Line data Source code
1 : /* Copyright (C) 2000 The PARI group.
2 :
3 : This file is part of the PARI/GP package.
4 :
5 : PARI/GP is free software; you can redistribute it and/or modify it under the
6 : terms of the GNU General Public License as published by the Free Software
7 : Foundation; either version 2 of the License, or (at your option) any later
8 : version. It is distributed in the hope that it will be useful, but WITHOUT
9 : ANY WARRANTY WHATSOEVER.
10 : Check the License for details. You should have received a copy of it, along
11 : with the package; see the file 'COPYING'. If not, write to the Free Software
12 : Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. */
13 :
14 : /*******************************************************************/
15 : /* */
16 : /* MAXIMAL ORDERS */
17 : /* */
18 : /*******************************************************************/
19 : #include "pari.h"
20 : #include "paripriv.h"
21 :
22 : #define DEBUGLEVEL DEBUGLEVEL_nf
23 :
24 : /* allow p = -1 from factorizations, avoid oo loop on p = 1 */
25 : static long
26 13923 : safe_Z_pvalrem(GEN x, GEN p, GEN *z)
27 : {
28 13923 : if (is_pm1(p))
29 : {
30 28 : if (signe(p) > 0) return gvaluation(x,p); /*error*/
31 21 : *z = absi(x); return 1;
32 : }
33 13895 : return Z_pvalrem(x, p, z);
34 : }
35 : /* D an integer, P a ZV, return a factorization matrix for D over P, removing
36 : * entries with 0 exponent. */
37 : static GEN
38 4018 : fact_from_factors(GEN D, GEN P, long flag)
39 : {
40 4018 : long i, l = lg(P), iq = 1;
41 4018 : GEN Q = cgetg(l+1,t_COL);
42 4018 : GEN E = cgetg(l+1,t_COL);
43 17934 : for (i=1; i<l; i++)
44 : {
45 13923 : GEN p = gel(P,i);
46 : long k;
47 13923 : if (flag && !equalim1(p))
48 : {
49 14 : p = gcdii(p, D);
50 14 : if (is_pm1(p)) continue;
51 : }
52 13923 : k = safe_Z_pvalrem(D, p, &D);
53 13916 : if (k) { gel(Q,iq) = p; gel(E,iq) = utoipos(k); iq++; }
54 : }
55 4011 : D = absi_shallow(D);
56 4011 : if (!equali1(D))
57 : {
58 812 : long k = Z_isanypower(D, &D);
59 812 : if (!k) k = 1;
60 812 : gel(Q,iq) = D; gel(E,iq) = utoipos(k); iq++;
61 : }
62 4011 : setlg(Q,iq);
63 4011 : setlg(E,iq); return mkmat2(Q,E);
64 : }
65 :
66 : /* d a t_INT; f a t_MAT factorisation of some t_INT sharing some divisors
67 : * with d, or a prime (t_INT). Return a factorization F of d: "primes"
68 : * entries in f _may_ be composite, and are included as is in d. */
69 : static GEN
70 2320 : update_fact(GEN d, GEN f)
71 : {
72 : GEN P;
73 2320 : switch (typ(f))
74 : {
75 2306 : case t_INT: case t_VEC: case t_COL: return f;
76 14 : case t_MAT:
77 14 : if (lg(f) == 3) { P = gel(f,1); break; }
78 : /*fall through*/
79 : default:
80 7 : pari_err_TYPE("nfbasis [factorization expected]",f);
81 : return NULL;/*LCOV_EXCL_LINE*/
82 : }
83 7 : return fact_from_factors(d, P, 1);
84 : }
85 :
86 : /* T = C T0(X/L); C = L^d / lt(T0), d = deg(T)
87 : * disc T = C^2(d - 1) L^-(d(d-1)) disc T0 = (L^d / lt(T0)^2)^(d-1) disc T0 */
88 : static GEN
89 812944 : set_disc(nfmaxord_t *S)
90 : {
91 : GEN L, dT;
92 : long d;
93 812944 : if (S->T0 == S->T) return ZX_disc(S->T);
94 249104 : d = degpol(S->T0);
95 249113 : L = S->unscale;
96 249113 : if (typ(L) == t_FRAC && abscmpii(gel(L,1), gel(L,2)) < 0)
97 11747 : dT = ZX_disc(S->T); /* more efficient */
98 : else
99 : {
100 237366 : GEN l0 = leading_coeff(S->T0);
101 237366 : GEN a = gpowgs(gdiv(gpowgs(L, d), sqri(l0)), d-1);
102 237365 : dT = gmul(a, ZX_disc(S->T0)); /* more efficient */
103 : }
104 249104 : return S->dT = dT;
105 : }
106 :
107 : /* dT != 0 */
108 : static GEN
109 788750 : poldiscfactors_i(GEN T, GEN dT, long flag)
110 : {
111 : GEN U, fa, Z, E, P, Tp;
112 : long i, l;
113 :
114 788750 : fa = absZ_factor_limit_strict(dT, minuu(tridiv_bound(dT), maxprime()), &U);
115 788778 : if (!U) return fa;
116 777 : Z = mkcol(gel(U,1)); P = gel(fa,1); Tp = NULL;
117 1680 : while (lg(Z) != 1)
118 : { /* pop and handle last element of Z */
119 903 : GEN p = veclast(Z), r;
120 903 : setlg(Z, lg(Z)-1);
121 903 : if (!Tp) /* first time: p is composite and not a power */
122 777 : Tp = ZX_deriv(T);
123 : else
124 : {
125 126 : (void)Z_isanypower(p, &p);
126 126 : if ((flag || lgefint(p)==3) && BPSW_psp(p))
127 96 : { P = vec_append(P, p); continue; }
128 : }
129 807 : r = FpX_gcd_check(T, Tp, p);
130 807 : if (r)
131 63 : Z = shallowconcat(Z, Z_cba(r, diviiexact(p,r)));
132 744 : else if (flag)
133 7 : P = shallowconcat(P, gel(Z_factor(p),1));
134 : else
135 737 : P = vec_append(P, p);
136 : }
137 777 : ZV_sort_inplace(P); l = lg(P); E = cgetg(l, t_COL);
138 7086 : for (i = 1; i < l; i++) gel(E,i) = utoipos(Z_pvalrem(dT, gel(P,i), &dT));
139 777 : return mkmat2(P,E);
140 : }
141 :
142 : GEN
143 42 : poldiscfactors(GEN T, long flag)
144 : {
145 42 : pari_sp av = avma;
146 : GEN dT;
147 42 : if (typ(T) != t_POL || !RgX_is_ZX(T)) pari_err_TYPE("poldiscfactors",T);
148 42 : if (flag < 0 || flag > 1) pari_err_FLAG("poldiscfactors");
149 42 : dT = ZX_disc(T);
150 42 : if (!signe(dT)) retmkvec2(gen_0, Z_factor(gen_0));
151 35 : return gc_GEN(av, mkvec2(dT, poldiscfactors_i(T, dT, flag)));
152 : }
153 :
154 : static void
155 812972 : nfmaxord_check_args(nfmaxord_t *S, GEN T, long flag)
156 : {
157 812972 : GEN dT, L, E, P, fa = NULL;
158 : pari_timer t;
159 812972 : long l, ty = typ(T);
160 :
161 812972 : if (DEBUGLEVEL) timer_start(&t);
162 812972 : if (ty == t_VEC) {
163 24178 : if (lg(T) != 3) pari_err_TYPE("nfmaxord",T);
164 24178 : fa = gel(T,2); T = gel(T,1); ty = typ(T);
165 : }
166 812972 : if (ty != t_POL) pari_err_TYPE("nfmaxord",T);
167 812972 : T = Q_primpart(T);
168 812950 : if (degpol(T) <= 0) pari_err_CONSTPOL("nfmaxord");
169 812949 : RgX_check_ZX(T, "nfmaxord");
170 812950 : S->T0 = T;
171 812950 : S->T = T = ZX_Q_normalize(T, &L);
172 812944 : S->unscale = L;
173 812944 : S->dT = dT = set_disc(S);
174 812893 : S->certify = 1;
175 812893 : if (!signe(dT)) pari_err_IRREDPOL("nfmaxord",T);
176 812893 : if (fa)
177 : {
178 24178 : const long MIN = 100; /* include at least all p < 101 */
179 24178 : GEN P0 = NULL, U;
180 24178 : S->certify = 0;
181 24178 : if (!isint1(L)) fa = update_fact(dT, fa);
182 24171 : switch(typ(fa))
183 : {
184 224 : case t_MAT:
185 224 : if (!is_Z_factornon0(fa)) pari_err_TYPE("nfmaxord",fa);
186 217 : P0 = gel(fa,1); /* fall through */
187 4011 : case t_VEC: case t_COL:
188 4011 : if (!P0)
189 : {
190 3794 : if (!RgV_is_ZV(fa)) pari_err_TYPE("nfmaxord",fa);
191 3794 : P0 = fa;
192 : }
193 4011 : P = gel(absZ_factor_limit_strict(dT, MIN, &U), 1);
194 4011 : if (lg(P) != 0) { settyp(P, typ(P0)); P0 = shallowconcat(P0,P); }
195 4011 : P0 = ZV_sort_uniq_shallow(P0);
196 4011 : fa = fact_from_factors(dT, P0, 0);
197 4004 : break;
198 20146 : case t_INT:
199 20146 : fa = absZ_factor_limit(dT, (signe(fa) <= 0)? 1: maxuu(itou(fa), MIN));
200 20146 : break;
201 7 : default:
202 7 : pari_err_TYPE("nfmaxord",fa);
203 : }
204 : }
205 : else
206 : {
207 788715 : S->certify = !(flag & nf_PARTIALFACT);
208 788715 : fa = poldiscfactors_i(T, dT, 0);
209 : }
210 812893 : P = gel(fa,1); l = lg(P);
211 812893 : E = gel(fa,2);
212 812893 : if (l > 1 && is_pm1(gel(P,1)))
213 : {
214 21 : l--;
215 21 : P = vecslice(P, 2, l);
216 21 : E = vecslice(E, 2, l);
217 : }
218 812888 : S->dTP = P;
219 812888 : S->dTE = vec_to_vecsmall(E);
220 812853 : if (DEBUGLEVEL>2) timer_printf(&t, "disc. factorisation");
221 812853 : }
222 :
223 : static int
224 221243 : fnz(GEN x,long j)
225 : {
226 : long i;
227 709423 : for (i=1; i<j; i++)
228 538735 : if (signe(gel(x,i))) return 0;
229 170688 : return 1;
230 : }
231 : /* return list u[i], 2 by 2 coprime with the same prime divisors as ab */
232 : static GEN
233 294 : get_coprimes(GEN a, GEN b)
234 : {
235 294 : long i, k = 1;
236 294 : GEN u = cgetg(3, t_COL);
237 294 : gel(u,1) = a;
238 294 : gel(u,2) = b;
239 : /* u1,..., uk 2 by 2 coprime */
240 1071 : while (k+1 < lg(u))
241 : {
242 777 : GEN d, c = gel(u,k+1);
243 777 : if (is_pm1(c)) { k++; continue; }
244 1309 : for (i=1; i<=k; i++)
245 : {
246 840 : GEN ui = gel(u,i);
247 840 : if (is_pm1(ui)) continue;
248 483 : d = gcdii(c, ui);
249 483 : if (d == gen_1) continue;
250 483 : c = diviiexact(c, d);
251 483 : gel(u,i) = diviiexact(ui, d);
252 483 : u = vec_append(u, d);
253 : }
254 469 : gel(u,++k) = c;
255 : }
256 1365 : for (i = k = 1; i < lg(u); i++)
257 1071 : if (!is_pm1(gel(u,i))) gel(u,k++) = gel(u,i);
258 294 : setlg(u, k); return u;
259 : }
260 :
261 : /*******************************************************************/
262 : /* */
263 : /* ROUND 4 */
264 : /* */
265 : /*******************************************************************/
266 : typedef struct {
267 : /* constants */
268 : long pisprime; /* -1: unknown, 1: prime, 0: composite */
269 : GEN p, f; /* goal: factor f p-adically */
270 : long df;
271 : GEN pdf; /* p^df = reduced discriminant of f */
272 : long mf; /* */
273 : GEN psf, pmf; /* stability precision for f, wanted precision for f */
274 : long vpsf; /* v_p(p_f) */
275 : /* these are updated along the way */
276 : GEN phi; /* a p-integer, in Q[X] */
277 : GEN phi0; /* a p-integer, in Q[X] from testb2 / testc2, to be composed with
278 : * phi when correct precision is known */
279 : GEN chi; /* characteristic polynomial of phi (mod psc) in Z[X] */
280 : GEN nu; /* irreducible divisor of chi mod p, in Z[X] */
281 : GEN invnu; /* numerator ( 1/ Mod(nu, chi) mod pmr ) */
282 : GEN Dinvnu;/* denominator ( ... ) */
283 : long vDinvnu; /* v_p(Dinvnu) */
284 : GEN prc, psc; /* reduced discriminant of chi, stability precision for chi */
285 : long vpsc; /* v_p(p_c) */
286 : GEN ns, nsf, precns; /* cached Newton sums for nsf and their precision */
287 : } decomp_t;
288 : static GEN maxord_i(decomp_t *S, GEN p, GEN f, long mf, GEN w, long flag);
289 : static GEN dbasis(GEN p, GEN f, long mf, GEN alpha, GEN U);
290 : static GEN maxord(GEN p,GEN f,long mf);
291 : static GEN ZX_Dedekind(GEN F, GEN *pg, GEN p);
292 :
293 : static void
294 498 : fix_PE(GEN *pP, GEN *pE, long i, GEN u, GEN N)
295 : {
296 : GEN P, E;
297 498 : long k, l = lg(u), lP = lg(*pP);
298 : pari_sp av;
299 :
300 498 : *pP = P = shallowconcat(*pP, vecslice(u, 2, l-1));
301 498 : *pE = E = vecsmall_lengthen(*pE, lP + l-2);
302 498 : gel(P,i) = gel(u,1); av = avma;
303 498 : E[i] = Z_pvalrem(N, gel(P,i), &N);
304 1003 : for (k=lP, lP=lg(P); k < lP; k++) E[k] = Z_pvalrem(N, gel(P,k), &N);
305 498 : set_avma(av);
306 498 : }
307 : static long
308 685904 : diag_denomval(GEN M, GEN p)
309 : {
310 : long j, v, l;
311 685904 : if (typ(M) != t_MAT) return 0;
312 463404 : v = 0; l = lg(M);
313 2072739 : for (j=1; j<l; j++)
314 : {
315 1609336 : GEN t = gcoeff(M,j,j);
316 1609336 : if (typ(t) == t_FRAC) v += Z_pval(gel(t,2), p);
317 : }
318 463403 : return v;
319 : }
320 :
321 : /* n > 1 is composite, not a pure power, and has no prime divisor < 2^14;
322 : * return a BPSW divisor of n and smallest k-th root of largest coprime cofactor */
323 : static GEN
324 183 : Z_fac(GEN n)
325 : {
326 183 : GEN p = icopy(n), part = ifac_start(p, 0);
327 : long e;
328 183 : ifac_next(&part, &p, &e); n = diviiexact(n, powiu(p, e));
329 183 : (void)Z_isanypower(n, &n); return mkvec2(p, n);
330 : }
331 :
332 : /* Warning: data computed for T = ZX_Q_normalize(T0). If S.unscale !=
333 : * gen_1, caller must take steps to correct the components if it wishes
334 : * to stick to the original T0. Return a vector of p-maximal orders, for
335 : * those p s.t p^2 | disc(T) [ = S->dTP ]*/
336 : static GEN
337 812952 : get_maxord(nfmaxord_t *S, GEN T0, long flag)
338 : {
339 : GEN P, E;
340 : VOLATILE GEN O;
341 : VOLATILE long lP, i, k;
342 :
343 812952 : nfmaxord_check_args(S, T0, flag);
344 812853 : P = S->dTP; lP = lg(P);
345 812853 : E = S->dTE;
346 812853 : O = cgetg(1, t_VEC);
347 3144077 : for (i=1; i<lP; i++)
348 : {
349 : VOLATILE pari_sp av;
350 : /* includes the silly case where P[i] = -1 */
351 2331188 : if (E[i] <= 1)
352 : {
353 1303368 : if (S->certify)
354 : {
355 1295324 : GEN p = gel(P,i);
356 1295324 : if (signe(p) > 0 && !BPSW_psp(p))
357 : {
358 183 : fix_PE(&P, &E, i, Z_fac(p), S->dT);
359 183 : lP = lg(P); i--; continue;
360 : }
361 : }
362 1303183 : O = vec_append(O, gen_1); continue;
363 : }
364 1027820 : av = avma;
365 1027820 : pari_CATCH(CATCH_ALL) {
366 294 : GEN u, err = pari_err_last();
367 : long l;
368 294 : switch(err_get_num(err))
369 : {
370 294 : case e_INV:
371 : {
372 294 : GEN p, x = err_get_compo(err, 2);
373 294 : if (typ(x) == t_INTMOD)
374 : { /* caught false prime, update factorization */
375 294 : p = gcdii(gel(x,1), gel(x,2));
376 294 : u = diviiexact(gel(x,1),p);
377 294 : if (DEBUGLEVEL) pari_warn(warner,"impossible inverse: %Ps", x);
378 294 : (void)gc_all(av, 2, &p, &u);
379 :
380 294 : u = get_coprimes(p, u); l = lg(u);
381 : /* no small factors, but often a prime power */
382 882 : for (k = 1; k < l; k++) (void)Z_isanypower(gel(u,k), &gel(u,k));
383 294 : break;
384 : }
385 : /* fall through */
386 : }
387 : case e_PRIME: case e_IRREDPOL:
388 : { /* we're here because we failed BPSW_isprime(), no point in
389 : * reporting a possible counter-example to the BPSW test */
390 0 : GEN p = gel(P,i);
391 0 : set_avma(av);
392 0 : if (DEBUGLEVEL)
393 0 : pari_warn(warner,"large composite in nfmaxord:loop(), %Ps", p);
394 0 : if (expi(p) < 100)
395 0 : u = gel(Z_factor(p), 1); /* p < 2^100 should take ~20ms */
396 0 : else if (S->certify)
397 0 : u = Z_fac(p);
398 : else
399 0 : { /* give up, probably not maximal */
400 0 : GEN B, g, k = ZX_Dedekind(S->T, &g, p);
401 0 : k = FpX_normalize(k, p);
402 0 : B = dbasis(p, S->T, E[i], NULL, FpX_div(S->T,k,p));
403 0 : O = vec_append(O, B);
404 0 : pari_CATCH_reset(); continue;
405 : }
406 0 : break;
407 : }
408 0 : default: pari_err(0, err);
409 : return NULL;/*LCOV_EXCL_LINE*/
410 : }
411 294 : fix_PE(&P, &E, i, u, S->dT);
412 294 : lP = lg(P); av = avma;
413 1028115 : } pari_RETRY {
414 1028115 : GEN p = gel(P,i), O2;
415 1028115 : if (DEBUGLEVEL>2) err_printf("Treating p^k = %Ps^%ld\n",p,E[i]);
416 1028115 : O2 = maxord(p,S->T,E[i]);
417 1027864 : if (S->certify && (odd(E[i]) || E[i] != 2*diag_denomval(O2, p))
418 610097 : && !BPSW_psp(p))
419 : {
420 21 : fix_PE(&P, &E, i, gel(Z_factor(p), 1), S->dT);
421 21 : lP = lg(P); i--;
422 : }
423 : else
424 1027838 : O = vec_append(O, O2);
425 1027849 : } pari_ENDCATCH;
426 : }
427 812889 : S->dTP = P; S->dTE = E; return O;
428 : }
429 :
430 : /* M a QM, return denominator of diagonal. All denominators are powers of
431 : * a given integer */
432 : static GEN
433 99436 : diag_denom(GEN M)
434 : {
435 99436 : GEN d = gen_1;
436 99436 : long j, l = lg(M);
437 694238 : for (j=1; j<l; j++)
438 : {
439 594802 : GEN t = gcoeff(M,j,j);
440 594802 : if (typ(t) == t_INT) continue;
441 211949 : t = gel(t,2);
442 211949 : if (abscmpii(t,d) > 0) d = t;
443 : }
444 99436 : return d;
445 : }
446 : static void
447 746475 : setPE(GEN D, GEN P, GEN *pP, GEN *pE)
448 : {
449 746475 : long k, j, l = lg(P);
450 : GEN P2, E2;
451 746475 : *pP = P2 = cgetg(l, t_VEC);
452 746473 : *pE = E2 = cgetg(l, t_VECSMALL);
453 2870098 : for (k = j = 1; j < l; j++)
454 : {
455 2123597 : long v = Z_pvalrem(D, gel(P,j), &D);
456 2123619 : if (v) { gel(P2,k) = gel(P,j); E2[k] = v; k++; }
457 : }
458 746501 : setlg(P2, k);
459 746498 : setlg(E2, k);
460 746491 : }
461 : void
462 101725 : nfmaxord(nfmaxord_t *S, GEN T0, long flag)
463 : {
464 101725 : GEN O = get_maxord(S, T0, flag);
465 101724 : GEN f = S->T, P = S->dTP, a = NULL, da = NULL;
466 101724 : long n = degpol(f), lP = lg(P), i, j, k;
467 101725 : int centered = 0;
468 101725 : pari_sp av = avma;
469 : /* r1 & basden not initialized here */
470 101725 : S->r1 = -1;
471 101725 : S->basden = NULL;
472 356240 : for (i=1; i<lP; i++)
473 : {
474 254516 : GEN M, db, b = gel(O,i);
475 254516 : if (b == gen_1) continue;
476 99436 : db = diag_denom(b);
477 99436 : if (db == gen_1) continue;
478 :
479 : /* db = denom(b), (da,db) = 1. Compute da Im(b) + db Im(a) */
480 99436 : b = Q_muli_to_int(b,db);
481 99435 : if (!da) { da = db; a = b; }
482 : else
483 : { /* optimization: easy as long as both matrix are diagonal */
484 134397 : j=2; while (j<=n && fnz(gel(a,j),j) && fnz(gel(b,j),j)) j++;
485 50569 : k = j-1; M = cgetg(2*n-k+1,t_MAT);
486 184966 : for (j=1; j<=k; j++)
487 : {
488 134397 : gel(M,j) = gel(a,j);
489 134397 : gcoeff(M,j,j) = mulii(gcoeff(a,j,j),gcoeff(b,j,j));
490 : }
491 : /* could reduce mod M(j,j) but not worth it: usually close to da*db */
492 279258 : for ( ; j<=n; j++) gel(M,j) = ZC_Z_mul(gel(a,j), db);
493 279256 : for ( ; j<=2*n-k; j++) gel(M,j) = ZC_Z_mul(gel(b,j+k-n), da);
494 50567 : da = mulii(da,db);
495 50569 : a = ZM_hnfmodall_i(M, da, hnf_MODID|hnf_CENTER);
496 50568 : (void)gc_all(av, 2, &a, &da);
497 50569 : centered = 1;
498 : }
499 : }
500 101724 : if (da)
501 : {
502 48866 : GEN index = diviiexact(da, gcoeff(a,1,1));
503 231697 : for (j=2; j<=n; j++) index = mulii(index, diviiexact(da, gcoeff(a,j,j)));
504 48866 : if (!centered) a = ZM_hnfcenter(a);
505 48866 : a = RgM_Rg_div(a, da);
506 48867 : S->index = index;
507 48867 : S->dK = diviiexact(S->dT, sqri(index));
508 : }
509 : else
510 : {
511 52858 : S->index = gen_1;
512 52858 : S->dK = S->dT;
513 52858 : a = matid(n);
514 : }
515 101724 : setPE(S->dK, P, &S->dKP, &S->dKE);
516 101725 : S->basis = RgM_to_RgXV(a, varn(f));
517 101723 : }
518 : GEN
519 938 : nfbasis(GEN x, GEN *pdK)
520 : {
521 938 : pari_sp av = avma;
522 : nfmaxord_t S;
523 : GEN B;
524 938 : nfmaxord(&S, x, 0);
525 938 : B = RgXV_unscale(S.basis, S.unscale);
526 938 : if (pdK) *pdK = S.dK;
527 938 : return gc_all(av, pdK? 2: 1, &B, pdK);
528 : }
529 : /* field discriminant: faster than nfmaxord, use local data only */
530 : static GEN
531 711239 : maxord_disc(nfmaxord_t *S, GEN x)
532 : {
533 711239 : GEN O = get_maxord(S, x, 0), I = gen_1;
534 711201 : long n = degpol(S->T), lP = lg(O), i, j;
535 2787705 : for (i = 1; i < lP; i++)
536 : {
537 2076517 : GEN b = gel(O,i);
538 2076517 : if (b == gen_1) continue;
539 2707777 : for (j = 1; j <= n; j++)
540 : {
541 2116209 : GEN c = gcoeff(b,j,j);
542 2116209 : if (typ(c) == t_FRAC) I = mulii(I, gel(c,2)) ;
543 : }
544 : }
545 711188 : return diviiexact(S->dT, sqri(I));
546 : }
547 : GEN
548 66465 : nfdisc(GEN x)
549 : {
550 66465 : pari_sp av = avma;
551 : nfmaxord_t S;
552 66465 : return gc_INT(av, maxord_disc(&S, x));
553 : }
554 : GEN
555 644781 : nfdiscfactors(GEN x)
556 : {
557 644781 : pari_sp av = avma;
558 644781 : GEN E, P, D, nf = checknf_i(x);
559 644781 : if (nf)
560 : {
561 7 : D = nf_get_disc(nf);
562 7 : P = nf_get_ramified_primes(nf);
563 : }
564 : else
565 : {
566 : nfmaxord_t S;
567 644774 : D = maxord_disc(&S, x);
568 644743 : P = S.dTP;
569 : }
570 644750 : setPE(D, P, &P, &E); settyp(P, t_COL);
571 644766 : return gc_GEN(av, mkvec2(D, mkmat2(P, zc_to_ZC(E))));
572 : }
573 :
574 : static ulong
575 1599640 : Flx_checkdeflate(GEN x)
576 : {
577 1599640 : ulong d = 0, i, lx = (ulong)lg(x);
578 2557615 : for (i=3; i<lx; i++)
579 1710122 : if (x[i]) { d = ugcd(d,i-2); if (d == 1) break; }
580 1599639 : return d;
581 : }
582 :
583 : /* product of (monic) irreducible factors of f over Fp[X]
584 : * Assume f reduced mod p, otherwise valuation at x may be wrong */
585 : static GEN
586 1599633 : Flx_radical(GEN f, ulong p)
587 : {
588 1599633 : long v0 = Flx_valrem(f, &f);
589 : ulong du, d, e;
590 : GEN u;
591 :
592 1599638 : d = Flx_checkdeflate(f);
593 1599672 : if (!d) return v0? polx_Flx(f[1]): pol1_Flx(f[1]);
594 1002737 : if (u_lvalrem(d,p, &e)) f = Flx_deflate(f, d/e); /* f(x^p^i) -> f(x) */
595 1002743 : u = Flx_gcd(f, Flx_deriv(f, p), p); /* (f,f') */
596 1002734 : du = degpol(u);
597 1002736 : if (du)
598 : {
599 315940 : if (du == (ulong)degpol(f))
600 0 : f = Flx_radical(Flx_deflate(f,p), p);
601 : else
602 : {
603 315940 : u = Flx_normalize(u, p);
604 315941 : f = Flx_div(f, u, p);
605 315943 : if (p <= du)
606 : {
607 66699 : GEN w = (degpol(f) >= degpol(u))? Flx_rem(f, u, p): f;
608 66699 : w = Flxq_powu(w, du, u, p);
609 66699 : w = Flx_div(u, Flx_gcd(w,u,p), p); /* u / gcd(u, v^(deg u-1)) */
610 66700 : f = Flx_mul(f, Flx_radical(Flx_deflate(w,p), p), p);
611 : }
612 : }
613 : }
614 1002744 : if (v0) f = Flx_shift(f, 1);
615 1002740 : return f;
616 : }
617 : /* Assume f reduced mod p, otherwise valuation at x may be wrong */
618 : static GEN
619 5692 : FpX_radical(GEN f, GEN p)
620 : {
621 : GEN u;
622 : long v0;
623 5692 : if (lgefint(p) == 3)
624 : {
625 1755 : ulong q = p[2];
626 1755 : return Flx_to_ZX( Flx_radical(ZX_to_Flx(f, q), q) );
627 : }
628 3937 : v0 = ZX_valrem(f, &f);
629 3937 : u = FpX_gcd(f,FpX_deriv(f, p), p);
630 3649 : if (degpol(u)) f = FpX_div(f, u, p);
631 3649 : if (v0) f = RgX_shift(f, 1);
632 3649 : return f;
633 : }
634 : /* f / a */
635 : static GEN
636 1531230 : zx_z_div(GEN f, ulong a)
637 : {
638 1531230 : long i, l = lg(f);
639 1531230 : GEN g = cgetg(l, t_VECSMALL);
640 1531214 : g[1] = f[1];
641 5186039 : for (i = 2; i < l; i++) g[i] = f[i] / a;
642 1531214 : return g;
643 : }
644 : /* Dedekind criterion; return k = gcd(g,h, (f-gh)/p), where
645 : * f = \prod f_i^e_i, g = \prod f_i, h = \prod f_i^{e_i-1}
646 : * k = 1 iff Z[X]/(f) is p-maximal */
647 : static GEN
648 1536944 : ZX_Dedekind(GEN F, GEN *pg, GEN p)
649 : {
650 : GEN k, h, g, f, f2;
651 1536944 : ulong q = p[2];
652 1536944 : if (lgefint(p) == 3 && q < (1UL << BITS_IN_HALFULONG))
653 1531182 : {
654 1531255 : ulong q2 = q*q;
655 1531255 : f2 = ZX_to_Flx(F, q2);
656 1531237 : f = Flx_red(f2, q);
657 1531178 : g = Flx_radical(f, q);
658 1531219 : h = Flx_div(f, g, q);
659 1531206 : k = zx_z_div(Flx_sub(f2, Flx_mul(g,h,q2), q2), q);
660 1531213 : k = Flx_gcd(k, Flx_gcd(g,h,q), q);
661 1531224 : k = Flx_to_ZX(k);
662 1531168 : g = Flx_to_ZX(g);
663 : }
664 : else
665 : {
666 5689 : f2 = FpX_red(F, sqri(p));
667 5692 : f = FpX_red(f2, p);
668 5692 : g = FpX_radical(f, p);
669 5398 : h = FpX_div(f, g, p);
670 5398 : k = ZX_Z_divexact(ZX_sub(f2, ZX_mul(g,h)), p);
671 5398 : k = FpX_gcd(FpX_red(k, p), FpX_gcd(g,h,p), p);
672 : }
673 1536581 : *pg = g; return k;
674 : }
675 :
676 : /* p-maximal order of Z[x]/f; mf = v_p(Disc(f)) or < 0 [unknown].
