[i]) i--; /* C1(i) = 0. No way to recover C0(i) */
1943 : }
1944 59838 : Hi = Flv_Flm_polint(x, mkvec3(y,C0,C1), p, 0);
1945 59846 : Hp = gel(Hi,1); H0p = gel(Hi,2); H1p = gel(Hi,3);
1946 59846 : if (!H && degpol(Hp) != dres) continue;
1947 59846 : if (dp != 1) Hp = Flx_Fl_mul(Hp, Fl_powu(Fl_inv(dp,p), degA, p), p);
1948 59846 : if (checksqfree) {
1949 10651 : if (!Flx_is_squarefree(Hp, p)) goto INIT;
1950 10591 : if (DEBUGLEVEL>4) err_printf("Final lambda = %ld\n", lambda);
1951 10591 : checksqfree = 0;
1952 : }
1953 :
1954 59786 : if (!H)
1955 : { /* initialize */
1956 10591 : q = utoipos(p); stable = 0;
1957 10591 : H = ZX_init_CRT(Hp, p,vX);
1958 10591 : H0= ZX_init_CRT(H0p, p,vX);
1959 10591 : H1= ZX_init_CRT(H1p, p,vX);
1960 : }
1961 : else
1962 : {
1963 49195 : GEN qp = muliu(q,p);
1964 49195 : stable = ZX_incremental_CRT_raw(&H, Hp, q,qp, p)
1965 49196 : & ZX_incremental_CRT_raw(&H0,H0p, q,qp, p)
1966 49195 : & ZX_incremental_CRT_raw(&H1,H1p, q,qp, p);
1967 49195 : q = qp;
1968 : }
1969 : /* could make it probabilistic for H ? [e.g if stable twice, etc]
1970 : * Probabilistic anyway for H0, H1 */
1971 59786 : if (DEBUGLEVEL>5 && (stable || ++cnt==100))
1972 0 : { cnt=0; err_printf("%ld%%%s ",100*expi(q)/bound,stable?"s":""); }
1973 59786 : if (stable && (ulong)expi(q) >= bound) break; /* DONE */
1974 49195 : if (gc_needed(av,2))
1975 : {
1976 0 : if (DEBUGMEM>1) pari_warn(warnmem,"ZX_ZXY_rnfequation");
1977 0 : (void)gc_all(av2, 4, &H, &q, &H0, &H1);
1978 : }
1979 : }
1980 20580 : END:
1981 20580 : if (DEBUGLEVEL>5) err_printf(" done\n");
1982 20580 : setvarn(H, vX); (void)delete_var();
1983 20580 : *LERS = mkvec2(H0,H1);
1984 20580 : *plambda = lambda; return gc_all(av, 2, &H, LERS);
1985 : }
1986 :
1987 : GEN
1988 59633 : ZX_ZXY_resultant_all(GEN A, GEN B, long *plambda, GEN *LERS)
1989 : {
1990 59633 : if (LERS)
1991 : {
1992 21840 : if (!plambda)
1993 0 : pari_err_BUG("ZX_ZXY_resultant_all [LERS != NULL needs lambda]");
1994 21840 : return ZX_ZXY_resultant_LERS(A, B, plambda, LERS);
1995 : }
1996 37793 : return ZX_ZXY_rnfequation(A, B, plambda);
1997 : }
1998 :
1999 : /* If lambda = NULL, return caract(Mod(A, T)), T,A in Z[X].
2000 : * Otherwise find a small lambda such that caract (Mod(A + lambda X, T)) is
2001 : * squarefree */
2002 : GEN
2003 22594 : ZXQ_charpoly_sqf(GEN A, GEN T, long *lambda, long v)
2004 : {
2005 22594 : pari_sp av = avma;
2006 : GEN R, a;
2007 : long dA;
2008 : int delvar;
2009 :
2010 22594 : if (v < 0) v = 0;
2011 22594 : switch (typ(A))
2012 : {
2013 22594 : case t_POL: dA = degpol(A); if (dA > 0) break;
2014 0 : A = constant_coeff(A);
2015 0 : default:
2016 0 : if (lambda) { A = scalar_ZX_shallow(A,varn(T)); dA = 0; break;}
2017 0 : return gc_upto(av, gpowgs(gsub(pol_x(v), A), degpol(T)));
2018 : }
2019 22594 : delvar = 0;
2020 22594 : if (varncmp(varn(T), 0) <= 0)
2021 : {
2022 3681 : long v0 = fetch_var(); delvar = 1;
2023 3681 : T = leafcopy(T); setvarn(T,v0);
2024 3681 : A = leafcopy(A); setvarn(A,v0);
2025 : }
2026 22594 : R = ZX_ZXY_rnfequation(T, deg1pol_shallow(gen_1, gneg_i(A), 0), lambda);
2027 22594 : if (delvar) (void)delete_var();
2028 22594 : setvarn(R, v); a = leading_coeff(T);
2029 22594 : if (!gequal1(a)) R = gdiv(R, powiu(a, dA));
2030 22594 : return gc_upto(av, R);
2031 : }
2032 :
2033 : /* charpoly(Mod(A,T)), A may be in Q[X], but assume T and result are integral */
2034 : GEN
2035 995247 : ZXQ_charpoly(GEN A, GEN T, long v)
2036 : {
2037 995247 : return (degpol(T) < 16) ? RgXQ_charpoly_i(A,T,v): ZXQ_charpoly_sqf(A,T, NULL, v);
2038 : }
2039 :
2040 : GEN
2041 9772 : QXQ_charpoly(GEN A, GEN T, long v)
2042 : {
2043 9772 : pari_sp av = avma;
2044 9772 : GEN den, B = Q_remove_denom(A, &den);
2045 9772 : GEN P = ZXQ_charpoly(B, T, v);
2046 9772 : return gc_GEN(av, den ? RgX_rescale(P, ginv(den)): P);
2047 : }
2048 :
2049 : static ulong
2050 3863712 : ZX_resultant_prime(GEN a, GEN b, GEN dB, long degA, long degB, ulong p)
2051 : {
2052 3863712 : long dropa = degA - degpol(a), dropb = degB - degpol(b);
2053 : ulong H, dp;
2054 3863607 : if (dropa && dropb) return 0; /* p | lc(A), p | lc(B) */
2055 3863607 : H = Flx_resultant(a, b, p);
2056 3863385 : if (dropa)
2057 : { /* multiply by ((-1)^deg B lc(B))^(deg A - deg a) */
2058 0 : ulong c = b[degB+2]; /* lc(B) */
2059 0 : if (odd(degB)) c = p - c;
2060 0 : c = Fl_powu(c, dropa, p);
2061 0 : if (c != 1) H = Fl_mul(H, c, p);
2062 : }
2063 3863385 : else if (dropb)
2064 : { /* multiply by lc(A)^(deg B - deg b) */
2065 0 : ulong c = a[degA+2]; /* lc(A) */
2066 0 : c = Fl_powu(c, dropb, p);
2067 0 : if (c != 1) H = Fl_mul(H, c, p);
2068 : }
2069 3863385 : dp = dB ? umodiu(dB, p): 1;
2070 3863384 : if (dp != 1) H = Fl_mul(H, Fl_powu(Fl_inv(dp,p), degA, p), p);
2071 3863388 : return H;
2072 : }
2073 :
2074 : /* If B=NULL, assume B=A' */
2075 : static GEN
2076 1494393 : ZX_resultant_slice(GEN A, GEN B, GEN dB, GEN P, GEN *mod)
2077 : {
2078 1494393 : pari_sp av = avma, av2;
2079 1494393 : long degA, degB, i, n = lg(P)-1;
2080 : GEN H, T;
2081 :
2082 1494393 : degA = degpol(A);
2083 1494391 : degB = B? degpol(B): degA - 1;
2084 1494392 : if (n == 1)
2085 : {
2086 810604 : ulong Hp, p = uel(P,1);
2087 810604 : GEN a = ZX_to_Flx(A, p), b = B? ZX_to_Flx(B, p): Flx_deriv(a, p);
2088 810588 : Hp = ZX_resultant_prime(a, b, dB, degA, degB, p);
2089 810616 : set_avma(av); *mod = utoipos(p); return utoi(Hp);
2090 : }
2091 683788 : T = ZV_producttree(P);
2092 683785 : A = ZX_nv_mod_tree(A, P, T);
2093 683781 : if (B) B = ZX_nv_mod_tree(B, P, T);
2094 683781 : H = cgetg(n+1, t_VECSMALL); av2 = avma;
2095 3736635 : for(i=1; i <= n; i++, set_avma(av2))
2096 : {
2097 3052853 : ulong p = P[i];
2098 3052853 : GEN a = gel(A,i), b = B? gel(B,i): Flx_deriv(a, p);
2099 3053114 : H[i] = ZX_resultant_prime(a, b, dB, degA, degB, p);
2100 : }
2101 683782 : H = ZV_chinese_tree(H, P, T, ZV_chinesetree(P,T));
2102 683787 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2103 : }
2104 :
2105 : GEN
2106 1494405 : ZX_resultant_worker(GEN P, GEN A, GEN B, GEN dB)
2107 : {
2108 1494405 : GEN V = cgetg(3, t_VEC);
2109 1494393 : if (typ(B) == t_INT) B = NULL;
2110 1494393 : if (!signe(dB)) dB = NULL;
2111 1494393 : gel(V,1) = ZX_resultant_slice(A, B, dB, P, &gel(V,2));
2112 1494401 : return V;
2113 : }
2114 :
2115 : /* Compute Res(A, B/dB) in Z, assuming A,B in Z[X], dB in Z or NULL (= 1)
2116 : * If B=NULL, take B = A' and assume deg A > 1 and 'bound' is set */
2117 : GEN
2118 1351094 : ZX_resultant_all(GEN A, GEN B, GEN dB, ulong bound)
2119 : {
2120 1351094 : pari_sp av = avma;
2121 : forprime_t S;
2122 : GEN H, worker;
2123 1351094 : if (!B && degpol(A)==2)
