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
11 Check the License for details. You should have received a copy of it, along
12 with the package; see the file 'COPYING'. If not, write to the Free Software
13 Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. */
14
15 /*******************************************************************/
16 /* */
17 /* BASIC NF OPERATIONS */
18 /* (continued) */
19 /* */
20 /*******************************************************************/
21 #include "pari.h"
22 #include "paripriv.h"
23
24 /*******************************************************************/
25 /* */
26 /* IDEAL OPERATIONS */
27 /* */
28 /*******************************************************************/
29
30 /* A valid ideal is either principal (valid nf_element), or prime, or a matrix
31 * on the integer basis in HNF.
32 * A prime ideal is of the form [p,a,e,f,b], where the ideal is p.Z_K+a.Z_K,
33 * p is a rational prime, a belongs to Z_K, e=e(P/p), f=f(P/p), and b
34 * is Lenstra's constant, such that p.P^(-1)= p Z_K + b Z_K.
35 *
36 * An extended ideal is a couple [I,F] where I is an ideal and F is either an
37 * algebraic number, or a factorization matrix attached to an algebraic number.
38 * All routines work with either extended ideals or ideals (an omitted F is
39 * assumed to be factor(1)). All ideals are output in HNF form. */
40
41 /* types and conversions */
42
43 long
idealtyp(GEN * ideal,GEN * arch)44 idealtyp(GEN *ideal, GEN *arch)
45 {
46 GEN x = *ideal;
47 long t,lx,tx = typ(x);
48
49 if (tx!=t_VEC || lg(x)!=3) *arch = NULL;
50 else
51 {
52 GEN a = gel(x,2);
53 if (typ(a) == t_MAT && lg(a) != 3)
54 { /* allow [;] */
55 if (lg(a) != 1) pari_err_TYPE("idealtyp [extended ideal]",x);
56 a = trivial_fact();
57 }
58 *arch = a;
59 x = gel(x,1); tx = typ(x);
60 }
61 switch(tx)
62 {
63 case t_MAT: lx = lg(x);
64 if (lx == 1) { t = id_PRINCIPAL; x = gen_0; break; }
65 if (lx != lgcols(x)) pari_err_TYPE("idealtyp [nonsquare t_MAT]",x);
66 t = id_MAT;
67 break;
68
69 case t_VEC: if (lg(x)!=6) pari_err_TYPE("idealtyp",x);
70 t = id_PRIME; break;
71
72 case t_POL: case t_POLMOD: case t_COL:
73 case t_INT: case t_FRAC:
74 t = id_PRINCIPAL; break;
75 default:
76 pari_err_TYPE("idealtyp",x);
77 return 0; /*LCOV_EXCL_LINE*/
78 }
79 *ideal = x; return t;
80 }
81
82 /* true nf; v = [a,x,...], a in Z. Return (a,x) */
83 GEN
idealhnf_two(GEN nf,GEN v)84 idealhnf_two(GEN nf, GEN v)
85 {
86 GEN p = gel(v,1), pi = gel(v,2), m = zk_scalar_or_multable(nf, pi);
87 if (typ(m) == t_INT) return scalarmat(gcdii(m,p), nf_get_degree(nf));
88 return ZM_hnfmodid(m, p);
89 }
90 /* true nf */
91 GEN
pr_hnf(GEN nf,GEN pr)92 pr_hnf(GEN nf, GEN pr)
93 {
94 GEN p = pr_get_p(pr), m;
95 if (pr_is_inert(pr)) return scalarmat(p, nf_get_degree(nf));
96 m = zk_scalar_or_multable(nf, pr_get_gen(pr));
97 return ZM_hnfmodprime(m, p);
98 }
99
100 GEN
idealhnf_principal(GEN nf,GEN x)101 idealhnf_principal(GEN nf, GEN x)
102 {
103 GEN cx;
104 x = nf_to_scalar_or_basis(nf, x);
105 switch(typ(x))
106 {
107 case t_COL: break;
108 case t_INT: if (!signe(x)) return cgetg(1,t_MAT);
109 return scalarmat(absi_shallow(x), nf_get_degree(nf));
110 case t_FRAC:
111 return scalarmat(Q_abs_shallow(x), nf_get_degree(nf));
112 default: pari_err_TYPE("idealhnf",x);
113 }
114 x = Q_primitive_part(x, &cx);
115 RgV_check_ZV(x, "idealhnf");
116 x = zk_multable(nf, x);
117 x = ZM_hnfmodid(x, zkmultable_capZ(x));
118 return cx? ZM_Q_mul(x,cx): x;
119 }
120
121 /* x integral ideal in t_MAT form, nx columns */
122 static GEN
vec_mulid(GEN nf,GEN x,long nx,long N)123 vec_mulid(GEN nf, GEN x, long nx, long N)
124 {
125 GEN m = cgetg(nx*N + 1, t_MAT);
126 long i, j, k;
127 for (i=k=1; i<=nx; i++)
128 for (j=1; j<=N; j++) gel(m, k++) = zk_ei_mul(nf, gel(x,i),j);
129 return m;
130 }
131 /* true nf */
132 GEN
idealhnf_shallow(GEN nf,GEN x)133 idealhnf_shallow(GEN nf, GEN x)
134 {
135 long tx = typ(x), lx = lg(x), N;
136
137 /* cannot use idealtyp because here we allow nonsquare matrices */
138 if (tx == t_VEC && lx == 3) { x = gel(x,1); tx = typ(x); lx = lg(x); }
139 if (tx == t_VEC && lx == 6) return pr_hnf(nf,x); /* PRIME */
140 switch(tx)
141 {
142 case t_MAT:
143 {
144 GEN cx;
145 long nx = lx-1;
146 N = nf_get_degree(nf);
147 if (nx == 0) return cgetg(1, t_MAT);
148 if (nbrows(x) != N) pari_err_TYPE("idealhnf [wrong dimension]",x);
149 if (nx == 1) return idealhnf_principal(nf, gel(x,1));
150
151 if (nx == N && RgM_is_ZM(x) && ZM_ishnf(x)) return x;
152 x = Q_primitive_part(x, &cx);
153 if (nx < N) x = vec_mulid(nf, x, nx, N);
154 x = ZM_hnfmod(x, ZM_detmult(x));
155 return cx? ZM_Q_mul(x,cx): x;
156 }
157 case t_QFI:
158 case t_QFR:
159 {
160 pari_sp av = avma;
161 GEN u, D = nf_get_disc(nf), T = nf_get_pol(nf), f = nf_get_index(nf);
162 GEN A = gel(x,1), B = gel(x,2);
163 N = nf_get_degree(nf);
164 if (N != 2)
165 pari_err_TYPE("idealhnf [Qfb for nonquadratic fields]", x);
166 if (!equalii(qfb_disc(x), D))
167 pari_err_DOMAIN("idealhnf [Qfb]", "disc(q)", "!=", D, x);
168 /* x -> A Z + (-B + sqrt(D)) / 2 Z
169 K = Q[t]/T(t), t^2 + ut + v = 0, u^2 - 4v = Df^2
170 => t = (-u + sqrt(D) f)/2
171 => sqrt(D)/2 = (t + u/2)/f */
172 u = gel(T,3);
173 B = deg1pol_shallow(ginv(f),
174 gsub(gdiv(u, shifti(f,1)), gdiv(B,gen_2)),
175 varn(T));
176 return gerepileupto(av, idealhnf_two(nf, mkvec2(A,B)));
177 }
178 default: return idealhnf_principal(nf, x); /* PRINCIPAL */
179 }
180 }
181 GEN
idealhnf(GEN nf,GEN x)182 idealhnf(GEN nf, GEN x)
183 {
184 pari_sp av = avma;
185 GEN y = idealhnf_shallow(checknf(nf), x);
186 return (avma == av)? gcopy(y): gerepileupto(av, y);
187 }
188
189 /* GP functions */
190
191 GEN
idealtwoelt0(GEN nf,GEN x,GEN a)192 idealtwoelt0(GEN nf, GEN x, GEN a)
193 {
194 if (!a) return idealtwoelt(nf,x);
195 return idealtwoelt2(nf,x,a);
196 }
197
198 GEN
idealpow0(GEN nf,GEN x,GEN n,long flag)199 idealpow0(GEN nf, GEN x, GEN n, long flag)
200 {
201 if (flag) return idealpowred(nf,x,n);
202 return idealpow(nf,x,n);
203 }
204
205 GEN
idealmul0(GEN nf,GEN x,GEN y,long flag)206 idealmul0(GEN nf, GEN x, GEN y, long flag)
207 {
208 if (flag) return idealmulred(nf,x,y);
209 return idealmul(nf,x,y);
210 }
211
212 GEN
idealdiv0(GEN nf,GEN x,GEN y,long flag)213 idealdiv0(GEN nf, GEN x, GEN y, long flag)
214 {
215 switch(flag)
216 {
217 case 0: return idealdiv(nf,x,y);
218 case 1: return idealdivexact(nf,x,y);
219 default: pari_err_FLAG("idealdiv");
220 }
221 return NULL; /* LCOV_EXCL_LINE */
222 }
223
224 GEN
idealaddtoone0(GEN nf,GEN arg1,GEN arg2)225 idealaddtoone0(GEN nf, GEN arg1, GEN arg2)
226 {
227 if (!arg2) return idealaddmultoone(nf,arg1);
228 return idealaddtoone(nf,arg1,arg2);
229 }
230
231 /* b not a scalar */
232 static GEN
hnf_Z_ZC(GEN nf,GEN a,GEN b)233 hnf_Z_ZC(GEN nf, GEN a, GEN b) { return hnfmodid(zk_multable(nf,b), a); }
234 /* b not a scalar */
235 static GEN
hnf_Z_QC(GEN nf,GEN a,GEN b)236 hnf_Z_QC(GEN nf, GEN a, GEN b)
237 {
238 GEN db;
239 b = Q_remove_denom(b, &db);
240 if (db) a = mulii(a, db);
241 b = hnf_Z_ZC(nf,a,b);
242 return db? RgM_Rg_div(b, db): b;
243 }
244 /* b not a scalar (not point in trying to optimize for this case) */
245 static GEN
hnf_Q_QC(GEN nf,GEN a,GEN b)246 hnf_Q_QC(GEN nf, GEN a, GEN b)
247 {
248 GEN da, db;
249 if (typ(a) == t_INT) return hnf_Z_QC(nf, a, b);
250 da = gel(a,2);
251 a = gel(a,1);
252 b = Q_remove_denom(b, &db);
253 /* write da = d*A, db = d*B, gcd(A,B) = 1
254 * gcd(a/(d A), b/(d B)) = gcd(a B, A b) / A B d = gcd(a B, b) / A B d */
255 if (db)
256 {
257 GEN d = gcdii(da,db);
258 if (!is_pm1(d)) db = diviiexact(db,d); /* B */
259 if (!is_pm1(db))
260 {