677 : * Return gen_1 if p-maximal */
678 : static GEN
679 1536944 : maxord(GEN p, GEN f, long mf)
680 : {
681 1536944 : const pari_sp av = avma;
682 1536944 : GEN res, g, k = ZX_Dedekind(f, &g, p);
683 1536584 : long dk = degpol(k);
684 1536581 : if (DEBUGLEVEL>2) err_printf(" ZX_Dedekind: gcd has degree %ld\n", dk);
685 1536627 : if (!dk) { set_avma(av); return gen_1; }
686 874400 : if (mf < 0) mf = ZpX_disc_val(f, p);
687 874401 : k = FpX_normalize(k, p);
688 874403 : if (2*dk >= mf-1)
689 420363 : res = dbasis(p, f, mf, NULL, FpX_div(f,k,p));
690 : else
691 : {
692 : GEN w, F1, F2;
693 : decomp_t S;
694 454040 : F1 = FpX_factor(k,p);
695 454068 : F2 = FpX_factor(FpX_div(g,k,p),p);
696 454071 : w = merge_sort_uniq(gel(F1,1),gel(F2,1),(void*)cmpii,&gen_cmp_RgX);
697 454067 : res = maxord_i(&S, p, f, mf, w, 0);
698 : }
699 874437 : return gc_GEN(av,res);
700 : }
701 : /* T monic separable ZX, p prime */
702 : GEN
703 0 : ZpX_primedec(GEN T, GEN p)
704 : {
705 0 : const pari_sp av = avma;
706 0 : GEN w, F1, F2, res, g, k = ZX_Dedekind(T, &g, p);
707 : decomp_t S;
708 0 : if (!degpol(k)) return zm_to_ZM(FpX_degfact(T, p));
709 0 : k = FpX_normalize(k, p);
710 0 : F1 = FpX_factor(k,p);
711 0 : F2 = FpX_factor(FpX_div(g,k,p),p);
712 0 : w = merge_sort_uniq(gel(F1,1),gel(F2,1),(void*)cmpii,&gen_cmp_RgX);
713 0 : res = maxord_i(&S, p, T, ZpX_disc_val(T, p), w, -1);
714 0 : if (!res)
715 : {
716 0 : long f = degpol(S.nu), e = degpol(T) / f;
717 0 : set_avma(av); retmkmat2(mkcols(f), mkcols(e));
718 : }
719 0 : return gc_GEN(av,res);
720 : }
721 :
722 : static GEN
723 4670893 : Zlx_sylvester_echelon(GEN f1, GEN f2, long early_abort, ulong p, ulong pm)
724 : {
725 4670893 : long j, n = degpol(f1);
726 4670881 : GEN h, a = cgetg(n+1,t_MAT);
727 4670863 : f1 = Flx_get_red(f1, pm);
728 4670848 : h = Flx_rem(f2,f1,pm);
729 16393394 : for (j=1;; j++)
730 : {
731 16393394 : gel(a,j) = Flx_to_Flv(h, n);
732 16392494 : if (j == n) break;
733 11721760 : h = Flx_rem(Flx_shift(h, 1), f1, pm);
734 : }
735 4670734 : return zlm_echelon(a, early_abort, p, pm);
736 : }
737 : /* Sylvester's matrix, mod p^m (assumes f1 monic). If early_abort
738 : * is set, return NULL if one pivot is 0 mod p^m */
739 : static GEN
740 74028 : ZpX_sylvester_echelon(GEN f1, GEN f2, long early_abort, GEN p, GEN pm)
741 : {
742 74028 : long j, n = degpol(f1);
743 74028 : GEN h, a = cgetg(n+1,t_MAT);
744 74028 : h = FpXQ_red(f2,f1,pm);
745 422177 : for (j=1;; j++)
746 : {
747 422177 : gel(a,j) = RgX_to_RgC(h, n);
748 422178 : if (j == n) break;
749 348150 : h = FpX_rem(RgX_shift_shallow(h, 1), f1, pm);
750 : }
751 74028 : return ZpM_echelon(a, early_abort, p, pm);
752 : }
753 :
754 : /* polynomial gcd mod p^m (assumes f1 monic). Return a QpX ! */
755 : static GEN
756 246198 : Zlx_gcd(GEN f1, GEN f2, ulong p, ulong pm)
757 : {
758 246198 : pari_sp av = avma;
759 246198 : GEN a = Zlx_sylvester_echelon(f1,f2,0,p,pm);
760 246199 : long c, l = lg(a), sv = f1[1];
761 754558 : for (c = 1; c < l; c++)
762 : {
763 754558 : ulong t = ucoeff(a,c,c);
764 754558 : if (t)
765 : {
766 246199 : a = Flx_to_ZX(Flv_to_Flx(gel(a,c), sv));
767 246194 : if (t == 1) return gc_GEN(av, a);
768 74803 : return gc_upto(av, RgX_Rg_div(a, utoipos(t)));
769 : }
770 : }
771 0 : set_avma(av);
772 0 : a = cgetg(2,t_POL); a[1] = sv; return a;
773 : }
774 : GEN
775 254713 : ZpX_gcd(GEN f1, GEN f2, GEN p, GEN pm)
776 : {
777 254713 : pari_sp av = avma;
778 : GEN a;
779 : long c, l, v;
780 254713 : if (lgefint(pm) == 3)
781 : {
782 246198 : ulong q = pm[2];
783 246198 : return Zlx_gcd(ZX_to_Flx(f1, q), ZX_to_Flx(f2,q), p[2], q);
784 : }
785 8515 : a = ZpX_sylvester_echelon(f1,f2,0,p,pm);
786 8515 : l = lg(a); v = varn(f1);
787 53130 : for (c = 1; c < l; c++)
788 : {
789 53130 : GEN t = gcoeff(a,c,c);
790 53130 : if (signe(t))
791 : {
792 8515 : a = RgV_to_RgX(gel(a,c), v);
793 8515 : if (equali1(t)) return gc_GEN(av, a);
794 2469 : return gc_upto(av, RgX_Rg_div(a, t));
795 : }
796 : }
797 0 : set_avma(av); return pol_0(v);
798 : }
799 :
800 : /* Return m > 0, such that p^m ~ 2^16 for initial value of m; assume p prime */
801 : static long
802 4382816 : init_m(GEN p)
803 : {
804 : ulong pp;
805 4382816 : if (lgefint(p) > 3) return 1;
806 4381679 : pp = p[2]; /* m ~ 16 / log2(pp) */
807 4381679 : if (pp < 41) switch(pp)
808 : {
809 1193210 : case 2: return 16;
810 351285 : case 3: return 10;
811 245284 : case 5: return 6;
812 153072 : case 7: return 5;
813 209264 : case 11: case 13: return 4;
814 300211 : default: return 3;
815 : }
816 1929353 : return pp < 257? 2: 1;
817 : }
818 :
819 : /* reduced resultant mod p^m (assumes x monic) */
820 : GEN
821 993142 : ZpX_reduced_resultant(GEN x, GEN y, GEN p, GEN pm)
822 : {
823 993142 : pari_sp av = avma;
824 : GEN z;
825 993142 : if (lgefint(pm) == 3)
826 : {
827 981332 : ulong q = pm[2];
828 981332 : z = Zlx_sylvester_echelon(ZX_to_Flx(x,q), ZX_to_Flx(y,q),0,p[2],q);
829 981397 : if (lg(z) > 1)
830 : {
831 981397 : ulong c = ucoeff(z,1,1);
832 981397 : if (c) return gc_utoipos(av, c);
833 : }
834 : }
835 : else
836 : {
837 11810 : z = ZpX_sylvester_echelon(x,y,0,p,pm);
838 11810 : if (lg(z) > 1)
839 : {
840 11810 : GEN c = gcoeff(z,1,1);
841 11810 : if (signe(c)) return gc_INT(av, c);
842 : }
843 : }
844 128702 : set_avma(av); return gen_0;
845 : }
846 : /* Assume Res(f,g) divides p^M. Return Res(f, g), using dynamic p-adic
847 : * precision (until result is nonzero or p^M). */
848 : GEN
849 931995 : ZpX_reduced_resultant_fast(GEN f, GEN g, GEN p, long M)
850 : {
851 931995 : GEN R, q = NULL;
852 : long m;
853 931995 : m = init_m(p); if (m < 1) m = 1;
854 61146 : for(;; m <<= 1) {
855 993134 : if (M < 2*m) break;
856 94061 : q = q? sqri(q): powiu(p, m); /* p^m */
857 94062 : R = ZpX_reduced_resultant(f,g, p, q); if (signe(R)) return R;
858 : }
859 899073 : q = powiu(p, M);
860 899083 : R = ZpX_reduced_resultant(f,g, p, q); return signe(R)? R: q;
861 : }
862 :
863 : /* v_p(Res(x,y) mod p^m), assumes (lc(x),p) = 1 */
864 : static long
865 3497000 : ZpX_resultant_val_i(GEN x, GEN y, GEN p, GEN pm)
866 : {
867 3497000 : pari_sp av = avma;
868 : GEN z;
869 : long i, l, v;
870 3497000 : if (lgefint(pm) == 3)
871 : {
872 3443296 : ulong q = pm[2], pp = p[2];
873 3443296 : z = Zlx_sylvester_echelon(ZX_to_Flx(x,q), ZX_to_Flx(y,q), 1, pp, q);
874 3443463 : if (!z) return gc_long(av,-1); /* failure */
875 3256914 : v = 0; l = lg(z);
876 13566336 : for (i = 1; i < l; i++) v += u_lval(ucoeff(z,i,i), pp);
877 : }
878 : else
879 : {
880 53704 : z = ZpX_sylvester_echelon(x, y, 1, p, pm);
881 53703 : if (!z) return gc_long(av,-1); /* failure */
882 52886 : v = 0; l = lg(z);
883 194474 : for (i = 1; i < l; i++) v += Z_pval(gcoeff(z,i,i), p);
884 : }
885 3309794 : return v;
886 : }
887 :
888 : /* assume (lc(f),p) = 1; no assumption on g */
889 : long
890 3450889 : ZpX_resultant_val(GEN f, GEN g, GEN p, long M)
891 : {
892 3450889 : pari_sp av = avma;
893 3450889 : GEN q = NULL;
894 : long v, m;
895 3450889 : m = init_m(p); if (m < 2) m = 2;
896 46069 : for(;; m <<= 1) {
897 3496952 : if (m > M) m = M;
898 3496952 : q = q? sqri(q): powiu(p, m); /* p^m */
899 3497003 : v = ZpX_resultant_val_i(f,g, p, q); if (v >= 0) return gc_long(av,v);
900 187363 : if (m == M) return gc_long(av,M);
901 : }
902 : }
903 :
904 : /* assume f separable and (lc(f),p) = 1 */
905 : long
906 184548 : ZpX_disc_val(GEN f, GEN p)
907 : {
908 184548 : pari_sp av = avma;
909 : long v;
910 184548 : if (degpol(f) == 1) return 0;
911 184546 : v = ZpX_resultant_val(f, ZX_deriv(f), p, LONG_MAX);
912 184549 : return gc_long(av,v);
913 : }
914 :
915 : /* *e a ZX, *d, *z in Z, *d = p^(*vd). Simplify e / d by cancelling a
916 : * common factor p^v; if z!=NULL, update it by cancelling the same power of p */
917 : static void
918 3570117 : update_den(GEN p, GEN *e, GEN *d, long *vd, GEN *z)
919 : {
920 : GEN newe;
921 3570117 : long ve = ZX_pvalrem(*e, p, &newe);
922 3570167 : if (ve) {
923 : GEN newd;
924 1756792 : long v = minss(*vd, ve);
925 1756781 : if (v) {
926 1756856 : if (v == *vd)
927 : { /* rare, denominator cancelled */
928 382226 : if (ve != v) newe = ZX_Z_mul(newe, powiu(p, ve - v));
929 382226 : newd = gen_1;
930 382226 : *vd = 0;
931 382226 : if (z) *z =diviiexact(*z, powiu(p, v));
932 : }
933 : else
934 : { /* v = ve < vd, generic case */
935 1374630 : GEN q = powiu(p, v);
936 1374688 : newd = diviiexact(*d, q);
937 1374509 : *vd -= v;
938 1374509 : if (z) *z = diviiexact(*z, q);
939 : }
940 1756716 : *e = newe;
941 1756716 : *d = newd;
942 : }
943 : }
944 3570016 : }
945 :
946 : /* return denominator, a power of p */
947 : static GEN
948 2749888 : QpX_denom(GEN x)
949 : {
950 2749888 : long i, l = lg(x);
951 2749888 : GEN maxd = gen_1;
952 9499439 : for (i=2; i<l; i++)
953 : {
954 6749551 : GEN d = gel(x,i);
955 6749551 : if (typ(d) == t_FRAC && cmpii(gel(d,2), maxd) > 0) maxd = gel(d,2);
956 : }
957 2749888 : return maxd;
958 : }
959 : static GEN
960 508822 : QpXV_denom(GEN x)
961 : {
962 508822 : long l = lg(x), i;
963 508822 : GEN maxd = gen_1;
964 1515603 : for (i = 1; i < l; i++)
965 : {
966 1006781 : GEN d = QpX_denom(gel(x,i));
967 1006781 : if (cmpii(d, maxd) > 0) maxd = d;
968 : }
969 508822 : return maxd;
970 : }
971 :
972 : static GEN
973 1743121 : QpX_remove_denom(GEN x, GEN p, GEN *pdx, long *pv)
974 : {
975 1743121 : *pdx = QpX_denom(x);
976 1743114 : if (*pdx == gen_1) { *pv = 0; *pdx = NULL; }
977 : else {
978 1266827 : x = Q_muli_to_int(x,*pdx);
979 1266736 : *pv = Z_pval(*pdx, p);
980 : }
981 1743033 : return x;
982 : }
983 :
984 : /* p^v * f o g mod (T,q). q = p^vq */
985 : static GEN
986 287188 : compmod(GEN p, GEN f, GEN g, GEN T, GEN q, long v)
987 : {
988 287188 : GEN D = NULL, z, df, dg, qD;
989 287188 : long vD = 0, vdf, vdg;
990 :
991 287188 : f = QpX_remove_denom(f, p, &df, &vdf);
992 287191 : if (typ(g) == t_VEC) /* [num,den,v_p(den)] */
993 0 : { vdg = itos(gel(g,3)); dg = gel(g,2); g = gel(g,1); }
994 : else
995 287191 : g = QpX_remove_denom(g, p, &dg, &vdg);
996 287192 : if (df) { D = df; vD = vdf; }
997 287192 : if (dg) {
998 56082 : long degf = degpol(f);
999 56082 : D = mul_content(D, powiu(dg, degf));
1000 56082 : vD += degf * vdg;
1001 : }
1002 287192 : qD = D ? mulii(q, D): q;
1003 287184 : if (dg) f = FpX_rescale(f, dg, qD);
1004 287184 : z = FpX_FpXQ_eval(f, g, T, qD);
1005 287187 : if (!D) {
1006 0 : if (v) {
1007 0 : if (v > 0)
1008 0 : z = ZX_Z_mul(z, powiu(p, v));
1009 : else
1010 0 : z = RgX_Rg_div(z, powiu(p, -v));
1011 : }
1012 0 : return z;
1013 : }
1014 287187 : update_den(p, &z, &D, &vD, NULL);
1015 287192 : qD = mulii(D,q);
1016 287186 : if (v) vD -= v;
1017 287186 : z = FpX_center_i(z, qD, shifti(qD,-1));
1018 287189 : if (vD > 0)
1019 287189 : z = RgX_Rg_div(z, powiu(p, vD));
1020 0 : else if (vD < 0)
1021 0 : z = ZX_Z_mul(z, powiu(p, -vD));
1022 287193 : return z;
1023 : }
1024 :
1025 : /* fast implementation of ZM_hnfmodid(M, D) / D, D = p^k */
1026 : static GEN
1027 454067 : ZpM_hnfmodid(GEN M, GEN p, GEN D)
1028 : {
1029 454067 : long i, l = lg(M);
1030 454067 : M = RgM_Rg_div(ZpM_echelon(M,0,p,D), D);
1031 2029572 : for (i = 1; i < l; i++)
1032 1575507 : if (gequal0(gcoeff(M,i,i))) gcoeff(M,i,i) = gen_1;
1033 454065 : return M;
1034 : }
1035 :
1036 : /* Return Z-basis for Z[a] + U(a)/p Z[a] in Z[t]/(f), mf = v_p(disc f), U
1037 : * a ZX. Special cases: a = t is coded as NULL, U = 0 is coded as NULL */
1038 : static GEN
1039 620016 : dbasis(GEN p, GEN f, long mf, GEN a, GEN U)
1040 : {
1041 620016 : long n = degpol(f), i, dU;
1042 : GEN b, h;
1043 :
1044 620016 : if (n == 1) return matid(1);
1045 620016 : if (a && gequalX(a)) a = NULL;
1046 620015 : if (DEBUGLEVEL>5)
1047 : {
1048 0 : err_printf(" entering Dedekind Basis with parameters p=%Ps\n",p);
1049 0 : err_printf(" f = %Ps,\n a = %Ps\n",f, a? a: pol_x(varn(f)));
1050 : }
1051 620020 : if (a)
1052 : {
1053 199656 : GEN pd = powiu(p, mf >> 1);
1054 199656 : GEN da, pdp = mulii(pd,p), D = pdp;
1055 : long vda;
1056 199657 : dU = U ? degpol(U): 0;
1057 199657 : b = cgetg(n+1, t_MAT);
1058 199655 : h = scalarpol(pd, varn(f));
1059 199657 : a = QpX_remove_denom(a, p, &da, &vda);
1060 199657 : if (da) D = mulii(D, da);
1061 199656 : gel(b,1) = scalarcol_shallow(pd, n);
1062 568737 : for (i=2; i<=n; i++)
1063 : {
1064 369081 : if (i == dU+1)
1065 0 : h = compmod(p, U, mkvec3(a,da,stoi(vda)), f, pdp, (mf>>1) - 1);
1066 : else
1067 : {
1068 369081 : h = FpXQ_mul(h, a, f, D);
1069 369075 : if (da) h = ZX_Z_divexact(h, da);
1070 : }
1071 369062 : gel(b,i) = RgX_to_RgC(h,n);
1072 : }
1073 199656 : return ZpM_hnfmodid(b, p, pd);
1074 : }
1075 : else
1076 : {
1077 420364 : if (!U) return matid(n);
1078 420364 : dU = degpol(U);
1079 420364 : if (dU == n) return matid(n);
1080 420364 : U = FpX_normalize(U, p);
1081 420368 : b = cgetg(n+1, t_MAT);
1082 1628825 : for (i = 1; i <= dU; i++) gel(b,i) = vec_ei(n, i);
1083 420370 : h = RgX_Rg_div(U, p);
1084 472483 : for ( ; i <= n; i++)
1085 : {
1086 472483 : gel(b, i) = RgX_to_RgC(h,n);
1087 472482 : if (i == n) break;
1088 52110 : h = RgX_shift_shallow(h,1);
1089 : }
1090 420372 : return b;
1091 : }
1092 : }
1093 :
1094 : static GEN
1095 508826 : get_partial_order_as_pols(GEN p, GEN f)
1096 : {
1097 508826 : GEN O = maxord(p, f, -1);
1098 508817 : long v = varn(f);
1099 508817 : return O == gen_1? pol_x_powers(degpol(f), v): RgM_to_RgXV(O, v);
1100 : }
1101 :
1102 : static long
1103 2209 : p_is_prime(decomp_t *S)
1104 : {
1105 2209 : if (S->pisprime < 0) S->pisprime = BPSW_psp(S->p);
1106 2209 : return S->pisprime;
1107 : }
1108 : static GEN ZpX_monic_factor_squarefree(GEN f, GEN p, long prec);
1109 :
1110 : /* if flag = 0, maximal order, else factorization to precision r = flag */
1111 : static GEN
1112 254705 : Decomp(decomp_t *S, long flag)
1113 : {
1114 254705 : pari_sp av = avma;
1115 : GEN fred, pr2, pr, pk, ph2, ph, b1, b2, a, e, de, f1, f2, dt, th, chip;
1116 254705 : GEN p = S->p;
1117 254705 : long vde, vdt, k, r = maxss(flag, 2*S->df + 1);
1118 :
1119 254705 : if (DEBUGLEVEL>5) err_printf(" entering Decomp: %Ps^%ld\n f = %Ps\n",
1120 : p, r, S->f);
1121 254705 : else if (DEBUGLEVEL>2) err_printf(" entering Decomp\n");
1122 254705 : chip = FpX_red(S->chi, p);
1123 254708 : if (!FpX_valrem(chip, S->nu, p, &b1))
1124 : {
1125 0 : if (!p_is_prime(S)) pari_err_PRIME("Decomp",p);
1126 0 : pari_err_BUG("Decomp (not a factor)");
1127 : }
1128 254713 : b2 = FpX_div(chip, b1, p);
1129 254706 : a = FpX_mul(FpXQ_inv(b2, b1, p), b2, p);
1130 : /* E = e / de, e in Z[X], de in Z, E = a(phi) mod (f, p) */
1131 254707 : th = QpX_remove_denom(S->phi, p, &dt, &vdt);
1132 254704 : if (dt)
1133 : {
1134 122875 : long dega = degpol(a);
1135 122875 : vde = dega * vdt;