2124 : {
2125 113998 : GEN a = gel(A,4), b = gel(A,3), c = gel(A,2);
2126 113998 : H = mulii(a, subii(shifti(mulii(a, c), 2), sqri(b)));
2127 113986 : if (dB) H = diviiexact(H, sqri(dB));
2128 113986 : return gc_INT(av, H);
2129 : }
2130 1237094 : if (B)
2131 : {
2132 155204 : long a = degpol(A), b = degpol(B);
2133 155204 : if (a < 0 || b < 0) return gen_0;
2134 155174 : if (!a) return powiu(gel(A,2), b);
2135 155174 : if (!b) return powiu(gel(B,2), a);
2136 153429 : if (minss(a, b) <= 1)
2137 : {
2138 76735 : H = RgX_resultant_all(A, B, NULL);
2139 76736 : if (dB) H = diviiexact(H, powiu(dB, a));
2140 76736 : return gc_INT(av, H);
2141 : }
2142 76694 : if (!bound) bound = ZX_ZXY_ResBound(A, B, dB);
2143 : }
2144 1158590 : worker = snm_closure(is_entry("_ZX_resultant_worker"),
2145 : mkvec3(A, B? B: gen_0, dB? dB: gen_0));
2146 1158643 : init_modular_big(&S);
2147 1158611 : H = gen_crt("ZX_resultant_all", worker, &S, dB, bound, 0, NULL,
2148 : ZV_chinese_center, Fp_center);
2149 1158636 : return gc_INT(av, H);
2150 : }
2151 :
2152 : /* A0 and B0 in Q[X] */
2153 : GEN
2154 56 : QX_resultant(GEN A0, GEN B0)
2155 : {
2156 : GEN s, a, b, A, B;
2157 56 : pari_sp av = avma;
2158 :
2159 56 : A = Q_primitive_part(A0, &a);
2160 56 : B = Q_primitive_part(B0, &b);
2161 56 : s = ZX_resultant(A, B);
2162 56 : if (!signe(s)) { set_avma(av); return gen_0; }
2163 56 : if (a) s = gmul(s, gpowgs(a,degpol(B)));
2164 56 : if (b) s = gmul(s, gpowgs(b,degpol(A)));
2165 56 : return gc_upto(av, s);
2166 : }
2167 :
2168 : GEN
2169 57309 : ZX_resultant(GEN A, GEN B) { return ZX_resultant_all(A,B,NULL,0); }
2170 :
2171 : GEN
2172 0 : QXQ_intnorm(GEN A, GEN B)
2173 : {
2174 : GEN c, n, R, lB;
2175 0 : long dA = degpol(A), dB = degpol(B);
2176 0 : pari_sp av = avma;
2177 0 : if (dA < 0) return gen_0;
2178 0 : A = Q_primitive_part(A, &c);
2179 0 : if (!c || typ(c) == t_INT) {
2180 0 : n = c;
2181 0 : R = ZX_resultant(B, A);
2182 : } else {
2183 0 : n = gel(c,1);
2184 0 : R = ZX_resultant_all(B, A, gel(c,2), 0);
2185 : }
2186 0 : if (n && !equali1(n)) R = mulii(R, powiu(n, dB));
2187 0 : lB = leading_coeff(B);
2188 0 : if (!equali1(lB)) R = diviiexact(R, powiu(lB, dA));
2189 0 : return gc_INT(av, R);
2190 : }
2191 :
2192 : GEN
2193 19418 : QXQ_norm(GEN A, GEN B)
2194 : {
2195 : GEN c, R, lB;
2196 19418 : long dA = degpol(A), dB = degpol(B);
2197 19418 : pari_sp av = avma;
2198 19418 : if (dA < 0) return gen_0;
2199 19418 : A = Q_primitive_part(A, &c);
2200 19418 : R = ZX_resultant(B, A);
2201 19418 : if (c) R = gmul(R, gpowgs(c, dB));
2202 19418 : lB = leading_coeff(B);
2203 19418 : if (!equali1(lB)) R = gdiv(R, gpowgs(lB, dA));
2204 19418 : return gc_upto(av, R);
2205 : }
2206 :
2207 : /* assume x has integral coefficients */
2208 : GEN
2209 1199227 : ZX_disc_all(GEN x, ulong bound)
2210 : {
2211 1199227 : pari_sp av = avma;
2212 1199227 : long s, d = degpol(x);
2213 : GEN l, R;
2214 :
2215 1199225 : if (d <= 1) return d == 1? gen_1: gen_0;
2216 1195932 : s = (d & 2) ? -1: 1;
2217 1195932 : l = leading_coeff(x);
2218 1195928 : if (!bound) bound = ZX_discbound(x);
2219 1195869 : R = ZX_resultant_all(x, NULL, NULL, bound);
2220 1195902 : if (is_pm1(l))
2221 1016881 : { if (signe(l) < 0) s = -s; }
2222 : else
2223 179023 : R = diviiexact(R,l);
2224 1195904 : if (s == -1) togglesign_safe(&R);
2225 1195903 : return gc_INT(av,R);
2226 : }
2227 :
2228 : GEN
2229 1137382 : ZX_disc(GEN x) { return ZX_disc_all(x,0); }
2230 :
2231 : static GEN
2232 10784 : ZXQX_resultant_prime(GEN a, GEN b, GEN dB, long degA, long degB, GEN T, ulong p)
2233 : {
2234 10784 : long dropa = degA - degpol(a), dropb = degB - degpol(b);
2235 : GEN H, dp;
2236 10783 : if (dropa && dropb) return pol0_Flx(T[1]); /* p | lc(A), p | lc(B) */
2237 10783 : H = FlxqX_saferesultant(a, b, T, p);
2238 10783 : if (!H) return NULL;
2239 10783 : if (dropa)
2240 : { /* multiply by ((-1)^deg B lc(B))^(deg A - deg a) */
2241 0 : GEN c = gel(b,degB+2); /* lc(B) */
2242 0 : if (odd(degB)) c = Flx_neg(c, p);
2243 0 : c = Flxq_powu(c, dropa, T, p);
2244 0 : if (!Flx_equal1(c)) H = Flxq_mul(H, c, T, p);
2245 : }
2246 10783 : else if (dropb)
2247 : { /* multiply by lc(A)^(deg B - deg b) */
2248 0 : GEN c = gel(a,degA+2); /* lc(A) */
2249 0 : c = Flxq_powu(c, dropb, T, p);
2250 0 : if (!Flx_equal1(c)) H = Flxq_mul(H, c, T, p);
2251 : }
2252 10783 : dp = dB ? ZX_to_Flx(dB, p): pol1_Flx(T[1]);
2253 10782 : if (!Flx_equal1(dp))
2254 : {
2255 0 : GEN idp = Flxq_invsafe(dp, T, p);
2256 0 : if (!idp) return NULL;
2257 0 : H = Flxq_mul(H, Flxq_powu(idp, degA, T, p), T, p);
2258 : }
2259 10782 : return H;
2260 : }
2261 :
2262 : /* If B=NULL, assume B=A' */
2263 : static GEN
2264 4687 : ZXQX_resultant_slice(GEN A, GEN B, GEN U, GEN dB, GEN P, GEN *mod)
2265 : {
2266 4687 : pari_sp av = avma;
2267 4687 : long degA, degB, i, n = lg(P)-1;
2268 : GEN H, T;
2269 4687 : long v = varn(U), redo = 0;
2270 :
2271 4687 : degA = degpol(A);
2272 4687 : degB = B? degpol(B): degA - 1;
2273 4687 : if (n == 1)
2274 : {
2275 2953 : ulong p = uel(P,1);
2276 2953 : GEN a = ZXX_to_FlxX(A, p, v), b = B? ZXX_to_FlxX(B, p, v): FlxX_deriv(a, p);
2277 2953 : GEN u = ZX_to_Flx(U, p);
2278 2953 : GEN Hp = ZXQX_resultant_prime(a, b, dB, degA, degB, u, p);
2279 2953 : if (!Hp) { set_avma(av); *mod = gen_1; return pol_0(v); }
2280 2953 : Hp = gc_upto(av, Flx_to_ZX(Hp)); *mod = utoipos(p); return Hp;
2281 : }
2282 1734 : T = ZV_producttree(P);
2283 1734 : A = ZXX_nv_mod_tree(A, P, T, v);
2284 1734 : if (B) B = ZXX_nv_mod_tree(B, P, T, v);
2285 1734 : U = ZX_nv_mod_tree(U, P, T);
2286 1734 : H = cgetg(n+1, t_VEC);
2287 9565 : for(i=1; i <= n; i++)
2288 : {
2289 7831 : ulong p = P[i];
2290 7831 : GEN a = gel(A,i), b = B? gel(B,i): FlxX_deriv(a, p), u = gel(U, i);
2291 7831 : GEN h = ZXQX_resultant_prime(a, b, dB, degA, degB, u, p);
2292 7829 : if (!h)
2293 : {
2294 0 : gel(H,i) = pol_0(v);
2295 0 : P[i] = 1; redo = 1;
2296 : }
2297 : else
2298 7829 : gel(H,i) = h;
2299 : }
2300 1734 : if (redo) T = ZV_producttree(P);
2301 1734 : H = nxV_chinese_center_tree(H, P, T, ZV_chinesetree(P, T));
2302 1734 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2303 : }
2304 :
2305 : GEN
2306 4687 : ZXQX_resultant_worker(GEN P, GEN A, GEN B, GEN T, GEN dB)
2307 : {
2308 4687 : GEN V = cgetg(3, t_VEC);
2309 4687 : if (isintzero(B)) B = NULL;
2310 4687 : if (!signe(dB)) dB = NULL;
2311 4687 : gel(V,1) = ZXQX_resultant_slice(A, B, T, dB, P, &gel(V,2));
2312 4687 : return V;
2313 : }
2314 :
2315 : static ulong
2316 4091 : ZXQX_resultant_bound_i(GEN nf, GEN A, GEN B, GEN (*f)(GEN,GEN,long))
2317 : {
2318 4091 : pari_sp av = avma;
2319 4091 : GEN r, M = nf_L2_bound(nf, NULL, &r);