261 a = mulii(a, db); /* a B */
262 da = mulii(da, db); /* A B d = lcm(denom(a),denom(b)) */
263 }
264 }
265 return RgM_Rg_div(hnf_Z_ZC(nf,a,b), da);
266 }
267 static GEN
hnf_QC_QC(GEN nf,GEN a,GEN b)268 hnf_QC_QC(GEN nf, GEN a, GEN b)
269 {
270 GEN da, db, d, x;
271 a = Q_remove_denom(a, &da);
272 b = Q_remove_denom(b, &db);
273 if (da) b = ZC_Z_mul(b, da);
274 if (db) a = ZC_Z_mul(a, db);
275 d = mul_denom(da, db);
276 a = zk_multable(nf,a); da = zkmultable_capZ(a);
277 b = zk_multable(nf,b); db = zkmultable_capZ(b);
278 x = ZM_hnfmodid(shallowconcat(a,b), gcdii(da,db));
279 return d? RgM_Rg_div(x, d): x;
280 }
281 static GEN
hnf_Q_Q(GEN nf,GEN a,GEN b)282 hnf_Q_Q(GEN nf, GEN a, GEN b) {return scalarmat(Q_gcd(a,b), nf_get_degree(nf));}
283 GEN
idealhnf0(GEN nf,GEN a,GEN b)284 idealhnf0(GEN nf, GEN a, GEN b)
285 {
286 long ta, tb;
287 pari_sp av;
288 GEN x;
289 if (!b) return idealhnf(nf,a);
290
291 /* HNF of aZ_K+bZ_K */
292 av = avma; nf = checknf(nf);
293 a = nf_to_scalar_or_basis(nf,a); ta = typ(a);
294 b = nf_to_scalar_or_basis(nf,b); tb = typ(b);
295 if (ta == t_COL)
296 x = (tb==t_COL)? hnf_QC_QC(nf, a,b): hnf_Q_QC(nf, b,a);
297 else
298 x = (tb==t_COL)? hnf_Q_QC(nf, a,b): hnf_Q_Q(nf, a,b);
299 return gerepileupto(av, x);
300 }
301
302 /*******************************************************************/
303 /* */
304 /* TWO-ELEMENT FORM */
305 /* */
306 /*******************************************************************/
307 static GEN idealapprfact_i(GEN nf, GEN x, int nored);
308
309 static int
ok_elt(GEN x,GEN xZ,GEN y)310 ok_elt(GEN x, GEN xZ, GEN y)
311 {
312 pari_sp av = avma;
313 return gc_bool(av, ZM_equal(x, ZM_hnfmodid(y, xZ)));
314 }
315
316 static GEN
addmul_col(GEN a,long s,GEN b)317 addmul_col(GEN a, long s, GEN b)
318 {
319 long i,l;
320 if (!s) return a? leafcopy(a): a;
321 if (!a) return gmulsg(s,b);
322 l = lg(a);
323 for (i=1; i<l; i++)
324 if (signe(gel(b,i))) gel(a,i) = addii(gel(a,i), mulsi(s, gel(b,i)));
325 return a;
326 }
327
328 /* a <-- a + s * b, all coeffs integers */
329 static GEN
addmul_mat(GEN a,long s,GEN b)330 addmul_mat(GEN a, long s, GEN b)
331 {
332 long j,l;
333 /* copy otherwise next call corrupts a */
334 if (!s) return a? RgM_shallowcopy(a): a;
335 if (!a) return gmulsg(s,b);
336 l = lg(a);
337 for (j=1; j<l; j++)
338 (void)addmul_col(gel(a,j), s, gel(b,j));
339 return a;
340 }
341
342 static GEN
get_random_a(GEN nf,GEN x,GEN xZ)343 get_random_a(GEN nf, GEN x, GEN xZ)
344 {
345 pari_sp av;
346 long i, lm, l = lg(x);
347 GEN a, z, beta, mul;
348
349 beta= cgetg(l, t_VEC);
350 mul = cgetg(l, t_VEC); lm = 1; /* = lg(mul) */
351 /* look for a in x such that a O/xZ = x O/xZ */
352 for (i = 2; i < l; i++)
353 {
354 GEN xi = gel(x,i);
355 GEN t = FpM_red(zk_multable(nf,xi), xZ); /* ZM, cannot be a scalar */
356 if (gequal0(t)) continue;
357 if (ok_elt(x,xZ, t)) return xi;
358 gel(beta,lm) = xi;
359 /* mul[i] = { canonical generators for x[i] O/xZ as Z-module } */
360 gel(mul,lm) = t; lm++;
361 }
362 setlg(mul, lm);
363 setlg(beta,lm);
364 z = cgetg(lm, t_VECSMALL);
365 for(av = avma;; set_avma(av))
366 {
367 for (a=NULL,i=1; i<lm; i++)
368 {
369 long t = random_bits(4) - 7; /* in [-7,8] */
370 z[i] = t;
371 a = addmul_mat(a, t, gel(mul,i));
372 }
373 /* a = matrix (NOT HNF) of ideal generated by beta.z in O/xZ */
374 if (a && ok_elt(x,xZ, a)) break;
375 }
376 for (a=NULL,i=1; i<lm; i++)
377 a = addmul_col(a, z[i], gel(beta,i));
378 return a;
379 }
380
381 /* x square matrix, assume it is HNF */
382 static GEN
mat_ideal_two_elt(GEN nf,GEN x)383 mat_ideal_two_elt(GEN nf, GEN x)
384 {
385 GEN y, a, cx, xZ;
386 long N = nf_get_degree(nf);
387 pari_sp av, tetpil;
388
389 if (lg(x)-1 != N) pari_err_DIM("idealtwoelt");
390 if (N == 2) return mkvec2copy(gcoeff(x,1,1), gel(x,2));
391
392 y = cgetg(3,t_VEC); av = avma;
393 cx = Q_content(x);
394 xZ = gcoeff(x,1,1);
395 if (gequal(xZ, cx)) /* x = (cx) */
396 {
397 gel(y,1) = cx;
398 gel(y,2) = gen_0; return y;
399 }
400 if (equali1(cx)) cx = NULL;
401 else
402 {
403 x = Q_div_to_int(x, cx);
404 xZ = gcoeff(x,1,1);
405 }
406 if (N < 6)
407 a = get_random_a(nf, x, xZ);
408 else
409 {
410 const long FB[] = { _evallg(15+1) | evaltyp(t_VECSMALL),
411 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
412 };
413 GEN P, E, a1 = Z_smoothen(xZ, (GEN)FB, &P, &E);
414 if (!a1) /* factors completely */
415 a = idealapprfact_i(nf, idealfactor(nf,x), 1);
416 else if (lg(P) == 1) /* no small factors */
417 a = get_random_a(nf, x, xZ);
418 else /* general case */
419 {
420 GEN A0, A1, a0, u0, u1, v0, v1, pi0, pi1, t, u;
421 a0 = diviiexact(xZ, a1);
422 A0 = ZM_hnfmodid(x, a0); /* smooth part of x */
423 A1 = ZM_hnfmodid(x, a1); /* cofactor */
424 pi0 = idealapprfact_i(nf, idealfactor(nf,A0), 1);
425 pi1 = get_random_a(nf, A1, a1);
426 (void)bezout(a0, a1, &v0,&v1);
427 u0 = mulii(a0, v0);
428 u1 = mulii(a1, v1);
429 if (typ(pi0) != t_COL) t = addmulii(u0, pi0, u1);
430 else
431 { t = ZC_Z_mul(pi0, u1); gel(t,1) = addii(gel(t,1), u0); }
432 u = ZC_Z_mul(pi1, u0); gel(u,1) = addii(gel(u,1), u1);
433 a = nfmuli(nf, centermod(u, xZ), centermod(t, xZ));
434 }
435 }
436 if (cx)
437 {
438 a = centermod(a, xZ);
439 tetpil = avma;
440 if (typ(cx) == t_INT)
441 {
442 gel(y,1) = mulii(xZ, cx);
443 gel(y,2) = ZC_Z_mul(a, cx);
444 }
445 else
446 {
447 gel(y,1) = gmul(xZ, cx);
448 gel(y,2) = RgC_Rg_mul(a, cx);
449 }
450 }
451 else
452 {
453 tetpil = avma;
454 gel(y,1) = icopy(xZ);
455 gel(y,2) = centermod(a, xZ);
456 }
457 gerepilecoeffssp(av,tetpil,y+1,2); return y;
458 }
459
460 /* Given an ideal x, returns [a,alpha] such that a is in Q,
461 * x = a Z_K + alpha Z_K, alpha in K^*
462 * a = 0 or alpha = 0 are possible, but do not try to determine whether
463 * x is principal. */
464 GEN
idealtwoelt(GEN nf,GEN x)465 idealtwoelt(GEN nf, GEN x)
466 {
467 pari_sp av;
468 GEN z;
469 long tx = idealtyp(&x,&z);
470 nf = checknf(nf);
471 if (tx == id_MAT) return mat_ideal_two_elt(nf,x);
472 if (tx == id_PRIME) return mkvec2copy(gel(x,1), gel(x,2));
473 /* id_PRINCIPAL */
474 av = avma; x = nf_to_scalar_or_basis(nf, x);
475 return gerepilecopy(av, typ(x)==t_COL? mkvec2(gen_0,x):
476 mkvec2(Q_abs_shallow(x),gen_0));
477 }
478
479 /*******************************************************************/
480 /* */
481 /* FACTORIZATION */
482 /* */
483 /*******************************************************************/
484 /* x integral ideal in HNF, Zval = v_p(x \cap Z) > 0; return v_p(Nx) */
485 static long
idealHNF_norm_pval(GEN x,GEN p,long Zval)486 idealHNF_norm_pval(GEN x, GEN p, long Zval)
487 {
488 long i, v = Zval, l = lg(x);
489 for (i = 2; i < l; i++) v += Z_pval(gcoeff(x,i,i), p);
490 return v;
491 }
492
493 /* x integral in HNF, f0 = partial factorization of a multiple of
494 * x[1,1] = x\cap Z */
495 GEN
idealHNF_Z_factor_i(GEN x,GEN f0,GEN * pvN,GEN * pvZ)496 idealHNF_Z_factor_i(GEN x, GEN f0, GEN *pvN, GEN *pvZ)
497 {
498 GEN P, E, vN, vZ, xZ = gcoeff(x,1,1), f = f0? f0: Z_factor(xZ);
499 long i, l;
500 P = gel(f,1); l = lg(P);
501 E = gel(f,2);
502 *pvN = vN = cgetg(l, t_VECSMALL);
503 *pvZ = vZ = cgetg(l, t_VECSMALL);
504 for (i = 1; i < l; i++)
505 {
506 GEN p = gel(P,i);
507 vZ[i] = f0? Z_pval(xZ, p): (long) itou(gel(E,i));
508 vN[i] = idealHNF_norm_pval(x,p, vZ[i]);
509 }
510 return P;
511 }
512 /* return P, primes dividing Nx and xZ = x\cap Z, set v_p(Nx), v_p(xZ);
513 * x integral in HNF */
514 GEN
idealHNF_Z_factor(GEN x,GEN * pvN,GEN * pvZ)515 idealHNF_Z_factor(GEN x, GEN *pvN, GEN *pvZ)
516 { return idealHNF_Z_factor_i(x, NULL, pvN, pvZ); }
517
518 /* v_P(A)*f(P) <= Nval [e.g. Nval = v_p(Norm A)], Zval = v_p(A \cap Z).