1136 122875 : de = powiu(dt, dega);
1137 122875 : pr = mulii(p, de);
1138 122876 : a = FpX_rescale(a, dt, pr);
1139 : }
1140 : else
1141 : {
1142 131829 : vde = 0;
1143 131829 : de = gen_1;
1144 131829 : pr = p;
1145 : }
1146 254706 : e = FpX_FpXQ_eval(a, th, S->f, pr);
1147 254707 : update_den(p, &e, &de, &vde, NULL);
1148 :
1149 254710 : pk = p; k = 1;
1150 : /* E, (1 - E) tend to orthogonal idempotents in Zp[X]/(f) */
1151 1178978 : while (k < r + vde)
1152 : { /* E <-- E^2(3-2E) mod p^2k, with E = e/de */
1153 : GEN D;
1154 924266 : pk = sqri(pk); k <<= 1;
1155 924257 : e = ZX_mul(ZX_sqr(e), Z_ZX_sub(mului(3,de), gmul2n(e,1)));
1156 924275 : de= mulii(de, sqri(de));
1157 924244 : vde *= 3;
1158 924244 : D = mulii(pk, de);
1159 924256 : e = FpX_rem(e, centermod(S->f, D), D); /* e/de defined mod pk */
1160 924252 : update_den(p, &e, &de, &vde, NULL);
1161 : }
1162 : /* required precision of the factors */
1163 254712 : pr = powiu(p, r); pr2 = shifti(pr, -1);
1164 254713 : ph = mulii(de,pr);ph2 = shifti(ph, -1);
1165 254711 : e = FpX_center_i(FpX_red(e, ph), ph, ph2);
1166 254713 : fred = FpX_red(S->f, ph);
1167 :
1168 254711 : f1 = ZpX_gcd(fred, Z_ZX_sub(de, e), p, ph); /* p-adic gcd(f, 1-e) */
1169 254712 : if (!is_pm1(de))
1170 : {
1171 122878 : fred = FpX_red(fred, pr);
1172 122879 : f1 = FpX_red(f1, pr);
1173 : }
1174 254712 : f2 = FpX_div(fred,f1, pr);
1175 254711 : f1 = FpX_center_i(f1, pr, pr2);
1176 254713 : f2 = FpX_center_i(f2, pr, pr2);
1177 :
1178 254713 : if (DEBUGLEVEL>5)
1179 0 : err_printf(" leaving Decomp: f1 = %Ps\nf2 = %Ps\ne = %Ps\nde= %Ps\n", f1,f2,e,de);
1180 :
1181 254713 : if (flag < 0)
1182 : {
1183 0 : GEN m = vconcat(ZpX_primedec(f1, p), ZpX_primedec(f2, p));
1184 0 : return sort_factor(m, (void*)&cmpii, &cmp_nodata);
1185 : }
1186 254713 : else if (flag)
1187 : {
1188 301 : (void)gc_all(av, 2, &f1, &f2);
1189 301 : return shallowconcat(ZpX_monic_factor_squarefree(f1, p, flag),
1190 : ZpX_monic_factor_squarefree(f2, p, flag));
1191 : } else {
1192 : GEN D, d1, d2, B1, B2, M;
1193 : long n, n1, n2, i;
1194 254412 : (void)gc_all(av, 4, &f1, &f2, &e, &de);
1195 254413 : D = de;
1196 254413 : B1 = get_partial_order_as_pols(p,f1); n1 = lg(B1)-1;
1197 254413 : B2 = get_partial_order_as_pols(p,f2); n2 = lg(B2)-1; n = n1+n2;
1198 254411 : d1 = QpXV_denom(B1);
1199 254411 : d2 = QpXV_denom(B2); if (cmpii(d1, d2) < 0) d1 = d2;
1200 254411 : if (d1 != gen_1) {
1201 157058 : B1 = Q_muli_to_int(B1, d1);
1202 157058 : B2 = Q_muli_to_int(B2, d1);
1203 157057 : D = mulii(d1, D);
1204 : }
1205 254410 : fred = centermod_i(S->f, D, shifti(D,-1));
1206 254409 : M = cgetg(n+1, t_MAT);
1207 806668 : for (i=1; i<=n1; i++)
1208 552261 : gel(M,i) = RgX_to_RgC(FpX_rem(FpX_mul(gel(B1,i),e,D), fred, D), n);
1209 254407 : e = Z_ZX_sub(de, e); B2 -= n1;
1210 708917 : for ( ; i<=n; i++)
1211 454509 : gel(M,i) = RgX_to_RgC(FpX_rem(FpX_mul(gel(B2,i),e,D), fred, D), n);
1212 254408 : return ZpM_hnfmodid(M, p, D);
1213 : }
1214 : }
1215 :
1216 : /* minimum extension valuation: L/E */
1217 : static void
1218 624238 : vstar(GEN p,GEN h, long *L, long *E)
1219 : {
1220 624238 : long first, j, k, v, w, m = degpol(h);
1221 :
1222 624238 : first = 1; k = 1; v = 0;
1223 2580103 : for (j=1; j<=m; j++)
1224 : {
1225 1955857 : GEN c = gel(h, m-j+2);
1226 1955857 : if (signe(c))
1227 : {
1228 1881147 : w = Z_pval(c,p);
1229 1881155 : if (first || w*k < v*j) { v = w; k = j; }
1230 1881155 : first = 0;
1231 : }
1232 : }
1233 : /* v/k = min_j ( v_p(h_{m-j}) / j ) */
1234 624246 : w = (long)ugcd(v,k);
1235 624246 : *L = v/w;
1236 624246 : *E = k/w;
1237 624246 : }
1238 :
1239 : static GEN
1240 64353 : redelt_i(GEN a, GEN N, GEN p, GEN *pda, long *pvda)
1241 : {
1242 : GEN z;
1243 64353 : a = Q_remove_denom(a, pda);
1244 64351 : *pvda = 0;
1245 64351 : if (*pda)
1246 : {
1247 64351 : long v = Z_pvalrem(*pda, p, &z);
1248 64351 : if (v) {
1249 64351 : *pda = powiu(p, v);
1250 64351 : *pvda = v;
1251 64351 : N = mulii(*pda, N);
1252 : }
1253 : else
1254 0 : *pda = NULL;
1255 64350 : if (!is_pm1(z)) a = ZX_Z_mul(a, Fp_inv(z, N));
1256 : }
1257 64350 : return centermod(a, N);
1258 : }
1259 : /* reduce the element a modulo N [ a power of p ], taking first care of the
1260 : * denominators */
1261 : static GEN
1262 48546 : redelt(GEN a, GEN N, GEN p)
1263 : {
1264 : GEN da;
1265 : long vda;
1266 48546 : a = redelt_i(a, N, p, &da, &vda);
1267 48545 : if (da) a = RgX_Rg_div(a, da);
1268 48546 : return a;
1269 : }
1270 :
1271 : /* compute the c first Newton sums modulo pp of the
1272 : characteristic polynomial of a/d mod chi, d > 0 power of p (NULL = gen_1),
1273 : a, chi in Zp[X], vda = v_p(da)
1274 : ns = Newton sums of chi */
1275 : static GEN
1276 706222 : newtonsums(GEN p, GEN a, GEN da, long vda, GEN chi, long c, GEN pp, GEN ns)
1277 : {
1278 : GEN va, pa, dpa, s;
1279 706222 : long j, k, vdpa, lns = lg(ns);
1280 : pari_sp av;
1281 :
1282 706222 : a = centermod(a, pp); av = avma;
1283 706197 : dpa = pa = NULL; /* -Wall */
1284 706197 : vdpa = 0;
1285 706197 : va = zerovec(c);
1286 2915855 : for (j = 1; j <= c; j++)
1287 : { /* pa/dpa = (a/d)^(j-1) mod (chi, pp), dpa = p^vdpa */
1288 : long l;
1289 2216458 : pa = j == 1? a: FpXQ_mul(pa, a, chi, pp);
1290 2216508 : l = lg(pa); if (l == 2) break;
1291 2216508 : if (lns < l) l = lns;
1292 :
1293 2216508 : if (da) {
1294 2081344 : dpa = j == 1? da: mulii(dpa, da);
1295 2081238 : vdpa += vda;
1296 2081238 : update_den(p, &pa, &dpa, &vdpa, &pp);
1297 : }
1298 2216357 : s = mulii(gel(pa,2), gel(ns,2)); /* k = 2 */
1299 10957259 : for (k = 3; k < l; k++) s = addii(s, mulii(gel(pa,k), gel(ns,k)));
1300 2216207 : if (da) {
1301 : GEN r;
1302 2081064 : s = dvmdii(s, dpa, &r);
1303 2080994 : if (r != gen_0) return NULL;
1304 : }
1305 2209380 : gel(va,j) = centermodii(s, pp, shifti(pp,-1));
1306 :
1307 2209418 : if (gc_needed(av, 1))
1308 : {
1309 7 : if(DEBUGMEM>1) pari_warn(warnmem, "newtonsums");
1310 7 : (void)gc_all(av, dpa?4:3, &pa, &va, &pp, &dpa);
1311 : }
1312 : }
1313 699397 : for (; j <= c; j++) gel(va,j) = gen_0;
1314 699397 : return va;
1315 : }
1316 :
1317 : /* compute the characteristic polynomial of a/da mod chi (a in Z[X]), given
1318 : * by its Newton sums to a precision of pp using Newton sums */
1319 : static GEN
1320 699398 : newtoncharpoly(GEN pp, GEN p, GEN NS)
1321 : {
1322 699398 : long n = lg(NS)-1, j, k;
1323 699398 : GEN c = cgetg(n + 2, t_VEC), pp2 = shifti(pp,-1);
1324 :
1325 699405 : gel(c,1) = (n & 1 ? gen_m1: gen_1);
1326 2898548 : for (k = 2; k <= n+1; k++)
1327 : {
1328 2199221 : pari_sp av2 = avma;
1329 2199221 : GEN s = gen_0;
1330 : ulong z;
1331 2199221 : long v = u_pvalrem(k - 1, p, &z);
1332 9282256 : for (j = 1; j < k; j++)
1333 : {
1334 7083572 : GEN t = mulii(gel(NS,j), gel(c,k-j));
1335 7082979 : if (!odd(j)) t = negi(t);
1336 7083047 : s = addii(s, t);
1337 : }
1338 2198684 : if (v) {
1339 842254 : s = gdiv(s, powiu(p, v));
1340 842257 : if (typ(s) != t_INT) return NULL;
1341 : }
1342 2198589 : s = mulii(s, Fp_inv(utoipos(z), pp));
1343 2198862 : gel(c,k) = gc_INT(av2, Fp_center_i(s, pp, pp2));
1344 : }
1345 1862886 : for (k = odd(n)? 1: 2; k <= n+1; k += 2) gel(c,k) = negi(gel(c,k));
1346 699330 : return gtopoly(c, 0);
1347 : }
1348 :
1349 : static void
1350 706159 : manage_cache(decomp_t *S, GEN f, GEN pp)
1351 : {
1352 706159 : GEN t = S->precns;
1353 :
1354 706159 : if (!t) t = mulii(S->pmf, powiu(S->p, S->df));
1355 706155 : if (cmpii(t, pp) < 0) t = pp;
1356 :
1357 706147 : if (!S->precns || !RgX_equal(f, S->nsf) || cmpii(S->precns, t) < 0)
1358 : {
1359 521629 : if (DEBUGLEVEL>4)
1360 0 : err_printf(" Precision for cached Newton sums for %Ps: %Ps -> %Ps\n",
1361 0 : f, S->precns? S->precns: gen_0, t);
1362 521629 : S->nsf = f;
1363 521629 : S->ns = FpX_Newton(f, degpol(f), t);
1364 521651 : S->precns = t;
1365 : }
1366 706195 : }
1367 :
1368 : /* return NULL if a mod f is not an integer
1369 : * The denominator of any integer in Zp[X]/(f) divides pdr */
1370 : static GEN
1371 706203 : mycaract(decomp_t *S, GEN f, GEN a, GEN pp, GEN pdr)
1372 : {
1373 : pari_sp av;
1374 : GEN d, chi, prec1, prec2, prec3, ns;
1375 706203 : long vd, n = degpol(f);
1376 :
1377 706205 : if (gequal0(a)) return pol_0(varn(f));
1378 :
1379 706204 : a = QpX_remove_denom(a, S->p, &d, &vd);
1380 706178 : prec1 = pp;
1381 706178 : if (lgefint(S->p) == 3)
1382 706127 : prec1 = mulii(prec1, powiu(S->p, factorial_lval(n, itou(S->p))));
1383 706158 : if (d)
1384 : {
1385 641352 : GEN p1 = powiu(d, n);
1386 641384 : prec2 = mulii(prec1, p1);
1387 641362 : prec3 = mulii(prec1, gmin_shallow(mulii(p1, d), pdr));
1388 : }
1389 : else
1390 64806 : prec2 = prec3 = prec1;
1391 706164 : manage_cache(S, f, prec3);
1392 :
1393 706222 : av = avma;
1394 706222 : ns = newtonsums(S->p, a, d, vd, f, n, prec2, S->ns);
1395 706153 : if (!ns) return NULL;
1396 699396 : chi = newtoncharpoly(prec1, S->p, ns);
1397 699453 : if (!chi) return NULL;
1398 699355 : setvarn(chi, varn(f));
1399 699355 : return gc_upto(av, centermod(chi, pp));
1400 : }
1401 :
1402 : static GEN
1403 641237 : get_nu(GEN chi, GEN p, long *ptl)
1404 : { /* split off powers of x first for efficiency */
1405 641237 : long v = ZX_valrem(FpX_red(chi,p), &chi), n;
1406 : GEN P;
1407 641226 : if (!degpol(chi)) { *ptl = 1; return pol_x(varn(chi)); }
1408 475703 : P = gel(FpX_factor(chi,p), 1); n = lg(P)-1;
1409 475720 : *ptl = v? n+1: n; return gel(P,n);
1410 : }
1411 :
1412 : /* Factor characteristic polynomial chi of phi mod p. If it splits, update
1413 : * S->{phi, chi, nu} and return 1. In any case, set *nu to an irreducible
1414 : * factor mod p of chi */
1415 : static int
1416 480494 : split_char(decomp_t *S, GEN chi, GEN phi, GEN phi0, GEN *nu)
1417 : {
1418 : long l;
1419 480494 : *nu = get_nu(chi, S->p, &l);
1420 480500 : if (l == 1) return 0; /* single irreducible factor: doesn't split */
1421 : /* phi o phi0 mod (p, f) */
1422 122876 : S->phi = compmod(S->p, phi, phi0, S->f, S->p, 0);
1423 122878 : S->chi = chi;
1424 122878 : S->nu = *nu; return 1;
1425 : }
1426 :
1427 : /* Return the prime element in Zp[phi], a t_INT (iff *Ep = 1) or QX;
1428 : * nup, chip are ZX. phi = NULL codes X
1429 : * If *Ep < oE or Ep divides Ediv (!=0) return NULL (uninteresting) */
1430 : static GEN
1431 563576 : getprime(decomp_t *S, GEN phi, GEN chip, GEN nup, long *Lp, long *Ep,
1432 : long oE, long Ediv)
1433 : {
1434 : GEN z, chin, q, qp;
1435 : long r, s;
1436 :
1437 563576 : if (phi && dvdii(constant_coeff(chip), S->psc))
1438 : {
1439 1696 : chip = mycaract(S, S->chi, phi, S->pmf, S->prc);
1440 1696 : if (dvdii(constant_coeff(chip), S->pmf))
1441 1248 : chip = ZXQ_charpoly(phi, S->chi, varn(chip));
1442 : }
1443 563573 : if (degpol(nup) == 1)
1444 : {
1445 524664 : GEN c = gel(nup,2); /* nup = X + c */
1446 524664 : chin = signe(c)? RgX_translate(chip, negi(c)): chip;
1447 : }
1448 : else
1449 38906 : chin = ZXQ_charpoly(nup, chip, varn(chip));
1450 :
1451 563578 : vstar(S->p, chin, Lp, Ep);
1452 563590 : if (*Ep < oE || (Ediv && Ediv % *Ep == 0)) return NULL;
1453 :
1454 443014 : if (*Ep == 1) return S->p;
1455 305566 : (void)cbezout(*Lp, -*Ep, &r, &s); /* = 1 */
1456 305568 : if (r <= 0)
1457 : {
1458 60122 : long t = 1 + ((-r) / *Ep);
1459 60122 : r += t * *Ep;
1460 60122 : s += t * *Lp;
1461 : }
1462 : /* r > 0 minimal such that r L/E - s = 1/E
1463 : * pi = nu^r / p^s is an element of valuation 1/E,
1464 : * so is pi + O(p) since 1/E < 1. May compute nu^r mod p^(s+1) */
1465 305568 : q = powiu(S->p, s); qp = mulii(q, S->p);
1466 305561 : nup = FpXQ_powu(nup, r, S->chi, qp);
1467 305559 : if (!phi) return RgX_Rg_div(nup, q); /* phi = X : no composition */
1468 48546 : z = compmod(S->p, nup, phi, S->chi, qp, -s);
1469 48546 : return signe(z)? z: NULL;
1470 : }
1471 :
1472 : static int
1473 276512 : update_phi(decomp_t *S)
1474 : {
1475 276512 : GEN PHI = NULL, prc, psc, X = pol_x(varn(S->f));
1476 : long k, vpsc;
1477 276512 : for (k = 1;; k++)
1478 : {
1479 278784 : prc = ZpX_reduced_resultant_fast(S->chi, ZX_deriv(S->chi), S->p, S->vpsc);
1480 : /* if prc == S->psc then either chi is not separable or
1481 : the reduced discriminant of chi is too large */
1482 278785 : if (!equalii(prc, S->psc)) break;
1483 :
1484 : /* increase precision */
1485 2272 : S->vpsc = maxss(S->vpsf, S->vpsc + 1);
1486 2272 : S->psc = (S->vpsc == S->vpsf)? S->psf: mulii(S->psc, S->p);
1487 :
1488 2272 : PHI = S->phi;
1489 2272 : if (S->phi0) PHI = compmod(S->p, PHI, S->phi0, S->f, S->psc, 0);
1490 : /* change phi (in case not separable) */
1491 2272 : PHI = gadd(PHI, ZX_Z_mul(X, mului(k, S->p)));
1492 2272 : S->chi = mycaract(S, S->f, PHI, S->psc, S->pdf);
1493 : }
1494 276513 : psc = mulii(sqri(prc), S->p);
1495 276505 : vpsc = 2*Z_pval(prc, S->p) + 1;
1496 :
1497 276504 : if (!PHI) /* break out of above loop immediately (k = 1) */
1498 : {
1499 274233 : PHI = S->phi;
1500 274233 : if (S->phi0) PHI = compmod(S->p, PHI, S->phi0, S->f, psc, 0);
1501 274235 : if (S->phi0 || cmpii(psc,S->psc) > 0)
1502 : {
1503 : for(;;)
1504 : {
1505 114076 : S->chi = mycaract(S, S->f, PHI, psc, S->pdf);
1506 114080 : prc = ZpX_reduced_resultant_fast(S->chi, ZX_deriv(S->chi), S->p, vpsc);
1507 114080 : if (!equalii(prc, psc)) break;
1508 497 : psc = mulii(psc, S->p); vpsc++;
1509 : /* it can happen that S->chi is never squarefree: then change PHI */
1510 497 : if (vpsc > 2*S->mf) PHI = gadd(PHI, ZX_Z_mul(X, S->p));
1511 : }
1512 113583 : psc = mulii(sqri(prc), S->p);
1513 113579 : vpsc = 2*Z_pval(prc, S->p) + 1;
1514 : }
1515 : }
1516 276506 : S->phi = PHI;
1517 276506 : S->chi = FpX_red(S->chi, psc);
1518 :
1519 : /* may happen if p is unramified */
1520 276511 : if (is_pm1(prc)) return 0;
1521 232079 : S->prc = prc;
1522 232079 : S->psc = psc;
1523 232079 : S->vpsc = vpsc; return 1;
1524 : }
1525 :
1526 : /* return 1 if at least 2 factors mod p ==> chi splits
1527 : * Replace S->phi such that F increases (to D) */
1528 : static int
1529 67225 : testb2(decomp_t *S, long D, GEN theta)
1530 : {
1531 67225 : long v = varn(S->chi), dlim = degpol(S->chi)-1;
1532 67225 : GEN T0 = S->phi, chi, phi, nu;
1533 67225 : if (DEBUGLEVEL>4) err_printf(" Increasing Fa\n");
1534 : for (;;)
1535 : {
1536 67279 : phi = gadd(theta, random_FpX(dlim, v, S->p));
1537 67279 : chi = mycaract(S, S->chi, phi, S->psf, S->prc);
1538 : /* phi nonprimary ? */
1539 67277 : if (split_char(S, chi, phi, T0, &nu)) return 1;
1540 67279 : if (degpol(nu) == D) break;
1541 : }
1542 : /* F_phi=lcm(F_alpha, F_theta)=D and E_phi=E_alpha */
1543 67225 : S->phi0 = T0;
1544 67225 : S->chi = chi;
1545 67225 : S->phi = phi;
1546 67225 : S->nu = nu; return 0;
1547 : }
1548 :
1549 : /* return 1 if at least 2 factors mod p ==> chi can be split.