2320 4091 : long v = nf_get_varn(nf), i, l = lg(r);
2321 4091 : GEN a = cgetg(l, t_COL);
2322 12662 : for (i = 1; i < l; i++)
2323 8571 : gel(a, i) = f(gsubst(A, v, gel(r,i)), gsubst(B, v, gel(r,i)), DEFAULTPREC);
2324 4091 : return gc_ulong(av, (ulong) dbllog2(gmul(M,RgC_fpnorml2(a, DEFAULTPREC))));
2325 : }
2326 : static ulong
2327 3776 : ZXQX_resultant_bound(GEN nf, GEN A, GEN B)
2328 3776 : { return ZXQX_resultant_bound_i(nf, A, B, &RgX_RgXY_ResBound); }
2329 :
2330 : static GEN
2331 56 : _ZXQ_powu(GEN x, ulong u, GEN T)
2332 56 : { return typ(x) == t_INT? powiu(x, u): ZXQ_powu(x, u, T); }
2333 :
2334 : /* Compute Res(A, B/dB) in Z[X]/T, assuming A,B in Z[X,Y], dB in Z or NULL (= 1)
2335 : * If B=NULL, take B = A' and assume deg A > 1 */
2336 : static GEN
2337 3773 : ZXQX_resultant_all(GEN A, GEN B, GEN T, GEN dB, ulong bound)
2338 : {
2339 3773 : pari_sp av = avma;
2340 : forprime_t S;
2341 : GEN H, worker;
2342 3773 : if (B)
2343 : {
2344 63 : long a = degpol(A), b = degpol(B);
2345 63 : if (a < 0 || b < 0) return gen_0;
2346 63 : if (!a) return _ZXQ_powu(gel(A,2), b, T);
2347 63 : if (!b) return _ZXQ_powu(gel(B,2), a, T);
2348 : } else
2349 3710 : if (!bound) B = RgX_deriv(A);
2350 3773 : if (!bound) bound = ZXQX_resultant_bound(nfinit(T, DEFAULTPREC), A, B);
2351 3773 : worker = snm_closure(is_entry("_ZXQX_resultant_worker"),
2352 : mkvec4(A, B? B: gen_0, T, dB? dB: gen_0));
2353 3773 : init_modular_big(&S);
2354 3773 : H = gen_crt("ZXQX_resultant_all", worker, &S, dB, bound, 0, NULL,
2355 : nxV_chinese_center, FpX_center);
2356 3773 : if (DEBUGLEVEL)
2357 0 : err_printf("ZXQX_resultant_all: a priori bound: %lu, a posteriori: %lu\n",
2358 : bound, expi(gsupnorm(H, DEFAULTPREC)));
2359 3773 : return gc_upto(av, H);
2360 : }
2361 :
2362 : GEN
2363 119 : nfX_resultant(GEN nf, GEN x, GEN y)
2364 : {
2365 119 : pari_sp av = avma;
2366 119 : GEN cx, cy, D, T = nf_get_pol(nf);
2367 119 : long dx = degpol(x), dy = degpol(y);
2368 119 : if (dx < 0 || dy < 0) return gen_0;
2369 119 : x = Q_primitive_part(x, &cx); if (cx) cx = gpowgs(cx, dy);
2370 119 : y = Q_primitive_part(y, &cy); if (cy) cy = gpowgs(cy, dx);
2371 119 : if (!dx) D = _ZXQ_powu(gel(x,2), dy, T);
2372 119 : else if (!dy) D = _ZXQ_powu(gel(y,2), dx, T);
2373 : else
2374 : {
2375 63 : ulong bound = ZXQX_resultant_bound(nf, x, y);
2376 63 : D = ZXQX_resultant_all(x, y, T, NULL, bound);
2377 : }
2378 119 : cx = mul_content(cx, cy); if (cx) D = gmul(D, cx);
2379 119 : return gc_upto(av, D);
2380 : }
2381 :
2382 : static GEN
2383 252 : to_ZX(GEN a, long v) { return typ(a)==t_INT? scalarpol(a,v): a; }
2384 :
2385 : static GEN
2386 3710 : ZXQX_disc_all(GEN x, GEN T, ulong bound)
2387 : {
2388 3710 : pari_sp av = avma;
2389 3710 : long s, d = degpol(x), v = varn(T);
2390 : GEN l, R;
2391 :
2392 3710 : if (d <= 1) return d == 1? pol_1(v): pol_0(v);
2393 3710 : s = (d & 2) ? -1: 1;
2394 3710 : l = leading_coeff(x);
2395 3710 : R = ZXQX_resultant_all(x, NULL, T, NULL, bound);
2396 3710 : if (!gequal1(l)) R = QXQ_div(R, to_ZX(l,v), T);
2397 3710 : if (s == -1) R = RgX_neg(R);
2398 3710 : return gc_upto(av, R);
2399 : }
2400 :
2401 : GEN
2402 7 : QX_disc(GEN x)
2403 : {
2404 7 : pari_sp av = avma;
2405 7 : GEN c, d = ZX_disc( Q_primitive_part(x, &c) );
2406 7 : if (c) d = gmul(d, gpowgs(c, 2*degpol(x) - 2));
2407 7 : return gc_upto(av, d);
2408 : }
2409 :
2410 : GEN
2411 3941 : nfX_disc(GEN nf, GEN x)
2412 : {
2413 3941 : pari_sp av = avma;
2414 3941 : GEN c, D, T = nf_get_pol(nf);
2415 : ulong bound;
2416 3941 : long d = degpol(x), v = varn(T);
2417 3941 : if (d <= 1) return d == 1? pol_1(v): pol_0(v);
2418 3710 : x = Q_primitive_part(x, &c);
2419 3710 : bound = ZXQX_resultant_bound(nf, x, RgX_deriv(x));
2420 3710 : D = ZXQX_disc_all(x, T, bound);
2421 3710 : if (c) D = gmul(D, gpowgs(c, 2*d - 2));
2422 3710 : return gc_upto(av, D);
2423 : }
2424 :
2425 : GEN
2426 837278 : QXQ_mul(GEN x, GEN y, GEN T)
2427 : {
2428 837278 : GEN dx, nx = Q_primitive_part(x, &dx);
2429 837276 : GEN dy, ny = Q_primitive_part(y, &dy);
2430 837277 : GEN z = ZXQ_mul(nx, ny, T);
2431 837279 : if (dx || dy)
2432 : {
2433 834479 : GEN d = dx ? dy ? gmul(dx, dy): dx : dy;
2434 834478 : if (!gequal1(d)) z = ZX_Q_mul(z, d);
2435 : }
2436 837279 : return z;
2437 : }
2438 :
2439 : GEN
2440 399479 : QXQ_sqr(GEN x, GEN T)
2441 : {
2442 399479 : GEN dx, nx = Q_primitive_part(x, &dx);
2443 399479 : GEN z = ZXQ_sqr(nx, T);
2444 399479 : if (dx)
2445 397743 : z = ZX_Q_mul(z, gsqr(dx));
2446 399479 : return z;
2447 : }
2448 :
2449 : static GEN
2450 212561 : QXQ_inv_slice(GEN A, GEN B, GEN P, GEN *mod)
2451 : {
2452 212561 : pari_sp av = avma;
2453 212561 : long i, n = lg(P)-1, v = varn(A), redo = 0;
2454 : GEN H, T;
2455 212561 : if (n == 1)
2456 : {
2457 165557 : ulong p = uel(P,1);
2458 165557 : GEN a = ZX_to_Flx(A, p), b = ZX_to_Flx(B, p);
2459 165557 : GEN U = Flxq_invsafe(a, b, p);
2460 165557 : if (!U)
2461 : {
2462 24 : set_avma(av);
2463 24 : *mod = gen_1; return pol_0(v);
2464 : }
2465 165533 : H = gc_GEN(av, Flx_to_ZX(U));
2466 165533 : *mod = utoipos(p); return H;
2467 : }
2468 47004 : T = ZV_producttree(P);
2469 47004 : A = ZX_nv_mod_tree(A, P, T);
2470 47004 : B = ZX_nv_mod_tree(B, P, T);
2471 47004 : H = cgetg(n+1, t_VEC);
2472 237735 : for(i=1; i <= n; i++)
2473 : {
2474 190731 : ulong p = P[i];
2475 190731 : GEN a = gel(A,i), b = gel(B,i);
2476 190731 : GEN U = Flxq_invsafe(a, b, p);
2477 190727 : if (!U)
2478 : {
2479 601 : gel(H,i) = pol_0(v);
2480 601 : P[i] = 1; redo = 1;
2481 : }
2482 : else
2483 190126 : gel(H,i) = U;
2484 : }
2485 47004 : if (redo) T = ZV_producttree(P);
2486 47004 : H = nxV_chinese_center_tree(H, P, T, ZV_chinesetree(P, T));
2487 47004 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2488 : }
2489 :
2490 : GEN
2491 212561 : QXQ_inv_worker(GEN P, GEN A, GEN B)
2492 : {
2493 212561 : GEN V = cgetg(3, t_VEC);
2494 212561 : gel(V,1) = QXQ_inv_slice(A, B, P, &gel(V,2));
2495 212561 : return V;
2496 : }
2497 :
2498 : /* lift(1 / Mod(A,B)). B a ZX, A a scalar or a QX */
2499 : GEN
2500 145954 : QXQ_inv(GEN A, GEN B)
2501 : {
2502 : GEN D, Ap, Bp;
2503 : ulong pp;
2504 145954 : pari_sp av2, av = avma;
2505 : forprime_t S;
2506 145954 : GEN worker, U, H = NULL, mod = gen_1;
2507 : pari_timer ti;
2508 : long k, dA, dB;
2509 145954 : if (is_scalar_t(typ(A))) return scalarpol(ginv(A), varn(B));
2510 : /* A a QX, B a ZX */
2511 145954 : A = Q_primitive_part(A, &D);
2512 145954 : dA = degpol(A); dB= degpol(B);
2513 : /* A, B in Z[X] */
2514 145954 : init_modular_small(&S);
2515 : do {
2516 145954 : pp = u_forprime_next(&S);
2517 145954 : Ap = ZX_to_Flx(A, pp);
2518 145954 : Bp = ZX_to_Flx(B, pp);