519 * Return v_P(A) */
520 static long
idealHNF_val(GEN A,GEN P,long Nval,long Zval)521 idealHNF_val(GEN A, GEN P, long Nval, long Zval)
522 {
523 long f = pr_get_f(P), vmax, v, e, i, j, k, l;
524 GEN mul, B, a, y, r, p, pk, cx, vals;
525 pari_sp av;
526
527 if (Nval < f) return 0;
528 p = pr_get_p(P);
529 e = pr_get_e(P);
530 /* v_P(A) <= max [ e * v_p(A \cap Z), floor[v_p(Nix) / f ] */
531 vmax = minss(Zval * e, Nval / f);
532 mul = pr_get_tau(P);
533 l = lg(mul);
534 B = cgetg(l,t_MAT);
535 /* B[1] not needed: v_pr(A[1]) = v_pr(A \cap Z) is known already */
536 gel(B,1) = gen_0; /* dummy */
537 for (j = 2; j < l; j++)
538 {
539 GEN x = gel(A,j);
540 gel(B,j) = y = cgetg(l, t_COL);
541 for (i = 1; i < l; i++)
542 { /* compute a = (x.t0)_i, A in HNF ==> x[j+1..l-1] = 0 */
543 a = mulii(gel(x,1), gcoeff(mul,i,1));
544 for (k = 2; k <= j; k++) a = addii(a, mulii(gel(x,k), gcoeff(mul,i,k)));
545 /* p | a ? */
546 gel(y,i) = dvmdii(a,p,&r); if (signe(r)) return 0;
547 }
548 }
549 vals = cgetg(l, t_VECSMALL);
550 /* vals[1] not needed */
551 for (j = 2; j < l; j++)
552 {
553 gel(B,j) = Q_primitive_part(gel(B,j), &cx);
554 vals[j] = cx? 1 + e * Q_pval(cx, p): 1;
555 }
556 pk = powiu(p, ceildivuu(vmax, e));
557 av = avma; y = cgetg(l,t_COL);
558 /* can compute mod p^ceil((vmax-v)/e) */
559 for (v = 1; v < vmax; v++)
560 { /* we know v_pr(Bj) >= v for all j */
561 if (e == 1 || (vmax - v) % e == 0) pk = diviiexact(pk, p);
562 for (j = 2; j < l; j++)
563 {
564 GEN x = gel(B,j); if (v < vals[j]) continue;
565 for (i = 1; i < l; i++)
566 {
567 pari_sp av2 = avma;
568 a = mulii(gel(x,1), gcoeff(mul,i,1));
569 for (k = 2; k < l; k++) a = addii(a, mulii(gel(x,k), gcoeff(mul,i,k)));
570 /* a = (x.t_0)_i; p | a ? */
571 a = dvmdii(a,p,&r); if (signe(r)) return v;
572 if (lgefint(a) > lgefint(pk)) a = remii(a, pk);
573 gel(y,i) = gerepileuptoint(av2, a);
574 }
575 gel(B,j) = y; y = x;
576 if (gc_needed(av,3))
577 {
578 if(DEBUGMEM>1) pari_warn(warnmem,"idealval");
579 gerepileall(av,3, &y,&B,&pk);
580 }
581 }
582 }
583 return v;
584 }
585 /* true nf, x != 0 integral ideal in HNF, cx t_INT or NULL,
586 * FA integer factorization matrix or NULL. Return partial factorization of
587 * cx * x above primes in FA (complete factorization if !FA)*/
588 static GEN
idealHNF_factor_i(GEN nf,GEN x,GEN cx,GEN FA)589 idealHNF_factor_i(GEN nf, GEN x, GEN cx, GEN FA)
590 {
591 const long N = lg(x)-1;
592 long i, j, k, l, v;
593 GEN vN, vZ, vP, vE, vp = idealHNF_Z_factor_i(x, FA, &vN,&vZ);
594
595 l = lg(vp);
596 i = cx? expi(cx)+1: 1;
597 vP = cgetg((l+i-2)*N+1, t_COL);
598 vE = cgetg((l+i-2)*N+1, t_COL);
599 for (i = k = 1; i < l; i++)
600 {
601 GEN L, p = gel(vp,i);
602 long Nval = vN[i], Zval = vZ[i], vc = cx? Z_pvalrem(cx,p,&cx): 0;
603 if (vc)
604 {
605 L = idealprimedec(nf,p);
606 if (is_pm1(cx)) cx = NULL;
607 }
608 else
609 L = idealprimedec_limit_f(nf,p,Nval);
610 for (j = 1; Nval && j < lg(L); j++) /* !Nval => only cx contributes */
611 {
612 GEN P = gel(L,j);
613 pari_sp av = avma;
614 v = idealHNF_val(x, P, Nval, Zval);
615 set_avma(av);
616 Nval -= v*pr_get_f(P);
617 v += vc * pr_get_e(P); if (!v) continue;
618 gel(vP,k) = P;
619 gel(vE,k) = utoipos(v); k++;
620 }
621 if (vc) for (; j<lg(L); j++)
622 {
623 GEN P = gel(L,j);
624 gel(vP,k) = P;
625 gel(vE,k) = utoipos(vc * pr_get_e(P)); k++;
626 }
627 }
628 if (cx && !FA)
629 { /* complete factorization */
630 GEN f = Z_factor(cx), cP = gel(f,1), cE = gel(f,2);
631 long lc = lg(cP);
632 for (i=1; i<lc; i++)
633 {
634 GEN p = gel(cP,i), L = idealprimedec(nf,p);
635 long vc = itos(gel(cE,i));
636 for (j=1; j<lg(L); j++)
637 {
638 GEN P = gel(L,j);
639 gel(vP,k) = P;
640 gel(vE,k) = utoipos(vc * pr_get_e(P)); k++;
641 }
642 }
643 }
644 setlg(vP, k);
645 setlg(vE, k); return mkmat2(vP, vE);
646 }
647 /* true nf, x integral ideal */
648 static GEN
idealHNF_factor(GEN nf,GEN x,ulong lim)649 idealHNF_factor(GEN nf, GEN x, ulong lim)
650 {
651 GEN cx, F = NULL;
652 if (lim)
653 {
654 GEN P, E;
655 long i;
656 /* strict useless because of prime table */
657 F = absZ_factor_limit(gcoeff(x,1,1), lim);
658 P = gel(F,1);
659 E = gel(F,2);
660 /* filter out entries > lim */
661 for (i = lg(P)-1; i; i--)
662 if (cmpiu(gel(P,i), lim) <= 0) break;
663 setlg(P, i+1);
664 setlg(E, i+1);
665 }
666 x = Q_primitive_part(x, &cx);
667 return idealHNF_factor_i(nf, x, cx, F);
668 }
669 /* c * vector(#L,i,L[i].e), assume results fit in ulong */
670 static GEN
prV_e_muls(GEN L,long c)671 prV_e_muls(GEN L, long c)
672 {
673 long j, l = lg(L);
674 GEN z = cgetg(l, t_COL);
675 for (j = 1; j < l; j++) gel(z,j) = stoi(c * pr_get_e(gel(L,j)));
676 return z;
677 }
678 /* true nf, y in Q */
679 static GEN
Q_nffactor(GEN nf,GEN y,ulong lim)680 Q_nffactor(GEN nf, GEN y, ulong lim)
681 {
682 GEN f, P, E;
683 long l, i;
684 if (typ(y) == t_INT)
685 {
686 if (!signe(y)) pari_err_DOMAIN("idealfactor", "ideal", "=",gen_0,y);
687 if (is_pm1(y)) return trivial_fact();
688 }
689 y = Q_abs_shallow(y);
690 if (!lim) f = Q_factor(y);
691 else
692 {
693 f = Q_factor_limit(y, lim);
694 P = gel(f,1);
695 E = gel(f,2);
696 for (i = lg(P)-1; i > 0; i--)
697 if (abscmpiu(gel(P,i), lim) < 0) break;
698 setlg(P,i+1); setlg(E,i+1);
699 }
700 P = gel(f,1); l = lg(P); if (l == 1) return f;
701 E = gel(f,2);
702 for (i = 1; i < l; i++)
703 {
704 gel(P,i) = idealprimedec(nf, gel(P,i));
705 gel(E,i) = prV_e_muls(gel(P,i), itos(gel(E,i)));
706 }
707 settyp(P,t_VEC); P = shallowconcat1(P);
708 settyp(E,t_VEC); E = shallowconcat1(E);
709 gel(f,1) = P; settyp(P, t_COL);
710 gel(f,2) = E; return f;
711 }
712
713 GEN
idealfactor_partial(GEN nf,GEN x,GEN L)714 idealfactor_partial(GEN nf, GEN x, GEN L)
715 {
716 pari_sp av = avma;
717 long i, j, l;
718 GEN P, E;
719 if (!L) return idealfactor(nf, x);
720 if (typ(L) == t_INT) return idealfactor_limit(nf, x, itou(L));
721 l = lg(L); if (l == 1) return trivial_fact();
722 P = cgetg(l, t_VEC);
723 for (i = 1; i < l; i++)
724 {
725 GEN p = gel(L,i);
726 gel(P,i) = typ(p) == t_INT? idealprimedec(nf, p): mkvec(p);
727 }
728 P = shallowconcat1(P); settyp(P, t_COL);
729 P = gen_sort_uniq(P, (void*)&cmp_prime_ideal, &cmp_nodata);
730 E = cgetg_copy(P, &l);
731 for (i = j = 1; i < l; i++)
732 {
733 long v = idealval(nf, x, gel(P,i));
734 if (v) { gel(P,j) = gel(P,i); gel(E,j) = stoi(v); j++; }
735 }
736 setlg(P,j);
737 setlg(E,j); return gerepilecopy(av, mkmat2(P, E));
738 }
739 GEN
idealfactor_limit(GEN nf,GEN x,ulong lim)740 idealfactor_limit(GEN nf, GEN x, ulong lim)
741 {
742 pari_sp av = avma;
743 GEN fa, y;
744 long tx = idealtyp(&x,&y);
745
746 if (tx == id_PRIME)
747 {
748 if (lim && abscmpiu(pr_get_p(x), lim) >= 0) return trivial_fact();
749 retmkmat2(mkcolcopy(x), mkcol(gen_1));
750 }
751 nf = checknf(nf);
752 if (tx == id_PRINCIPAL)
753 {
754 y = nf_to_scalar_or_basis(nf, x);
755 if (typ(y) != t_COL) return gerepilecopy(av, Q_nffactor(nf, y, lim));
756 }
757 y = idealnumden(nf, x);
758 fa = idealHNF_factor(nf, gel(y,1), lim);
759 if (!isint1(gel(y,2)))
760 fa = famat_div_shallow(fa, idealHNF_factor(nf, gel(y,2), lim));
761 fa = gerepilecopy(av, fa);
762 return sort_factor(fa, (void*)&cmp_prime_ideal, &cmp_nodata);
763 }
764 GEN
idealfactor(GEN nf,GEN x)765 idealfactor(GEN nf, GEN x) { return idealfactor_limit(nf, x, 0); }
766 GEN
gpidealfactor(GEN nf,GEN x,GEN lim)767 gpidealfactor(GEN nf, GEN x, GEN lim)
768 {
769 ulong L = 0;
770 if (lim)
771 {
772 if (typ(lim) != t_INT || signe(lim) < 0) pari_err_FLAG("idealfactor");
773 L = itou(lim);
774 }
775 return idealfactor_limit(nf, x, L);
776 }
777
778 static GEN
ramified_root(GEN nf,GEN R,GEN A,long n)779 ramified_root(GEN nf, GEN R, GEN A, long n)
780 {
781 GEN v, P = gel(idealfactor(nf, R), 1);
782 long i, l = lg(P);
783 v = cgetg(l, t_VECSMALL);
784 for (i = 1; i < l; i++)
785 {
786 long w = idealval(nf, A, gel(P,i));
787 if (w % n) return NULL;
788 v[i] = w / n;
789 }
790 return idealfactorback(nf, P, v, 0);
791 }
792 static int
ramified_root_simple(GEN nf,long n,GEN P,GEN v)793 ramified_root_simple(GEN nf, long n, GEN P, GEN v)
794 {
795 long i, l = lg(v);
796 for (i = 1; i < l; i++) if (v[i])
797 {
798 GEN vpr = idealprimedec(nf, gel(P,i));
799 long lpr = lg(vpr), j;
800 for (j = 1; j < lpr; j++)
801 {
802 long e = pr_get_e(gel(vpr,j));
803 if ((e * v[i]) % n) return 0;
804 }
805 }
806 return 1;
807 }
808 /* true nf; A is assumed to be the n-th power of an integral ideal,
809 * return its n-th root; n > 1 */
810 static long
idealsqrtn_int(GEN nf,GEN A,long n,GEN * pB)811 idealsqrtn_int(GEN nf, GEN A, long n, GEN *pB)
812 {
813 GEN C, root;
814 long i, l;
815
816 if (typ(A) == t_INT) /* > 0 */
817 {
818 GEN P = nf_get_ramified_primes(nf), v, q;
819 l = lg(P); v = cgetg(l, t_VECSMALL);
820 for (i = 1; i < l; i++) v[i] = Z_pvalrem(A, gel(P,i), &A);
821 C = gen_1;
822 if (!isint1(A) && !Z_ispowerall(A, n, pB? &C: NULL)) return 0;
823 if (!pB) return ramified_root_simple(nf, n, P, v);
824 q = factorback2(P, v);
825 root = ramified_root(nf, q, q, n);
826 if (!root) return 0;
827 if (!equali1(C)) root = isint1(root)? C: ZM_Z_mul(root, C);