1550 : * compute a new S->phi such that E = lcm(Ea, Et);
1551 : * A a ZX, T a t_INT (iff Et = 1, probably impossible ?) or QX */
1552 : static int
1553 48546 : testc2(decomp_t *S, GEN A, long Ea, GEN T, long Et)
1554 : {
1555 48546 : GEN c, chi, phi, nu, T0 = S->phi;
1556 :
1557 48546 : if (DEBUGLEVEL>4) err_printf(" Increasing Ea\n");
1558 48546 : if (Et == 1) /* same as other branch, split for efficiency */
1559 0 : c = A; /* Et = 1 => s = 1, r = 0, t = 0 */
1560 : else
1561 : {
1562 : long r, s, t;
1563 48546 : (void)cbezout(Ea, Et, &r, &s); t = 0;
1564 48644 : while (r < 0) { r = r + Et; t++; }
1565 48756 : while (s < 0) { s = s + Ea; t++; }
1566 :
1567 : /* A^s T^r / p^t */
1568 48546 : c = RgXQ_mul(RgXQ_powu(A, s, S->chi), RgXQ_powu(T, r, S->chi), S->chi);
1569 48546 : c = RgX_Rg_div(c, powiu(S->p, t));
1570 48546 : c = redelt(c, S->psc, S->p);
1571 : }
1572 48546 : phi = RgX_add(c, pol_x(varn(S->chi)));
1573 48546 : chi = mycaract(S, S->chi, phi, S->psf, S->prc);
1574 48546 : if (split_char(S, chi, phi, T0, &nu)) return 1;
1575 : /* E_phi = lcm(E_alpha,E_theta) */
1576 48545 : S->phi0 = T0;
1577 48545 : S->chi = chi;
1578 48545 : S->phi = phi;
1579 48545 : S->nu = nu; return 0;
1580 : }
1581 :
1582 : /* Return h^(-degpol(P)) P(x * h) if result is integral, NULL otherwise */
1583 : static GEN
1584 59925 : ZX_rescale_inv(GEN P, GEN h)
1585 : {
1586 59925 : long i, l = lg(P);
1587 59925 : GEN Q = cgetg(l,t_POL), hi = h;
1588 59926 : gel(Q,l-1) = gel(P,l-1);
1589 174186 : for (i=l-2; i>=2; i--)
1590 : {
1591 : GEN r;
1592 174186 : gel(Q,i) = dvmdii(gel(P,i), hi, &r);
1593 174186 : if (signe(r)) return NULL;
1594 174186 : if (i == 2) break;
1595 114261 : hi = mulii(hi,h);
1596 : }
1597 59925 : Q[1] = P[1]; return Q;
1598 : }
1599 :
1600 : /* x p^-eq nu^-er mod p */
1601 : static GEN
1602 303408 : get_gamma(decomp_t *S, GEN x, long eq, long er)
1603 : {
1604 303408 : GEN q, g = x, Dg = powiu(S->p, eq);
1605 303404 : long vDg = eq;
1606 303404 : if (er)
1607 : {
1608 22949 : if (!S->invnu)
1609 : {
1610 15807 : while (gdvd(S->chi, S->nu)) S->nu = RgX_Rg_add(S->nu, S->p);
1611 15807 : S->invnu = QXQ_inv(S->nu, S->chi);
1612 15807 : S->invnu = redelt_i(S->invnu, S->psc, S->p, &S->Dinvnu, &S->vDinvnu);
1613 : }
1614 22949 : if (S->Dinvnu) {
1615 22949 : Dg = mulii(Dg, powiu(S->Dinvnu, er));
1616 22949 : vDg += er * S->vDinvnu;
1617 : }
1618 22949 : q = mulii(S->p, Dg);
1619 22949 : g = ZX_mul(g, FpXQ_powu(S->invnu, er, S->chi, q));
1620 22949 : g = FpX_rem(g, S->chi, q);
1621 22949 : update_den(S->p, &g, &Dg, &vDg, NULL);
1622 22949 : g = centermod(g, mulii(S->p, Dg));
1623 : }
1624 303404 : if (!is_pm1(Dg)) g = RgX_Rg_div(g, Dg);
1625 303408 : return g;
1626 : }
1627 : static GEN
1628 356479 : get_g(decomp_t *S, long Ea, long L, long E, GEN beta, GEN *pchig,
1629 : long *peq, long *per)
1630 : {
1631 : long eq, er;
1632 356479 : GEN g, chig, chib = NULL;
1633 : for(;;) /* at most twice */
1634 : {
1635 363334 : if (L < 0)
1636 : {
1637 60656 : chib = ZXQ_charpoly(beta, S->chi, varn(S->chi));
1638 60656 : vstar(S->p, chib, &L, &E);
1639 : }
1640 363336 : eq = L / E; er = L*Ea / E - eq*Ea;
1641 : /* floor(L Ea/E) = eq Ea + er */
1642 363336 : if (er || !chib)
1643 : { /* g might not be an integer ==> chig = NULL */
1644 303410 : g = get_gamma(S, beta, eq, er);
1645 303407 : chig = mycaract(S, S->chi, g, S->psc, S->prc);
1646 : }
1647 : else
1648 : { /* g = beta/p^eq, special case of the above */
1649 59926 : GEN h = powiu(S->p, eq);
1650 59926 : g = RgX_Rg_div(beta, h);
1651 59925 : chig = ZX_rescale_inv(chib, h); /* chib(x h) / h^N */
1652 59925 : if (chig) chig = FpX_red(chig, S->pmf);
1653 : }
1654 : /* either success or second consecutive failure */
1655 363336 : if (chig || chib) break;
1656 : /* if g fails the v*-test, v(beta) was wrong. Retry once */
1657 6855 : L = -1;
1658 : }
1659 356481 : *pchig = chig; *peq = eq; *per = er; return g;
1660 : }
1661 :
1662 : /* return 1 if at least 2 factors mod p ==> chi can be split */
1663 : static int
1664 238648 : loop(decomp_t *S, long Ea)
1665 : {
1666 238648 : pari_sp av = avma;
1667 238648 : GEN beta = FpXQ_powu(S->nu, Ea, S->chi, S->p);
1668 238644 : long N = degpol(S->f), v = varn(S->f);
1669 238644 : S->invnu = NULL;
1670 : for (;;)
1671 117833 : { /* beta tends to a factor of chi */
1672 : long L, i, Fg, eq, er;
1673 356477 : GEN chig = NULL, d, g, nug;
1674 :
1675 356477 : if (DEBUGLEVEL>4) err_printf(" beta = %Ps\n", beta);
1676 356477 : L = ZpX_resultant_val(S->chi, beta, S->p, S->mf+1);
1677 356479 : if (L > S->mf) L = -1; /* from scratch */
1678 356479 : g = get_g(S, Ea, L, N, beta, &chig, &eq, &er);
1679 356481 : if (DEBUGLEVEL>4) err_printf(" (eq,er) = (%ld,%ld)\n", eq,er);
1680 : /* g = beta p^-eq nu^-er (a unit), chig = charpoly(g) */
1681 474291 : if (split_char(S, chig, g,S->phi, &nug)) return 1;
1682 :
1683 235645 : Fg = degpol(nug);
1684 235645 : if (Fg == 1)
1685 : { /* frequent special case nug = x - d */
1686 : long Le, Ee;
1687 : GEN chie, nue, e, pie;
1688 160225 : d = negi(gel(nug,2));
1689 160225 : chie = RgX_translate(chig, d);
1690 160225 : nue = pol_x(v);
1691 160224 : e = RgX_Rg_sub(g, d);
1692 160223 : pie = getprime(S, e, chie, nue, &Le, &Ee, 0,Ea);
1693 160225 : if (pie) return testc2(S, S->nu, Ea, pie, Ee);
1694 : }
1695 : else
1696 : {
1697 75420 : long Fa = degpol(S->nu), vdeng;
1698 : GEN deng, numg, nume;
1699 78906 : if (Fa % Fg) return testb2(S, ulcm(Fa,Fg), g);
1700 : /* nu & nug irreducible mod p, deg nug | deg nu. To improve beta, look
1701 : * for a root d of nug in Fp[phi] such that v_p(g - d) > 0 */
1702 8195 : if (ZX_equal(nug, S->nu))
1703 5986 : d = pol_x(v);
1704 : else
1705 : {
1706 2209 : if (!p_is_prime(S)) pari_err_PRIME("FpX_ffisom",S->p);
1707 2209 : d = FpX_ffisom(nug, S->nu, S->p);
1708 : }
1709 : /* write g = numg / deng, e = nume / deng */
1710 8195 : numg = QpX_remove_denom(g, S->p, &deng, &vdeng);
1711 12850 : for (i = 1; i <= Fg; i++)
1712 : {
1713 : GEN chie, nue, e;
1714 12850 : if (i != 1) d = FpXQ_pow(d, S->p, S->nu, S->p); /* next root */
1715 12850 : nume = ZX_sub(numg, ZX_Z_mul(d, deng));
1716 : /* test e = nume / deng */
1717 12850 : if (ZpX_resultant_val(S->chi, nume, S->p, vdeng*N+1) <= vdeng*N)
1718 4655 : continue;
1719 8195 : e = RgX_Rg_div(nume, deng);
1720 8195 : chie = mycaract(S, S->chi, e, S->psc, S->prc);
1721 9641 : if (split_char(S, chie, e,S->phi, &nue)) return 1;
1722 6155 : if (RgX_is_monomial(nue))
1723 : { /* v_p(e) = v_p(g - d) > 0 */
1724 : long Le, Ee;
1725 : GEN pie;
1726 6155 : pie = getprime(S, e, chie, nue, &Le, &Ee, 0,Ea);
1727 6155 : if (pie) return testc2(S, S->nu, Ea, pie, Ee);
1728 4709 : break;
1729 : }
1730 : }
1731 4709 : if (i > Fg)
1732 : {
1733 0 : if (!p_is_prime(S)) pari_err_PRIME("nilord",S->p);
1734 0 : pari_err_BUG("nilord (no root)");
1735 : }
1736 : }
1737 117834 : if (eq) d = gmul(d, powiu(S->p, eq));
1738 117833 : if (er) d = gmul(d, gpowgs(S->nu, er));
1739 117833 : beta = gsub(beta, d);
1740 :
1741 117833 : if (gc_needed(av,1))
1742 : {
1743 0 : if (DEBUGMEM > 1) pari_warn(warnmem, "nilord");
1744 0 : (void)gc_all(av, S->invnu? 6: 4, &beta, &(S->precns), &(S->ns), &(S->nsf), &(S->invnu), &(S->Dinvnu));
1745 : }
1746 : }
1747 : }
1748 :
1749 : /* E and F cannot decrease; return 1 if O = Zp[phi], 2 if we can get a
1750 : * decomposition and 0 otherwise */
1751 : static long
1752 394462 : progress(decomp_t *S, GEN *ppa, long *pE)
1753 : {
1754 394462 : long E = *pE, F;
1755 394462 : GEN pa = *ppa;
1756 394462 : S->phi0 = NULL; /* no delayed composition */
1757 : for(;;)
1758 2742 : {
1759 : long l, La, Ea; /* N.B If E = 0, getprime cannot return NULL */
1760 397204 : GEN pia = getprime(S, NULL, S->chi, S->nu, &La, &Ea, E,0);
1761 397211 : if (pia) { /* success, we break out in THIS loop */
1762 394469 : pa = (typ(pia) == t_POL)? RgX_RgXQ_eval(pia, S->phi, S->f): pia;
1763 394471 : E = Ea;
1764 394471 : if (La == 1) break; /* no need to change phi so that nu = pia */
1765 : }
1766 : /* phi += prime elt */
1767 65067 : S->phi = typ(pa) == t_INT? RgX_Rg_add_shallow(S->phi, pa)
1768 160744 : : RgX_add(S->phi, pa);
1769 : /* recompute char. poly. chi from scratch */
1770 160742 : S->chi = mycaract(S, S->f, S->phi, S->psf, S->pdf);
1771 160743 : S->nu = get_nu(S->chi, S->p, &l);
1772 160742 : if (l > 1) return 2;
1773 160742 : if (!update_phi(S)) return 1; /* unramified */
1774 160743 : if (pia) break;
1775 : }
1776 394470 : *pE = E; *ppa = pa; F = degpol(S->nu);
1777 394467 : if (DEBUGLEVEL>4) err_printf(" (E, F) = (%ld,%ld)\n", E, F);
1778 394467 : if (E * F == degpol(S->f)) return 1;
1779 238648 : if (loop(S, E)) return 2;
1780 115769 : if (!update_phi(S)) return 1;
1781 71337 : return 0;
1782 : }
1783 :
1784 : /* flag != 0 iff we're looking for the p-adic factorization,
1785 : in which case it is the p-adic precision we want */
1786 : static GEN
1787 454962 : maxord_i(decomp_t *S, GEN p, GEN f, long mf, GEN w, long flag)
1788 : {
1789 454962 : long oE, n = lg(w)-1; /* factor of largest degree */
1790 454962 : GEN opa, D = ZpX_reduced_resultant_fast(f, ZX_deriv(f), p, mf);
1791 454959 : S->pisprime = -1;
1792 454959 : S->p = p;
1793 454959 : S->mf = mf;
1794 454959 : S->nu = gel(w,n);
1795 454959 : S->df = Z_pval(D, p);
1796 454958 : S->pdf = powiu(p, S->df);
1797 454950 : S->phi = pol_x(varn(f));
1798 454947 : S->chi = S->f = f;
1799 454947 : if (n > 1) return Decomp(S, flag); /* FIXME: use bezout_lift_fact */
1800 :
1801 323120 : if (DEBUGLEVEL>4)
1802 0 : err_printf(" entering Nilord: %Ps^%ld\n f = %Ps, nu = %Ps\n",
1803 : p, S->df, S->f, S->nu);
1804 323120 : else if (DEBUGLEVEL>2) err_printf(" entering Nilord\n");
1805 323120 : S->psf = S->psc = mulii(sqri(D), p);
1806 323115 : S->vpsf = S->vpsc = 2*S->df + 1;
1807 323115 : S->prc = mulii(D, p);
1808 323120 : S->chi = FpX_red(S->f, S->psc);
1809 323125 : S->pmf = powiu(p, S->mf+1);
1810 323127 : S->precns = NULL;
1811 323127 : for(opa = NULL, oE = 0;;)
1812 71335 : {
1813 394462 : long n = progress(S, &opa, &oE);
1814 394465 : if (n == 1) return flag? NULL: dbasis(p, S->f, S->mf, S->phi, S->chi);
1815 194213 : if (n == 2) return Decomp(S, flag);
1816 : }
1817 : }
1818 :
1819 : static int
1820 889 : expo_is_squarefree(GEN e)
1821 : {
1822 889 : long i, l = lg(e);
1823 1260 : for (i=1; i<l; i++)
1824 1022 : if (e[i] != 1) return 0;
1825 238 : return 1;
1826 : }
1827 : /* pure round 4 */
1828 : static GEN
1829 896 : ZpX_round4(GEN f, GEN p, GEN w, long prec)
1830 : {
1831 : decomp_t S;
1832 896 : GEN L = maxord_i(&S, p, f, ZpX_disc_val(f,p), w, prec);
1833 896 : return L? L: mkvec(f);
1834 : }
1835 : /* f a squarefree ZX with leading_coeff 1, degree > 0. Return list of
1836 : * irreducible factors in Zp[X] (computed mod p^prec) */
1837 : static GEN
1838 1155 : ZpX_monic_factor_squarefree(GEN f, GEN p, long prec)
1839 : {
1840 1155 : pari_sp av = avma;
1841 : GEN L, fa, w, e;
1842 : long i, l;
1843 1155 : if (degpol(f) == 1) return mkvec(f);
1844 889 : fa = FpX_factor(f,p); w = gel(fa,1); e = gel(fa,2);
1845 : /* no repeated factors: Hensel lift */
1846 889 : if (expo_is_squarefree(e)) return ZpX_liftfact(f, w, powiu(p,prec), p, prec);
1847 651 : l = lg(w);
1848 651 : if (l == 2)
1849 : {
1850 392 : L = ZpX_round4(f,p,w,prec);
1851 392 : if (lg(L) == 2) { set_avma(av); return mkvec(f); }
1852 : }
1853 : else
1854 : { /* >= 2 factors mod p: partial Hensel lift */
1855 259 : GEN D = ZpX_reduced_resultant_fast(f, ZX_deriv(f), p, ZpX_disc_val(f,p));
1856 259 : long r = maxss(2*Z_pval(D,p)+1, prec);
1857 259 : GEN W = cgetg(l, t_VEC);
1858 833 : for (i = 1; i < l; i++)
1859 574 : gel(W,i) = e[i] == 1? gel(w,i): FpX_powu(gel(w,i), e[i], p);
1860 259 : L = ZpX_liftfact(f, W, powiu(p,r), p, r);
1861 833 : for (i = 1; i < l; i++)
1862 574 : gel(L,i) = e[i] == 1? mkvec(gel(L,i))
1863 574 : : ZpX_round4(gel(L,i), p, mkvec(gel(w,i)), prec);
1864 259 : L = shallowconcat1(L);
1865 : }
1866 399 : return gc_GEN(av, L);
1867 : }
1868 :
1869 : /* assume T a ZX with leading_coeff 1, degree > 0 */
1870 : GEN
1871 546 : ZpX_monic_factor(GEN T, GEN p, long prec)
1872 : {
1873 : GEN Q, P, E, F;
1874 : long L, l, i, v;
1875 :
1876 546 : if (degpol(T) == 1) return mkmat2(mkcol(T), mkcol(gen_1));
1877 546 : v = ZX_valrem(T, &T);
1878 546 : Q = ZX_squff(T, &F); l = lg(Q); L = v? l + 1: l;
1879 546 : P = cgetg(L, t_VEC);
1880 546 : E = cgetg(L, t_VEC);
1881 1099 : for (i = 1; i < l; i++)
1882 : {
1883 553 : GEN w = ZpX_monic_factor_squarefree(gel(Q,i), p, prec);
1884 553 : gel(P,i) = w; settyp(w, t_COL);
1885 553 : gel(E,i) = const_col(lg(w)-1, utoipos(F[i]));
1886 : }
1887 546 : if (v) { gel(P,i) = pol_x(varn(T)); gel(E,i) = utoipos(v); }
1888 546 : return mkmat2(shallowconcat1(P), shallowconcat1(E));
1889 : }
1890 :
1891 : /* DT = multiple of disc(T) or NULL
1892 : * Return a multiple of the denominator of an algebraic integer (in Q[X]/(T))
1893 : * when expressed in terms of the power basis */
1894 : GEN
1895 44088 : indexpartial(GEN T, GEN DT)
1896 : {
1897 44088 : pari_sp av = avma;
1898 : long i, nb;
1899 44088 : GEN fa, E, P, U, res = gen_1, dT = ZX_deriv(T);
1900 :
1901 44083 : if (!DT) DT = ZX_disc(T);
1902 44083 : fa = absZ_factor_limit_strict(DT, 0, &U);
1903 44089 : P = gel(fa,1);
1904 44089 : E = gel(fa,2); nb = lg(P)-1;
1905 211836 : for (i = 1; i <= nb; i++)
1906 : {
1907 167751 : long e = itou(gel(E,i)), e2 = e >> 1;
1908 167751 : GEN p = gel(P,i), q = p;
1909 167751 : if (e2 >= 2) q = ZpX_reduced_resultant_fast(T, dT, p, e2);
1910 167753 : res = mulii(res, q);
1911 : }
1912 44085 : if (U)
1913 : {
1914 1916 : long e = itou(gel(U,2)), e2 = e >> 1;
1915 1916 : GEN p = gel(U,1), q = powiu(p, odd(e)? e2+1: e2);
1916 1916 : res = mulii(res, q);
1917 : }
1918 44085 : return gc_INT(av,res);
1919 : }
1920 :
1921 : /*******************************************************************/
1922 : /* */
1923 : /* 2-ELT REPRESENTATION FOR PRIME IDEALS (dividing index) */
1924 : /* */
1925 : /*******************************************************************/
1926 : /* to compute norm of elt in basis form */
1927 : typedef struct {
1928 : long r1;
1929 : GEN M; /* via embed_norm */
1930 :
1931 : GEN D, w, T; /* via resultant if M = NULL */
1932 : } norm_S;
1933 :
1934 : static GEN
1935 502727 : get_norm(norm_S *S, GEN a)
1936 : {
1937 502727 : if (S->M)
1938 : {
1939 : long e;
1940 501321 : GEN N = grndtoi( embed_norm(RgM_RgC_mul(S->M, a), S->r1), &e );
1941 501370 : if (e > -5) pari_err_PREC( "get_norm");
1942 501370 : return N;
1943 : }
1944 1406 : if (S->w) a = RgV_RgC_mul(S->w, a);
1945 1406 : return ZX_resultant_all(S->T, a, S->D, 0);
1946 : }
1947 : static void
1948 214410 : init_norm(norm_S *S, GEN nf, GEN p)
1949 : {
1950 214410 : GEN T = nf_get_pol(nf), M = nf_get_M(nf);
1951 214412 : long N = degpol(T), ex = gexpo(M) + gexpo(mului(8 * N, p));
1952 :
1953 214411 : S->r1 = nf_get_r1(nf);
1954 214413 : if (N * ex <= gprecision(M) - 20)
1955 : { /* enough prec to use embed_norm */
1956 214240 : S->M = M;
1957 214240 : S->D = NULL;
1958 214240 : S->w = NULL;
1959 214240 : S->T = NULL;
1960 : }
1961 : else
1962 : {
1963 191 : GEN w = leafcopy(nf_get_zkprimpart(nf)), D = nf_get_zkden(nf), Dp = sqri(p);
1964 : long i;
1965 191 : if (!equali1(D))
1966 : {
1967 191 : GEN w1 = D;
1968 191 : long v = Z_pval(D, p);
1969 191 : D = powiu(p, v);
1970 191 : Dp = mulii(D, Dp);
1971 191 : gel(w, 1) = remii(w1, Dp);
1972 : }
1973 3969 : for (i=2; i<=N; i++) gel(w,i) = FpX_red(gel(w,i), Dp);
1974 191 : S->M = NULL;
1975 191 : S->D = D;
1976 191 : S->w = w;
1977 191 : S->T = T;
1978 : }
1979 214431 : }
1980 : /* f = f(pr/p), q = p^(f+1), a in pr.
1981 : * Return 1 if v_pr(a) = 1, and 0 otherwise */
1982 : static int
1983 502728 : is_uniformizer(GEN a, GEN q, norm_S *S) { return !dvdii(get_norm(S,a), q); }
1984 :
1985 : /* Return x * y, x, y are t_MAT (Fp-basis of in O_K/p), assume (x,y)=1.
1986 : * Either x or y may be NULL (= O_K), not both */
1987 : static GEN
1988 702398 : mul_intersect(GEN x, GEN y, GEN p)
1989 : {
1990 702398 : if (!x) return y;
1991 373520 : if (!y) return x;
1992 263894 : return FpM_intersect_i(x, y, p);
1993 : }
1994 : /* Fp-basis of (ZK/pr): applied to the primes found in primedec_aux()
1995 : * true nf */
1996 : static GEN
1997 307214 : Fp_basis(GEN nf, GEN pr)
1998 : {
1999 : long i, j, l;
2000 : GEN x, y;
2001 : /* already in basis form (from Buchman-Lenstra) ? */
2002 307214 : if (typ(pr) == t_MAT) return pr;
2003 : /* ordinary prid (from Kummer) */
2004 72776 : x = pr_hnf(nf, pr);
2005 72778 : l = lg(x);
2006 72778 : y = cgetg(l, t_MAT);
2007 613213 : for (i=j=1; i<l; i++)
2008 540435 : if (gequal1(gcoeff(x,i,i))) gel(y,j++) = gel(x,i);
2009 72778 : setlg(y, j); return y;
2010 : }
2011 : /* Let Ip = prod_{ P | p } P be the p-radical. The list L contains the
2012 : * P (mod Ip) seen as sub-Fp-vector spaces of ZK/Ip.
2013 : * Return the list of (Ip / P) (mod Ip).
2014 : * N.B: All ideal multiplications are computed as intersections of Fp-vector
2015 : * spaces. true nf */
2016 : static GEN
2017 214437 : get_LV(GEN nf, GEN L, GEN p, long N)
2018 : {
2019 214437 : long i, l = lg(L)-1;
2020 : GEN LV, LW, A, B;
2021 :
2022 214437 : LV = cgetg(l+1, t_VEC);
2023 214438 : if (l == 1) { gel(LV,1) = matid(N); return LV; }
2024 109626 : LW = cgetg(l+1, t_VEC);
2025 416839 : for (i=1; i<=l; i++) gel(LW,i) = Fp_basis(nf, gel(L,i));
2026 :
2027 : /* A[i] = L[1]...L[i-1], i = 2..l */
2028 109628 : A = cgetg(l+1, t_VEC); gel(A,1) = NULL;
2029 307216 : for (i=1; i < l; i++) gel(A,i+1) = mul_intersect(gel(A,i), gel(LW,i), p);
2030 : /* B[i] = L[i+1]...L[l], i = 1..(l-1) */
2031 109625 : B = cgetg(l+1, t_VEC); gel(B,l) = NULL;
2032 307217 : for (i=l; i>=2; i--) gel(B,i-1) = mul_intersect(gel(B,i), gel(LW,i), p);
2033 416843 : for (i=1; i<=l; i++) gel(LV,i) = mul_intersect(gel(A,i), gel(B,i), p);
2034 109626 : return LV;
2035 : }
2036 :
2037 : static void
2038 0 : errprime(GEN p) { pari_err_PRIME("idealprimedec",p); }
2039 :
2040 : /* P = Fp-basis (over O_K/p) for pr.
2041 : * V = Z-basis for I_p/pr. ramif != 0 iff some pr|p is ramified.