2519 145954 : } while (degpol(Ap) != dA || degpol(Bp) != dB);
2520 145954 : if (degpol(Flx_gcd(Ap, Bp, pp)) != 0 && degpol(ZX_gcd(A,B))!=0)
2521 14 : pari_err_INV("QXQ_inv",mkpolmod(A,B));
2522 145940 : worker = snm_closure(is_entry("_QXQ_inv_worker"), mkvec2(A, B));
2523 145940 : av2 = avma;
2524 145940 : for (k = 1; ;k *= 2)
2525 42431 : {
2526 : GEN res, b, N, den;
2527 188371 : gen_inccrt_i("QXQ_inv", worker, NULL, (k+1)>>1, 0, &S, &H, &mod,
2528 : nxV_chinese_center, FpX_center);
2529 188371 : (void)gc_all(av2, 2, &H, &mod);
2530 188371 : b = sqrti(shifti(mod,-1));
2531 188371 : if (DEBUGLEVEL>5) timer_start(&ti);
2532 188371 : U = FpX_ratlift(H, mod, b, b, NULL);
2533 188371 : if (DEBUGLEVEL>5) timer_printf(&ti,"QXQ_inv: ratlift");
2534 194069 : if (!U) continue;
2535 151638 : N = Q_remove_denom(U, &den); if (!den) den = gen_1;
2536 151637 : res = Flx_rem(Flx_Fl_sub(Flx_mul(Ap, ZX_to_Flx(N,pp), pp),
2537 : umodiu(den, pp), pp), Bp, pp);
2538 151638 : if (degpol(res) >= 0) continue;
2539 145940 : res = ZX_Z_sub(ZX_mul(A, N), den);
2540 145940 : res = ZX_is_monic(B) ? ZX_rem(res, B): RgX_pseudorem(res, B);
2541 145940 : if (DEBUGLEVEL>5) timer_printf(&ti,"QXQ_inv: final check");
2542 145940 : if (degpol(res)<0)
2543 : {
2544 145940 : if (D) U = RgX_Rg_div(U, D);
2545 145940 : return gc_GEN(av, U);
2546 : }
2547 : }
2548 : }
2549 :
2550 : static GEN
2551 120510 : QXQ_div_slice(GEN A, GEN B, GEN C, GEN P, GEN *mod)
2552 : {
2553 120510 : pari_sp av = avma;
2554 120510 : long i, n = lg(P)-1, v = varn(A), redo = 0;
2555 : GEN H, T;
2556 120510 : if (n == 1)
2557 : {
2558 44148 : ulong p = uel(P,1);
2559 44148 : GEN a = ZX_to_Flx(A, p), b = ZX_to_Flx(B, p), c = ZX_to_Flx(C, p);
2560 44148 : GEN bi = Flxq_invsafe(b, c, p), U;
2561 44148 : if (!bi)
2562 : {
2563 0 : set_avma(av);
2564 0 : *mod = gen_1; return pol_0(v);
2565 : }
2566 44148 : U = Flxq_mul(a, bi, c, p);
2567 44148 : H = gc_GEN(av, Flx_to_ZX(U));
2568 44148 : *mod = utoipos(p); return H;
2569 : }
2570 76362 : T = ZV_producttree(P);
2571 76362 : A = ZX_nv_mod_tree(A, P, T);
2572 76362 : B = ZX_nv_mod_tree(B, P, T);
2573 76361 : C = ZX_nv_mod_tree(C, P, T);
2574 76362 : H = cgetg(n+1, t_VEC);
2575 337314 : for(i=1; i <= n; i++)
2576 : {
2577 260952 : ulong p = P[i];
2578 260952 : GEN a = gel(A,i), b = gel(B,i), c = gel(C, i);
2579 260952 : GEN bi = Flxq_invsafe(b, c, p);
2580 260953 : if (!bi)
2581 : {
2582 4 : gel(H,i) = pol_0(v);
2583 4 : P[i] = 1; redo = 1;
2584 : }
2585 : else
2586 260949 : gel(H,i) = Flxq_mul(a, bi, c, p);
2587 : }
2588 76362 : if (redo) T = ZV_producttree(P);
2589 76362 : H = nxV_chinese_center_tree(H, P, T, ZV_chinesetree(P, T));
2590 76362 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2591 : }
2592 :
2593 : GEN
2594 120510 : QXQ_div_worker(GEN P, GEN A, GEN B, GEN C)
2595 : {
2596 120510 : GEN V = cgetg(3, t_VEC);
2597 120510 : gel(V,1) = QXQ_div_slice(A, B, C, P, &gel(V,2));
2598 120510 : return V;
2599 : }
2600 :
2601 : /* lift(Mod(A/B, C)). C a ZX, A, B a scalar or a QX */
2602 : GEN
2603 32759 : QXQ_div(GEN A, GEN B, GEN C)
2604 : {
2605 : GEN DA, DB, Ap, Bp, Cp;
2606 : ulong pp;
2607 32759 : pari_sp av2, av = avma;
2608 : forprime_t S;
2609 32759 : GEN worker, U, H = NULL, mod = gen_1;
2610 : pari_timer ti;
2611 : long k, dA, dB, dC;
2612 32759 : if (is_scalar_t(typ(A))) return scalarpol(ginv(A), varn(B));
2613 : /* A a QX, B a ZX */
2614 32759 : A = Q_primitive_part(A, &DA);
2615 32759 : B = Q_primitive_part(B, &DB);
2616 32759 : dA = degpol(A); dB = degpol(B); dC = degpol(C);
2617 : /* A, B in Z[X] */
2618 32759 : init_modular_small(&S);
2619 : do {
2620 32759 : pp = u_forprime_next(&S);
2621 32759 : Ap = ZX_to_Flx(A, pp);
2622 32759 : Bp = ZX_to_Flx(B, pp);
2623 32759 : Cp = ZX_to_Flx(C, pp);
2624 32759 : } while (degpol(Ap) != dA || degpol(Bp) != dB || degpol(Cp) != dC);
2625 32759 : if (degpol(Flx_gcd(Bp, Cp, pp)) != 0 && degpol(ZX_gcd(B,C))!=0)
2626 0 : pari_err_INV("QXQ_div",mkpolmod(B,C));
2627 32759 : worker = snm_closure(is_entry("_QXQ_div_worker"), mkvec3(A, B, C));
2628 32759 : av2 = avma;
2629 32759 : for (k = 1; ;k *= 2)
2630 46575 : {
2631 : GEN res, b, N, den;
2632 79334 : gen_inccrt_i("QXQ_div", worker, NULL, (k+1)>>1, 0, &S, &H, &mod,
2633 : nxV_chinese_center, FpX_center);
2634 79334 : (void)gc_all(av2, 2, &H, &mod);
2635 79334 : b = sqrti(shifti(mod,-1));
2636 79334 : if (DEBUGLEVEL>5) timer_start(&ti);
2637 79334 : U = FpX_ratlift(H, mod, b, b, NULL);
2638 79334 : if (DEBUGLEVEL>5) timer_printf(&ti,"QXQ_div: ratlift");
2639 89944 : if (!U) continue;
2640 43369 : N = Q_remove_denom(U, &den); if (!den) den = gen_1;
2641 43369 : res = Flx_rem(Flx_sub(Flx_mul(Bp, ZX_to_Flx(N,pp), pp),
2642 : Flx_Fl_mul(Ap, umodiu(den, pp), pp), pp), Cp, pp);
2643 43369 : if (degpol(res) >= 0) continue;
2644 32759 : res = ZX_sub(ZX_mul(B, N), ZX_Z_mul(A,den));
2645 32759 : res = ZX_is_monic(C) ? ZX_rem(res, C): RgX_pseudorem(res, C);
2646 32759 : if (DEBUGLEVEL>5) timer_printf(&ti,"QXQ_div: final check");
2647 32759 : if (degpol(res)<0)
2648 : {
2649 32759 : if (DA && DB) U = RgX_Rg_mul(U, gdiv(DA,DB));
2650 27831 : else if (DA) U = RgX_Rg_mul(U, DA);
2651 15743 : else if (DB) U = RgX_Rg_div(U, DB);
2652 32759 : return gc_GEN(av, U);
2653 : }
2654 : }
2655 : }
2656 :
2657 : /************************************************************************
2658 : * *
2659 : * ZXQ_minpoly *
2660 : * *
2661 : ************************************************************************/
2662 :
2663 : static GEN
2664 3523 : ZXQ_minpoly_slice(GEN A, GEN B, long d, GEN P, GEN *mod)
2665 : {
2666 3523 : pari_sp av = avma;
2667 3523 : long i, n = lg(P)-1, v = evalvarn(varn(B));
2668 : GEN H, T;
2669 3523 : if (n == 1)
2670 : {
2671 716 : ulong p = uel(P,1);
2672 716 : GEN a = ZX_to_Flx(A, p), b = ZX_to_Flx(B, p);
2673 716 : GEN Hp = Flxq_minpoly(a, b, p);
2674 716 : if (degpol(Hp) != d) { p = 1; Hp = pol0_Flx(v); }
2675 716 : H = gc_upto(av, Flx_to_ZX(Hp));
2676 716 : *mod = utoipos(p); return H;
2677 : }
2678 2807 : T = ZV_producttree(P);
2679 2807 : A = ZX_nv_mod_tree(A, P, T);
2680 2807 : B = ZX_nv_mod_tree(B, P, T);
2681 2807 : H = cgetg(n+1, t_VEC);
2682 16838 : for(i=1; i <= n; i++)
2683 : {
2684 14031 : ulong p = P[i];
2685 14031 : GEN a = gel(A,i), b = gel(B,i);
2686 14031 : GEN m = Flxq_minpoly(a, b, p);
2687 14031 : if (degpol(m) != d) { P[i] = 1; m = pol0_Flx(v); }
2688 14031 : gel(H, i) = m;
2689 : }
2690 2807 : H = nxV_chinese_center_tree(H, P, T, ZV_chinesetree(P, T));
2691 2807 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2692 : }
2693 :
2694 : GEN
2695 3523 : ZXQ_minpoly_worker(GEN P, GEN A, GEN B, long d)
2696 : {