828 *pB = root; return 1;
829 }
830 /* compute valuations at ramified primes */
831 root = ramified_root(nf, idealadd(nf, nf_get_diff(nf), A), A, n);
832 if (!root) return 0;
833 /* remove ramified primes */
834 if (isint1(root))
835 root = matid(nf_get_degree(nf));
836 else
837 A = idealdivexact(nf, A, idealpows(nf,root,n));
838 A = Q_primitive_part(A, &C);
839 if (C)
840 {
841 if (!Z_ispowerall(C,n,&C)) return 0;
842 if (pB) root = ZM_Z_mul(root, C);
843 }
844
845 /* compute final n-th root, at most degree(nf)-1 iterations */
846 for (i = 0;; i++)
847 {
848 GEN J, b, a = gcoeff(A,1,1); /* A \cap Z */
849 if (is_pm1(a)) break;
850 if (!Z_ispowerall(a,n,&b)) return 0;
851 J = idealadd(nf, b, A);
852 A = idealdivexact(nf, idealpows(nf,J,n), A);
853 /* div and not divexact here */
854 if (pB) root = odd(i)? idealdiv(nf, root, J): idealmul(nf, root, J);
855 }
856 if (pB) *pB = root;
857 return 1;
858 }
859
860 /* A is assumed to be the n-th power of an ideal in nf
861 returns its n-th root. */
862 long
idealispower(GEN nf,GEN A,long n,GEN * pB)863 idealispower(GEN nf, GEN A, long n, GEN *pB)
864 {
865 pari_sp av = avma;
866 GEN v, N, D;
867 nf = checknf(nf);
868 if (n <= 0) pari_err_DOMAIN("idealispower", "n", "<=", gen_0, stoi(n));
869 if (n == 1) { if (pB) *pB = idealhnf(nf,A); return 1; }
870 v = idealnumden(nf,A);
871 if (gequal0(gel(v,1))) { set_avma(av); if (pB) *pB = cgetg(1,t_MAT); return 1; }
872 if (!idealsqrtn_int(nf, gel(v,1), n, pB? &N: NULL)) return 0;
873 if (!idealsqrtn_int(nf, gel(v,2), n, pB? &D: NULL)) return 0;
874 if (pB) *pB = gerepileupto(av, idealdiv(nf,N,D)); else set_avma(av);
875 return 1;
876 }
877
878 /* x t_INT or integral nonzero ideal in HNF */
879 static GEN
idealredmodpower_i(GEN nf,GEN x,ulong k,ulong B)880 idealredmodpower_i(GEN nf, GEN x, ulong k, ulong B)
881 {
882 GEN cx, y, U, N, F, Q;
883 if (typ(x) == t_INT)
884 {
885 if (!signe(x) || is_pm1(x)) return gen_1;
886 F = Z_factor_limit(x, B);
887 gel(F,2) = gdiventgs(gel(F,2), k);
888 return ginv(factorback(F));
889 }
890 N = gcoeff(x,1,1); if (is_pm1(N)) return gen_1;
891 F = absZ_factor_limit_strict(N, B, &U);
892 if (U)
893 {
894 GEN M = powii(gel(U,1), gel(U,2));
895 y = hnfmodid(x, M); /* coprime part to B! */
896 if (!idealispower(nf, y, k, &U)) U = NULL;
897 x = hnfmodid(x, diviiexact(N, M));
898 }
899 /* x = B-smooth part of initial x */
900 x = Q_primitive_part(x, &cx);
901 F = idealHNF_factor_i(nf, x, cx, F);
902 gel(F,2) = gdiventgs(gel(F,2), k);
903 Q = idealfactorback(nf, gel(F,1), gel(F,2), 0);
904 if (U) Q = idealmul(nf,Q,U);
905 if (typ(Q) == t_INT) return Q;
906 y = idealred_elt(nf, idealHNF_inv_Z(nf, Q));
907 return gdiv(y, gcoeff(Q,1,1));
908 }
909 GEN
idealredmodpower(GEN nf,GEN x,ulong n,ulong B)910 idealredmodpower(GEN nf, GEN x, ulong n, ulong B)
911 {
912 pari_sp av = avma;
913 GEN a, b;
914 nf = checknf(nf);
915 if (!n) pari_err_DOMAIN("idealredmodpower","n", "=", gen_0, gen_0);
916 x = idealnumden(nf, x);
917 a = gel(x,1);
918 if (isintzero(a)) { set_avma(av); return gen_1; }
919 a = idealredmodpower_i(nf, gel(x,1), n, B);
920 b = idealredmodpower_i(nf, gel(x,2), n, B);
921 if (!isint1(b)) a = nf_to_scalar_or_basis(nf, nfdiv(nf, a, b));
922 return gerepilecopy(av, a);
923 }
924
925 /* P prime ideal in idealprimedec format. Return valuation(A) at P */
926 long
idealval(GEN nf,GEN A,GEN P)927 idealval(GEN nf, GEN A, GEN P)
928 {
929 pari_sp av = avma;
930 GEN a, p, cA;
931 long vcA, v, Zval, tx = idealtyp(&A,&a);
932
933 if (tx == id_PRINCIPAL) return nfval(nf,A,P);
934 checkprid(P);
935 if (tx == id_PRIME) return pr_equal(P, A)? 1: 0;
936 /* id_MAT */
937 nf = checknf(nf);
938 A = Q_primitive_part(A, &cA);
939 p = pr_get_p(P);
940 vcA = cA? Q_pval(cA,p): 0;
941 if (pr_is_inert(P)) return gc_long(av,vcA);
942 Zval = Z_pval(gcoeff(A,1,1), p);
943 if (!Zval) v = 0;
944 else
945 {
946 long Nval = idealHNF_norm_pval(A, p, Zval);
947 v = idealHNF_val(A, P, Nval, Zval);
948 }
949 return gc_long(av, vcA? v + vcA*pr_get_e(P): v);
950 }
951 GEN
gpidealval(GEN nf,GEN ix,GEN P)952 gpidealval(GEN nf, GEN ix, GEN P)
953 {
954 long v = idealval(nf,ix,P);
955 return v == LONG_MAX? mkoo(): stoi(v);
956 }
957
958 /* gcd and generalized Bezout */
959
960 GEN
idealadd(GEN nf,GEN x,GEN y)961 idealadd(GEN nf, GEN x, GEN y)
962 {
963 pari_sp av = avma;
964 long tx, ty;
965 GEN z, a, dx, dy, dz;
966
967 tx = idealtyp(&x,&z);
968 ty = idealtyp(&y,&z); nf = checknf(nf);
969 if (tx != id_MAT) x = idealhnf_shallow(nf,x);
970 if (ty != id_MAT) y = idealhnf_shallow(nf,y);
971 if (lg(x) == 1) return gerepilecopy(av,y);
972 if (lg(y) == 1) return gerepilecopy(av,x); /* check for 0 ideal */
973 dx = Q_denom(x);
974 dy = Q_denom(y); dz = lcmii(dx,dy);
975 if (is_pm1(dz)) dz = NULL; else {
976 x = Q_muli_to_int(x, dz);
977 y = Q_muli_to_int(y, dz);
978 }
979 a = gcdii(gcoeff(x,1,1), gcoeff(y,1,1));
980 if (is_pm1(a))
981 {
982 long N = lg(x)-1;
983 if (!dz) { set_avma(av); return matid(N); }
984 return gerepileupto(av, scalarmat(ginv(dz), N));
985 }
986 z = ZM_hnfmodid(shallowconcat(x,y), a);
987 if (dz) z = RgM_Rg_div(z,dz);
988 return gerepileupto(av,z);
989 }
990
991 static GEN
trivial_merge(GEN x)992 trivial_merge(GEN x)
993 { return (lg(x) == 1 || !is_pm1(gcoeff(x,1,1)))? NULL: gen_1; }
994 /* true nf */
995 static GEN
_idealaddtoone(GEN nf,GEN x,GEN y,long red)996 _idealaddtoone(GEN nf, GEN x, GEN y, long red)
997 {
998 GEN a;
999 long tx = idealtyp(&x, &a/*junk*/);
1000 long ty = idealtyp(&y, &a/*junk*/);
1001 long ea;
1002 if (tx != id_MAT) x = idealhnf_shallow(nf, x);
1003 if (ty != id_MAT) y = idealhnf_shallow(nf, y);
1004 if (lg(x) == 1)
1005 a = trivial_merge(y);
1006 else if (lg(y) == 1)
1007 a = trivial_merge(x);
1008 else
1009 a = hnfmerge_get_1(x, y);
1010 if (!a) pari_err_COPRIME("idealaddtoone",x,y);
1011 if (red && (ea = gexpo(a)) > 10)
1012 {
1013 GEN b = (typ(a) == t_COL)? a: scalarcol_shallow(a, nf_get_degree(nf));
1014 b = ZC_reducemodlll(b, idealHNF_mul(nf,x,y));
1015 if (gexpo(b) < ea) a = b;
1016 }
1017 return a;
1018 }
1019 /* true nf */
1020 GEN
idealaddtoone_i(GEN nf,GEN x,GEN y)1021 idealaddtoone_i(GEN nf, GEN x, GEN y)
1022 { return _idealaddtoone(nf, x, y, 1); }
1023 /* true nf */
1024 GEN
idealaddtoone_raw(GEN nf,GEN x,GEN y)1025 idealaddtoone_raw(GEN nf, GEN x, GEN y)
1026 { return _idealaddtoone(nf, x, y, 0); }
1027
1028 GEN
idealaddtoone(GEN nf,GEN x,GEN y)1029 idealaddtoone(GEN nf, GEN x, GEN y)
1030 {
1031 GEN z = cgetg(3,t_VEC), a;
1032 pari_sp av = avma;
1033 nf = checknf(nf);
1034 a = gerepileupto(av, idealaddtoone_i(nf,x,y));
1035 gel(z,1) = a;
1036 gel(z,2) = typ(a) == t_COL? Z_ZC_sub(gen_1,a): subui(1,a);
1037 return z;
1038 }
1039
1040 /* assume elements of list are integral ideals */
1041 GEN
idealaddmultoone(GEN nf,GEN list)1042 idealaddmultoone(GEN nf, GEN list)
1043 {
1044 pari_sp av = avma;
1045 long N, i, l, nz, tx = typ(list);
1046 GEN H, U, perm, L;
1047
1048 nf = checknf(nf); N = nf_get_degree(nf);
1049 if (!is_vec_t(tx)) pari_err_TYPE("idealaddmultoone",list);
1050 l = lg(list);
1051 L = cgetg(l, t_VEC);
1052 if (l == 1)
1053 pari_err_DOMAIN("idealaddmultoone", "sum(ideals)", "!=", gen_1, L);
1054 nz = 0; /* number of nonzero ideals in L */
1055 for (i=1; i<l; i++)
1056 {
1057 GEN I = gel(list,i);
1058 if (typ(I) != t_MAT) I = idealhnf_shallow(nf,I);
1059 if (lg(I) != 1)
1060 {
1061 nz++; RgM_check_ZM(I,"idealaddmultoone");
1062 if (lgcols(I) != N+1) pari_err_TYPE("idealaddmultoone [not an ideal]", I);
1063 }
1064 gel(L,i) = I;
1065 }
1066 H = ZM_hnfperm(shallowconcat1(L), &U, &perm);
1067 if (lg(H) == 1 || !equali1(gcoeff(H,1,1)))
1068 pari_err_DOMAIN("idealaddmultoone", "sum(ideals)", "!=", gen_1, L);
1069 for (i=1; i<=N; i++)
1070 if (perm[i] == 1) break;
1071 U = gel(U,(nz-1)*N + i); /* (L[1]|...|L[nz]) U = 1 */
1072 nz = 0;
1073 for (i=1; i<l; i++)
1074 {
1075 GEN c = gel(L,i);
1076 if (lg(c) == 1)
1077 c = gen_0;
1078 else {
1079 c = ZM_ZC_mul(c, vecslice(U, nz*N + 1, (nz+1)*N));
1080 nz++;
1081 }
1082 gel(L,i) = c;
1083 }
1084 return gerepilecopy(av, L);
1085 }
1086
1087 /* multiplication */
1088
1089 /* x integral ideal (without archimedean component) in HNF form
1090 * y = [a,alpha] corresponds to the integral ideal aZ_K+alpha Z_K, a in Z,
1091 * alpha a ZV or a ZM (multiplication table). Multiply them */
1092 static GEN
idealHNF_mul_two(GEN nf,GEN x,GEN y)1093 idealHNF_mul_two(GEN nf, GEN x, GEN y)
1094 {
1095 GEN m, a = gel(y,1), alpha = gel(y,2);
1096 long i, N;
1097
1098 if (typ(alpha) != t_MAT)
1099 {
1100 alpha = zk_scalar_or_multable(nf, alpha);
1101 if (typ(alpha) == t_INT) /* e.g. y inert ? 0 should not (but may) occur */
1102 return signe(a)? ZM_Z_mul(x, gcdii(a, alpha)): cgetg(1,t_MAT);
1103 }
1104 N = lg(x)-1; m = cgetg((N<<1)+1,t_MAT);
1105 for (i=1; i<=N; i++) gel(m,i) = ZM_ZC_mul(alpha,gel(x,i));
1106 for (i=1; i<=N; i++) gel(m,i+N) = ZC_Z_mul(gel(x,i), a);
1107 return ZM_hnfmodid(m, mulii(a, gcoeff(x,1,1)));
1108 }
1109
1110 /* Assume x and y are integral in HNF form [NOT extended]. Not memory clean.