2042 : * Return a p-uniformizer for pr. Assume pr not inert, i.e. m > 0 */
2043 : static GEN
2044 296754 : uniformizer(GEN nf, norm_S *S, GEN P, GEN V, GEN p, int ramif)
2045 : {
2046 296754 : long i, l, f, m = lg(P)-1, N = nf_get_degree(nf);
2047 : GEN u, Mv, x, q;
2048 :
2049 296754 : f = N - m; /* we want v_p(Norm(x)) = p^f */
2050 296754 : q = powiu(p,f+1);
2051 :
2052 296732 : u = FpM_FpC_invimage(shallowconcat(P, V), col_ei(N,1), p);
2053 296757 : setlg(u, lg(P));
2054 296757 : u = centermod(ZM_ZC_mul(P, u), p);
2055 296740 : if (is_uniformizer(u, q, S)) return u;
2056 158047 : if (signe(gel(u,1)) <= 0) /* make sure u[1] in ]-p,p] */
2057 133992 : gel(u,1) = addii(gel(u,1), p); /* try u + p */
2058 : else
2059 24055 : gel(u,1) = subii(gel(u,1), p); /* try u - p */
2060 158042 : if (!ramif || is_uniformizer(u, q, S)) return u;
2061 :
2062 : /* P/p ramified, u in P^2, not in Q for all other Q|p */
2063 85993 : Mv = zk_multable(nf, Z_ZC_sub(gen_1,u));
2064 85998 : l = lg(P);
2065 115317 : for (i=1; i<l; i++)
2066 : {
2067 115317 : x = centermod(ZC_add(u, ZM_ZC_mul(Mv, gel(P,i))), p);
2068 115319 : if (is_uniformizer(x, q, S)) return x;
2069 : }
2070 0 : errprime(p);
2071 : return NULL; /* LCOV_EXCL_LINE */
2072 : }
2073 :
2074 : /*******************************************************************/
2075 : /* */
2076 : /* BUCHMANN-LENSTRA ALGORITHM */
2077 : /* */
2078 : /*******************************************************************/
2079 : static GEN
2080 4257092 : mk_pr(GEN p, GEN u, long e, long f, GEN t)
2081 4257092 : { return mkvec5(p, u, utoipos(e), utoipos(f), t); }
2082 :
2083 : /* nf a true nf, u in Z[X]/(T); pr = p Z_K + u Z_K of ramification index e */
2084 : GEN
2085 3799846 : idealprimedec_kummer(GEN nf,GEN u,long e,GEN p)
2086 : {
2087 3799846 : GEN t, T = nf_get_pol(nf);
2088 3799848 : long f = degpol(u), N = degpol(T);
2089 :
2090 3799848 : if (f == N)
2091 : { /* inert */
2092 624972 : u = scalarcol_shallow(p,N);
2093 624980 : t = gen_1;
2094 : }
2095 : else
2096 : {
2097 3174876 : t = centermod(poltobasis(nf, FpX_div(T, u, p)), p);
2098 3174595 : u = centermod(poltobasis(nf, u), p);
2099 3174594 : if (e == 1)
2100 : { /* make sure v_pr(u) = 1 (automatic if e>1) */
2101 2896891 : GEN cw, w = Q_primitive_part(nf_to_scalar_or_alg(nf, u), &cw);
2102 2897040 : long v = cw? f - Q_pval(cw, p) * N: f;
2103 2897042 : if (ZpX_resultant_val(T, w, p, v + 1) > v)
2104 : {
2105 107686 : GEN c = gel(u,1);
2106 107686 : gel(u,1) = signe(c) > 0? subii(c, p): addii(c, p);
2107 : }
2108 : }
2109 3174856 : t = zk_multable(nf, t);
2110 : }
2111 3799712 : return mk_pr(p,u,e,f,t);
2112 : }
2113 :
2114 : typedef struct {
2115 : GEN nf, p;
2116 : long I;
2117 : } eltmod_muldata;
2118 :
2119 : static GEN
2120 822044 : sqr_mod(void *data, GEN x)
2121 : {
2122 822044 : eltmod_muldata *D = (eltmod_muldata*)data;
2123 822044 : return FpC_red(nfsqri(D->nf, x), D->p);
2124 : }
2125 : static GEN
2126 347699 : ei_msqr_mod(void *data, GEN x)
2127 : {
2128 347699 : GEN x2 = sqr_mod(data, x);
2129 347690 : eltmod_muldata *D = (eltmod_muldata*)data;
2130 347690 : return FpC_red(zk_ei_mul(D->nf, x2, D->I), D->p);
2131 : }
2132 : /* nf a true nf; compute lift(nf.zk[I]^p mod p) */
2133 : static GEN
2134 738630 : pow_ei_mod_p(GEN nf, long I, GEN p)
2135 : {
2136 738630 : pari_sp av = avma;
2137 : eltmod_muldata D;
2138 738630 : long N = nf_get_degree(nf);
2139 738628 : GEN y = col_ei(N,I);
2140 738637 : if (I == 1) return y;
2141 521866 : D.nf = nf;
2142 521866 : D.p = p;
2143 521866 : D.I = I;
2144 521866 : y = gen_pow_fold(y, p, (void*)&D, &sqr_mod, &ei_msqr_mod);
2145 521870 : return gc_upto(av,y);
2146 : }
2147 :
2148 : /* nf a true nf; return a Z basis of Z_K's p-radical, phi = x--> x^p-x */
2149 : static GEN
2150 214436 : pradical(GEN nf, GEN p, GEN *phi)
2151 : {
2152 214436 : long i, N = nf_get_degree(nf);
2153 : GEN q,m,frob,rad;
2154 :
2155 : /* matrix of Frob: x->x^p over Z_K/p */
2156 214436 : frob = cgetg(N+1,t_MAT);
2157 943431 : for (i=1; i<=N; i++) gel(frob,i) = pow_ei_mod_p(nf,i,p);
2158 :
2159 214440 : m = frob; q = p;
2160 304413 : while (abscmpiu(q,N) < 0) { q = mulii(q,p); m = FpM_mul(m, frob, p); }
2161 214439 : rad = FpM_ker(m, p); /* m = Frob^k, s.t p^k >= N */
2162 943398 : for (i=1; i<=N; i++) gcoeff(frob,i,i) = subiu(gcoeff(frob,i,i), 1);
2163 214409 : *phi = frob; return rad;
2164 : }
2165 :
2166 : /* return powers of a: a^0, ... , a^d, d = dim A */
2167 : static GEN
2168 159593 : get_powers(GEN mul, GEN p)
2169 : {
2170 159593 : long i, d = lgcols(mul);
2171 159593 : GEN z, pow = cgetg(d+2,t_MAT), P = pow+1;
2172 :
2173 159593 : gel(P,0) = scalarcol_shallow(gen_1, d-1);
2174 159599 : z = gel(mul,1);
2175 756758 : for (i=1; i<=d; i++)
2176 : {
2177 597166 : gel(P,i) = z; /* a^i */
2178 597166 : if (i!=d) z = FpM_FpC_mul(mul, z, p);
2179 : }
2180 159592 : return pow;
2181 : }
2182 :
2183 : /* minimal polynomial of a in A (dim A = d).
2184 : * mul = multiplication table by a in A */
2185 : static GEN
2186 117710 : pol_min(GEN mul, GEN p)
2187 : {
2188 117710 : pari_sp av = avma;
2189 117710 : GEN z = FpM_deplin(get_powers(mul, p), p);
2190 117710 : return gc_GEN(av, RgV_to_RgX(z,0));
2191 : }
2192 :
2193 : static GEN
2194 411702 : get_pr(GEN nf, norm_S *S, GEN p, GEN P, GEN V, int ramif, long N, long flim)
2195 : {
2196 : GEN u, t;
2197 : long e, f;
2198 :
2199 411702 : if (typ(P) == t_VEC)
2200 : { /* already done (Kummer) */
2201 72779 : f = pr_get_f(P);
2202 72779 : if (flim > 0 && f > flim) return NULL;
2203 71887 : if (flim == -2) return (GEN)f;
2204 71880 : return P;
2205 : }
2206 338923 : f = N - (lg(P)-1);
2207 338923 : if (flim > 0 && f > flim) return NULL;
2208 337324 : if (flim == -2) return (GEN)f;
2209 : /* P = (p,u) prime. t is an anti-uniformizer: Z_K + t/p Z_K = P^(-1),
2210 : * so that v_P(t) = e(P/p)-1 */
2211 336946 : if (f == N) {
2212 40193 : u = scalarcol_shallow(p,N);
2213 40194 : t = gen_1;
2214 40194 : e = 1;
2215 : } else {
2216 : GEN mt;
2217 296753 : u = uniformizer(nf, S, P, V, p, ramif);
2218 296721 : t = FpM_deplin(zk_multable(nf,u), p);
2219 296751 : mt = zk_multable(nf, t);
2220 296751 : e = ramif? 1 + ZC_nfval(t,mk_pr(p,u,0,0,mt)): 1;
2221 296732 : t = mt;
2222 : }
2223 336926 : return mk_pr(p,u,e,f,t);
2224 : }
2225 :
2226 : /* true nf */
2227 : static GEN
2228 214437 : primedec_end(GEN nf, GEN L, GEN p, long flim)
2229 : {
2230 214437 : long i, j, l = lg(L), N = nf_get_degree(nf);
2231 214437 : GEN LV = get_LV(nf, L,p,N);
2232 214438 : int ramif = dvdii(nf_get_disc(nf), p);
2233 214410 : norm_S S; init_norm(&S, nf, p);
2234 625732 : for (i = j = 1; i < l; i++)
2235 : {
2236 411700 : GEN P = get_pr(nf, &S, p, gel(L,i), gel(LV,i), ramif, N, flim);
2237 411696 : if (!P) continue;
2238 409205 : gel(L,j++) = P;
2239 409205 : if (flim == -1) return P;
2240 : }
2241 214032 : setlg(L, j); return L;
2242 : }
2243 :
2244 : /* prime ideal decomposition of p; if flim>0, restrict to f(P,p) <= flim
2245 : * if flim = -1 return only the first P
2246 : * if flim = -2 return only the f(P/p) in a t_VECSMALL; true nf */
2247 : static GEN
2248 2707619 : primedec_aux(GEN nf, GEN p, long flim)
2249 : {
2250 2707619 : const long TYP = (flim == -2)? t_VECSMALL: t_VEC;
2251 2707619 : GEN E, F, L, Ip, phi, f, g, h, UN, T = nf_get_pol(nf);
2252 : long i, k, c, iL, N;
2253 : int kummer;
2254 :
2255 2707611 : F = FpX_factor(T, p);
2256 2707715 : E = gel(F,2);
2257 2707715 : F = gel(F,1);
2258 :
2259 2707715 : k = lg(F); if (k == 1) errprime(p);
2260 2707715 : if ( !dvdii(nf_get_index(nf),p) ) /* p doesn't divide index */
2261 : {
2262 2491697 : L = cgetg(k, TYP);
2263 6075249 : for (i=1; i<k; i++)
2264 : {
2265 4139391 : GEN t = gel(F,i);
2266 4139391 : long f = degpol(t);
2267 4139392 : if (flim > 0 && f > flim) { setlg(L, i); break; }
2268 3588229 : if (flim == -2)
2269 0 : L[i] = f;
2270 : else
2271 3588229 : gel(L,i) = idealprimedec_kummer(nf, t, E[i],p);
2272 3588343 : if (flim == -1) return gel(L,1);
2273 : }
2274 2487022 : return L;
2275 : }
2276 :
2277 215827 : kummer = 0;
2278 215827 : g = FpXV_prod(F, p);
2279 215823 : h = FpX_div(T,g,p);
2280 215823 : f = FpX_red(ZX_Z_divexact(ZX_sub(ZX_mul(g,h), T), p), p);
2281 :
2282 215818 : N = degpol(T);
2283 215819 : L = cgetg(N+1,TYP);
2284 215820 : iL = 1;
2285 524992 : for (i=1; i<k; i++)
2286 310556 : if (E[i] == 1 || signe(FpX_rem(f,gel(F,i),p)))
2287 72779 : {
2288 74172 : GEN t = gel(F,i);
2289 74172 : kummer = 1;
2290 74172 : gel(L,iL++) = idealprimedec_kummer(nf, t, E[i],p);
2291 74172 : if (flim == -1) return gel(L,1);
2292 : }
2293 : else /* F[i] | (f,g,h), happens at least once by Dedekind criterion */
2294 236393 : E[i] = 0;
2295 :
2296 : /* phi matrix of x -> x^p - x in algebra Z_K/p */
2297 214436 : Ip = pradical(nf,p,&phi);
2298 :
2299 : /* split etale algebra Z_K / (p,Ip) */
2300 214428 : h = cgetg(N+1,t_VEC);
2301 214426 : if (kummer)
2302 : { /* split off Kummer factors */
2303 46160 : GEN mb, b = NULL;
2304 172593 : for (i=1; i<k; i++)
2305 126433 : if (!E[i]) b = b? FpX_mul(b, gel(F,i), p): gel(F,i);
2306 46160 : if (!b) errprime(p);
2307 46160 : b = FpC_red(poltobasis(nf,b), p);
2308 46160 : mb = FpM_red(zk_multable(nf,b), p);
2309 : /* Fp-base of ideal (Ip, b) in ZK/p */
2310 46157 : gel(h,1) = FpM_image(shallowconcat(mb,Ip), p);
2311 : }
2312 : else
2313 168266 : gel(h,1) = Ip;
2314 :
2315 214426 : UN = col_ei(N, 1);
2316 468348 : for (c=1; c; c--)
2317 : { /* Let A:= (Z_K/p) / Ip etale; split A2 := A / Im H ~ Im M2
2318 : H * ? + M2 * Mi2 = Id_N ==> M2 * Mi2 projector A --> A2 */
2319 253910 : GEN M, Mi, M2, Mi2, phi2, mat1, H = gel(h,c); /* maximal rank */
2320 253910 : long dim, r = lg(H)-1;
2321 :
2322 253910 : M = FpM_suppl(shallowconcat(H,UN), p);
2323 253909 : Mi = FpM_inv(M, p);
2324 253903 : M2 = vecslice(M, r+1,N); /* M = (H|M2) invertible */
2325 253904 : Mi2 = rowslice(Mi,r+1,N);
2326 : /* FIXME: FpM_mul(,M2) could be done with vecpermute */
2327 253909 : phi2 = FpM_mul(Mi2, FpM_mul(phi,M2, p), p);
2328 253908 : mat1 = FpM_ker(phi2, p);
2329 253912 : dim = lg(mat1)-1; /* A2 product of 'dim' fields */
2330 253912 : if (dim > 1)
2331 : { /* phi2 v = 0 => a = M2 v in Ker phi, a not in Fp.1 + H */
2332 117710 : GEN R, a, mula, mul2, v = gel(mat1,2);
2333 : long n;
2334 :
2335 117710 : a = FpM_FpC_mul(M2,v, p); /* not a scalar */
2336 117709 : mula = FpM_red(zk_multable(nf,a), p);
2337 117707 : mul2 = FpM_mul(Mi2, FpM_mul(mula,M2, p), p);
2338 117710 : R = FpX_roots(pol_min(mul2,p), p); /* totally split mod p */
2339 117711 : n = lg(R)-1;
2340 360234 : for (i=1; i<=n; i++)
2341 : {
2342 242523 : GEN I = RgM_Rg_sub_shallow(mula, gel(R,i));
2343 242519 : gel(h,c++) = FpM_image(shallowconcat(H, I), p);
2344 : }
2345 117711 : if (n == dim)
2346 303071 : for (i=1; i<=n; i++) gel(L,iL++) = gel(h,--c);
2347 : }
2348 : else /* A2 field ==> H maximal, f = N-r = dim(A2) */
2349 136202 : gel(L,iL++) = H;
2350 : }
2351 214438 : setlg(L, iL);
2352 214438 : return primedec_end(nf, L, p, flim);
2353 : }
2354 :
2355 : GEN
2356 2700718 : idealprimedec_limit_f(GEN nf, GEN p, long f)
2357 : {
2358 2700718 : pari_sp av = avma;
2359 : GEN v;
2360 2700718 : if (typ(p) != t_INT) pari_err_TYPE("idealprimedec",p);
2361 2700718 : if (f < 0) pari_err_DOMAIN("idealprimedec", "f", "<", gen_0, stoi(f));
2362 2700718 : v = primedec_aux(checknf(nf), p, f);
2363 2700603 : v = gen_sort(v, (void*)&cmp_prime_over_p, &cmp_nodata);
2364 2700660 : return gc_upto(av,v);
2365 : }
2366 : /* true nf */
2367 : GEN
2368 6552 : idealprimedec_galois(GEN nf, GEN p)
2369 : {
2370 6552 : pari_sp av = avma;
2371 6552 : GEN v = primedec_aux(nf, p, -1);
2372 6552 : return gc_GEN(av,v);
2373 : }
2374 : /* true nf */
2375 : GEN
2376 371 : idealprimedec_degrees(GEN nf, GEN p)
2377 : {
2378 371 : pari_sp av = avma;
2379 371 : GEN v = primedec_aux(nf, p, -2);
2380 371 : vecsmall_sort(v); return gc_uptoleaf(av, v);
2381 : }
2382 : GEN
2383 505224 : idealprimedec_limit_norm(GEN nf, GEN p, GEN B)
2384 505224 : { return idealprimedec_limit_f(nf, p, logint(B,p)); }
2385 : GEN
2386 1273330 : idealprimedec(GEN nf, GEN p)
2387 1273330 : { return idealprimedec_limit_f(nf, p, 0); }
2388 : static GEN
2389 26614 : nf_pV_to_prVV(GEN nf, GEN x)
2390 89649 : { pari_APPLY_same(idealprimedec(nf, gel(x,i))); }
2391 : GEN
2392 38604 : nf_pV_to_prV(GEN nf, GEN x)
2393 : {
2394 38604 : if (lg(x) == 1) return leafcopy(x);
2395 26614 : return shallowconcat1(nf_pV_to_prVV(nf, x));
2396 : }
2397 :
2398 : /* return [Fp[x]: Fp] */
2399 : static long
2400 4109 : ffdegree(GEN x, GEN frob, GEN p)
2401 : {
2402 4109 : pari_sp av = avma;
2403 4109 : long d, f = lg(frob)-1;
2404 4109 : GEN y = x;
2405 :
2406 13209 : for (d=1; d < f; d++)
2407 : {
2408 10878 : y = FpM_FpC_mul(frob, y, p);
2409 10878 : if (ZV_equal(y, x)) break;
2410 : }
2411 4109 : return gc_long(av,d);
2412 : }
2413 :
2414 : static GEN
2415 92189 : lift_to_zk(GEN v, GEN c, long N)
2416 : {
2417 92189 : GEN w = zerocol(N);
2418 92189 : long i, l = lg(c);
2419 307535 : for (i=1; i<l; i++) gel(w,c[i]) = gel(v,i);
2420 92189 : return w;
2421 : }
2422 :
2423 : /* return t = 1 mod pr, t = 0 mod p / pr^e(pr/p) */
2424 : GEN
2425 966795 : pr_anti_uniformizer(GEN nf, GEN pr)
2426 : {
2427 966795 : long N = nf_get_degree(nf), e = pr_get_e(pr);
2428 : GEN p, b, z;
2429 :
2430 966778 : if (e * pr_get_f(pr) == N) return gen_1;
2431 459272 : p = pr_get_p(pr);
2432 459271 : b = pr_get_tau(pr); /* ZM */
2433 459267 : if (e != 1)
2434 : {
2435 22771 : GEN q = powiu(pr_get_p(pr), e-1);
2436 22771 : b = ZM_Z_divexact(ZM_powu(b,e), q);
2437 : }
2438 : /* b = tau^e / p^(e-1), v_pr(b) = 0, v_Q(b) >= e(Q/p) for other Q | p */
2439 459267 : z = ZM_hnfmodid(FpM_red(b,p), p); /* ideal (p) / pr^e, coprime to pr */
2440 459293 : z = idealaddtoone_raw(nf, pr, z);
2441 459256 : return Z_ZC_sub(gen_1, FpC_center(FpC_red(z,p), p, shifti(p,-1)));
2442 : }
2443 :
2444 : #define mpr_TAU 1
2445 : #define mpr_FFP 2
2446 : #define mpr_NFP 5
2447 : #define SMALLMODPR 4
2448 : #define LARGEMODPR 6
2449 : static GEN
2450 3517444 : modpr_TAU(GEN modpr)
2451 : {
2452 3517444 : GEN tau = gel(modpr,mpr_TAU);
2453 3517444 : return isintzero(tau)? NULL: tau;
2454 : }
2455 :
2456 : /* prh = HNF matrix, which is identity but for the first line. Return a
2457 : * projector to ZK / prh ~ Z/prh[1,1] */
2458 : GEN
2459 1030284 : dim1proj(GEN prh)
2460 : {
2461 1030284 : long i, N = lg(prh)-1;
2462 1030284 : GEN ffproj = cgetg(N+1, t_VEC);
2463 1030278 : GEN x, q = gcoeff(prh,1,1);
2464 1030278 : gel(ffproj,1) = gen_1;
2465 2375226 : for (i=2; i<=N; i++)
2466 : {
2467 1344994 : x = gcoeff(prh,1,i);
2468 1344994 : if (signe(x)) x = subii(q,x);
2469 1344948 : gel(ffproj,i) = x;
2470 : }
2471 1030232 : return ffproj;
2472 : }
2473 :
2474 : /* p not necessarily prime, but coprime to denom(basis) */
2475 : GEN
2476 203 : QXQV_to_FpM(GEN basis, GEN T, GEN p)
2477 : {
2478 203 : long i, l = lg(basis), f = degpol(T);
2479 203 : GEN z = cgetg(l, t_MAT);
2480 4515 : for (i = 1; i < l; i++)
2481 : {
2482 4312 : GEN w = gel(basis,i);
2483 4312 : if (typ(w) == t_INT)
2484 0 : w = scalarcol_shallow(w, f);
2485 : else
2486 : {
2487 : GEN dx;
2488 4312 : w = Q_remove_denom(w, &dx);
2489 4312 : w = FpXQ_red(w, T, p);
2490 4312 : if (dx)
2491 : {
2492 0 : dx = Fp_inv(dx, p);
2493 0 : if (!equali1(dx)) w = FpX_Fp_mul(w, dx, p);
2494 : }
2495 4312 : w = RgX_to_RgC(w, f);
2496 : }
2497 4312 : gel(z,i) = w; /* w_i mod (T,p) */
2498 : }
2499 203 : return z;
2500 : }
2501 :
2502 : /* initialize reduction mod pr; if zk = 1, will only init data required to
2503 : * reduce *integral* element. Realize (O_K/pr) as Fp[X] / (T), for a
2504 : * *monic* T; use variable vT for varn(T) */
2505 : static GEN
2506 1158188 : modprinit(GEN nf, GEN pr, int zk, long vT)
2507 : {
2508 1158188 : pari_sp av = avma;
2509 : GEN res, tau, mul, x, p, T, pow, ffproj, nfproj, prh, c;
2510 : long N, i, k, f;
2511 :
2512 1158188 : nf = checknf(nf); checkprid(pr);
2513 1158157 : if (vT < 0) vT = nf_get_varn(nf);
2514 1158147 : f = pr_get_f(pr);
2515 1158140 : N = nf_get_degree(nf);
2516 1158130 : prh = pr_hnf(nf, pr);
2517 1158184 : tau = zk? gen_0: pr_anti_uniformizer(nf, pr);
2518 1158153 : p = pr_get_p(pr);
2519 :
2520 1158147 : if (f == 1)
2521 : {
2522 1012103 : res = cgetg(SMALLMODPR, t_COL);
2523 1012112 : gel(res,mpr_TAU) = tau;
2524 1012112 : gel(res,mpr_FFP) = dim1proj(prh);
2525 1012061 : gel(res,3) = pr; return gc_GEN(av, res);
2526 : }
2527 :
2528 146044 : c = cgetg(f+1, t_VECSMALL);
2529 146055 : ffproj = cgetg(N+1, t_MAT);
2530 600719 : for (k=i=1; i<=N; i++)
2531 : {
2532 454664 : x = gcoeff(prh, i,i);
2533 454664 : if (!is_pm1(x)) { c[k] = i; gel(ffproj,i) = col_ei(N, i); k++; }
2534 : else
2535 128660 : gel(ffproj,i) = ZC_neg(gel(prh,i));
2536 : }
2537 146055 : ffproj = rowpermute(ffproj, c);
2538 146055 : if (! dvdii(nf_get_index(nf), p))
2539 : {
2540 104171 : GEN basis = nf_get_zkprimpart(nf), D = nf_get_zkden(nf);
2541 104171 : if (N == f)
2542 : { /* pr inert */
2543 45297 : T = nf_get_pol(nf);
2544 45297 : T = FpX_red(T,p);
2545 45297 : ffproj = RgV_to_RgM(basis, lg(basis)-1);
2546 : }
2547 : else
2548 : {
2549 58874 : T = RgV_RgC_mul(basis, pr_get_gen(pr));
2550 58874 : T = FpX_normalize(FpX_red(T,p),p);
2551 58874 : basis = FqV_red(vecpermute(basis,c), T, p);
2552 58874 : basis = RgV_to_RgM(basis, lg(basis)-1);
2553 58874 : ffproj = ZM_mul(basis, ffproj);
2554 : }
2555 104171 : setvarn(T, vT);
2556 104171 : ffproj = FpM_red(ffproj, p);
2557 104171 : if (!equali1(D))
2558 : {
2559 33266 : D = modii(D,p);
2560 33266 : if (!equali1(D)) ffproj = FpM_Fp_mul(ffproj, Fp_inv(D,p), p);
2561 : }
2562 :
2563 104171 : res = cgetg(SMALLMODPR+1, t_COL);
2564 104171 : gel(res,mpr_TAU) = tau;
2565 104171 : gel(res,mpr_FFP) = ffproj;
2566 104171 : gel(res,3) = pr;
2567 104171 : gel(res,4) = T; return gc_GEN(av, res);
2568 : }
2569 :
2570 41884 : if (uisprime(f))
2571 : {
2572 39553 : mul = ei_multable(nf, c[2]);
2573 39553 : mul = vecpermute(mul, c);
2574 : }
2575 : else
2576 : {
2577 : GEN v, u, u2, frob;
2578 : long deg,deg1,deg2;
2579 :
2580 : /* matrix of Frob: x->x^p over Z_K/pr = < w[c1], ..., w[cf] > over Fp */
2581 2331 : frob = cgetg(f+1, t_MAT);
2582 11963 : for (i=1; i<=f; i++)
2583 : {