2697 3523 : GEN V = cgetg(3, t_VEC);
2698 3523 : gel(V,1) = ZXQ_minpoly_slice(A, B, d, P, &gel(V,2));
2699 3523 : return V;
2700 : }
2701 :
2702 : GEN
2703 1701 : ZXQ_minpoly(GEN A, GEN B, long d, ulong bound)
2704 : {
2705 1701 : pari_sp av = avma;
2706 : GEN worker, H, dB;
2707 : forprime_t S;
2708 1701 : B = Q_remove_denom(B, &dB);
2709 1701 : worker = strtoclosure("_ZXQ_minpoly_worker", 3, A, B, stoi(d));
2710 1701 : init_modular_big(&S);
2711 1701 : H = gen_crt("ZXQ_minpoly", worker, &S, dB, bound, 0, NULL,
2712 : nxV_chinese_center, FpX_center_i);
2713 1701 : return gc_GEN(av, H);
2714 : }
2715 :
2716 : /************************************************************************
2717 : * *
2718 : * ZX_ZXY_resultant *
2719 : * *
2720 : ************************************************************************/
2721 :
2722 : static GEN
2723 364818 : ZX_ZXY_resultant_prime(GEN a, GEN b, ulong dp, ulong p,
2724 : long degA, long degB, long dres, long sX)
2725 : {
2726 364818 : long dropa = degA - degpol(a), dropb = degB - degpol(b);
2727 364815 : ulong pi = SMALL_ULONG(p)? 0: get_Fl_red(p);
2728 364815 : GEN Hp = Flx_FlxY_resultant_polint(a, b, p, pi, dres, sX);
2729 364822 : if (dropa && dropb)
2730 0 : Hp = zero_Flx(sX);
2731 : else {
2732 364822 : if (dropa)
2733 : { /* multiply by ((-1)^deg B lc(B))^(deg A - deg a) */
2734 0 : GEN c = gel(b,degB+2); /* lc(B) */
2735 0 : if (odd(degB)) c = Flx_neg(c, p);
2736 0 : if (!Flx_equal1(c)) {
2737 0 : c = Flx_powu_pre(c, dropa, p, pi);
2738 0 : if (!Flx_equal1(c)) Hp = Flx_mul_pre(Hp, c, p, pi);
2739 : }
2740 : }
2741 364822 : else if (dropb)
2742 : { /* multiply by lc(A)^(deg B - deg b) */
2743 0 : ulong c = uel(a, degA+2); /* lc(A) */
2744 0 : c = Fl_powu(c, dropb, p);
2745 0 : if (c != 1) Hp = Flx_Fl_mul_pre(Hp, c, p, pi);
2746 : }
2747 : }
2748 364822 : if (dp != 1) Hp = Flx_Fl_mul_pre(Hp, Fl_powu_pre(Fl_inv(dp,p), degA, p, pi), p, pi);
2749 364822 : return Hp;
2750 : }
2751 :
2752 : static GEN
2753 124940 : ZX_ZXY_resultant_slice(GEN A, GEN B, GEN dB, long degA, long degB, long dres,
2754 : GEN P, GEN *mod, long sX, long vY)
2755 : {
2756 124940 : pari_sp av = avma;
2757 124940 : long i, n = lg(P)-1;
2758 : GEN H, T, D;
2759 124940 : if (n == 1)
2760 : {
2761 40163 : ulong p = uel(P,1);
2762 40163 : ulong dp = dB ? umodiu(dB, p): 1;
2763 40163 : GEN a = ZX_to_Flx(A, p), b = ZXX_to_FlxX(B, p, vY);
2764 40163 : GEN Hp = ZX_ZXY_resultant_prime(a, b, dp, p, degA, degB, dres, sX);
2765 40164 : H = gc_upto(av, Flx_to_ZX(Hp));
2766 40164 : *mod = utoipos(p); return H;
2767 : }
2768 84777 : T = ZV_producttree(P);
2769 84777 : A = ZX_nv_mod_tree(A, P, T);
2770 84777 : B = ZXX_nv_mod_tree(B, P, T, vY);
2771 84777 : D = dB ? Z_ZV_mod_tree(dB, P, T): NULL;
2772 84777 : H = cgetg(n+1, t_VEC);
2773 364108 : for(i=1; i <= n; i++)
2774 : {
2775 279331 : ulong p = P[i];
2776 279331 : GEN a = gel(A,i), b = gel(B,i);
2777 279331 : ulong dp = D ? uel(D, i): 1;
2778 279331 : gel(H,i) = ZX_ZXY_resultant_prime(a, b, dp, p, degA, degB, dres, sX);
2779 : }
2780 84777 : H = nxV_chinese_center_tree(H, P, T, ZV_chinesetree(P, T));
2781 84777 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2782 : }
2783 :
2784 : GEN
2785 124940 : ZX_ZXY_resultant_worker(GEN P, GEN A, GEN B, GEN dB, GEN v)
2786 : {
2787 124940 : GEN V = cgetg(3, t_VEC);
2788 124940 : if (isintzero(dB)) dB = NULL;
2789 124940 : gel(V,1) = ZX_ZXY_resultant_slice(A, B, dB, v[1], v[2], v[3], P, &gel(V,2), v[4], v[5]);
2790 124940 : return V;
2791 : }
2792 :
2793 : GEN
2794 79188 : ZX_ZXY_resultant(GEN A, GEN B)
2795 : {
2796 79188 : pari_sp av = avma;
2797 : forprime_t S;
2798 : ulong bound;
2799 79188 : long v = fetch_var_higher();
2800 79188 : long degA = degpol(A), degB, dres = degA * degpol(B);
2801 79188 : long vX = varn(B), vY = varn(A); /* assume vY has lower priority */
2802 79188 : long sX = evalvarn(vX);
2803 : GEN worker, H, dB;
2804 79188 : B = Q_remove_denom(B, &dB);
2805 79188 : if (!dB) B = leafcopy(B);
2806 79188 : A = leafcopy(A); setvarn(A,v);
2807 79188 : B = swap_vars(B, vY, v); degB = degpol(B);
2808 79188 : bound = ZX_ZXY_ResBound(A, B, dB);
2809 79188 : if (DEBUGLEVEL>4) err_printf("bound for resultant coeffs: 2^%ld\n",bound);
2810 158377 : worker = snm_closure(is_entry("_ZX_ZXY_resultant_worker"),
2811 79189 : mkvec4(A, B, dB? dB: gen_0,
2812 : mkvecsmall5(degA, degB, dres, sX, vY)));
2813 79189 : init_modular_big(&S);
2814 79188 : H = gen_crt("ZX_ZXY_resultant_all", worker, &S, dB, bound, 0, NULL,
2815 : nxV_chinese_center, FpX_center_i);
2816 79188 : setvarn(H, vX); (void)delete_var();
2817 79188 : return gc_GEN(av, H);
2818 : }
2819 :
2820 : static long
2821 40516 : ZX_ZXY_rnfequation_lambda(GEN A, GEN B0, long lambda)
2822 : {
2823 40516 : pari_sp av = avma;
2824 40516 : long degA = degpol(A), degB, dres = degA*degpol(B0);
2825 40516 : long v = fetch_var_higher();
2826 40516 : long vX = varn(B0), vY = varn(A); /* assume vY has lower priority */
2827 40516 : long sX = evalvarn(vX);
2828 : GEN dB, B, a, b, Hp;
2829 : forprime_t S;
2830 :
2831 40516 : B0 = Q_remove_denom(B0, &dB);
2832 40515 : if (!dB) B0 = leafcopy(B0);
2833 40516 : A = leafcopy(A);
2834 40516 : B = B0;
2835 40516 : setvarn(A,v);
2836 45328 : INIT:
2837 45328 : if (lambda) B = RgX_translate(B0, monomial(stoi(lambda), 1, vY));
2838 45328 : B = swap_vars(B, vY, v);
2839 : /* B0(lambda v + x, v) */
2840 45327 : if (DEBUGLEVEL>4) err_printf("Trying lambda = %ld\n", lambda);
2841 :
2842 45327 : degB = degpol(B);
2843 45327 : init_modular_big(&S);
2844 : while (1)
2845 0 : {
2846 45327 : ulong p = u_forprime_next(&S);
2847 45327 : ulong dp = dB ? umodiu(dB, p): 1;
2848 45327 : if (!dp) continue;
2849 45327 : a = ZX_to_Flx(A, p);
2850 45328 : b = ZXX_to_FlxX(B, p, v);
2851 45326 : Hp = ZX_ZXY_resultant_prime(a, b, dp, p, degA, degB, dres, sX);
2852 45327 : if (degpol(Hp) != dres) continue;
2853 45327 : if (dp != 1) Hp = Flx_Fl_mul(Hp, Fl_powu(Fl_inv(dp,p), degA, p), p);
2854 45327 : if (!Flx_is_squarefree(Hp, p)) { lambda = next_lambda(lambda); goto INIT; }
2855 40515 : if (DEBUGLEVEL>4) err_printf("Final lambda = %ld\n", lambda);
2856 40515 : (void)delete_var(); return gc_long(av,lambda);
2857 : }
2858 : }
2859 :
2860 : GEN
2861 60555 : ZX_ZXY_rnfequation(GEN A, GEN B, long *lambda)
2862 : {
2863 60555 : if (lambda)
2864 : {
2865 40516 : *lambda = ZX_ZXY_rnfequation_lambda(A, B, *lambda);
2866 40515 : if (*lambda) B = RgX_translate(B, monomial(stoi(*lambda), 1, varn(A)));
2867 : }
2868 60554 : return ZX_ZXY_resultant(A,B);
2869 : }
2870 :
2871 : static GEN
2872 10349 : ZX_composedsum_slice(GEN A, GEN B, GEN P, GEN *mod)
2873 : {
2874 10349 : pari_sp av = avma;