1111 * HACK: ideal in y can be of the form [a,b], a in Z, b in Z_K */
1112 GEN
idealHNF_mul(GEN nf,GEN x,GEN y)1113 idealHNF_mul(GEN nf, GEN x, GEN y)
1114 {
1115 GEN z;
1116 if (typ(y) == t_VEC)
1117 z = idealHNF_mul_two(nf,x,y);
1118 else
1119 { /* reduce one ideal to two-elt form. The smallest */
1120 GEN xZ = gcoeff(x,1,1), yZ = gcoeff(y,1,1);
1121 if (cmpii(xZ, yZ) < 0)
1122 {
1123 if (is_pm1(xZ)) return gcopy(y);
1124 z = idealHNF_mul_two(nf, y, mat_ideal_two_elt(nf,x));
1125 }
1126 else
1127 {
1128 if (is_pm1(yZ)) return gcopy(x);
1129 z = idealHNF_mul_two(nf, x, mat_ideal_two_elt(nf,y));
1130 }
1131 }
1132 return z;
1133 }
1134
1135 /* operations on elements in factored form */
1136
1137 GEN
famat_mul_shallow(GEN f,GEN g)1138 famat_mul_shallow(GEN f, GEN g)
1139 {
1140 if (typ(f) != t_MAT) f = to_famat_shallow(f,gen_1);
1141 if (typ(g) != t_MAT) g = to_famat_shallow(g,gen_1);
1142 if (lgcols(f) == 1) return g;
1143 if (lgcols(g) == 1) return f;
1144 return mkmat2(shallowconcat(gel(f,1), gel(g,1)),
1145 shallowconcat(gel(f,2), gel(g,2)));
1146 }
1147 GEN
famat_mulpow_shallow(GEN f,GEN g,GEN e)1148 famat_mulpow_shallow(GEN f, GEN g, GEN e)
1149 {
1150 if (!signe(e)) return f;
1151 return famat_mul_shallow(f, famat_pow_shallow(g, e));
1152 }
1153
1154 GEN
famat_mulpows_shallow(GEN f,GEN g,long e)1155 famat_mulpows_shallow(GEN f, GEN g, long e)
1156 {
1157 if (e==0) return f;
1158 return famat_mul_shallow(f, famat_pows_shallow(g, e));
1159 }
1160
1161 GEN
famat_div_shallow(GEN f,GEN g)1162 famat_div_shallow(GEN f, GEN g)
1163 { return famat_mul_shallow(f, famat_inv_shallow(g)); }
1164
1165 GEN
to_famat(GEN x,GEN y)1166 to_famat(GEN x, GEN y) { retmkmat2(mkcolcopy(x), mkcolcopy(y)); }
1167 GEN
to_famat_shallow(GEN x,GEN y)1168 to_famat_shallow(GEN x, GEN y) { return mkmat2(mkcol(x), mkcol(y)); }
1169
1170 /* concat the single elt x; not gconcat since x may be a t_COL */
1171 static GEN
append(GEN v,GEN x)1172 append(GEN v, GEN x)
1173 {
1174 long i, l = lg(v);
1175 GEN w = cgetg(l+1, typ(v));
1176 for (i=1; i<l; i++) gel(w,i) = gcopy(gel(v,i));
1177 gel(w,i) = gcopy(x); return w;
1178 }
1179 /* add x^1 to famat f */
1180 static GEN
famat_add(GEN f,GEN x)1181 famat_add(GEN f, GEN x)
1182 {
1183 GEN h = cgetg(3,t_MAT);
1184 if (lgcols(f) == 1)
1185 {
1186 gel(h,1) = mkcolcopy(x);
1187 gel(h,2) = mkcol(gen_1);
1188 }
1189 else
1190 {
1191 gel(h,1) = append(gel(f,1), x);
1192 gel(h,2) = gconcat(gel(f,2), gen_1);
1193 }
1194 return h;
1195 }
1196 /* add x^-1 to famat f */
1197 static GEN
famat_sub(GEN f,GEN x)1198 famat_sub(GEN f, GEN x)
1199 {
1200 GEN h = cgetg(3,t_MAT);
1201 if (lgcols(f) == 1)
1202 {
1203 gel(h,1) = mkcolcopy(x);
1204 gel(h,2) = mkcol(gen_m1);
1205 }
1206 else
1207 {
1208 gel(h,1) = append(gel(f,1), x);
1209 gel(h,2) = gconcat(gel(f,2), gen_m1);
1210 }
1211 return h;
1212 }
1213
1214 GEN
famat_mul(GEN f,GEN g)1215 famat_mul(GEN f, GEN g)
1216 {
1217 GEN h;
1218 if (typ(g) != t_MAT) {
1219 if (typ(f) == t_MAT) return famat_add(f, g);
1220 h = cgetg(3, t_MAT);
1221 gel(h,1) = mkcol2(gcopy(f), gcopy(g));
1222 gel(h,2) = mkcol2(gen_1, gen_1);
1223 }
1224 if (typ(f) != t_MAT) return famat_add(g, f);
1225 if (lgcols(f) == 1) return gcopy(g);
1226 if (lgcols(g) == 1) return gcopy(f);
1227 h = cgetg(3,t_MAT);
1228 gel(h,1) = gconcat(gel(f,1), gel(g,1));
1229 gel(h,2) = gconcat(gel(f,2), gel(g,2));
1230 return h;
1231 }
1232
1233 GEN
famat_div(GEN f,GEN g)1234 famat_div(GEN f, GEN g)
1235 {
1236 GEN h;
1237 if (typ(g) != t_MAT) {
1238 if (typ(f) == t_MAT) return famat_sub(f, g);
1239 h = cgetg(3, t_MAT);
1240 gel(h,1) = mkcol2(gcopy(f), gcopy(g));
1241 gel(h,2) = mkcol2(gen_1, gen_m1);
1242 }
1243 if (typ(f) != t_MAT) return famat_sub(g, f);
1244 if (lgcols(f) == 1) return famat_inv(g);
1245 if (lgcols(g) == 1) return gcopy(f);
1246 h = cgetg(3,t_MAT);
1247 gel(h,1) = gconcat(gel(f,1), gel(g,1));
1248 gel(h,2) = gconcat(gel(f,2), gneg(gel(g,2)));
1249 return h;
1250 }
1251
1252 GEN
famat_sqr(GEN f)1253 famat_sqr(GEN f)
1254 {
1255 GEN h;
1256 if (typ(f) != t_MAT) return to_famat(f,gen_2);
1257 if (lgcols(f) == 1) return gcopy(f);
1258 h = cgetg(3,t_MAT);
1259 gel(h,1) = gcopy(gel(f,1));
1260 gel(h,2) = gmul2n(gel(f,2),1);
1261 return h;
1262 }
1263
1264 GEN
famat_inv_shallow(GEN f)1265 famat_inv_shallow(GEN f)
1266 {
1267 if (typ(f) != t_MAT) return to_famat_shallow(f,gen_m1);
1268 if (lgcols(f) == 1) return f;
1269 return mkmat2(gel(f,1), ZC_neg(gel(f,2)));
1270 }
1271 GEN
famat_inv(GEN f)1272 famat_inv(GEN f)
1273 {
1274 if (typ(f) != t_MAT) return to_famat(f,gen_m1);
1275 if (lgcols(f) == 1) return gcopy(f);
1276 retmkmat2(gcopy(gel(f,1)), ZC_neg(gel(f,2)));
1277 }
1278 GEN
famat_pow(GEN f,GEN n)1279 famat_pow(GEN f, GEN n)
1280 {
1281 if (typ(f) != t_MAT) return to_famat(f,n);
1282 if (lgcols(f) == 1) return gcopy(f);
1283 retmkmat2(gcopy(gel(f,1)), ZC_Z_mul(gel(f,2),n));
1284 }
1285 GEN
famat_pow_shallow(GEN f,GEN n)1286 famat_pow_shallow(GEN f, GEN n)
1287 {
1288 if (is_pm1(n)) return signe(n) > 0? f: famat_inv_shallow(f);
1289 if (typ(f) != t_MAT) return to_famat_shallow(f,n);
1290 if (lgcols(f) == 1) return f;
1291 return mkmat2(gel(f,1), ZC_Z_mul(gel(f,2),n));
1292 }
1293
1294 GEN
famat_pows_shallow(GEN f,long n)1295 famat_pows_shallow(GEN f, long n)
1296 {
1297 if (n==1) return f;
1298 if (n==-1) return famat_inv_shallow(f);
1299 if (typ(f) != t_MAT) return to_famat_shallow(f, stoi(n));
1300 if (lgcols(f) == 1) return f;
1301 return mkmat2(gel(f,1), ZC_z_mul(gel(f,2),n));
1302 }
1303
1304 GEN
famat_Z_gcd(GEN M,GEN n)1305 famat_Z_gcd(GEN M, GEN n)
1306 {
1307 pari_sp av=avma;
1308 long i, j, l=lgcols(M);
1309 GEN F=cgetg(3,t_MAT);
1310 gel(F,1)=cgetg(l,t_COL);
1311 gel(F,2)=cgetg(l,t_COL);
1312 for (i=1, j=1; i<l; i++)
1313 {
1314 GEN p = gcoeff(M,i,1);
1315 GEN e = gminsg(Z_pval(n,p),gcoeff(M,i,2));
1316 if (signe(e))
1317 {
1318 gcoeff(F,j,1)=p;
1319 gcoeff(F,j,2)=e;
1320 j++;
1321 }
1322 }
1323 setlg(gel(F,1),j); setlg(gel(F,2),j);
1324 return gerepilecopy(av,F);
1325 }
1326
1327 /* x assumed to be a t_MATs (factorization matrix), or compatible with
1328 * the element_* functions. */
1329 static GEN
ext_sqr(GEN nf,GEN x)1330 ext_sqr(GEN nf, GEN x)
1331 { return (typ(x)==t_MAT)? famat_sqr(x): nfsqr(nf, x); }
1332 static GEN
ext_mul(GEN nf,GEN x,GEN y)1333 ext_mul(GEN nf, GEN x, GEN y)
1334 { return (typ(x)==t_MAT)? famat_mul(x,y): nfmul(nf, x, y); }
1335 static GEN
ext_inv(GEN nf,GEN x)1336 ext_inv(GEN nf, GEN x)
1337 { return (typ(x)==t_MAT)? famat_inv(x): nfinv(nf, x); }
1338 static GEN
ext_pow(GEN nf,GEN x,GEN n)1339 ext_pow(GEN nf, GEN x, GEN n)
1340 { return (typ(x)==t_MAT)? famat_pow(x,n): nfpow(nf, x, n); }
1341
1342 GEN
famat_to_nf(GEN nf,GEN f)1343 famat_to_nf(GEN nf, GEN f)
1344 {
1345 GEN t, x, e;
1346 long i;
1347 if (lgcols(f) == 1) return gen_1;
1348 x = gel(f,1);
1349 e = gel(f,2);
1350 t = nfpow(nf, gel(x,1), gel(e,1));
1351 for (i=lg(x)-1; i>1; i--)
1352 t = nfmul(nf, t, nfpow(nf, gel(x,i), gel(e,i)));
1353 return t;
1354 }
1355
1356 GEN
famat_idealfactor(GEN nf,GEN x)1357 famat_idealfactor(GEN nf, GEN x)
1358 {
1359 long i, l;
1360 GEN g = gel(x,1), e = gel(x,2), h = cgetg_copy(g, &l);
1361 for (i = 1; i < l; i++) gel(h,i) = idealfactor(nf, gel(g,i));
1362 h = famat_reduce(famatV_factorback(h,e));
1363 return sort_factor(h, (void*)&cmp_prime_ideal, &cmp_nodata);
1364 }
1365
1366 GEN
famat_reduce(GEN fa)1367 famat_reduce(GEN fa)
1368 {
1369 GEN E, G, L, g, e;
1370 long i, k, l;
1371
1372 if (typ(fa) != t_MAT || lgcols(fa) == 1) return fa;
1373 g = gel(fa,1); l = lg(g);
1374 e = gel(fa,2);
1375 L = gen_indexsort(g, (void*)&cmp_universal, &cmp_nodata);
1376 G = cgetg(l, t_COL);
1377 E = cgetg(l, t_COL);
1378 /* merge */
1379 for (k=i=1; i<l; i++,k++)
1380 {
1381 gel(G,k) = gel(g,L[i]);
1382 gel(E,k) = gel(e,L[i]);
1383 if (k > 1 && gidentical(gel(G,k), gel(G,k-1)))
1384 {
1385 gel(E,k-1) = addii(gel(E,k), gel(E,k-1));
1386 k--;
1387 }
1388 }
1389 /* kill 0 exponents */
1390 l = k;
1391 for (k=i=1; i<l; i++)
1392 if (!gequal0(gel(E,i)))
1393 {
1394 gel(G,k) = gel(G,i);
1395 gel(E,k) = gel(E,i); k++;
1396 }
1397 setlg(G, k);
1398 setlg(E, k); return mkmat2(G,E);
1399 }
1400 GEN
matreduce(GEN f)1401 matreduce(GEN f)
1402 { pari_sp av = avma;
1403 switch(typ(f))
1404 {
1405 case t_VEC: case t_COL:
1406 {
1407 GEN e; f = vec_reduce(f, &e); settyp(f, t_COL);
1408 return gerepilecopy(av, mkmat2(f, zc_to_ZC(e)));
1409 }
1410 case t_MAT:
1411 if (lg(f) == 3) break;
1412 default:
1413 pari_err_TYPE("matreduce", f);
1414 }
1415 if (typ(gel(f,1)) == t_VECSMALL)
1416 f = famatsmall_reduce(f);
1417 else
1418 {
1419 if (!RgV_is_ZV(gel(f,2))) pari_err_TYPE("matreduce",f);
1420 f = famat_reduce(f);