2584 9632 : x = pow_ei_mod_p(nf,c[i],p);
2585 9632 : gel(frob,i) = FpM_FpC_mul(ffproj, x, p);
2586 : }
2587 2331 : u = col_ei(f,2); k = 2;
2588 2331 : deg1 = ffdegree(u, frob, p);
2589 4102 : while (deg1 < f)
2590 : {
2591 1771 : k++; u2 = col_ei(f, k);
2592 1771 : deg2 = ffdegree(u2, frob, p);
2593 1771 : deg = ulcm(deg1,deg2);
2594 1771 : if (deg == deg1) continue;
2595 1764 : if (deg == deg2) { deg1 = deg2; u = u2; continue; }
2596 7 : u = ZC_add(u, u2);
2597 7 : while (ffdegree(u, frob, p) < deg) u = ZC_add(u, u2);
2598 7 : deg1 = deg;
2599 : }
2600 2331 : v = lift_to_zk(u,c,N);
2601 :
2602 2331 : mul = cgetg(f+1,t_MAT);
2603 2331 : gel(mul,1) = v; /* assume w_1 = 1 */
2604 9632 : for (i=2; i<=f; i++) gel(mul,i) = zk_ei_mul(nf,v,c[i]);
2605 : }
2606 :
2607 : /* Z_K/pr = Fp(v), mul = mul by v */
2608 41884 : mul = FpM_red(mul, p);
2609 41882 : mul = FpM_mul(ffproj, mul, p);
2610 :
2611 41884 : pow = get_powers(mul, p);
2612 41884 : T = RgV_to_RgX(FpM_deplin(pow, p), vT);
2613 41884 : nfproj = cgetg(f+1, t_MAT);
2614 131742 : for (i=1; i<=f; i++) gel(nfproj,i) = lift_to_zk(gel(pow,i), c, N);
2615 :
2616 41884 : setlg(pow, f+1);
2617 41884 : ffproj = FpM_mul(FpM_inv(pow, p), ffproj, p);
2618 :
2619 41883 : res = cgetg(LARGEMODPR, t_COL);
2620 41883 : gel(res,mpr_TAU) = tau;
2621 41883 : gel(res,mpr_FFP) = ffproj;
2622 41883 : gel(res,3) = pr;
2623 41883 : gel(res,4) = T;
2624 41883 : gel(res,mpr_NFP) = nfproj; return gc_GEN(av, res);
2625 : }
2626 :
2627 : GEN
2628 7 : nfmodprinit(GEN nf, GEN pr) { return modprinit(nf, pr, 0, -1); }
2629 : GEN
2630 175383 : zkmodprinit(GEN nf, GEN pr) { return modprinit(nf, pr, 1, -1); }
2631 : GEN
2632 77 : nfmodprinit0(GEN nf, GEN pr, long v) { return modprinit(nf, pr, 0, v); }
2633 :
2634 : /* x may be a modpr */
2635 : static int
2636 4367604 : ok_modpr(GEN x)
2637 4367604 : { return typ(x) == t_COL && lg(x) >= SMALLMODPR && lg(x) <= LARGEMODPR; }
2638 : void
2639 210 : checkmodpr(GEN x)
2640 : {
2641 210 : if (!ok_modpr(x)) pari_err_TYPE("checkmodpr [use nfmodprinit]", x);
2642 210 : checkprid(modpr_get_pr(x));
2643 210 : }
2644 : GEN
2645 137753 : get_modpr(GEN x)
2646 137753 : { return ok_modpr(x)? x: NULL; }
2647 :
2648 : int
2649 22875463 : checkprid_i(GEN x)
2650 : {
2651 22056651 : return (typ(x) == t_VEC && lg(x) == 6
2652 21983612 : && typ(gel(x,2)) == t_COL && typ(gel(x,3)) == t_INT
2653 44932114 : && typ(gel(x,5)) != t_COL); /* tau changed to t_MAT/t_INT in 2.6 */
2654 : }
2655 : void
2656 17905248 : checkprid(GEN x)
2657 17905248 : { if (!checkprid_i(x)) pari_err_TYPE("checkprid",x); }
2658 : GEN
2659 939491 : get_prid(GEN x)
2660 : {
2661 939491 : long lx = lg(x);
2662 939491 : if (lx == 3 && typ(x) == t_VEC) x = gel(x,1);
2663 939491 : if (checkprid_i(x)) return x;
2664 694673 : if (ok_modpr(x)) {
2665 108465 : x = modpr_get_pr(x);
2666 108465 : if (checkprid_i(x)) return x;
2667 : }
2668 586208 : return NULL;
2669 : }
2670 :
2671 : static GEN
2672 3534969 : to_ff_init(GEN nf, GEN *pr, GEN *T, GEN *p, int zk)
2673 : {
2674 3534969 : GEN modpr = ok_modpr(*pr)? *pr: modprinit(nf, *pr, zk, -1);
2675 3535088 : *T = modpr_get_T(modpr);
2676 3535036 : *pr = modpr_get_pr(modpr);
2677 3535019 : *p = pr_get_p(*pr); return modpr;
2678 : }
2679 :
2680 : /* Return an element of O_K which is set to x Mod T */
2681 : GEN
2682 4508 : modpr_genFq(GEN modpr)
2683 : {
2684 4508 : switch(lg(modpr))
2685 : {
2686 917 : case SMALLMODPR: /* Fp */
2687 917 : return gen_1;
2688 1568 : case LARGEMODPR: /* painful case, p \mid index */
2689 1568 : return gmael(modpr,mpr_NFP, 2);
2690 2023 : default: /* trivial case : p \nmid index */
2691 : {
2692 2023 : long v = varn( modpr_get_T(modpr) );
2693 2023 : return pol_x(v);
2694 : }
2695 : }
2696 : }
2697 :
2698 : GEN
2699 3516183 : nf_to_Fq_init(GEN nf, GEN *pr, GEN *T, GEN *p) {
2700 3516183 : GEN modpr = to_ff_init(nf,pr,T,p,0);
2701 3516255 : GEN tau = modpr_TAU(modpr);
2702 3516228 : if (!tau) gel(modpr,mpr_TAU) = pr_anti_uniformizer(nf, *pr);
2703 3516228 : return modpr;
2704 : }
2705 : GEN
2706 18760 : zk_to_Fq_init(GEN nf, GEN *pr, GEN *T, GEN *p) {
2707 18760 : return to_ff_init(nf,pr,T,p,1);
2708 : }
2709 :
2710 : /* assume x in 'basis' form (t_COL) */
2711 : GEN
2712 6394996 : zk_to_Fq(GEN x, GEN modpr)
2713 : {
2714 6394996 : GEN pr = modpr_get_pr(modpr), p = pr_get_p(pr);
2715 6394983 : GEN ffproj = gel(modpr,mpr_FFP);
2716 6394983 : GEN T = modpr_get_T(modpr);
2717 6394986 : return T? FpM_FpC_mul_FpX(ffproj,x, p, varn(T)): FpV_dotproduct(ffproj,x, p);
2718 : }
2719 :
2720 : /* REDUCTION Modulo a prime ideal */
2721 :
2722 : /* nf a true nf, not GC-clean, OK for gc_upto */
2723 : static GEN
2724 13906336 : nf_to_Fq_i(GEN nf, GEN x0, GEN modpr)
2725 : {
2726 13906336 : GEN x = x0, den, pr = modpr_get_pr(modpr), p = pr_get_p(pr);
2727 13906334 : long tx = typ(x);
2728 :
2729 13906334 : if (tx == t_POLMOD) { x = gel(x,2); tx = typ(x); }
2730 13906334 : switch(tx)
2731 : {
2732 7600435 : case t_INT: return modii(x, p);
2733 5574 : case t_FRAC: return Rg_to_Fp(x, p);
2734 204211 : case t_POL:
2735 204211 : switch(lg(x))
2736 : {
2737 224 : case 2: return gen_0;
2738 25798 : case 3: return Rg_to_Fp(gel(x,2), p);
2739 : }
2740 178189 : x = Q_remove_denom(x, &den);
2741 178185 : x = poltobasis(nf, x);
2742 : /* content(x) and den may not be coprime */
2743 178060 : break;
2744 6096158 : case t_COL:
2745 6096158 : x = Q_remove_denom(x, &den);
2746 : /* content(x) and den are coprime */
2747 6096153 : if (lg(x)-1 == nf_get_degree(nf)) break;
2748 48 : default: pari_err_TYPE("Rg_to_ff",x);
2749 : return NULL;/*LCOV_EXCL_LINE*/
2750 : }
2751 6274157 : if (den)
2752 : {
2753 49013 : long v = Z_pvalrem(den, p, &den);
2754 49013 : if (v)
2755 : {
2756 1799 : if (tx == t_POL) v -= ZV_pvalrem(x, p, &x);
2757 : /* now v = valuation(true denominator of x) */
2758 1799 : if (v > 0)
2759 : {
2760 1197 : GEN tau = modpr_TAU(modpr);
2761 1197 : if (!tau) pari_err_TYPE("zk_to_ff", x0);
2762 1197 : x = nfmuli(nf,x, nfpow_u(nf, tau, v));
2763 1197 : v -= ZV_pvalrem(x, p, &x);
2764 : }
2765 1799 : if (v > 0) pari_err_INV("Rg_to_ff", mkintmod(gen_0,p));
2766 1771 : if (v) return gen_0;
2767 1232 : if (is_pm1(den)) den = NULL;
2768 : }
2769 48446 : x = FpC_red(x, p);
2770 : }
2771 6273590 : x = zk_to_Fq(x, modpr);
2772 6273587 : if (den)
2773 : {
2774 47920 : GEN c = Fp_inv(den, p);
2775 47921 : x = typ(x) == t_INT? Fp_mul(x,c,p): FpX_Fp_mul(x,c,p);
2776 : }
2777 6273588 : return x;
2778 : }
2779 :
2780 : GEN
2781 210 : nfreducemodpr(GEN nf, GEN x, GEN modpr)
2782 : {
2783 210 : pari_sp av = avma;
2784 210 : nf = checknf(nf); checkmodpr(modpr);
2785 210 : return gc_upto(av, algtobasis(nf, Fq_to_nf(nf_to_Fq_i(nf,x,modpr),modpr)));
2786 : }
2787 :
2788 : GEN
2789 350 : nfmodpr(GEN nf, GEN x, GEN pr)
2790 : {
2791 350 : pari_sp av = avma;
2792 : GEN T, p, modpr;
2793 350 : nf = checknf(nf);
2794 350 : modpr = nf_to_Fq_init(nf, &pr, &T, &p);
2795 343 : if (typ(x) == t_MAT && lg(x) == 3)
2796 42 : {
2797 49 : GEN y, v = famat_nfvalrem(nf, x, pr, &y);
2798 49 : long s = signe(v);
2799 49 : if (s < 0) pari_err_INV("nfmodpr", mkintmod(gen_0,p));
2800 42 : if (s > 0)
2801 28 : x = gen_0;
2802 : else
2803 14 : x = FqV_factorback(nfV_to_FqV(gel(y,1), nf, modpr), gel(y,2), T, p);
2804 : }
2805 : else
2806 294 : x = nf_to_Fq_i(nf, x, modpr);
2807 224 : if (!T) return gc_upto(av, Fp_to_mod(x, p));
2808 56 : x = Fq_to_FF(x, Tp_to_FF(T,p));
2809 56 : return gc_GEN(av, x);
2810 : }
2811 : GEN
2812 77 : nfmodprlift(GEN nf, GEN x, GEN pr)
2813 : {
2814 77 : pari_sp av = avma;
2815 : GEN T, p, modpr;
2816 : long d;
2817 77 : nf = checknf(nf);
2818 77 : switch(typ(x))
2819 : {
2820 7 : case t_INT: return icopy(x);
2821 28 : case t_INTMOD: return icopy(gel(x,2));
2822 14 : case t_FFELT: break;
2823 28 : case t_VEC: case t_COL: case t_MAT:
2824 63 : pari_APPLY_same(nfmodprlift(nf,gel(x,i),pr));
2825 0 : default: pari_err_TYPE("nfmodprlit",x);
2826 : }
2827 14 : x = FF_to_FpXQ(x);
2828 14 : setvarn(x, nf_get_varn(nf));
2829 14 : d = degpol(x);
2830 14 : if (d <= 0) { set_avma(av); return d? gen_0: icopy(gel(x,2)); }
2831 14 : modpr = nf_to_Fq_init(nf, &pr, &T, &p);
2832 14 : return gc_GEN(av, Fq_to_nf(x, modpr));
2833 : }
2834 :
2835 : /* lift A from residue field to nf */
2836 : GEN
2837 3086540 : Fq_to_nf(GEN A, GEN modpr)
2838 : {
2839 : long dA;
2840 3086540 : if (typ(A) == t_INT || lg(modpr) < LARGEMODPR) return A;
2841 44774 : dA = degpol(A);
2842 44774 : if (dA <= 0) return dA ? gen_0: gel(A,2);
2843 40581 : return ZM_ZX_mul(gel(modpr,mpr_NFP), A);
2844 : }
2845 : GEN
2846 0 : FqV_to_nfV(GEN x, GEN modpr)
2847 0 : { pari_APPLY_same(Fq_to_nf(gel(x,i), modpr)) }
2848 : GEN
2849 2934 : FqM_to_nfM(GEN A, GEN modpr)
2850 : {
2851 2934 : long i,j,h,l = lg(A);
2852 2934 : GEN B = cgetg(l, t_MAT);
2853 :
2854 2934 : if (l == 1) return B;
2855 2633 : h = lgcols(A);
2856 11010 : for (j=1; j<l; j++)
2857 : {
2858 8377 : GEN Aj = gel(A,j), Bj = cgetg(h,t_COL); gel(B,j) = Bj;
2859 52732 : for (i=1; i<h; i++) gel(Bj,i) = Fq_to_nf(gel(Aj,i), modpr);
2860 : }
2861 2633 : return B;
2862 : }
2863 : GEN
2864 10184 : FqX_to_nfX(GEN x, GEN modpr)
2865 : {
2866 10184 : if (typ(x) != t_POL) return icopy(x); /* scalar */
2867 42998 : pari_APPLY_pol(Fq_to_nf(gel(x,i), modpr));
2868 : }
2869 :
2870 : /* true nf */
2871 : static GEN
2872 13905824 : gc_nf_to_Fq(GEN nf, GEN A, GEN modpr)
2873 : {
2874 13905824 : pari_sp av = avma;
2875 13905824 : return gc_upto(av, nf_to_Fq_i(nf, A, modpr));
2876 : }
2877 : GEN
2878 12639488 : nf_to_Fq(GEN nf, GEN A, GEN modpr)
2879 12639488 : { return gc_nf_to_Fq(checknf(nf), A, modpr); }
2880 : /* A t_VEC/t_COL */
2881 : GEN
2882 167410 : nfV_to_FqV(GEN x, GEN nf, GEN modpr)
2883 : {
2884 167410 : nf = checknf(nf);
2885 916830 : pari_APPLY_same(gc_nf_to_Fq(nf, gel(x,i), modpr));
2886 : }
2887 : /* A t_MAT */
2888 : GEN
2889 1872 : nfM_to_FqM(GEN A, GEN nf, GEN modpr)
2890 : {
2891 1872 : long i,j,h,l = lg(A);
2892 1872 : GEN B = cgetg(l,t_MAT);
2893 :
2894 1872 : if (l == 1) return B;
2895 1872 : h = lgcols(A); nf = checknf(nf);
2896 42624 : for (j=1; j<l; j++)
2897 : {
2898 40752 : GEN Aj = gel(A,j), Bj = cgetg(h,t_COL); gel(B,j) = Bj;
2899 305810 : for (i=1; i<h; i++) gel(Bj,i) = gc_nf_to_Fq(nf, gel(Aj,i), modpr);
2900 : }
2901 1872 : return B;
2902 : }
2903 : /* A t_POL */
2904 : GEN
2905 9373 : nfX_to_FqX(GEN x, GEN nf, GEN modpr)
2906 : {
2907 9373 : nf = checknf(nf);
2908 51659 : pari_APPLY_pol(gc_nf_to_Fq(nf, gel(x,i), modpr));
2909 : }
2910 :
2911 : /*******************************************************************/
2912 : /* */
2913 : /* RELATIVE ROUND 2 */
2914 : /* */
2915 : /*******************************************************************/
2916 : /* Shallow functions */
2917 : /* FIXME: use a bb_field and export the nfX_* routines */
2918 : static GEN
2919 4123 : nfX_sub(GEN nf, GEN x, GEN y)
2920 : {
2921 4123 : long i, lx = lg(x), ly = lg(y);
2922 : GEN z;
2923 4123 : if (ly <= lx) {
2924 4123 : z = cgetg(lx,t_POL); z[1] = x[1];
2925 24640 : for (i=2; i < ly; i++) gel(z,i) = nfsub(nf,gel(x,i),gel(y,i));
2926 4123 : for ( ; i < lx; i++) gel(z,i) = gel(x,i);
2927 4123 : z = normalizepol_lg(z, lx);
2928 : } else {
2929 0 : z = cgetg(ly,t_POL); z[1] = y[1];
2930 0 : for (i=2; i < lx; i++) gel(z,i) = nfsub(nf,gel(x,i),gel(y,i));
2931 0 : for ( ; i < ly; i++) gel(z,i) = gneg(gel(y,i));
2932 0 : z = normalizepol_lg(z, ly);
2933 : }
2934 4123 : return z;
2935 : }
2936 : /* FIXME: quadratic multiplication */
2937 : static GEN
2938 20646 : nfX_mul(GEN nf, GEN a, GEN b)
2939 : {
2940 20646 : long da = degpol(a), db = degpol(b), dc, lc, k;
2941 : GEN c;
2942 20646 : if (da < 0 || db < 0) return gen_0;
2943 20646 : dc = da + db;
2944 20646 : if (dc == 0) return nfmul(nf, gel(a,2),gel(b,2));
2945 20646 : lc = dc+3;
2946 20646 : c = cgetg(lc, t_POL); c[1] = a[1];
2947 169839 : for (k = 0; k <= dc; k++)
2948 : {
2949 149193 : long i, I = minss(k, da);
2950 149193 : GEN d = NULL;
2951 541142 : for (i = maxss(k-db, 0); i <= I; i++)
2952 : {
2953 391949 : GEN e = nfmul(nf, gel(a, i+2), gel(b, k-i+2));
2954 391949 : d = d? nfadd(nf, d, e): e;
2955 : }
2956 149193 : gel(c, k+2) = d;
2957 : }
2958 20646 : return normalizepol_lg(c, lc);
2959 : }
2960 : /* assume b monic */
2961 : static GEN
2962 16523 : nfX_rem(GEN nf, GEN a, GEN b)
2963 : {
2964 16523 : long da = degpol(a), db = degpol(b);
2965 16523 : if (da < 0) return gen_0;
2966 16523 : a = leafcopy(a);
2967 39759 : while (da >= db)
2968 : {
2969 23236 : long i, k = da;
2970 23236 : GEN A = gel(a, k+2);
2971 190325 : for (i = db-1, k--; i >= 0; i--, k--)
2972 167089 : gel(a,k+2) = nfsub(nf, gel(a,k+2), nfmul(nf, A, gel(b,i+2)));
2973 23236 : a = normalizepol_lg(a, lg(a)-1);
2974 23236 : da = degpol(a);
2975 : }
2976 16523 : return a;
2977 : }
2978 : static GEN
2979 16523 : nfXQ_mul(GEN nf, GEN a, GEN b, GEN T)
2980 : {
2981 16523 : GEN c = nfX_mul(nf, a, b);
2982 16523 : if (typ(c) != t_POL) return c;
2983 16523 : return nfX_rem(nf, c, T);
2984 : }
2985 :
2986 : static void
2987 4222 : fill(long l, GEN H, GEN Hx, GEN I, GEN Ix)
2988 : {
2989 : long i;
2990 4222 : if (typ(Ix) == t_VEC) /* standard */
2991 15335 : for (i=1; i<l; i++) { gel(H,i) = gel(Hx,i); gel(I,i) = gel(Ix,i); }
2992 : else /* constant ideal */
2993 3830 : for (i=1; i<l; i++) { gel(H,i) = gel(Hx,i); gel(I,i) = Ix; }
2994 4222 : }
2995 :
2996 : /* given MODULES x and y by their pseudo-bases, returns a pseudo-basis of the
2997 : * module generated by x and y. */
2998 : static GEN
2999 2111 : rnfjoinmodules_i(GEN nf, GEN Hx, GEN Ix, GEN Hy, GEN Iy)
3000 : {
3001 2111 : long lx = lg(Hx), ly = lg(Hy), l = lx+ly-1;
3002 2111 : GEN H = cgetg(l, t_MAT), I = cgetg(l, t_VEC);
3003 2111 : fill(lx, H , Hx, I , Ix);
3004 2111 : fill(ly, H+lx-1, Hy, I+lx-1, Iy); return nfhnf(nf, mkvec2(H, I));
3005 : }
3006 : static GEN
3007 1329 : rnfjoinmodules(GEN nf, GEN x, GEN y)
3008 : {
3009 1329 : if (!x) return y;
3010 644 : if (!y) return x;
3011 644 : return rnfjoinmodules_i(nf, gel(x,1), gel(x,2), gel(y,1), gel(y,2));
3012 : }
3013 :
3014 : typedef struct {
3015 : GEN multab, T,p;
3016 : long h;
3017 : } rnfeltmod_muldata;
3018 :
3019 : static GEN
3020 16190 : _sqr(void *data, GEN x)
3021 : {
3022 16190 : rnfeltmod_muldata *D = (rnfeltmod_muldata *) data;
3023 10356 : GEN z = x? tablesqr(D->multab,x)
3024 16190 : : tablemul_ei_ej(D->multab,D->h,D->h);
3025 16190 : return FqC_red(z,D->T,D->p);
3026 : }
3027 : static GEN
3028 4158 : _msqr(void *data, GEN x)
3029 : {
3030 4158 : GEN x2 = _sqr(data, x), z;
3031 4158 : rnfeltmod_muldata *D = (rnfeltmod_muldata *) data;
3032 4158 : z = tablemul_ei(D->multab, x2, D->h);
3033 4158 : return FqC_red(z,D->T,D->p);
3034 : }
3035 :
3036 : /* Compute W[h]^n mod (T,p) in the extension, assume n >= 0. T a ZX */
3037 : static GEN
3038 5834 : rnfeltid_powmod(GEN multab, long h, GEN n, GEN T, GEN p)
3039 : {
3040 5834 : pari_sp av = avma;
3041 : GEN y;
3042 : rnfeltmod_muldata D;
3043 :
3044 5834 : if (!signe(n)) return gen_1;
3045 :
3046 5834 : D.multab = multab;
3047 5834 : D.h = h;
3048 5834 : D.T = T;
3049 5834 : D.p = p;
3050 5834 : y = gen_pow_fold(NULL, n, (void*)&D, &_sqr, &_msqr);
3051 5834 : return gc_GEN(av, y);
3052 : }
3053 :
3054 : /* P != 0 has at most degpol(P) roots. Look for an element in Fq which is not
3055 : * a root, cf repres() */
3056 : static GEN
3057 21 : FqX_non_root(GEN P, GEN T, GEN p)
3058 : {
3059 21 : long dP = degpol(P), f, vT;
3060 : long i, j, k, pi, pp;
3061 : GEN v;
3062 :
3063 21 : if (dP == 0) return gen_1;
3064 21 : pp = is_bigint(p) ? dP+1: itos(p);
3065 21 : v = cgetg(dP + 2, t_VEC);
3066 21 : gel(v,1) = gen_0;
3067 21 : if (T)
3068 0 : { f = degpol(T); vT = varn(T); }
3069 : else
3070 21 : { f = 1; vT = 0; }
3071 42 : for (i=pi=1; i<=f; i++,pi*=pp)
3072 : {
3073 21 : GEN gi = i == 1? gen_1: pol_xn(i-1, vT), jgi = gi;
3074 42 : for (j=1; j<pp; j++)
3075 : {
3076 42 : for (k=1; k<=pi; k++)
3077 : {
3078 21 : GEN z = Fq_add(gel(v,k), jgi, T,p);
3079 21 : if (!gequal0(FqX_eval(P, z, T,p))) return z;
3080 21 : gel(v, j*pi+k) = z;
3081 : }
3082 21 : if (j < pp-1) jgi = Fq_add(jgi, gi, T,p); /* j*g[i] */
3083 : }
3084 : }
3085 21 : return NULL;
3086 : }
3087 :
3088 : /* true nf, x t_POL */
3089 : static int
3090 8225 : nfpolisintegral_i(GEN nf, GEN x)
3091 : {
3092 8225 : GEN d, T = nf_get_pol(nf);
3093 8225 : long l = lg(x);
3094 8225 : if (varn(x) != varn(T)) pari_err_VAR("nfisintegral", x,T);
3095 8225 : if (l >= lg(T)) { x = RgX_rem(x, T); l = lg(x); }
3096 8225 : if (l == 2) return 1;
3097 8225 : if (l == 3)
3098 : {
3099 0 : switch(typ(gel(x,2)))
3100 : {
3101 0 : case t_INT: return 1;
3102 0 : case t_FRAC: return 0;
3103 0 : default: pari_err_TYPE("nfisintegral",x);
3104 : }
3105 : }
3106 8225 : x = Q_remove_denom(x, &d);
3107 8225 : if (!RgX_is_ZX(x)) pari_err_TYPE("nfisintegral",x);
3108 8225 : if (!d) return 1;
3109 672 : x = ZM_ZX_mul(nf_get_invzk(nf), x);
3110 672 : return ZV_Z_dvd(x, d);
3111 : }
3112 : static int
3113 8225 : nfpolisintegral(GEN nf, GEN x)
3114 8225 : { pari_sp av = avma; return gc_int(av, nfpolisintegral_i(nf, x)); }
3115 :
3116 : /* true nf */
3117 : static int
3118 31283 : nfisintegral(GEN nf, GEN x)
3119 : {
3120 31283 : switch(typ(x))
3121 : {
3122 23051 : case t_INT: return 1;
3123 7 : case t_FRAC: return 0;
3124 0 : case t_POLMOD:
3125 0 : x = checknfelt_mod(nf,x,"nfisintegral");
3126 0 : switch(typ(x))
3127 : {
3128 0 : case t_INT: return 1;
3129 0 : case t_FRAC: return 0;
3130 0 : case t_POL: return nfpolisintegral(nf,x);
3131 : }
3132 0 : break;
3133 8225 : case t_POL: return nfpolisintegral(nf,x);
3134 0 : case t_COL:
3135 0 : if (lg(x)-1 != nf_get_degree(nf)) break;
3136 0 : return RgV_is_ZV(x);
3137 : }
3138 0 : pari_err_TYPE("nfisintegral",x);
3139 : return 0; /* LCOV_EXCL_LINE */
3140 : }
3141 : /* true nf */
3142 : static int
3143 7077 : nfXisintegral(GEN nf, GEN x)
3144 : {
3145 7077 : long i, l = lg(x);
3146 38346 : for (i = 2; i < l; i++)
3147 31283 : if (!nfisintegral(nf, gel(x,i))) return 0;
3148 7063 : return 1;
3149 : }
3150 :
3151 : /* Relative Dedekind criterion over (true) nf, applied to the order defined by a
3152 : * root of monic irreducible polynomial P, modulo the prime ideal pr. Assume
3153 : * vdisc = v_pr( disc(P) ).