2875 10349 : long i, n = lg(P)-1;
2876 : GEN H, T;
2877 10349 : if (n == 1)
2878 : {
2879 9847 : ulong p = uel(P,1);
2880 9847 : GEN a = ZX_to_Flx(A, p), b = ZX_to_Flx(B, p);
2881 9845 : GEN Hp = Flx_composedsum(a, b, p);
2882 9846 : H = gc_upto(av, Flx_to_ZX(Hp));
2883 9844 : *mod = utoipos(p); return H;
2884 : }
2885 502 : T = ZV_producttree(P);
2886 502 : A = ZX_nv_mod_tree(A, P, T);
2887 502 : B = ZX_nv_mod_tree(B, P, T);
2888 502 : H = cgetg(n+1, t_VEC);
2889 4526 : for(i=1; i <= n; i++)
2890 : {
2891 4024 : ulong p = P[i];
2892 4024 : GEN a = gel(A,i), b = gel(B,i);
2893 4024 : gel(H,i) = Flx_composedsum(a, b, p);
2894 : }
2895 502 : H = nxV_chinese_center_tree(H, P, T, ZV_chinesetree(P, T));
2896 502 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2897 : }
2898 :
2899 : GEN
2900 10349 : ZX_composedsum_worker(GEN P, GEN A, GEN B)
2901 : {
2902 10349 : GEN V = cgetg(3, t_VEC);
2903 10349 : gel(V,1) = ZX_composedsum_slice(A, B, P, &gel(V,2));
2904 10346 : return V;
2905 : }
2906 :
2907 : static GEN
2908 10085 : ZX_composedsum_i(GEN A, GEN B, GEN lead)
2909 : {
2910 10085 : pari_sp av = avma;
2911 : forprime_t S;
2912 : ulong bound;
2913 : GEN H, worker, mod;
2914 10085 : if (degpol(A) < degpol(B)) swap(A, B);
2915 10082 : if (!lead) lead = mulii(leading_coeff(A),leading_coeff(B));
2916 10082 : bound = ZX_ZXY_ResBound_1(A, B);
2917 10085 : worker = snm_closure(is_entry("_ZX_composedsum_worker"), mkvec2(A,B));
2918 10087 : init_modular_big(&S);
2919 10085 : H = gen_crt("ZX_composedsum", worker, &S, lead, bound, 0, &mod,
2920 : nxV_chinese_center, FpX_center);
2921 10085 : return gc_upto(av, H);
2922 : }
2923 :
2924 : static long
2925 9694 : ZX_compositum_lambda(GEN A, GEN B, GEN lead, long lambda)
2926 : {
2927 9694 : pari_sp av = avma;
2928 : forprime_t S;
2929 : ulong p;
2930 9694 : init_modular_big(&S);
2931 9695 : p = u_forprime_next(&S);
2932 : while (1)
2933 112 : {
2934 : GEN Hp, a;
2935 9807 : if (DEBUGLEVEL>4) err_printf("Trying lambda = %ld\n", lambda);
2936 9807 : if (lead && dvdiu(lead,p)) { p = u_forprime_next(&S); continue; }
2937 9800 : a = ZX_to_Flx(ZX_rescale(A, stoi(-lambda)), p);
2938 9804 : Hp = Flx_composedsum(a, ZX_to_Flx(B, p), p);
2939 9799 : if (!Flx_is_squarefree(Hp, p)) { lambda = next_lambda(lambda); continue; }
2940 9693 : if (DEBUGLEVEL>4) err_printf("Final lambda = %ld\n", lambda);
2941 9693 : return gc_long(av, lambda);
2942 : }
2943 : }
2944 :
2945 : GEN
2946 9698 : ZX_compositum(GEN A, GEN B, long *lambda)
2947 : {
2948 9698 : GEN lead = mulii(leading_coeff(A),leading_coeff(B));
2949 9694 : if (lambda)
2950 : {
2951 9694 : *lambda = ZX_compositum_lambda(A, B, lead, *lambda);
2952 9692 : A = ZX_rescale(A, stoi(-*lambda));
2953 : }
2954 9700 : return ZX_composedsum_i(A, B, lead);
2955 : }
2956 :
2957 : GEN
2958 385 : ZX_composedsum(GEN A, GEN B)
2959 385 : { return ZX_composedsum_i(A, B, NULL); }
2960 :
2961 : static GEN
2962 359 : ZXQX_composedsum_slice(GEN A, GEN B, GEN C, GEN P, GEN *mod)
2963 : {
2964 359 : pari_sp av = avma;
2965 359 : long i, n = lg(P)-1, dC = degpol(C), v = varn(C);
2966 : GEN H, T;
2967 359 : if (n == 1)
2968 : {
2969 181 : ulong p = uel(P,1);
2970 181 : GEN a = ZXX_to_FlxX(A, p, v), b = ZXX_to_FlxX(B, p, v);
2971 181 : GEN c = ZX_to_Flx(C, p);
2972 181 : GEN Hp = FlxX_to_Flm(FlxqX_composedsum(a, b, c, p), dC);
2973 181 : H = gc_upto(av, Flm_to_ZM(Hp));
2974 181 : *mod = utoipos(p); return H;
2975 : }
2976 178 : T = ZV_producttree(P);
2977 178 : A = ZXX_nv_mod_tree(A, P, T, v);
2978 178 : B = ZXX_nv_mod_tree(B, P, T, v);
2979 178 : C = ZX_nv_mod_tree(C, P, T);
2980 178 : H = cgetg(n+1, t_VEC);
2981 660 : for(i=1; i <= n; i++)
2982 : {
2983 482 : ulong p = P[i];
2984 482 : GEN a = gel(A,i), b = gel(B,i), c = gel(C,i);
2985 482 : gel(H,i) = FlxX_to_Flm(FlxqX_composedsum(a, b, c, p), dC);
2986 : }
2987 178 : H = nmV_chinese_center_tree_seq(H, P, T, ZV_chinesetree(P, T));
2988 178 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
2989 : }
2990 :
2991 : GEN
2992 359 : ZXQX_composedsum_worker(GEN P, GEN A, GEN B, GEN C)
2993 : {
2994 359 : GEN V = cgetg(3, t_VEC);
2995 359 : gel(V,1) = ZXQX_composedsum_slice(A, B, C, P, &gel(V,2));
2996 359 : return V;
2997 : }
2998 :
2999 : static GEN
3000 315 : ZXQX_composedsum(GEN A, GEN B, GEN T, ulong bound)
3001 : {
3002 315 : pari_sp av = avma;
3003 : forprime_t S;
3004 : GEN H, worker, mod;
3005 315 : GEN lead = mulii(Q_content(leading_coeff(A)), Q_content(leading_coeff(B)));
3006 315 : worker = snm_closure(is_entry("_ZXQX_composedsum_worker")
3007 : , mkvec3(A,B,T));
3008 315 : init_modular_big(&S);
3009 315 : H = gen_crt("ZXQX_composedsum", worker, &S, lead, bound, 0, &mod,
3010 : nmV_chinese_center, FpM_center);
3011 315 : if (DEBUGLEVEL > 4)
3012 0 : err_printf("nfcompositum: a priori bound: %lu, a posteriori: %lu\n",
3013 : bound, expi(gsupnorm(H, DEFAULTPREC)));
3014 315 : return gc_GEN(av, RgM_to_RgXX(H, varn(A), varn(T)));
3015 : }
3016 :
3017 : static long
3018 315 : ZXQX_composedsum_bound(GEN nf, GEN A, GEN B)
3019 315 : { return ZXQX_resultant_bound_i(nf, A, B, &RgX_RgXY_ResBound_1); }
3020 :
3021 : GEN
3022 315 : nf_direct_compositum(GEN nf, GEN A, GEN B)
3023 : {
3024 315 : ulong bnd = ZXQX_composedsum_bound(nf, A, B);
3025 315 : return ZXQX_composedsum(A, B, nf_get_pol(nf), bnd);
3026 : }
3027 :
3028 : /************************************************************************
3029 : * *
3030 : * IRREDUCIBLE POLYNOMIAL / Fp *
3031 : * *
3032 : ************************************************************************/
3033 :
3034 : /* irreducible (unitary) polynomial of degree n over Fp */
3035 : GEN
3036 0 : ffinit_rand(GEN p,long n)
3037 : {
3038 0 : for(;;) {
3039 0 : pari_sp av = avma;
3040 0 : GEN pol = ZX_add(pol_xn(n, 0), random_FpX(n-1,0, p));
3041 0 : if (FpX_is_irred(pol, p)) return pol;
3042 0 : set_avma(av);
3043 : }
3044 : }
3045 :
3046 : /* return an extension of degree 2^l of F_2, assume l > 0
3047 : * Not stack clean. */
3048 : static GEN
3049 605 : ffinit_Artin_Schreier_2(long l)
3050 : {
3051 : GEN Q, T, S;
3052 : long i, v;
3053 :
3054 605 : if (l == 1) return mkvecsmall4(0,1,1,1); /*x^2 + x + 1*/
3055 556 : v = fetch_var_higher();
3056 556 : S = mkvecsmall5(0, 0, 0, 1, 1); /* y(y^2 + y) */
3057 556 : Q = mkpoln(3, pol1_Flx(0), pol1_Flx(0), S); /* x^2 + x + y(y^2+y) */
3058 556 : setvarn(Q, v);
3059 :
3060 : /* x^4+x+1, irred over F_2, minimal polynomial of a root of Q */
3061 556 : T = mkvecsmalln(6,evalvarn(v),1UL,1UL,0UL,0UL,1UL);
3062 : /* Q = x^2 + x + a(y) irred. over K = F2[y] / (T(y))
3063 : * ==> x^2 + x + a(y) b irred. over K for any root b of Q
3064 : * ==> x^2 + x + (b^2+b)b */