1421 }
1422 return gerepilecopy(av, f);
1423 }
1424
1425 GEN
famatsmall_reduce(GEN fa)1426 famatsmall_reduce(GEN fa)
1427 {
1428 GEN E, G, L, g, e;
1429 long i, k, l;
1430 if (lgcols(fa) == 1) return fa;
1431 g = gel(fa,1); l = lg(g);
1432 e = gel(fa,2);
1433 L = vecsmall_indexsort(g);
1434 G = cgetg(l, t_VECSMALL);
1435 E = cgetg(l, t_VECSMALL);
1436 /* merge */
1437 for (k=i=1; i<l; i++,k++)
1438 {
1439 G[k] = g[L[i]];
1440 E[k] = e[L[i]];
1441 if (k > 1 && G[k] == G[k-1])
1442 {
1443 E[k-1] += E[k];
1444 k--;
1445 }
1446 }
1447 /* kill 0 exponents */
1448 l = k;
1449 for (k=i=1; i<l; i++)
1450 if (E[i])
1451 {
1452 G[k] = G[i];
1453 E[k] = E[i]; k++;
1454 }
1455 setlg(G, k);
1456 setlg(E, k); return mkmat2(G,E);
1457 }
1458
1459 GEN
famat_remove_trivial(GEN fa)1460 famat_remove_trivial(GEN fa)
1461 {
1462 GEN P, E, p = gel(fa,1), e = gel(fa,2);
1463 long j, k, l = lg(p);
1464 P = cgetg(l, t_COL);
1465 E = cgetg(l, t_COL);
1466 for (j = k = 1; j < l; j++)
1467 if (signe(gel(e,j))) { gel(P,k) = gel(p,j); gel(E,k++) = gel(e,j); }
1468 setlg(P, k); setlg(E, k); return mkmat2(P,E);
1469 }
1470
1471 GEN
famatV_factorback(GEN v,GEN e)1472 famatV_factorback(GEN v, GEN e)
1473 {
1474 long i, l = lg(e);
1475 GEN V;
1476 if (l == 1) return trivial_fact();
1477 V = signe(gel(e,1))? famat_pow_shallow(gel(v,1), gel(e,1)): trivial_fact();
1478 for (i = 2; i < l; i++) V = famat_mulpow_shallow(V, gel(v,i), gel(e,i));
1479 return V;
1480 }
1481
1482 GEN
famatV_zv_factorback(GEN v,GEN e)1483 famatV_zv_factorback(GEN v, GEN e)
1484 {
1485 long i, l = lg(e);
1486 GEN V;
1487 if (l == 1) return trivial_fact();
1488 V = uel(e,1)? famat_pows_shallow(gel(v,1), uel(e,1)): trivial_fact();
1489 for (i = 2; i < l; i++) V = famat_mulpows_shallow(V, gel(v,i), uel(e,i));
1490 return V;
1491 }
1492
1493 GEN
ZM_famat_limit(GEN fa,GEN limit)1494 ZM_famat_limit(GEN fa, GEN limit)
1495 {
1496 pari_sp av;
1497 GEN E, G, g, e, r;
1498 long i, k, l, n, lG;
1499
1500 if (lgcols(fa) == 1) return fa;
1501 g = gel(fa,1); l = lg(g);
1502 e = gel(fa,2);
1503 for(n=0, i=1; i<l; i++)
1504 if (cmpii(gel(g,i),limit)<=0) n++;
1505 lG = n<l-1 ? n+2 : n+1;
1506 G = cgetg(lG, t_COL);
1507 E = cgetg(lG, t_COL);
1508 av = avma;
1509 for (i=1, k=1, r = gen_1; i<l; i++)
1510 {
1511 if (cmpii(gel(g,i),limit)<=0)
1512 {
1513 gel(G,k) = gel(g,i);
1514 gel(E,k) = gel(e,i);
1515 k++;
1516 } else r = mulii(r, powii(gel(g,i), gel(e,i)));
1517 }
1518 if (k<i)
1519 {
1520 gel(G, k) = gerepileuptoint(av, r);
1521 gel(E, k) = gen_1;
1522 }
1523 return mkmat2(G,E);
1524 }
1525
1526 /* assume pr has degree 1 and coprime to Q_denom(x) */
1527 static GEN
to_Fp_coprime(GEN nf,GEN x,GEN modpr)1528 to_Fp_coprime(GEN nf, GEN x, GEN modpr)
1529 {
1530 GEN d, r, p = modpr_get_p(modpr);
1531 x = nf_to_scalar_or_basis(nf,x);
1532 if (typ(x) != t_COL) return Rg_to_Fp(x,p);
1533 x = Q_remove_denom(x, &d);
1534 r = zk_to_Fq(x, modpr);
1535 if (d) r = Fp_div(r, d, p);
1536 return r;
1537 }
1538
1539 /* pr coprime to all denominators occurring in x */
1540 static GEN
famat_to_Fp_coprime(GEN nf,GEN x,GEN modpr)1541 famat_to_Fp_coprime(GEN nf, GEN x, GEN modpr)
1542 {
1543 GEN p = modpr_get_p(modpr);
1544 GEN t = NULL, g = gel(x,1), e = gel(x,2), q = subiu(p,1);
1545 long i, l = lg(g);
1546 for (i = 1; i < l; i++)
1547 {
1548 GEN n = modii(gel(e,i), q);
1549 if (signe(n))
1550 {
1551 GEN h = to_Fp_coprime(nf, gel(g,i), modpr);
1552 h = Fp_pow(h, n, p);
1553 t = t? Fp_mul(t, h, p): h;
1554 }
1555 }
1556 return t? modii(t, p): gen_1;
1557 }
1558
1559 /* cf famat_to_nf_modideal_coprime, modpr attached to prime of degree 1 */
1560 GEN
nf_to_Fp_coprime(GEN nf,GEN x,GEN modpr)1561 nf_to_Fp_coprime(GEN nf, GEN x, GEN modpr)
1562 {
1563 return typ(x)==t_MAT? famat_to_Fp_coprime(nf, x, modpr)
1564 : to_Fp_coprime(nf, x, modpr);
1565 }
1566
1567 static long
zk_pvalrem(GEN x,GEN p,GEN * py)1568 zk_pvalrem(GEN x, GEN p, GEN *py)
1569 { return (typ(x) == t_INT)? Z_pvalrem(x, p, py): ZV_pvalrem(x, p, py); }
1570 /* x a QC or Q. Return a ZC or Z, whose content is coprime to Z. Set v, dx
1571 * such that x = p^v (newx / dx); dx = NULL if 1 */
1572 static GEN
nf_remove_denom_p(GEN nf,GEN x,GEN p,GEN * pdx,long * pv)1573 nf_remove_denom_p(GEN nf, GEN x, GEN p, GEN *pdx, long *pv)
1574 {
1575 long vcx;
1576 GEN dx;
1577 x = nf_to_scalar_or_basis(nf, x);
1578 x = Q_remove_denom(x, &dx);
1579 if (dx)
1580 {
1581 vcx = - Z_pvalrem(dx, p, &dx);
1582 if (!vcx) vcx = zk_pvalrem(x, p, &x);
1583 if (isint1(dx)) dx = NULL;
1584 }
1585 else
1586 {
1587 vcx = zk_pvalrem(x, p, &x);
1588 dx = NULL;
1589 }
1590 *pv = vcx;
1591 *pdx = dx; return x;
1592 }
1593 /* x = b^e/p^(e-1) in Z_K; x = 0 mod p/pr^e, (x,pr) = 1. Return NULL
1594 * if p inert (instead of 1) */
1595 static GEN
p_makecoprime(GEN pr)1596 p_makecoprime(GEN pr)
1597 {
1598 GEN B = pr_get_tau(pr), b;
1599 long i, e;
1600
1601 if (typ(B) == t_INT) return NULL;
1602 b = gel(B,1); /* B = multiplication table by b */
1603 e = pr_get_e(pr);
1604 if (e == 1) return b;
1605 /* one could also divide (exactly) by p in each iteration */
1606 for (i = 1; i < e; i++) b = ZM_ZC_mul(B, b);
1607 return ZC_Z_divexact(b, powiu(pr_get_p(pr), e-1));
1608 }
1609
1610 /* Compute A = prod g[i]^e[i] mod pr^k, assuming (A, pr) = 1.
1611 * Method: modify each g[i] so that it becomes coprime to pr,
1612 * g[i] *= (b/p)^v_pr(g[i]), where b/p = pr^(-1) times something integral
1613 * and prime to p; globally, we multiply by (b/p)^v_pr(A) = 1.
1614 * Optimizations:
1615 * 1) remove all powers of p from contents, and consider extra generator p^vp;
1616 * modified as p * (b/p)^e = b^e / p^(e-1)
1617 * 2) remove denominators, coprime to p, by multiplying by inverse mod prk\cap Z
1618 *
1619 * EX = multiple of exponent of (O_K / pr^k)^* used to reduce the product in
1620 * case the e[i] are large */
1621 GEN
famat_makecoprime(GEN nf,GEN g,GEN e,GEN pr,GEN prk,GEN EX)1622 famat_makecoprime(GEN nf, GEN g, GEN e, GEN pr, GEN prk, GEN EX)
1623 {
1624 GEN G, E, t, vp = NULL, p = pr_get_p(pr), prkZ = gcoeff(prk, 1,1);
1625 long i, l = lg(g);
1626
1627 G = cgetg(l+1, t_VEC);
1628 E = cgetg(l+1, t_VEC); /* l+1: room for "modified p" */
1629 for (i=1; i < l; i++)
1630 {
1631 long vcx;
1632 GEN dx, x = nf_remove_denom_p(nf, gel(g,i), p, &dx, &vcx);
1633 if (vcx) /* = v_p(content(g[i])) */
1634 {
1635 GEN a = mulsi(vcx, gel(e,i));
1636 vp = vp? addii(vp, a): a;
1637 }
1638 /* x integral, content coprime to p; dx coprime to p */
1639 if (typ(x) == t_INT)
1640 { /* x coprime to p, hence to pr */
1641 x = modii(x, prkZ);
1642 if (dx) x = Fp_div(x, dx, prkZ);
1643 }
1644 else
1645 {
1646 (void)ZC_nfvalrem(x, pr, &x); /* x *= (b/p)^v_pr(x) */
1647 x = ZC_hnfrem(FpC_red(x,prkZ), prk);
1648 if (dx) x = FpC_Fp_mul(x, Fp_inv(dx,prkZ), prkZ);
1649 }
1650 gel(G,i) = x;
1651 gel(E,i) = gel(e,i);
1652 }
1653
1654 t = vp? p_makecoprime(pr): NULL;
1655 if (!t)
1656 { /* no need for extra generator */
1657 setlg(G,l);
1658 setlg(E,l);
1659 }
1660 else
1661 {
1662 gel(G,i) = FpC_red(t, prkZ);
1663 gel(E,i) = vp;
1664 }
1665 return famat_to_nf_modideal_coprime(nf, G, E, prk, EX);
1666 }
1667
1668 /* simplified version of famat_makecoprime for X = SUnits[1] */
1669 GEN
sunits_makecoprime(GEN X,GEN pr,GEN prk)1670 sunits_makecoprime(GEN X, GEN pr, GEN prk)
1671 {
1672 GEN G, p = pr_get_p(pr), prkZ = gcoeff(prk,1,1);
1673 long i, l = lg(X);
1674
1675 G = cgetg(l, t_VEC);
1676 for (i = 1; i < l; i++)
1677 {
1678 GEN x = gel(X,i);
1679 if (typ(x) == t_INT) /* a prime */
1680 x = equalii(x,p)? p_makecoprime(pr): modii(x, prkZ);
1681 else
1682 {
1683 (void)ZC_nfvalrem(x, pr, &x); /* x *= (b/p)^v_pr(x) */
1684 x = ZC_hnfrem(FpC_red(x,prkZ), prk);
1685 }
1686 gel(G,i) = x;
1687 }
1688 return G;
1689 }
1690
1691 /* prod g[i]^e[i] mod bid, assume (g[i], id) = 1 and 1 < lg(g) <= lg(e) */
1692 GEN
famat_to_nf_moddivisor(GEN nf,GEN g,GEN e,GEN bid)1693 famat_to_nf_moddivisor(GEN nf, GEN g, GEN e, GEN bid)
1694 {
1695 GEN t, cyc = bid_get_cyc(bid);
1696 if (lg(cyc) == 1)
1697 t = gen_1;
1698 else
1699 t = famat_to_nf_modideal_coprime(nf, g, e, bid_get_ideal(bid),
1700 cyc_get_expo(cyc));
1701 return set_sign_mod_divisor(nf, mkmat2(g,e), t, bid_get_sarch(bid));
1702 }
1703
1704 GEN
vecmul(GEN x,GEN y)1705 vecmul(GEN x, GEN y)
1706 {
1707 if (is_scalar_t(typ(x))) return gmul(x,y);
1708 pari_APPLY_same(vecmul(gel(x,i), gel(y,i)))
1709 }
1710
1711 GEN
vecinv(GEN x)1712 vecinv(GEN x)
1713 {
1714 if (is_scalar_t(typ(x))) return ginv(x);
1715 pari_APPLY_same(vecinv(gel(x,i)))
1716 }
1717
1718 GEN
vecpow(GEN x,GEN n)1719 vecpow(GEN x, GEN n)
1720 {
1721 if (is_scalar_t(typ(x))) return powgi(x,n);
1722 pari_APPLY_same(vecpow(gel(x,i), n))