3154 : * Return NULL if nf[X]/P is pr-maximal. Otherwise, return [flag, O, v]:
3155 : * O = enlarged order, given by a pseudo-basis
3156 : * flag = 1 if O is proven pr-maximal (may be 0 and O nevertheless pr-maximal)
3157 : * v = v_pr(disc(O)). */
3158 : static GEN
3159 4151 : rnfdedekind_i(GEN nf, GEN P, GEN pr, long vdisc, long only_maximal)
3160 : {
3161 : GEN Ppr, A, I, p, tau, g, h, k, base, T, gzk, hzk, prinvp, pal, nfT, modpr;
3162 : long m, vt, r, d, i, j, mpr;
3163 :
3164 4151 : if (vdisc < 0) pari_err_TYPE("rnfdedekind [non integral pol]", P);
3165 4144 : if (vdisc == 1) return NULL; /* pr-maximal */
3166 4144 : if (!only_maximal && !gequal1(leading_coeff(P)))
3167 0 : pari_err_IMPL( "the full Dedekind criterion in the nonmonic case");
3168 4144 : if (!nfXisintegral(nf, P))
3169 0 : pari_err_IMPL("non integral polynomial in rnfdedekind");
3170 : /* either monic OR only_maximal = 1 */
3171 4144 : m = degpol(P);
3172 4144 : nfT = nf_get_pol(nf);
3173 4144 : modpr = nf_to_Fq_init(nf,&pr, &T, &p);
3174 4144 : Ppr = nfX_to_FqX(P, nf, modpr);
3175 4144 : mpr = degpol(Ppr);
3176 4144 : if (mpr < m) /* nonmonic => only_maximal = 1 */
3177 : {
3178 21 : if (mpr < 0) return NULL;
3179 21 : if (! RgX_valrem(Ppr, &Ppr))
3180 : { /* nonzero constant coefficient */
3181 0 : Ppr = RgX_shift_shallow(RgX_recip_i(Ppr), m - mpr);
3182 0 : P = RgX_recip_i(P);
3183 : }
3184 : else
3185 : {
3186 21 : GEN z = FqX_non_root(Ppr, T, p);
3187 21 : if (!z) pari_err_IMPL( "Dedekind in the difficult case");
3188 0 : z = Fq_to_nf(z, modpr);
3189 0 : if (typ(z) == t_INT)
3190 0 : P = RgX_translate(P, z);
3191 : else
3192 0 : P = RgXQX_translate(P, z, T);
3193 0 : P = RgX_recip_i(P);
3194 0 : Ppr = nfX_to_FqX(P, nf, modpr); /* degpol(P) = degpol(Ppr) = m */
3195 : }
3196 : }
3197 4123 : A = gel(FqX_factor(Ppr,T,p),1);
3198 4123 : r = lg(A); /* > 1 */
3199 4123 : g = gel(A,1);
3200 7245 : for (i=2; i<r; i++) g = FqX_mul(g, gel(A,i), T, p);
3201 4123 : h = FqX_div(Ppr,g, T, p);
3202 4123 : gzk = FqX_to_nfX(g, modpr);
3203 4123 : hzk = FqX_to_nfX(h, modpr);
3204 4123 : k = nfX_sub(nf, P, nfX_mul(nf, gzk,hzk));
3205 4123 : tau = pr_get_tau(pr);
3206 4123 : switch(typ(tau))
3207 : {
3208 2086 : case t_INT: k = gdiv(k, p); break;
3209 2037 : case t_MAT: k = RgX_Rg_div(tablemulvec(NULL,tau, k), p); break;
3210 : }
3211 4123 : k = nfX_to_FqX(k, nf, modpr);
3212 4123 : k = FqX_normalize(FqX_gcd(FqX_gcd(g,h, T,p), k, T,p), T,p);
3213 4123 : d = degpol(k); /* <= m */
3214 4123 : if (!d) return NULL; /* pr-maximal */
3215 1952 : if (only_maximal) return gen_0; /* not maximal */
3216 :
3217 1931 : A = cgetg(m+d+1,t_MAT);
3218 1931 : I = cgetg(m+d+1,t_VEC); base = mkvec2(A, I);
3219 : /* base[2] temporarily multiplied by p, for the final nfhnfmod,
3220 : * which requires integral ideals */
3221 1931 : prinvp = pr_inv_p(pr); /* again multiplied by p */
3222 10427 : for (j=1; j<=m; j++)
3223 : {
3224 8496 : gel(A,j) = col_ei(m, j);
3225 8496 : gel(I,j) = p;
3226 : }
3227 1931 : pal = FqX_to_nfX(FqX_div(Ppr,k, T,p), modpr);
3228 4198 : for ( ; j<=m+d; j++)
3229 : {
3230 2267 : gel(A,j) = RgX_to_RgC(pal,m);
3231 2267 : gel(I,j) = prinvp;
3232 2267 : if (j < m+d) pal = RgXQX_rem(RgX_shift_shallow(pal,1),P,nfT);
3233 : }
3234 : /* the modulus is integral */
3235 1931 : base = nfhnfmod(nf,base, idealmulpowprime(nf, powiu(p,m), pr, utoineg(d)));
3236 1931 : gel(base,2) = gdiv(gel(base,2), p); /* cancel the factor p */
3237 1931 : vt = vdisc - 2*d;
3238 1931 : return mkvec3(vt < 2? gen_1: gen_0, base, stoi(vt));
3239 : }
3240 :
3241 : /* [L:K] = n */
3242 : static GEN
3243 1506 : triv_order(long n)
3244 : {
3245 1506 : GEN z = cgetg(3, t_VEC);
3246 1506 : gel(z,1) = matid(n);
3247 1506 : gel(z,2) = const_vec(n, gen_1); return z;
3248 : }
3249 :
3250 : /* if flag is set, return gen_1 (resp. gen_0) if the order K[X]/(P)
3251 : * is pr-maximal (resp. not pr-maximal). */
3252 : GEN
3253 91 : rnfdedekind(GEN nf, GEN P, GEN pr, long flag)
3254 : {
3255 91 : pari_sp av = avma;
3256 : GEN z, dP;
3257 : long v;
3258 :
3259 91 : nf = checknf(nf);
3260 91 : P = RgX_nffix("rnfdedekind", nf_get_pol(nf), P, 1);
3261 91 : dP = nfX_disc(nf, P);
3262 91 : if (gequal0(dP))
3263 7 : pari_err_DOMAIN("rnfdedekind","issquarefree(pol)","=",gen_0,P);
3264 84 : if (!pr)
3265 : {
3266 21 : GEN fa = idealfactor(nf, dP);
3267 21 : GEN Q = gel(fa,1), E = gel(fa,2);
3268 21 : pari_sp av2 = avma;
3269 21 : long i, l = lg(Q);
3270 21 : for (i = 1; i < l; i++, set_avma(av2))
3271 : {
3272 21 : v = itos(gel(E,i));
3273 21 : if (rnfdedekind_i(nf,P,gel(Q,i),v,1)) { set_avma(av); return gen_0; }
3274 0 : set_avma(av2);
3275 : }
3276 0 : set_avma(av); return gen_1;
3277 : }
3278 63 : else if (typ(pr) == t_VEC)
3279 : { /* flag = 1 is implicit */
3280 63 : if (lg(pr) == 1) { set_avma(av); return gen_1; }
3281 63 : if (typ(gel(pr,1)) == t_VEC)
3282 : { /* list of primes */
3283 14 : GEN Q = pr;
3284 14 : pari_sp av2 = avma;
3285 14 : long i, l = lg(Q);
3286 14 : for (i = 1; i < l; i++, set_avma(av2))
3287 : {
3288 14 : v = nfval(nf, dP, gel(Q,i));
3289 14 : if (rnfdedekind_i(nf,P,gel(Q,i),v,1)) { set_avma(av); return gen_0; }
3290 : }
3291 0 : set_avma(av); return gen_1;
3292 : }
3293 : }
3294 : /* single prime */
3295 49 : v = nfval(nf, dP, pr);
3296 49 : z = rnfdedekind_i(nf, P, pr, v, flag);
3297 42 : if (z)
3298 : {
3299 21 : if (flag) { set_avma(av); return gen_0; }
3300 14 : z = gc_GEN(av, z);
3301 : }
3302 : else
3303 : {
3304 21 : set_avma(av); if (flag) return gen_1;
3305 7 : z = cgetg(4, t_VEC);
3306 7 : gel(z,1) = gen_1;
3307 7 : gel(z,2) = triv_order(degpol(P));
3308 7 : gel(z,3) = stoi(v);
3309 : }
3310 21 : return z;
3311 : }
3312 :
3313 : static int
3314 8049 : ideal_is1(GEN x) {
3315 8049 : switch(typ(x))
3316 : {
3317 4119 : case t_INT: return is_pm1(x);
3318 3391 : case t_MAT: return RgM_isidentity(x);
3319 : }
3320 539 : return 0;
3321 : }
3322 :
3323 : /* return a in ideal A such that v_pr(a) = v_pr(A) */
3324 : static GEN
3325 3776 : minval(GEN nf, GEN A, GEN pr)
3326 : {
3327 3776 : GEN ab = idealtwoelt(nf,A), a = gel(ab,1), b = gel(ab,2);
3328 3776 : if (nfval(nf,a,pr) > nfval(nf,b,pr)) a = b;
3329 3776 : return a;
3330 : }
3331 :
3332 : /* nf a true nf. Return NULL if power order is pr-maximal */
3333 : static GEN
3334 4067 : rnfmaxord(GEN nf, GEN pol, GEN pr, long vdisc)
3335 : {
3336 4067 : pari_sp av = avma, av1;
3337 : long i, j, k, n, nn, vpol, cnt, sep;
3338 : GEN q, q1, p, T, modpr, W, I, p1;
3339 : GEN prhinv, mpi, Id;
3340 :
3341 4067 : if (DEBUGLEVEL>1) err_printf(" treating %Ps^%ld\n", pr, vdisc);
3342 4067 : modpr = nf_to_Fq_init(nf,&pr,&T,&p);
3343 4067 : av1 = avma;
3344 4067 : p1 = rnfdedekind_i(nf, pol, modpr, vdisc, 0);
3345 4067 : if (!p1) return gc_NULL(av);
3346 1917 : if (is_pm1(gel(p1,1))) return gc_GEN(av,gel(p1,2));
3347 832 : sep = itos(gel(p1,3));
3348 832 : W = gmael(p1,2,1);
3349 832 : I = gmael(p1,2,2);
3350 832 : (void)gc_all(av1, 2, &W, &I);
3351 :
3352 832 : mpi = zk_multable(nf, pr_get_gen(pr));
3353 832 : n = degpol(pol); nn = n*n;
3354 832 : vpol = varn(pol);
3355 832 : q1 = q = pr_norm(pr);
3356 1021 : while (abscmpiu(q1,n) < 0) q1 = mulii(q1,q);
3357 832 : Id = matid(n);
3358 832 : prhinv = pr_inv(pr);
3359 832 : av1 = avma;
3360 832 : for(cnt=1;; cnt++)
3361 1040 : {
3362 1872 : GEN I0 = leafcopy(I), W0 = leafcopy(W);
3363 : GEN Wa, Winv, Ip, A, MW, MWmod, F, pseudo, C, G;
3364 1872 : GEN Tauinv = cgetg(n+1, t_VEC), Tau = cgetg(n+1, t_VEC);
3365 :
3366 1872 : if (DEBUGLEVEL>1) err_printf(" pass no %ld\n",cnt);
3367 9578 : for (j=1; j<=n; j++)
3368 : {
3369 : GEN tau, tauinv;
3370 7706 : if (ideal_is1(gel(I,j)))
3371 : {
3372 3930 : gel(I,j) = gel(Tau,j) = gel(Tauinv,j) = gen_1;
3373 3930 : continue;
3374 : }
3375 3776 : gel(Tau,j) = tau = minval(nf, gel(I,j), pr);
3376 3776 : gel(Tauinv,j) = tauinv = nfinv(nf, tau);
3377 3776 : gel(W,j) = nfC_nf_mul(nf, gel(W,j), tau);
3378 3776 : gel(I,j) = idealmul(nf, tauinv, gel(I,j)); /* v_pr(I[j]) = 0 */
3379 : }
3380 : /* W = (Z_K/pr)-basis of O/pr. O = (W0,I0) ~ (W, I) */
3381 :
3382 : /* compute MW: W_i*W_j = sum MW_k,(i,j) W_k */
3383 1872 : Wa = RgM_to_RgXV(W,vpol);
3384 1872 : Winv = nfM_inv(nf, W);
3385 1872 : MW = cgetg(nn+1, t_MAT);
3386 : /* W_1 = 1 */
3387 9578 : for (j=1; j<=n; j++) gel(MW, j) = gel(MW, (j-1)*n+1) = gel(Id,j);
3388 7706 : for (i=2; i<=n; i++)
3389 22357 : for (j=i; j<=n; j++)
3390 : {
3391 16523 : GEN z = nfXQ_mul(nf, gel(Wa,i), gel(Wa,j), pol);
3392 16523 : if (typ(z) != t_POL)
3393 0 : z = nfC_nf_mul(nf, gel(Winv,1), z);
3394 : else
3395 : {
3396 16523 : z = RgX_to_RgC(z, lg(Winv)-1);
3397 16523 : z = nfM_nfC_mul(nf, Winv, z);
3398 : }
3399 16523 : gel(MW, (i-1)*n+j) = gel(MW, (j-1)*n+i) = z;
3400 : }
3401 :
3402 : /* compute Ip = pr-radical [ could use Ker(trace) if q large ] */
3403 1872 : MWmod = nfM_to_FqM(MW,nf,modpr);
3404 1872 : F = cgetg(n+1, t_MAT); gel(F,1) = gel(Id,1);
3405 7706 : for (j=2; j<=n; j++) gel(F,j) = rnfeltid_powmod(MWmod, j, q1, T,p);
3406 1872 : Ip = FqM_ker(F,T,p);
3407 1872 : if (lg(Ip) == 1) { W = W0; I = I0; break; }
3408 :
3409 : /* Fill C: W_k A_j = sum_i C_(i,j),k A_i */
3410 1467 : A = FqM_to_nfM(FqM_suppl(Ip,T,p), modpr);
3411 3795 : for (j = lg(Ip); j<=n; j++) gel(A,j) = nfC_multable_mul(gel(A,j), mpi);
3412 1467 : MW = nfM_mul(nf, nfM_inv(nf,A), MW);
3413 1467 : C = cgetg(n+1, t_MAT);
3414 7481 : for (k=1; k<=n; k++)
3415 : {
3416 6014 : GEN mek = vecslice(MW, (k-1)*n+1, k*n), Ck;
3417 6014 : gel(C,k) = Ck = cgetg(nn+1, t_COL);
3418 38020 : for (j = 1; j <= n; j++)
3419 : {
3420 32006 : GEN z = nfM_nfC_mul(nf, mek, gel(A,j));
3421 241576 : for (i = 1; i <= n; i++)
3422 209570 : gel(Ck, (j-1)*n+i) = gc_nf_to_Fq(nf,gel(z,i),modpr);
3423 : }
3424 : }
3425 1467 : G = FqM_to_nfM(FqM_ker(C,T,p), modpr);
3426 :
3427 1467 : pseudo = rnfjoinmodules_i(nf, G,prhinv, Id,I);
3428 : /* express W in terms of the power basis */
3429 1467 : W = nfM_mul(nf, W, gel(pseudo,1));
3430 1467 : I = gel(pseudo,2);
3431 : /* restore the HNF property W[i,i] = 1. NB: W upper triangular, with
3432 : * W[i,i] = Tau[i] */
3433 7481 : for (j=1; j<=n; j++)
3434 6014 : if (gel(Tau,j) != gen_1)
3435 : {
3436 2699 : gel(W,j) = nfC_nf_mul(nf, gel(W,j), gel(Tauinv,j));
3437 2699 : gel(I,j) = idealmul(nf, gel(Tau,j), gel(I,j));
3438 : }
3439 1467 : if (DEBUGLEVEL>3) err_printf(" new order:\n%Ps\n%Ps\n", W, I);
3440 1467 : if (sep <= 3 || gequal(I,I0)) break;
3441 :
3442 1040 : if (gc_needed(av1,2))
3443 : {
3444 0 : if(DEBUGMEM>1) pari_warn(warnmem,"rnfmaxord");
3445 0 : (void)gc_all(av1,2, &W,&I);
3446 : }
3447 : }
3448 832 : return gc_GEN(av, mkvec2(W, I));
3449 : }
3450 :
3451 : GEN
3452 945354 : Rg_nffix(const char *f, GEN T, GEN c, int lift)
3453 : {
3454 945354 : switch(typ(c))
3455 : {
3456 516629 : case t_INT: case t_FRAC: return c;
3457 68694 : case t_POL:
3458 68694 : if (lg(c) >= lg(T)) c = RgX_rem(c,T);
3459 68694 : break;
3460 360025 : case t_POLMOD:
3461 360025 : if (!RgX_equal_var(gel(c,1), T)) pari_err_MODULUS(f, gel(c,1),T);
3462 359444 : c = gel(c,2);
3463 359444 : switch(typ(c))
3464 : {
3465 313487 : case t_POL: break;
3466 45957 : case t_INT: case t_FRAC: return c;
3467 0 : default: pari_err_TYPE(f, c);
3468 : }
3469 313487 : break;
3470 6 : default: pari_err_TYPE(f,c);
3471 : }
3472 : /* typ(c) = t_POL */
3473 382181 : if (varn(c) != varn(T)) pari_err_VAR(f, c,T);
3474 382167 : switch(lg(c))
3475 : {
3476 13395 : case 2: return gen_0;
3477 30022 : case 3:
3478 30022 : c = gel(c,2); if (is_rational_t(typ(c))) return c;
3479 0 : pari_err_TYPE(f,c);
3480 : }
3481 338750 : RgX_check_QX(c, f);
3482 338729 : return lift? c: mkpolmod(c, T);
3483 : }
3484 : /* check whether x is a polynomials with coeffs in number field Q[y]/(T)
3485 : * and returned a normalized copy. If 'lift' is set return lifted coefs
3486 : * (t_POL/t_FRAC/t_INT) else t_POLMOD/t_FRAC/t_INT */
3487 : GEN
3488 322974 : RgX_nffix(const char *f, GEN T, GEN x, int lift)
3489 : {
3490 322974 : long vT = varn(T);
3491 322974 : if (typ(x) != t_POL) pari_err_TYPE(stack_strcat(f," [t_POL expected]"), x);
3492 322974 : if (varncmp(varn(x), vT) >= 0) pari_err_PRIORITY(f, x, ">=", vT);
3493 1202389 : pari_APPLY_pol_normalized(Rg_nffix(f, T, gel(x,i), lift));
3494 : }
3495 : GEN
3496 49 : RgV_nffix(const char *f, GEN T, GEN x, int lift)
3497 119 : { pari_APPLY_same(Rg_nffix(f, T, gel(x,i), lift)); }
3498 :
3499 : static GEN
3500 2989 : get_d(GEN nf, GEN d)
3501 : {
3502 2989 : GEN b = idealredmodpower(nf, d, 2, 100000);
3503 2989 : return nfmul(nf, d, nfsqr(nf,b));
3504 : }
3505 :
3506 : /* true nf */
3507 : static GEN
3508 4263 : pr_factorback(GEN nf, GEN fa)
3509 : {
3510 4263 : GEN P = gel(fa,1), E = gel(fa,2), z = gen_1;
3511 4263 : long i, l = lg(P);
3512 8322 : for (i = 1; i < l; i++) z = idealmulpowprime(nf, z, gel(P,i), gel(E,i));
3513 4263 : return z;
3514 : }
3515 : /* true nf */
3516 : static GEN
3517 4263 : pr_factorback_scal(GEN nf, GEN fa)
3518 : {
3519 4263 : GEN D = pr_factorback(nf,fa);
3520 4263 : if (typ(D) == t_MAT && RgM_isscalar(D,NULL)) D = gcoeff(D,1,1);
3521 4263 : return D;
3522 : }
3523 :
3524 : /* nf = base field K
3525 : * pol= monic polynomial in Z_K[X] defining a relative extension L = K[X]/(pol).