3065 3070 : for (i=2; i<l; i++) T = Flx_FlxY_resultant(T, Q, 2); /* minpoly(b) / F2*/
3066 556 : (void)delete_var(); T[1] = 0; return T;
3067 : }
3068 :
3069 : /* return an extension of degree p^l of F_p, assume l > 0
3070 : * Not stack clean. */
3071 : GEN
3072 962 : ffinit_Artin_Schreier(ulong p, long l)
3073 : {
3074 : long i, v;
3075 : GEN Q, R, S, T, xp;
3076 962 : if (p==2) return ffinit_Artin_Schreier_2(l);
3077 357 : xp = polxn_Flx(p,0); /* x^p */
3078 357 : T = Flx_sub(xp, mkvecsmall3(0,1,1),p); /* x^p - x - 1 */
3079 357 : if (l == 1) return T;
3080 :
3081 7 : v = evalvarn(fetch_var_higher());
3082 7 : xp[1] = v;
3083 7 : R = Flx_sub(polxn_Flx(2*p-1,0), polxn_Flx(p,0),p);
3084 7 : S = Flx_sub(xp, polx_Flx(0), p);
3085 7 : Q = FlxX_Flx_sub(Flx_to_FlxX(S, v), R, p); /* x^p - x - (y^(2p-1)-y^p) */
3086 14 : for (i = 2; i <= l; ++i) T = Flx_FlxY_resultant(T, Q, p);
3087 7 : (void)delete_var(); T[1] = 0; return T;
3088 : }
3089 :
3090 : static long
3091 149716 : flinit_check(ulong p, long n, long l)
3092 : {
3093 : ulong q;
3094 149716 : if (!uisprime(n)) return 0;
3095 102496 : q = p % n; if (!q) return 0;
3096 99885 : return ugcd((n-1)/Fl_order(q, n-1, n), l) == 1;
3097 : }
3098 :
3099 : static GEN
3100 31944 : flinit(ulong p, long l)
3101 : {
3102 31944 : ulong n = 1+l;
3103 96782 : while (!flinit_check(p,n,l)) n += l;
3104 31944 : if (DEBUGLEVEL>=4) err_printf("FFInit: using polsubcyclo(%ld, %ld)\n",n,l);
3105 31944 : return ZX_to_Flx(polsubcyclo(n,l,0), p);
3106 : }
3107 :
3108 : static GEN
3109 28996 : ffinit_fact_Flx(ulong p, long n)
3110 : {
3111 28996 : GEN P, F = factoru_pow(n), Fp = gel(F,1), Fe = gel(F,2), Fm = gel(F,3);
3112 28996 : long i, l = lg(Fm);
3113 28996 : P = cgetg(l, t_VEC);
3114 61902 : for (i = 1; i < l; i++)
3115 32906 : gel(P,i) = p==uel(Fp,i) ? ffinit_Artin_Schreier(p, Fe[i])
3116 32906 : : flinit(p, uel(Fm,i));
3117 28996 : return FlxV_composedsum(P, p);
3118 : }
3119 :
3120 : static GEN
3121 52941 : init_Flxq_i(ulong p, long n, long sv)
3122 : {
3123 : GEN P;
3124 52941 : if (!odd(p) && p != 2) pari_err_PRIME("ffinit", utoi(p));
3125 52934 : if (n == 1) return polx_Flx(sv);
3126 52934 : if (flinit_check(p, n+1, n))
3127 : {
3128 23938 : P = const_vecsmall(n+2,1);
3129 23938 : P[1] = sv; return P;
3130 : }
3131 28996 : P = ffinit_fact_Flx(p,n);
3132 28996 : P[1] = sv; return P;
3133 : }
3134 :
3135 : GEN
3136 0 : init_Flxq(ulong p, long n, long v)
3137 : {
3138 0 : pari_sp av = avma;
3139 0 : return gc_upto(av, init_Flxq_i(p, n, v));
3140 : }
3141 :
3142 : /* check if polsubcyclo(n,l,0) is irreducible modulo p */
3143 : static long
3144 8207 : fpinit_check(GEN p, long n, long l)
3145 : {
3146 : ulong q;
3147 8207 : if (!uisprime(n)) return 0;
3148 4842 : q = umodiu(p,n); if (!q) return 0;
3149 4842 : return ugcd((n-1)/Fl_order(q, n-1, n), l) == 1;
3150 : }
3151 :
3152 : /* let k=2 if p%4==1, and k=4 else and assume k*p does not divide l.
3153 : * Return an irreducible polynomial of degree l over F_p.
3154 : * Variant of Adleman and Lenstra "Finding irreducible polynomials over
3155 : * finite fields", ACM, 1986 (5) 350--355.
3156 : * Not stack clean */
3157 : static GEN
3158 1828 : fpinit(GEN p, long l)
3159 : {
3160 1828 : ulong n = 1+l;
3161 6168 : while (!fpinit_check(p,n,l)) n += l;
3162 1828 : if (DEBUGLEVEL>=4) err_printf("FFInit: using polsubcyclo(%ld, %ld)\n",n,l);
3163 1828 : return FpX_red(polsubcyclo(n,l,0),p);
3164 : }
3165 :
3166 : static GEN
3167 1637 : ffinit_fact(GEN p, long n)
3168 : {
3169 1637 : GEN P, F = factoru_pow(n), Fp = gel(F,1), Fe = gel(F,2), Fm = gel(F,3);
3170 1637 : long i, l = lg(Fm);
3171 1637 : P = cgetg(l, t_VEC);
3172 3465 : for (i = 1; i < l; ++i)
3173 3656 : gel(P,i) = absequaliu(p, Fp[i]) ?
3174 0 : Flx_to_ZX(ffinit_Artin_Schreier(Fp[i], Fe[i]))
3175 1828 : : fpinit(p, Fm[i]);
3176 1637 : return FpXV_composedsum(P, p);
3177 : }
3178 :
3179 : static GEN
3180 55246 : init_Fq_i(GEN p, long n, long v)
3181 : {
3182 : GEN P;
3183 55246 : if (n <= 0) pari_err_DOMAIN("ffinit", "degree", "<=", gen_0, stoi(n));
3184 55246 : if (typ(p) != t_INT) pari_err_TYPE("ffinit",p);
3185 55246 : if (cmpiu(p, 2) < 0) pari_err_PRIME("ffinit",p);
3186 55239 : if (v < 0) v = 0;
3187 55239 : if (n == 1) return pol_x(v);
3188 54987 : if (lgefint(p) == 3)
3189 52941 : return Flx_to_ZX(init_Flxq_i(p[2], n, evalvarn(v)));
3190 2046 : if (!mpodd(p)) pari_err_PRIME("ffinit", p);
3191 2039 : if (fpinit_check(p, n+1, n)) return polcyclo(n+1, v);
3192 1637 : P = ffinit_fact(p,n);
3193 1637 : setvarn(P, v); return P;
3194 : }
3195 : GEN
3196 54679 : init_Fq(GEN p, long n, long v)
3197 : {
3198 54679 : pari_sp av = avma;
3199 54679 : return gc_upto(av, init_Fq_i(p, n, v));
3200 : }
3201 : GEN
3202 567 : ffinit(GEN p, long n, long v)
3203 : {
3204 567 : pari_sp av = avma;
3205 567 : return gc_upto(av, FpX_to_mod(init_Fq_i(p, n, v), p));
3206 : }
3207 :
3208 : GEN
3209 3178 : ffnbirred(GEN p, long n)
3210 : {
3211 3178 : pari_sp av = avma;
3212 3178 : GEN s = powiu(p,n), F = factoru(n), D = divisorsu_moebius(gel(F, 1));
3213 3178 : long j, l = lg(D);
3214 6797 : for (j = 2; j < l; j++) /* skip d = 1 */
3215 : {
3216 3619 : long md = D[j]; /* mu(d) * d, d squarefree */
3217 3619 : GEN pd = powiu(p, n / labs(md)); /* p^{n/d} */
3218 3619 : s = md > 0? addii(s, pd): subii(s,pd);
3219 : }
3220 3178 : return gc_INT(av, diviuexact(s, n));
3221 : }
3222 :
3223 : GEN
3224 616 : ffsumnbirred(GEN p, long n)
3225 : {
3226 616 : pari_sp av = avma, av2;
3227 616 : GEN q, t = p, v = vecfactoru_i(1, n);
3228 : long i;
3229 616 : q = cgetg(n+1,t_VEC); gel(q,1) = p;
3230 1764 : for (i=2; i<=n; i++) gel(q,i) = mulii(gel(q,i-1), p);
3231 616 : av2 = avma;
3232 1764 : for (i=2; i<=n; i++)
3233 : {
3234 1148 : GEN s = gel(q,i), F = gel(v,i), D = divisorsu_moebius(gel(F,1));
3235 1148 : long j, l = lg(D);
3236 2534 : for (j = 2; j < l; j++) /* skip 1 */
3237 : {
3238 1386 : long md = D[j];
3239 1386 : GEN pd = gel(q, i / labs(md)); /* p^{i/d} */
3240 1386 : s = md > 0? addii(s, pd): subii(s, pd);
3241 : }
3242 1148 : t = gc_INT(av2, addii(t, diviuexact(s, i)));
3243 : }
3244 616 : return gc_INT(av, t);
3245 : }
3246 :
3247 : GEN
3248 140 : ffnbirred0(GEN p, long n, long flag)
3249 : {
3250 140 : if (typ(p) != t_INT) pari_err_TYPE("ffnbirred", p);
3251 140 : if (n <= 0) pari_err_DOMAIN("ffnbirred", "degree", "<=", gen_0, stoi(n));
3252 140 : switch(flag)
3253 : {
3254 70 : case 0: return ffnbirred(p, n);
3255 70 : case 1: return ffsumnbirred(p, n);
3256 : }
3257 0 : pari_err_FLAG("ffnbirred");
3258 : return NULL; /* LCOV_EXCL_LINE */
3259 : }
3260 :
3261 : static void
3262 2261 : checkmap(GEN m, const char *s)
3263 : {