1723 }
1724
1725 GEN
vecdiv(GEN x,GEN y)1726 vecdiv(GEN x, GEN y)
1727 {
1728 if (is_scalar_t(typ(x))) return gdiv(x,y);
1729 pari_APPLY_same(vecdiv(gel(x,i), gel(y,i)))
1730 }
1731
1732 /* A ideal as a square t_MAT */
1733 static GEN
idealmulelt(GEN nf,GEN x,GEN A)1734 idealmulelt(GEN nf, GEN x, GEN A)
1735 {
1736 long i, lx;
1737 GEN dx, dA, D;
1738 if (lg(A) == 1) return cgetg(1, t_MAT);
1739 x = nf_to_scalar_or_basis(nf,x);
1740 if (typ(x) != t_COL)
1741 return isintzero(x)? cgetg(1,t_MAT): RgM_Rg_mul(A, Q_abs_shallow(x));
1742 x = Q_remove_denom(x, &dx);
1743 A = Q_remove_denom(A, &dA);
1744 x = zk_multable(nf, x);
1745 D = mulii(zkmultable_capZ(x), gcoeff(A,1,1));
1746 x = zkC_multable_mul(A, x);
1747 settyp(x, t_MAT); lx = lg(x);
1748 /* x may contain scalars (at most 1 since the ideal is nonzero)*/
1749 for (i=1; i<lx; i++)
1750 if (typ(gel(x,i)) == t_INT)
1751 {
1752 if (i > 1) swap(gel(x,1), gel(x,i)); /* help HNF */
1753 gel(x,1) = scalarcol_shallow(gel(x,1), lx-1);
1754 break;
1755 }
1756 x = ZM_hnfmodid(x, D);
1757 dx = mul_denom(dx,dA);
1758 return dx? gdiv(x,dx): x;
1759 }
1760
1761 /* nf a true nf, tx <= ty */
1762 static GEN
idealmul_aux(GEN nf,GEN x,GEN y,long tx,long ty)1763 idealmul_aux(GEN nf, GEN x, GEN y, long tx, long ty)
1764 {
1765 GEN z, cx, cy;
1766 switch(tx)
1767 {
1768 case id_PRINCIPAL:
1769 switch(ty)
1770 {
1771 case id_PRINCIPAL:
1772 return idealhnf_principal(nf, nfmul(nf,x,y));
1773 case id_PRIME:
1774 {
1775 GEN p = pr_get_p(y), pi = pr_get_gen(y), cx;
1776 if (pr_is_inert(y)) return RgM_Rg_mul(idealhnf_principal(nf,x),p);
1777
1778 x = nf_to_scalar_or_basis(nf, x);
1779 switch(typ(x))
1780 {
1781 case t_INT:
1782 if (!signe(x)) return cgetg(1,t_MAT);
1783 return ZM_Z_mul(pr_hnf(nf,y), absi_shallow(x));
1784 case t_FRAC:
1785 return RgM_Rg_mul(pr_hnf(nf,y), Q_abs_shallow(x));
1786 }
1787 /* t_COL */
1788 x = Q_primitive_part(x, &cx);
1789 x = zk_multable(nf, x);
1790 z = shallowconcat(ZM_Z_mul(x,p), ZM_ZC_mul(x,pi));
1791 z = ZM_hnfmodid(z, mulii(p, zkmultable_capZ(x)));
1792 return cx? ZM_Q_mul(z, cx): z;
1793 }
1794 default: /* id_MAT */
1795 return idealmulelt(nf, x,y);
1796 }
1797 case id_PRIME:
1798 if (ty==id_PRIME)
1799 { y = pr_hnf(nf,y); cy = NULL; }
1800 else
1801 y = Q_primitive_part(y, &cy);
1802 y = idealHNF_mul_two(nf,y,x);
1803 return cy? ZM_Q_mul(y,cy): y;
1804
1805 default: /* id_MAT */
1806 {
1807 long N = nf_get_degree(nf);
1808 if (lg(x)-1 != N || lg(y)-1 != N) pari_err_DIM("idealmul");
1809 x = Q_primitive_part(x, &cx);
1810 y = Q_primitive_part(y, &cy); cx = mul_content(cx,cy);
1811 y = idealHNF_mul(nf,x,y);
1812 return cx? ZM_Q_mul(y,cx): y;
1813 }
1814 }
1815 }
1816
1817 /* output the ideal product x.y */
1818 GEN
idealmul(GEN nf,GEN x,GEN y)1819 idealmul(GEN nf, GEN x, GEN y)
1820 {
1821 pari_sp av;
1822 GEN res, ax, ay, z;
1823 long tx = idealtyp(&x,&ax);
1824 long ty = idealtyp(&y,&ay), f;
1825 if (tx>ty) { swap(ax,ay); swap(x,y); lswap(tx,ty); }
1826 f = (ax||ay); res = f? cgetg(3,t_VEC): NULL; /*product is an extended ideal*/
1827 av = avma;
1828 z = gerepileupto(av, idealmul_aux(checknf(nf), x,y, tx,ty));
1829 if (!f) return z;
1830 if (ax && ay)
1831 ax = ext_mul(nf, ax, ay);
1832 else
1833 ax = gcopy(ax? ax: ay);
1834 gel(res,1) = z; gel(res,2) = ax; return res;
1835 }
1836
1837 /* Return x, integral in 2-elt form, such that pr^2 = c * x. cf idealpowprime
1838 * nf = true nf */
1839 static GEN
idealsqrprime(GEN nf,GEN pr,GEN * pc)1840 idealsqrprime(GEN nf, GEN pr, GEN *pc)
1841 {
1842 GEN p = pr_get_p(pr), q, gen;
1843 long e = pr_get_e(pr), f = pr_get_f(pr);
1844
1845 q = (e == 1)? sqri(p): p;
1846 if (e <= 2 && e * f == nf_get_degree(nf))
1847 { /* pr^e = (p) */
1848 *pc = q;
1849 return mkvec2(gen_1,gen_0);
1850 }
1851 gen = nfsqr(nf, pr_get_gen(pr));
1852 gen = FpC_red(gen, q);
1853 *pc = NULL;
1854 return mkvec2(q, gen);
1855 }
1856 /* cf idealpow_aux */
1857 static GEN
idealsqr_aux(GEN nf,GEN x,long tx)1858 idealsqr_aux(GEN nf, GEN x, long tx)
1859 {
1860 GEN T = nf_get_pol(nf), m, cx, a, alpha;
1861 long N = degpol(T);
1862 switch(tx)
1863 {
1864 case id_PRINCIPAL:
1865 return idealhnf_principal(nf, nfsqr(nf,x));
1866 case id_PRIME:
1867 if (pr_is_inert(x)) return scalarmat(sqri(gel(x,1)), N);
1868 x = idealsqrprime(nf, x, &cx);
1869 x = idealhnf_two(nf,x);
1870 return cx? ZM_Z_mul(x, cx): x;
1871 default:
1872 x = Q_primitive_part(x, &cx);
1873 a = mat_ideal_two_elt(nf,x); alpha = gel(a,2); a = gel(a,1);
1874 alpha = nfsqr(nf,alpha);
1875 m = zk_scalar_or_multable(nf, alpha);
1876 if (typ(m) == t_INT) {
1877 x = gcdii(sqri(a), m);
1878 if (cx) x = gmul(x, gsqr(cx));
1879 x = scalarmat(x, N);
1880 }
1881 else
1882 { /* could use gcdii(sqri(a), zkmultable_capZ(m)), but costly */
1883 x = ZM_hnfmodid(m, sqri(a));
1884 if (cx) cx = gsqr(cx);
1885 if (cx) x = ZM_Q_mul(x, cx);
1886 }
1887 return x;
1888 }
1889 }
1890 GEN
idealsqr(GEN nf,GEN x)1891 idealsqr(GEN nf, GEN x)
1892 {
1893 pari_sp av;
1894 GEN res, ax, z;
1895 long tx = idealtyp(&x,&ax);
1896 res = ax? cgetg(3,t_VEC): NULL; /*product is an extended ideal*/
1897 av = avma;
1898 z = gerepileupto(av, idealsqr_aux(checknf(nf), x, tx));
1899 if (!ax) return z;
1900 gel(res,1) = z;
1901 gel(res,2) = ext_sqr(nf, ax); return res;
1902 }
1903
1904 /* norm of an ideal */
1905 GEN
idealnorm(GEN nf,GEN x)1906 idealnorm(GEN nf, GEN x)
1907 {
1908 pari_sp av;
1909 GEN y;
1910 long tx;
1911
1912 switch(idealtyp(&x,&y))
1913 {
1914 case id_PRIME: return pr_norm(x);
1915 case id_MAT: return RgM_det_triangular(x);
1916 }
1917 /* id_PRINCIPAL */
1918 nf = checknf(nf); av = avma;
1919 x = nfnorm(nf, x);
1920 tx = typ(x);
1921 if (tx == t_INT) return gerepileuptoint(av, absi(x));
1922 if (tx != t_FRAC) pari_err_TYPE("idealnorm",x);
1923 return gerepileupto(av, Q_abs(x));
1924 }
1925
1926 /* x \cap Z */
1927 GEN
idealdown(GEN nf,GEN x)1928 idealdown(GEN nf, GEN x)
1929 {
1930 pari_sp av = avma;
1931 GEN y, c;
1932 switch(idealtyp(&x,&y))
1933 {
1934 case id_PRIME: return icopy(pr_get_p(x));
1935 case id_MAT: return gcopy(gcoeff(x,1,1));
1936 }
1937 /* id_PRINCIPAL */
1938 nf = checknf(nf); av = avma;
1939 x = nf_to_scalar_or_basis(nf, x);
1940 if (is_rational_t(typ(x))) return Q_abs(x);
1941 x = Q_primitive_part(x, &c);
1942 y = zkmultable_capZ(zk_multable(nf, x));
1943 return gerepilecopy(av, mul_content(c, y));
1944 }
1945
1946 /* true nf */
1947 static GEN
idealismaximal_int(GEN nf,GEN p)1948 idealismaximal_int(GEN nf, GEN p)
1949 {
1950 GEN L;
1951 if (!BPSW_psp(p)) return NULL;
1952 if (!dvdii(nf_get_index(nf), p) &&
1953 !FpX_is_irred(FpX_red(nf_get_pol(nf),p), p)) return NULL;
1954 L = idealprimedec(nf, p);
1955 return lg(L) == 2? gel(L,1): NULL;
1956 }
1957 /* true nf */
1958 static GEN
idealismaximal_mat(GEN nf,GEN x)1959 idealismaximal_mat(GEN nf, GEN x)
1960 {
1961 GEN p, c, L;
1962 long i, l, f;
1963 x = Q_primitive_part(x, &c);
1964 p = gcoeff(x,1,1);
1965 if (c)
1966 {
1967 if (typ(c) == t_FRAC || !equali1(p)) return NULL;
1968 return idealismaximal_int(nf, p);
1969 }
1970 if (!BPSW_psp(p)) return NULL;
1971 l = lg(x); f = 1;
1972 for (i = 2; i < l; i++)
1973 {
1974 c = gcoeff(x,i,i);
1975 if (equalii(c, p)) f++; else if (!equali1(c)) return NULL;
1976 }
1977 L = idealprimedec_limit_f(nf, p, f);
1978 for (i = lg(L)-1; i; i--)
1979 {
1980 GEN pr = gel(L,i);
1981 if (pr_get_f(pr) != f) break;
1982 if (idealval(nf, x, pr) == 1) return pr;
1983 }
1984 return NULL;
1985 }
1986 /* true nf */
1987 static GEN
idealismaximal_i(GEN nf,GEN x)1988 idealismaximal_i(GEN nf, GEN x)
1989 {
1990 GEN L, p, pr, c;
1991 long i, l;
1992 switch(idealtyp(&x,&c))
1993 {
1994 case id_PRIME: return x;
1995 case id_MAT: return idealismaximal_mat(nf, x);
1996 }
1997 /* id_PRINCIPAL */
1998 nf = checknf(nf);
1999 x = nf_to_scalar_or_basis(nf, x);
2000 switch(typ(x))
2001 {
2002 case t_INT: return idealismaximal_int(nf, absi_shallow(x));
2003 case t_FRAC: return NULL;
2004 }
2005 x = Q_primitive_part(x, &c);
2006 if (c) return NULL;
2007 p = zkmultable_capZ(zk_multable(nf, x));
2008 L = idealprimedec(nf, p); l = lg(L); pr = NULL;
2009 for (i = 1; i < l; i++)
2010 {
2011 long v = ZC_nfval(x, gel(L,i));
2012 if (v > 1 || (v && pr)) return NULL;
2013 pr = gel(L,i);
2014 }
2015 return pr;
2016 }
2017 GEN
idealismaximal(GEN nf,GEN x)2018 idealismaximal(GEN nf, GEN x)
2019 {
2020 pari_sp av = avma;
2021 x = idealismaximal_i(checknf(nf), x);
2022 if (!x) { set_avma(av); return gen_0; }
2023 return gerepilecopy(av, x);
2024 }
2025
2026 /* I^(-1) = { x \in K, Tr(x D^(-1) I) \in Z }, D different of K/Q
2027 *
2028 * nf[5][6] = pp( D^(-1) ) = pp( HNF( T^(-1) ) ), T = (Tr(wi wj))
2029 * nf[5][7] = same in 2-elt form.