3526 : * Returns a pseudo-basis [A,I] of Z_L, set *pD to [D,d] and *pf to the
3527 : * index-ideal; rnf is used when lim != 0 and may be NULL */
3528 : GEN
3529 2940 : rnfallbase(GEN nf, GEN pol, GEN lim, GEN rnf, GEN *pD, GEN *pf, GEN *pDKP)
3530 : {
3531 : long i, j, jf, l;
3532 : GEN fa, E, P, Ef, Pf, z, disc;
3533 :
3534 2940 : nf = checknf(nf); pol = liftpol_shallow(pol);
3535 2940 : if (!gequal1(leading_coeff(pol)))
3536 7 : pari_err_IMPL("nonmonic relative polynomials in rnfallbase");
3537 2933 : if (!nfXisintegral(nf, pol))
3538 14 : pari_err_IMPL("non integral polynomial in rnfallbase");
3539 2919 : disc = nf_to_scalar_or_basis(nf, nfX_disc(nf, pol));
3540 2919 : if (gequal0(disc))
3541 7 : pari_err_DOMAIN("rnfpseudobasis","issquarefree(pol)","=",gen_0, pol);
3542 2912 : if (lim)
3543 : {
3544 : GEN rnfeq, zknf, dzknf, U, vU, dA, A, MB, dB, BdB, vj, B, Tabs;
3545 791 : GEN D = idealhnf_shallow(nf, disc), extendP = NULL;
3546 791 : long rU, m = nf_get_degree(nf), n = degpol(pol), N = n*m;
3547 : nfmaxord_t S;
3548 :
3549 791 : if (typ(lim) == t_INT)
3550 133 : P = ZV_union_shallow(nf_get_ramified_primes(nf),
3551 133 : gel(Z_factor_limit(gcoeff(D,1,1), itou(lim)), 1));
3552 : else
3553 : {
3554 658 : P = cgetg_copy(lim, &l);
3555 2219 : for (i = 1; i < l; i++)
3556 : {
3557 1561 : GEN p = gel(lim,i);
3558 1561 : if (typ(p) != t_INT) p = pr_get_p(p);
3559 1561 : gel(P,i) = p;
3560 : }
3561 658 : P = ZV_sort_uniq_shallow(P);
3562 : }
3563 791 : if (rnf)
3564 : {
3565 742 : rnfeq = rnf_get_map(rnf);
3566 742 : zknf = rnf_get_nfzk(rnf);
3567 : }
3568 : else
3569 : {
3570 49 : rnfeq = nf_rnfeq(nf, pol);
3571 49 : zknf = nf_nfzk(nf, rnfeq);
3572 : }
3573 791 : dzknf = gel(zknf,1);
3574 791 : if (gequal1(dzknf)) dzknf = NULL;
3575 658 : RESTART:
3576 812 : if (extendP)
3577 : {
3578 21 : GEN oldP = P;
3579 21 : if (typ(extendP)==t_POL)
3580 : {
3581 0 : long l = lg(extendP);
3582 0 : for (i = 2; i < l; i++)
3583 : {
3584 0 : GEN q = gel(extendP,i);
3585 0 : if (typ(q) == t_FRAC) P = ZV_cba_extend(P, gel(q,2));
3586 : }
3587 : } else /*t_FRAC*/
3588 21 : P = ZV_cba_extend(P, gel(extendP,2));
3589 21 : if (ZV_equal(P, oldP))
3590 0 : pari_err(e_MISC, "rnfpseudobasis fails, try increasing B");
3591 21 : extendP = NULL;
3592 : }
3593 812 : Tabs = gel(rnfeq,1);
3594 812 : nfmaxord(&S, mkvec2(Tabs,P), 0);
3595 812 : B = RgXV_unscale(S.basis, S.unscale);
3596 812 : BdB = Q_remove_denom(B, &dB);
3597 812 : MB = RgXV_to_RgM(BdB, N); /* HNF */
3598 :
3599 812 : vU = cgetg(N+1, t_VEC);
3600 812 : vj = cgetg(N+1, t_VECSMALL);
3601 812 : gel(vU,1) = U = cgetg(m+1, t_MAT);
3602 812 : gel(U,1) = col_ei(N, 1);
3603 812 : A = dB? (dzknf? gdiv(dB,dzknf): dB): NULL;
3604 812 : if (A)
3605 : {
3606 763 : if (typ(A) != t_INT) { extendP = A; goto RESTART; }
3607 742 : if (equali1(A)) A = NULL;
3608 : }
3609 1617 : for (j = 2; j <= m; j++)
3610 : {
3611 826 : GEN t = gel(zknf,j);
3612 826 : if (!RgX_is_ZX(t)) { extendP = t; goto RESTART; }
3613 826 : if (A) t = ZX_Z_mul(t, A);
3614 826 : gel(U,j) = hnf_solve(MB, RgX_to_RgC(t, N));
3615 : }
3616 5425 : for (i = 2; i <= N; i++)
3617 : {
3618 4634 : GEN b = gel(BdB,i);
3619 4634 : gel(vU,i) = U = cgetg(m+1, t_MAT);
3620 4634 : gel(U,1) = hnf_solve(MB, RgX_to_RgC(b, N));
3621 10164 : for (j = 2; j <= m; j++)
3622 : {
3623 5530 : GEN t = ZX_rem(ZX_mul(b, gel(zknf,j)), Tabs);
3624 5530 : if (dzknf)
3625 : {
3626 4914 : t = RgX_Rg_div(t, dzknf);
3627 4914 : if (!RgX_is_ZX(t)) { extendP = t; goto RESTART; }
3628 : }
3629 5530 : gel(U,j) = hnf_solve(MB, RgX_to_RgC(t, N));
3630 : }
3631 : }
3632 791 : vj[1] = 1; U = gel(vU,1); rU = m;
3633 1932 : for (i = j = 2; i <= N; i++)
3634 : {
3635 1925 : GEN V = shallowconcat(U, gel(vU,i));
3636 1925 : if (ZM_rank(V) != rU)
3637 : {
3638 1925 : U = V; rU += m; vj[j++] = i;
3639 1925 : if (rU == N) break;
3640 : }
3641 : }
3642 791 : if (dB) for(;;)
3643 1092 : {
3644 1834 : GEN c = gen_1, H = ZM_hnfmodid(U, dB);
3645 1834 : long ic = 0;
3646 17717 : for (i = 1; i <= N; i++)
3647 15883 : if (cmpii(gcoeff(H,i,i), c) > 0) { c = gcoeff(H,i,i); ic = i; }
3648 1834 : if (!ic) break;
3649 1092 : vj[j++] = ic;
3650 1092 : U = shallowconcat(H, gel(vU, ic));
3651 : }
3652 791 : setlg(vj, j);
3653 791 : B = vecpermute(B, vj);
3654 :
3655 791 : l = lg(B);
3656 791 : A = cgetg(l,t_MAT);
3657 4599 : for (j = 1; j < l; j++)
3658 : {
3659 3808 : GEN t = eltabstorel_lift(rnfeq, gel(B,j));
3660 3808 : gel(A,j) = Rg_to_RgC(t, n);
3661 : }
3662 791 : A = RgM_to_nfM(nf, A);
3663 791 : A = Q_remove_denom(A, &dA);
3664 791 : if (!dA)
3665 : { /* order is maximal */
3666 63 : z = triv_order(n);
3667 63 : if (pf) *pf = gen_1;
3668 : }
3669 : else
3670 : {
3671 : GEN fi;
3672 : /* the first n columns of A are probably in HNF already */
3673 728 : A = shallowconcat(vecslice(A,n+1,lg(A)-1), vecslice(A,1,n));
3674 728 : A = mkvec2(A, const_vec(l-1,gen_1));
3675 728 : if (DEBUGLEVEL > 2) err_printf("rnfallbase: nfhnf in dim %ld\n", l-1);
3676 728 : z = nfhnfmod(nf, A, nfdetint(nf,A));
3677 728 : gel(z,2) = gdiv(gel(z,2), dA);
3678 728 : fi = idealprod(nf,gel(z,2));
3679 728 : D = idealmul(nf, D, idealsqr(nf, fi));
3680 728 : if (pf) *pf = idealinv(nf, fi);
3681 : }
3682 791 : if (RgM_isscalar(D,NULL)) D = gcoeff(D,1,1);
3683 791 : if (pDKP) *pDKP = S.dKP;
3684 791 : *pD = mkvec2(D, get_d(nf, disc)); return z;
3685 : }
3686 2121 : fa = idealfactor(nf, disc);
3687 2121 : P = gel(fa,1); l = lg(P); z = NULL;
3688 2121 : E = gel(fa,2);
3689 2121 : Pf = cgetg(l, t_COL);
3690 2121 : Ef = cgetg(l, t_COL);
3691 5970 : for (i = j = jf = 1; i < l; i++)
3692 : {
3693 3849 : GEN pr = gel(P,i);
3694 3849 : long e = itos(gel(E,i));
3695 3849 : if (e > 1)
3696 : {
3697 2856 : GEN vD = rnfmaxord(nf, pol, pr, e);
3698 2856 : if (vD)
3699 : {
3700 1329 : long ef = idealprodval(nf, gel(vD,2), pr);
3701 1329 : z = rnfjoinmodules(nf, z, vD);
3702 1329 : if (ef) { gel(Pf, jf) = pr; gel(Ef, jf++) = stoi(-ef); }
3703 1329 : e += 2 * ef;
3704 : }
3705 : }
3706 3849 : if (e) { gel(P, j) = pr; gel(E, j++) = stoi(e); }
3707 : }
3708 2121 : setlg(P,j);
3709 2121 : setlg(E,j);
3710 2121 : if (pDKP) *pDKP = prV_primes(P);
3711 2121 : if (pf)
3712 : {
3713 2065 : setlg(Pf, jf);
3714 2065 : setlg(Ef, jf); *pf = pr_factorback_scal(nf, mkmat2(Pf,Ef));
3715 : }
3716 2121 : *pD = mkvec2(pr_factorback_scal(nf,fa), get_d(nf, disc));
3717 2121 : return z? z: triv_order(degpol(pol));
3718 : }
3719 :
3720 : static GEN
3721 1666 : RgX_to_algX(GEN nf, GEN x)
3722 10143 : { pari_APPLY_pol_normalized(nf_to_scalar_or_alg(nf, gel(x,i))); }
3723 :
3724 : GEN
3725 1680 : nfX_to_monic(GEN nf, GEN T, GEN *pL)
3726 : {
3727 : GEN lT, g, a;
3728 1680 : long i, l = lg(T);
3729 1680 : if (l == 2) return pol_0(varn(T));
3730 1680 : if (l == 3) return pol_1(varn(T));
3731 1680 : nf = checknf(nf);
3732 1680 : T = Q_primpart(RgX_to_nfX(nf, T));
3733 1680 : lT = leading_coeff(T); if (pL) *pL = lT;
3734 1680 : if (isint1(T)) return T;
3735 1680 : g = cgetg_copy(T, &l); g[1] = T[1]; a = lT;
3736 1680 : gel(g, l-1) = gen_1;
3737 1680 : gel(g, l-2) = gel(T,l-2);
3738 1680 : if (l == 4) { gel(g,l-2) = nf_to_scalar_or_alg(nf, gel(g,l-2)); return g; }
3739 1666 : if (typ(lT) == t_INT)
3740 : {
3741 1652 : gel(g, l-3) = gmul(a, gel(T,l-3));
3742 5124 : for (i = l-4; i > 1; i--) { a = mulii(a,lT); gel(g,i) = gmul(a, gel(T,i)); }
3743 : }
3744 : else
3745 : {
3746 14 : gel(g, l-3) = nfmul(nf, a, gel(T,l-3));
3747 35 : for (i = l-3; i > 1; i--)
3748 : {
3749 21 : a = nfmul(nf,a,lT);
3750 21 : gel(g,i) = nfmul(nf, a, gel(T,i));
3751 : }
3752 : }
3753 1666 : return RgX_to_algX(nf, g);
3754 : }
3755 :
3756 : GEN
3757 868 : rnfdisc_factored(GEN nf, GEN pol, GEN *pd)
3758 : {
3759 : long i, j, l;
3760 : GEN fa, E, P, disc, lim;
3761 :
3762 868 : pol = rnfdisc_get_T(nf, pol, &lim);
3763 868 : disc = nf_to_scalar_or_basis(nf, nfX_disc(nf, pol));
3764 868 : if (gequal0(disc))
3765 0 : pari_err_DOMAIN("rnfdisc","issquarefree(pol)","=",gen_0, pol);
3766 868 : pol = nfX_to_monic(nf, pol, NULL);
3767 868 : fa = idealfactor_partial(nf, disc, lim);
3768 868 : P = gel(fa,1); l = lg(P);
3769 868 : E = gel(fa,2);
3770 2352 : for (i = j = 1; i < l; i++)
3771 : {
3772 1484 : long e = itos(gel(E,i));
3773 1484 : GEN pr = gel(P,i);
3774 1484 : if (e > 1)
3775 : {
3776 1211 : GEN vD = rnfmaxord(nf, pol, pr, e);
3777 1211 : if (vD) e += 2*idealprodval(nf, gel(vD,2), pr);
3778 : }
3779 1484 : if (e) { gel(P, j) = pr; gel(E, j++) = stoi(e); }
3780 : }
3781 868 : if (pd) *pd = get_d(nf, disc);
3782 868 : setlg(P, j);
3783 868 : setlg(E, j); return fa;
3784 : }
3785 : GEN
3786 77 : rnfdiscf(GEN nf, GEN pol)
3787 : {
3788 77 : pari_sp av = avma;
3789 : GEN d, fa;
3790 77 : nf = checknf(nf); fa = rnfdisc_factored(nf, pol, &d);
3791 77 : return gc_GEN(av, mkvec2(pr_factorback_scal(nf,fa), d));
3792 : }
3793 :
3794 : GEN
3795 35 : gen_if_principal(GEN bnf, GEN x)
3796 : {
3797 35 : pari_sp av = avma;
3798 35 : GEN z = bnfisprincipal0(bnf,x, nf_GEN_IF_PRINCIPAL | nf_FORCE);
3799 35 : return isintzero(z)? gc_NULL(av): z;
3800 : }
3801 :
3802 : /* given bnf and a HNF pseudo-basis of a proj. module, simplify the HNF as
3803 : * much as possible. The resulting matrix will be upper triangular but the
3804 : * diagonal coefficients will not be equal to 1. The ideals are integral and
3805 : * primitive. */
3806 : GEN
3807 0 : rnfsimplifybasis(GEN bnf, GEN M)
3808 : {
3809 0 : pari_sp av = avma;
3810 : long i, l;
3811 : GEN y, Az, Iz, nf, A, I;
3812 :
3813 0 : bnf = checkbnf(bnf); nf = bnf_get_nf(bnf);
3814 0 : if (!check_ZKmodule_i(M)) pari_err_TYPE("rnfsimplifybasis",M);
3815 0 : A = gel(M,1);
3816 0 : I = gel(M,2); l = lg(I);
3817 0 : Az = cgetg(l, t_MAT);
3818 0 : Iz = cgetg(l, t_VEC); y = mkvec2(Az, Iz);
3819 0 : for (i = 1; i < l; i++)
3820 : {
3821 : GEN c, d;
3822 0 : if (ideal_is1(gel(I,i)))
3823 : {
3824 0 : gel(Iz,i) = gen_1;
3825 0 : gel(Az,i) = gel(A,i); continue;
3826 : }
3827 :
3828 0 : gel(Iz,i) = Q_primitive_part(gel(I,i), &c);
3829 0 : gel(Az,i) = c? RgC_Rg_mul(gel(A,i),c): gel(A,i);
3830 0 : if (c && ideal_is1(gel(Iz,i))) continue;
3831 :
3832 0 : d = gen_if_principal(bnf, gel(Iz,i));
3833 0 : if (d)
3834 : {
3835 0 : gel(Iz,i) = gen_1;
3836 0 : gel(Az,i) = nfC_nf_mul(nf, gel(Az,i), d);
3837 : }
3838 : }
3839 0 : return gc_GEN(av, y);
3840 : }
3841 :
3842 : static GEN
3843 63 : get_module(GEN nf, GEN O, const char *s)
3844 : {
3845 63 : if (typ(O) == t_POL) return rnfpseudobasis(nf, O);
3846 56 : if (!check_ZKmodule_i(O)) pari_err_TYPE(s, O);
3847 56 : return shallowcopy(O);
3848 : }
3849 :
3850 : GEN
3851 14 : rnfdet(GEN nf, GEN M)
3852 : {
3853 14 : pari_sp av = avma;
3854 : GEN D;
3855 14 : nf = checknf(nf);
3856 14 : M = get_module(nf, M, "rnfdet");
3857 14 : D = idealmul(nf, nfM_det(nf, gel(M,1)), idealprod(nf, gel(M,2)));
3858 14 : return gc_upto(av, D);
3859 : }
3860 :
3861 : /* Given two fractional ideals a and b, gives x in a, y in b, z in b^-1,
3862 : t in a^-1 such that xt-yz=1. In the present version, z is in Z. */
3863 : static void
3864 63 : nfidealdet1(GEN nf, GEN a, GEN b, GEN *px, GEN *py, GEN *pz, GEN *pt)
3865 : {
3866 : GEN x, uv, y, da, db;
3867 :
3868 63 : a = idealinv(nf,a);
3869 63 : a = Q_remove_denom(a, &da);
3870 63 : b = Q_remove_denom(b, &db);
3871 63 : x = idealcoprime(nf,a,b);
3872 63 : uv = idealaddtoone(nf, idealmul(nf,x,a), b);
3873 63 : y = gel(uv,2);
3874 63 : if (da) x = gmul(x,da);
3875 63 : if (db) y = gdiv(y,db);
3876 63 : *px = x;
3877 63 : *py = y;
3878 63 : *pz = db ? negi(db): gen_m1;
3879 63 : *pt = nfdiv(nf, gel(uv,1), x);
3880 63 : }
3881 :
3882 : /* given a pseudo-basis of a proj. module in HNF [A,I] (or [A,I,D,d]), gives
3883 : * an n x n matrix (not HNF) of a pseudo-basis and an ideal vector
3884 : * [1,...,1,I] such that M ~ Z_K^(n-1) x I. Uses the approximation theorem.*/
3885 : GEN
3886 28 : rnfsteinitz(GEN nf, GEN M)
3887 : {
3888 28 : pari_sp av = avma;
3889 : long i, n;
3890 : GEN A, I;
3891 :
3892 28 : nf = checknf(nf);
3893 28 : M = get_module(nf, M, "rnfsteinitz");
3894 28 : A = RgM_to_nfM(nf, gel(M,1));
3895 28 : I = leafcopy(gel(M,2)); n = lg(A)-1;
3896 189 : for (i = 1; i < n; i++)
3897 : {
3898 161 : GEN c1, c2, b, a = gel(I,i);
3899 161 : gel(I,i) = gen_1;
3900 161 : if (ideal_is1(a)) continue;
3901 :
3902 63 : c1 = gel(A,i);
3903 63 : c2 = gel(A,i+1);
3904 63 : b = gel(I,i+1);
3905 63 : if (ideal_is1(b))
3906 : {
3907 0 : gel(A,i) = c2;
3908 0 : gel(A,i+1) = gneg(c1);
3909 0 : gel(I,i+1) = a;
3910 : }
3911 : else
3912 : {
3913 63 : pari_sp av2 = avma;
3914 : GEN x, y, z, t, c;
3915 63 : nfidealdet1(nf,a,b, &x,&y,&z,&t);
3916 63 : x = RgC_add(nfC_nf_mul(nf, c1, x), nfC_nf_mul(nf, c2, y));
3917 63 : y = RgC_add(nfC_nf_mul(nf, c1, z), nfC_nf_mul(nf, c2, t));
3918 63 : (void)gc_all(av2, 2, &x,&y);
3919 63 : gel(A,i) = x;
3920 63 : gel(A,i+1) = y;
3921 63 : gel(I,i+1) = Q_primitive_part(idealmul(nf,a,b), &c);
3922 63 : if (c) gel(A,i+1) = nfC_nf_mul(nf, gel(A,i+1), c);
3923 : }
3924 : }
3925 28 : gel(M,1) = A;
3926 28 : gel(M,2) = I; return gc_GEN(av, M);
3927 : }
3928 :
3929 : /* Given bnf and a proj. module (or a t_POL -> rnfpseudobasis), and outputs a
3930 : * basis if it is free, an n+1-generating set if it is not */
3931 : GEN
3932 21 : rnfbasis(GEN bnf, GEN M)
3933 : {
3934 21 : pari_sp av = avma;
3935 : long j, n;
3936 : GEN nf, A, I, cl, col, a;
3937 :
3938 21 : bnf = checkbnf(bnf); nf = bnf_get_nf(bnf);
3939 21 : M = get_module(nf, M, "rnfbasis");
3940 21 : I = gel(M,2); n = lg(I)-1;
3941 98 : j = 1; while (j < n && ideal_is1(gel(I,j))) j++;
3942 21 : if (j < n) { M = rnfsteinitz(nf,M); I = gel(M,2); }
3943 21 : A = gel(M,1);
3944 21 : col= gel(A,n); A = vecslice(A, 1, n-1);
3945 21 : cl = gel(I,n);
3946 21 : a = gen_if_principal(bnf, cl);
3947 21 : if (!a)
3948 : {
3949 7 : GEN v = idealtwoelt(nf, cl);
3950 7 : A = vec_append(A, gmul(gel(v,1), col));
3951 7 : a = gel(v,2);
3952 : }
3953 21 : A = vec_append(A, nfC_nf_mul(nf, col, a));
3954 21 : return gc_GEN(av, A);
3955 : }
3956 :
3957 : /* Given a Z_K-module M (or a polynomial => rnfpseudobasis) outputs a
3958 : * Z_K-basis in HNF if it exists, zero if not */
3959 : GEN
3960 7 : rnfhnfbasis(GEN bnf, GEN M)
3961 : {
3962 7 : pari_sp av = avma;
3963 : long j, l;
3964 : GEN nf, A, I, a;
3965 :
3966 7 : bnf = checkbnf(bnf); nf = bnf_get_nf(bnf);
3967 7 : if (typ(M) == t_POL) M = rnfpseudobasis(nf, M);
3968 : else
3969 : {
3970 7 : if (typ(M) != t_VEC) pari_err_TYPE("rnfhnfbasis", M);
3971 7 : if (lg(M) == 5) M = mkvec2(gel(M,1), gel(M,2));
3972 7 : M = nfhnf(nf, M); /* in case M is not in HNF */
3973 : }
3974 7 : A = shallowcopy(gel(M,1));
3975 7 : I = gel(M,2); l = lg(A);
3976 42 : for (j = 1; j < l; j++)
3977 : {
3978 35 : if (ideal_is1(gel(I,j))) continue;
3979 14 : a = gen_if_principal(bnf, gel(I,j));
3980 14 : if (!a) return gc_const(av, gen_0);
3981 14 : gel(A,j) = nfC_nf_mul(nf, gel(A,j), a);
3982 : }
3983 7 : return gc_GEN(av,A);
3984 : }
3985 :
3986 : long
3987 7 : rnfisfree(GEN bnf, GEN M)
3988 : {
3989 7 : pari_sp av = avma;
3990 : GEN nf, P, I;
3991 : long l, j;
3992 :
3993 7 : bnf = checkbnf(bnf);
3994 7 : if (is_pm1( bnf_get_no(bnf) )) return 1;
3995 0 : nf = bnf_get_nf(bnf);
3996 0 : M = get_module(nf, M, "rnfisfree");
3997 0 : I = gel(M,2); l = lg(I); P = NULL;
3998 0 : for (j = 1; j < l; j++)
3999 0 : if (!ideal_is1(gel(I,j))) P = P? idealmul(nf, P, gel(I,j)): gel(I,j);
4000 0 : return gc_long(av, P? gequal0( isprincipal(bnf,P) ): 1);
4001 : }
4002 :
4003 : /**********************************************************************/
4004 : /** **/
4005 : /** COMPOSITUM OF TWO NUMBER FIELDS **/
4006 : /** **/
4007 : /**********************************************************************/
4008 : static GEN
4009 26561 : compositum_fix(GEN nf, GEN A)
4010 : {
4011 : int ok;
4012 26561 : if (nf)
4013 : {
4014 980 : A = RgXQX_red(A, nf_get_pol(nf));
4015 980 : A = Q_primpart(liftpol_shallow(A)); RgX_check_ZXX(A,"polcompositum");
4016 980 : ok = nfissquarefree(nf,A);
4017 : }
4018 : else
4019 : {
4020 25581 : A = Q_primpart(A); RgX_check_ZX(A,"polcompositum");
4021 25581 : ok = ZX_is_squarefree(A);
4022 : }
4023 26562 : if (!ok) pari_err_DOMAIN("polcompositum","issquarefree(arg)","=",gen_0,A);
4024 26555 : return A;
4025 : }
4026 : #define next_lambda(a) (a>0 ? -a : 1-a)
4027 :
4028 : static long
4029 511 : nfcompositum_lambda(GEN nf, GEN A, GEN B, long lambda)
4030 : {
4031 511 : pari_sp av = avma;
4032 : forprime_t S;
4033 511 : GEN T = nf_get_pol(nf);
4034 511 : long vT = varn(T);
4035 : ulong p;
4036 511 : init_modular_big(&S);
4037 511 : p = u_forprime_next(&S);
4038 : while (1)
4039 42 : {
4040 : GEN Hp, Tp, a;
4041 553 : if (DEBUGLEVEL>4) err_printf("Trying lambda = %ld\n", lambda);
4042 553 : a = ZXX_to_FlxX(RgX_rescale(A, stoi(-lambda)), p, vT);
4043 553 : Tp = ZX_to_Flx(T, p);
4044 553 : Hp = FlxqX_composedsum(a, ZXX_to_FlxX(B, p, vT), Tp, p);
4045 553 : if (!FlxqX_is_squarefree(Hp, Tp, p))
4046 42 : { lambda = next_lambda(lambda); continue; }
4047 511 : if (DEBUGLEVEL>4) err_printf("Final lambda = %ld\n", lambda);
4048 511 : return gc_long(av, lambda);
4049 : }
4050 : }
4051 :
4052 : /* modular version */
4053 : GEN
4054 13393 : nfcompositum(GEN nf, GEN A, GEN B, long flag)
4055 : {
4056 13393 : pari_sp av = avma;
4057 : int same;
4058 : long v, k;
4059 : GEN C, D, LPRS;
4060 :
4061 13393 : if (typ(A)!=t_POL) pari_err_TYPE("polcompositum",A);
4062 13393 : if (typ(B)!=t_POL) pari_err_TYPE("polcompositum",B);
4063 13393 : if (degpol(A)<=0 || degpol(B)<=0) pari_err_CONSTPOL("polcompositum");
4064 13393 : v = varn(A);
4065 13393 : if (varn(B) != v) pari_err_VAR("polcompositum", A,B);
4066 13393 : if (nf)
4067 : {
4068 553 : nf = checknf(nf);
4069 546 : if (varncmp(v,nf_get_varn(nf))>=0) pari_err_PRIORITY("polcompositum", nf, ">=", v);
4070 : }
4071 13351 : same = (A == B || RgX_equal(A,B));
4072 13351 : A = compositum_fix(nf,A);
4073 13347 : B = same ? A: compositum_fix(nf,B);
4074 :
4075 13345 : D = LPRS = NULL; /* -Wall */
4076 13345 : k = same? -1: 1;
4077 13345 : if (nf)
4078 : {
4079 511 : long v0 = fetch_var();
4080 511 : GEN q, T = nf_get_pol(nf);
4081 511 : A = liftpol_shallow(A);
4082 511 : B = liftpol_shallow(B);
4083 511 : k = nfcompositum_lambda(nf, A, B, k);
4084 511 : if (flag&1)
4085 : {
4086 : GEN H0, H1;
4087 196 : GEN chgvar = deg1pol_shallow(stoi(k),pol_x(v0),v);
4088 196 : GEN B1 = poleval(QXQX_to_mod_shallow(B, T), chgvar);
4089 196 : C = RgX_resultant_all(QXQX_to_mod_shallow(A, T), B1, &q);
4090 196 : C = gsubst(C,v0,pol_x(v));
4091 196 : C = lift_if_rational(C);
4092 196 : H0 = gsubst(gel(q,2),v0,pol_x(v));
4093 196 : H1 = gsubst(gel(q,3),v0,pol_x(v));
4094 196 : if (typ(H0) != t_POL) H0 = scalarpol_shallow(H0,v);
4095 196 : if (typ(H1) != t_POL) H1 = scalarpol_shallow(H1,v);
4096 196 : H0 = lift_if_rational(H0);
4097 196 : H1 = lift_if_rational(H1);
4098 196 : LPRS = mkvec2(H0,H1);
4099 : }
4100 : else
4101 : {
4102 315 : C = nf_direct_compositum(nf, RgX_rescale(A,stoi(-k)), B);
4103 315 : setvarn(C, v); C = QXQX_to_mod_shallow(C, T);
4104 : }
4105 511 : C = RgX_normalize(C);
4106 : }
4107 : else
4108 : {
4109 12834 : B = leafcopy(B); setvarn(B,fetch_var_higher());
4110 3136 : C = (flag&1)? ZX_ZXY_resultant_all(A, B, &k, &LPRS)
4111 12834 : : ZX_compositum(A, B, &k);
4112 12832 : setvarn(C, v);
4113 : }
4114 : /* C = Res_Y (A(Y), B(X + kY)) guaranteed squarefree */
4115 13343 : if (flag & 2)
4116 10242 : C = mkvec(C);
4117 : else
4118 : {
4119 3101 : if (same)
4120 : {
4121 105 : D = RgX_rescale(A, stoi(1 - k));
4122 105 : if (nf) D = RgX_normalize(QXQX_to_mod_shallow(D, nf_get_pol(nf)));
4123 105 : C = RgX_div(C, D);
4124 105 : if (degpol(C) <= 0)
4125 0 : C = mkvec(D);
4126 : else
4127 105 : C = shallowconcat(nf? gel(nffactor(nf,C),1): ZX_DDF(C), D);
4128 : }
4129 : else
4130 2996 : C = nf? gel(nffactor(nf,C),1): ZX_DDF(C);
4131 : }
4132 13343 : gen_sort_inplace(C, (void*)(nf?&cmp_RgX: &cmpii), &gen_cmp_RgX, NULL);
4133 13343 : if (flag&1)
4134 : { /* a,b,c root of A,B,C = compositum, c = b - k a */
4135 3332 : long i, l = lg(C);
4136 3332 : GEN a, b, mH0 = RgX_neg(gel(LPRS,1)), H1 = gel(LPRS,2);
4137 3332 : setvarn(mH0,v);
4138 3332 : setvarn(H1,v);
4139 6741 : for (i=1; i<l; i++)
4140 : {
4141 3409 : GEN D = gel(C,i);
4142 3409 : a = RgXQ_mul(mH0, nf? RgXQ_inv(H1,D): QXQ_inv(H1,D), D);
4143 3409 : b = gadd(pol_x(v), gmulsg(k,a));
4144 3409 : if (degpol(D) == 1) b = RgX_rem(b,D);
4145 3409 : gel(C,i) = mkvec4(D, mkpolmod(a,D), mkpolmod(b,D), stoi(-k));
4146 : }
4147 : }
4148 13343 : (void)delete_var();
4149 13343 : settyp(C, t_VEC);
4150 13343 : if (flag&2) C = gel(C,1);
4151 13343 : return gc_GEN(av, C);
4152 : }
4153 : GEN
4154 12840 : polcompositum0(GEN A, GEN B, long flag)
4155 12840 : { return nfcompositum(NULL,A,B,flag); }
4156 :
4157 : GEN
4158 91 : compositum(GEN pol1,GEN pol2) { return polcompositum0(pol1,pol2,0); }
4159 : GEN
4160 2877 : compositum2(GEN pol1,GEN pol2) { return polcompositum0(pol1,pol2,1); }
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