3264 2261 : if (typ(m)!=t_VEC || lg(m)!=3 || typ(gel(m,1))!=t_FFELT)
3265 0 : pari_err_TYPE(s,m);
3266 2261 : }
3267 :
3268 : GEN
3269 189 : ffembed(GEN a, GEN b)
3270 : {
3271 189 : pari_sp av = avma;
3272 189 : GEN p, Ta, Tb, g, r = NULL;
3273 189 : if (typ(a)!=t_FFELT) pari_err_TYPE("ffembed",a);
3274 189 : if (typ(b)!=t_FFELT) pari_err_TYPE("ffembed",b);
3275 189 : p = FF_p_i(a); g = FF_gen(a);
3276 189 : if (!equalii(p, FF_p_i(b))) pari_err_MODULUS("ffembed",a,b);
3277 189 : Ta = FF_mod(a);
3278 189 : Tb = FF_mod(b);
3279 189 : if (degpol(Tb)%degpol(Ta)!=0)
3280 7 : pari_err_DOMAIN("ffembed",GENtostr_raw(a),"is not a subfield of",b,a);
3281 182 : r = gel(FFX_roots(Ta, b), 1);
3282 182 : return gc_GEN(av, mkvec2(g,r));
3283 : }
3284 :
3285 : GEN
3286 91 : ffextend(GEN a, GEN P, long v)
3287 : {
3288 91 : pari_sp av = avma;
3289 : long n;
3290 : GEN p, T, R, g, m;
3291 91 : if (typ(a)!=t_FFELT) pari_err_TYPE("ffextend",a);
3292 91 : T = a; p = FF_p_i(a);
3293 91 : if (typ(P)!=t_POL || !RgX_is_FpXQX(P,&T,&p)) pari_err_TYPE("ffextend", P);
3294 49 : if (!FF_samefield(a, T)) pari_err_MODULUS("ffextend",a,T);
3295 49 : if (v < 0) v = varn(P);
3296 49 : n = FF_f(T) * degpol(P); R = ffinit(p, n, v); g = ffgen(R, v);
3297 49 : m = ffembed(a, g);
3298 49 : R = FFX_roots(ffmap(m, P),g);
3299 49 : return gc_GEN(av, mkvec2(gel(R,1), m));
3300 : }
3301 :
3302 : GEN
3303 42 : fffrobenius(GEN a, long n)
3304 : {
3305 42 : if (typ(a)!=t_FFELT) pari_err_TYPE("fffrobenius",a);
3306 42 : retmkvec2(FF_gen(a), FF_Frobenius(a, n));
3307 : }
3308 :
3309 : GEN
3310 133 : ffinvmap(GEN m)
3311 : {
3312 133 : pari_sp av = avma;
3313 : long i, l;
3314 133 : GEN T, F, a, g, r, f = NULL;
3315 133 : checkmap(m, "ffinvmap");
3316 133 : a = gel(m,1); r = gel(m,2);
3317 133 : if (typ(r) != t_FFELT)
3318 7 : pari_err_TYPE("ffinvmap", m);
3319 126 : g = FF_gen(a);
3320 126 : T = FF_mod(r);
3321 126 : F = gel(FFX_factor(T, a), 1);
3322 126 : l = lg(F);
3323 490 : for(i=1; i<l; i++)
3324 : {
3325 490 : GEN s = FFX_rem(FF_to_FpXQ_i(r), gel(F, i), a);
3326 490 : if (degpol(s)==0 && gequal(constant_coeff(s),g)) { f = gel(F, i); break; }
3327 : }
3328 126 : if (f==NULL) pari_err_TYPE("ffinvmap", m);
3329 126 : if (degpol(f)==1) f = FF_neg_i(gel(f,2));
3330 126 : return gc_GEN(av, mkvec2(FF_gen(r),f));
3331 : }
3332 :
3333 : static GEN
3334 1260 : ffpartmapimage(const char *s, GEN r)
3335 : {
3336 1260 : GEN a = NULL, p = NULL;
3337 1260 : if (typ(r)==t_POL && degpol(r) >= 1
3338 1260 : && RgX_is_FpXQX(r,&a,&p) && a && typ(a)==t_FFELT) return a;
3339 0 : pari_err_TYPE(s, r);
3340 : return NULL; /* LCOV_EXCL_LINE */
3341 : }
3342 :
3343 : static GEN
3344 2709 : ffeltmap_i(GEN m, GEN x)
3345 : {
3346 2709 : GEN r = gel(m,2);
3347 2709 : if (!FF_samefield(x, gel(m,1)))
3348 84 : pari_err_DOMAIN("ffmap","m","domain does not contain", x, r);
3349 2625 : if (typ(r)==t_FFELT)
3350 1659 : return FF_map(r, x);
3351 : else
3352 966 : return FFX_preimage(x, r, ffpartmapimage("ffmap", r));
3353 : }
3354 :
3355 : static GEN
3356 4459 : ffmap_i(GEN m, GEN x)
3357 : {
3358 : GEN y;
3359 4459 : long i, lx, tx = typ(x);
3360 4459 : switch(tx)
3361 : {
3362 2541 : case t_FFELT:
3363 2541 : return ffeltmap_i(m, x);
3364 1267 : case t_POL: case t_RFRAC: case t_SER:
3365 : case t_VEC: case t_COL: case t_MAT:
3366 1267 : y = cgetg_copy(x, &lx);
3367 1988 : for (i = 1; i < lontyp[tx]; i++) y[i] = x[i];
3368 4564 : for (; i < lx; i++)
3369 : {
3370 3339 : GEN yi = ffmap_i(m, gel(x,i));
3371 3297 : if (!yi) return NULL;
3372 3297 : gel(y,i) = yi;
3373 : }
3374 1225 : return y;
3375 : }
3376 651 : return gcopy(x);
3377 : }
3378 :
3379 : GEN
3380 1036 : ffmap(GEN m, GEN x)
3381 : {
3382 1036 : pari_sp ltop = avma;
3383 : GEN y;
3384 1036 : checkmap(m, "ffmap");
3385 1036 : y = ffmap_i(m, x);
3386 1036 : if (y) return y;
3387 42 : retgc_const(ltop, cgetg(1, t_VEC));
3388 : }
3389 :
3390 : static GEN
3391 252 : ffeltmaprel_i(GEN m, GEN x)
3392 : {
3393 252 : GEN g = gel(m,1), r = gel(m,2);
3394 252 : if (!FF_samefield(x, g))
3395 0 : pari_err_DOMAIN("ffmap","m","domain does not contain", x, r);
3396 252 : if (typ(r)==t_FFELT)
3397 84 : retmkpolmod(FF_map(r, x), pol_x(FF_var(g)));
3398 : else
3399 168 : retmkpolmod(FFX_preimagerel(x, r, ffpartmapimage("ffmap", r)), gcopy(r));
3400 : }
3401 :
3402 : static GEN
3403 252 : ffmaprel_i(GEN m, GEN x)
3404 : {
3405 252 : switch(typ(x))
3406 : {
3407 252 : case t_FFELT:
3408 252 : return ffeltmaprel_i(m, x);
3409 0 : case t_POL: pari_APPLY_pol_normalized(ffmaprel_i(m, gel(x,i)));
3410 0 : case t_SER: pari_APPLY_ser_normalized(ffmaprel_i(m, gel(x,i)));
3411 0 : case t_RFRAC: case t_VEC: case t_COL: case t_MAT:
3412 0 : pari_APPLY_same(ffmaprel_i(m, gel(x,i)));
3413 : }
3414 0 : return gcopy(x);
3415 : }
3416 : GEN
3417 252 : ffmaprel(GEN m, GEN x) { checkmap(m, "ffmaprel"); return ffmaprel_i(m, x); }
3418 :
3419 : static void
3420 84 : err_compo(GEN m, GEN n)
3421 84 : { pari_err_DOMAIN("ffcompomap","m","domain does not contain codomain of",n,m); }
3422 :
3423 : GEN
3424 420 : ffcompomap(GEN m, GEN n)
3425 : {
3426 420 : pari_sp av = avma;
3427 420 : GEN g = gel(n,1), r, m2, n2;
3428 420 : checkmap(m, "ffcompomap");
3429 420 : checkmap(n, "ffcompomap");
3430 420 : m2 = gel(m,2); n2 = gel(n,2);
3431 420 : switch((typ(m2)==t_POL)|((typ(n2)==t_POL)<<1))
3432 : {
3433 84 : case 0:
3434 84 : if (!FF_samefield(gel(m,1),n2)) err_compo(m,n);
3435 42 : r = FF_map(gel(m,2), n2);
3436 42 : break;
3437 84 : case 2:
3438 84 : r = ffmap_i(m, n2);
3439 42 : if (lg(r) == 1) err_compo(m,n);
3440 42 : break;
3441 168 : case 1:
3442 168 : r = ffeltmap_i(m, n2);
3443 126 : if (!r)
3444 : {
3445 : GEN a, A, R, M;
3446 : long dm, dn;
3447 42 : a = ffpartmapimage("ffcompomap",m2);
3448 42 : A = FF_to_FpXQ_i(FF_neg(n2));
3449 42 : setvarn(A, 1);
3450 42 : R = deg1pol(gen_1, A, 0);
3451 42 : setvarn(R, 0);
3452 42 : M = gcopy(m2);
3453 42 : setvarn(M, 1);
3454 42 : r = polresultant0(R, M, 1, 0);
3455 42 : dm = FF_f(gel(m,1)); dn = FF_f(gel(n,1));
3456 42 : if (dm % dn || !FFX_ispower(r, dm/dn, a, &r)) err_compo(m,n);
3457 42 : setvarn(r, varn(FF_mod(g)));
3458 : }
3459 126 : break;
3460 84 : case 3:
3461 : {
3462 : GEN M, R, T, p, a;
3463 84 : a = ffpartmapimage("ffcompomap",n2);
3464 84 : if (!FF_samefield(a, gel(m,1))) err_compo(m,n);
3465 42 : p = FF_p_i(gel(n,1));
3466 42 : T = FF_mod(gel(n,1));
3467 42 : setvarn(T, 1);
3468 42 : R = RgX_to_FpXQX(n2,T,p);
3469 42 : setvarn(R, 0);
3470 42 : M = gcopy(m2);
3471 42 : setvarn(M, 1);
3472 42 : r = polresultant0(R, M, 1, 0);
3473 42 : setvarn(r, varn(n2));
3474 : }
3475 : }
3476 252 : return gc_GEN(av, mkvec2(g,r));
3477 : }
|