2030 * Assume I integral. Return the integral ideal (I\cap Z) I^(-1) */
2031 GEN
idealHNF_inv_Z(GEN nf,GEN I)2032 idealHNF_inv_Z(GEN nf, GEN I)
2033 {
2034 GEN J, dual, IZ = gcoeff(I,1,1); /* I \cap Z */
2035 if (isint1(IZ)) return matid(lg(I)-1);
2036 J = idealHNF_mul(nf,I, gmael(nf,5,7));
2037 /* I in HNF, hence easily inverted; multiply by IZ to get integer coeffs
2038 * missing content cancels while solving the linear equation */
2039 dual = shallowtrans( hnf_divscale(J, gmael(nf,5,6), IZ) );
2040 return ZM_hnfmodid(dual, IZ);
2041 }
2042 /* I HNF with rational coefficients (denominator d). */
2043 GEN
idealHNF_inv(GEN nf,GEN I)2044 idealHNF_inv(GEN nf, GEN I)
2045 {
2046 GEN J, IQ = gcoeff(I,1,1); /* I \cap Q; d IQ = dI \cap Z */
2047 J = idealHNF_inv_Z(nf, Q_remove_denom(I, NULL)); /* = (dI)^(-1) * (d IQ) */
2048 return equali1(IQ)? J: RgM_Rg_div(J, IQ);
2049 }
2050
2051 /* return p * P^(-1) [integral] */
2052 GEN
pr_inv_p(GEN pr)2053 pr_inv_p(GEN pr)
2054 {
2055 if (pr_is_inert(pr)) return matid(pr_get_f(pr));
2056 return ZM_hnfmodid(pr_get_tau(pr), pr_get_p(pr));
2057 }
2058 GEN
pr_inv(GEN pr)2059 pr_inv(GEN pr)
2060 {
2061 GEN p = pr_get_p(pr);
2062 if (pr_is_inert(pr)) return scalarmat(ginv(p), pr_get_f(pr));
2063 return RgM_Rg_div(ZM_hnfmodid(pr_get_tau(pr),p), p);
2064 }
2065
2066 GEN
idealinv(GEN nf,GEN x)2067 idealinv(GEN nf, GEN x)
2068 {
2069 GEN res, ax;
2070 pari_sp av;
2071 long tx = idealtyp(&x,&ax), N;
2072
2073 res = ax? cgetg(3,t_VEC): NULL;
2074 nf = checknf(nf); av = avma;
2075 N = nf_get_degree(nf);
2076 switch (tx)
2077 {
2078 case id_MAT:
2079 if (lg(x)-1 != N) pari_err_DIM("idealinv");
2080 x = idealHNF_inv(nf,x); break;
2081 case id_PRINCIPAL:
2082 x = nf_to_scalar_or_basis(nf, x);
2083 if (typ(x) != t_COL)
2084 x = idealhnf_principal(nf,ginv(x));
2085 else
2086 { /* nfinv + idealhnf where we already know (x) \cap Z */
2087 GEN c, d;
2088 x = Q_remove_denom(x, &c);
2089 x = zk_inv(nf, x);
2090 x = Q_remove_denom(x, &d); /* true inverse is c/d * x */
2091 if (!d) /* x and x^(-1) integral => x a unit */
2092 x = c? scalarmat(c, N): matid(N);
2093 else
2094 {
2095 c = c? gdiv(c,d): ginv(d);
2096 x = zk_multable(nf, x);
2097 x = ZM_Q_mul(ZM_hnfmodid(x,d), c);
2098 }
2099 }
2100 break;
2101 case id_PRIME:
2102 x = pr_inv(x); break;
2103 }
2104 x = gerepileupto(av,x); if (!ax) return x;
2105 gel(res,1) = x;
2106 gel(res,2) = ext_inv(nf, ax); return res;
2107 }
2108
2109 /* write x = A/B, A,B coprime integral ideals */
2110 GEN
idealnumden(GEN nf,GEN x)2111 idealnumden(GEN nf, GEN x)
2112 {
2113 pari_sp av = avma;
2114 GEN x0, ax, c, d, A, B, J;
2115 long tx = idealtyp(&x,&ax);
2116 nf = checknf(nf);
2117 switch (tx)
2118 {
2119 case id_PRIME:
2120 retmkvec2(idealhnf(nf, x), gen_1);
2121 case id_PRINCIPAL:
2122 {
2123 GEN xZ, mx;
2124 x = nf_to_scalar_or_basis(nf, x);
2125 switch(typ(x))
2126 {
2127 case t_INT: return gerepilecopy(av, mkvec2(absi_shallow(x),gen_1));
2128 case t_FRAC:return gerepilecopy(av, mkvec2(absi_shallow(gel(x,1)), gel(x,2)));
2129 }
2130 /* t_COL */
2131 x = Q_remove_denom(x, &d);
2132 if (!d) return gerepilecopy(av, mkvec2(idealhnf(nf, x), gen_1));
2133 mx = zk_multable(nf, x);
2134 xZ = zkmultable_capZ(mx);
2135 x = ZM_hnfmodid(mx, xZ); /* principal ideal (x) */
2136 x0 = mkvec2(xZ, mx); /* same, for fast multiplication */
2137 break;
2138 }
2139 default: /* id_MAT */
2140 {
2141 long n = lg(x)-1;
2142 if (n == 0) return mkvec2(gen_0, gen_1);
2143 if (n != nf_get_degree(nf)) pari_err_DIM("idealnumden");
2144 x0 = x = Q_remove_denom(x, &d);
2145 if (!d) return gerepilecopy(av, mkvec2(x, gen_1));
2146 break;
2147 }
2148 }
2149 J = hnfmodid(x, d); /* = d/B */
2150 c = gcoeff(J,1,1); /* (d/B) \cap Z, divides d */
2151 B = idealHNF_inv_Z(nf, J); /* (d/B \cap Z) B/d */
2152 if (!equalii(c,d)) B = ZM_Z_mul(B, diviiexact(d,c)); /* = B ! */
2153 A = idealHNF_mul(nf, B, x0); /* d * (original x) * B = d A */
2154 A = ZM_Z_divexact(A, d); /* = A ! */
2155 return gerepilecopy(av, mkvec2(A, B));
2156 }
2157
2158 /* Return x, integral in 2-elt form, such that pr^n = c * x. Assume n != 0.
2159 * nf = true nf */
2160 static GEN
idealpowprime(GEN nf,GEN pr,GEN n,GEN * pc)2161 idealpowprime(GEN nf, GEN pr, GEN n, GEN *pc)
2162 {
2163 GEN p = pr_get_p(pr), q, gen;
2164
2165 *pc = NULL;
2166 if (is_pm1(n)) /* n = 1 special cased for efficiency */
2167 {
2168 q = p;
2169 if (typ(pr_get_tau(pr)) == t_INT) /* inert */
2170 {
2171 *pc = (signe(n) >= 0)? p: ginv(p);
2172 return mkvec2(gen_1,gen_0);
2173 }
2174 if (signe(n) >= 0) gen = pr_get_gen(pr);
2175 else
2176 {
2177 gen = pr_get_tau(pr); /* possibly t_MAT */
2178 *pc = ginv(p);
2179 }
2180 }
2181 else if (equalis(n,2)) return idealsqrprime(nf, pr, pc);
2182 else
2183 {
2184 long e = pr_get_e(pr), f = pr_get_f(pr);
2185 GEN r, m = truedvmdis(n, e, &r);
2186 if (e * f == nf_get_degree(nf))
2187 { /* pr^e = (p) */
2188 if (signe(m)) *pc = powii(p,m);
2189 if (!signe(r)) return mkvec2(gen_1,gen_0);
2190 q = p;
2191 gen = nfpow(nf, pr_get_gen(pr), r);
2192 }
2193 else
2194 {
2195 m = absi_shallow(m);
2196 if (signe(r)) m = addiu(m,1);
2197 q = powii(p,m); /* m = ceil(|n|/e) */
2198 if (signe(n) >= 0) gen = nfpow(nf, pr_get_gen(pr), n);
2199 else
2200 {
2201 gen = pr_get_tau(pr);
2202 if (typ(gen) == t_MAT) gen = gel(gen,1);
2203 n = negi(n);
2204 gen = ZC_Z_divexact(nfpow(nf, gen, n), powii(p, subii(n,m)));
2205 *pc = ginv(q);
2206 }
2207 }
2208 gen = FpC_red(gen, q);
2209 }
2210 return mkvec2(q, gen);
2211 }
2212
2213 /* x * pr^n. Assume x in HNF or scalar (possibly nonintegral) */
2214 GEN
idealmulpowprime(GEN nf,GEN x,GEN pr,GEN n)2215 idealmulpowprime(GEN nf, GEN x, GEN pr, GEN n)
2216 {
2217 GEN c, cx, y;
2218 long N;
2219
2220 nf = checknf(nf);
2221 N = nf_get_degree(nf);
2222 if (!signe(n)) return typ(x) == t_MAT? x: scalarmat_shallow(x, N);
2223
2224 /* inert, special cased for efficiency */
2225 if (pr_is_inert(pr))
2226 {
2227 GEN q = powii(pr_get_p(pr), n);
2228 return typ(x) == t_MAT? RgM_Rg_mul(x,q)
2229 : scalarmat_shallow(gmul(Q_abs(x),q), N);
2230 }
2231
2232 y = idealpowprime(nf, pr, n, &c);
2233 if (typ(x) == t_MAT)
2234 { x = Q_primitive_part(x, &cx); if (is_pm1(gcoeff(x,1,1))) x = NULL; }
2235 else
2236 { cx = x; x = NULL; }
2237 cx = mul_content(c,cx);
2238 if (x)
2239 x = idealHNF_mul_two(nf,x,y);
2240 else
2241 x = idealhnf_two(nf,y);
2242 if (cx) x = ZM_Q_mul(x,cx);
2243 return x;
2244 }
2245 GEN
idealdivpowprime(GEN nf,GEN x,GEN pr,GEN n)2246 idealdivpowprime(GEN nf, GEN x, GEN pr, GEN n)
2247 {
2248 return idealmulpowprime(nf,x,pr, negi(n));
2249 }
2250
2251 /* nf = true nf */
2252 static GEN
idealpow_aux(GEN nf,GEN x,long tx,GEN n)2253 idealpow_aux(GEN nf, GEN x, long tx, GEN n)
2254 {
2255 GEN T = nf_get_pol(nf), m, cx, n1, a, alpha;
2256 long N = degpol(T), s = signe(n);
2257 if (!s) return matid(N);
2258 switch(tx)
2259 {
2260 case id_PRINCIPAL:
2261 return idealhnf_principal(nf, nfpow(nf,x,n));
2262 case id_PRIME:
2263 if (pr_is_inert(x)) return scalarmat(powii(gel(x,1), n), N);
2264 x = idealpowprime(nf, x, n, &cx);
2265 x = idealhnf_two(nf,x);
2266 return cx? ZM_Q_mul(x, cx): x;
2267 default:
2268 if (is_pm1(n)) return (s < 0)? idealinv(nf, x): gcopy(x);
2269 n1 = (s < 0)? negi(n): n;
2270
2271 x = Q_primitive_part(x, &cx);
2272 a = mat_ideal_two_elt(nf,x); alpha = gel(a,2); a = gel(a,1);
2273 alpha = nfpow(nf,alpha,n1);
2274 m = zk_scalar_or_multable(nf, alpha);
2275 if (typ(m) == t_INT) {
2276 x = gcdii(powii(a,n1), m);
2277 if (s<0) x = ginv(x);
2278 if (cx) x = gmul(x, powgi(cx,n));
2279 x = scalarmat(x, N);
2280 }
2281 else
2282 { /* could use gcdii(powii(a,n1), zkmultable_capZ(m)), but costly */
2283 x = ZM_hnfmodid(m, powii(a,n1));
2284 if (cx) cx = powgi(cx,n);
2285 if (s<0) {
2286 GEN xZ = gcoeff(x,1,1);
2287 cx = cx ? gdiv(cx, xZ): ginv(xZ);
2288 x = idealHNF_inv_Z(nf,x);
2289 }
2290 if (cx) x = ZM_Q_mul(x, cx);
2291 }
2292 return x;
2293 }
2294 }
2295
2296 /* raise the ideal x to the power n (in Z) */
2297 GEN
idealpow(GEN nf,GEN x,GEN n)2298 idealpow(GEN nf, GEN x, GEN n)
2299 {
2300 pari_sp av;
2301 long tx;
2302 GEN res, ax;
2303
2304 if (typ(n) != t_INT) pari_err_TYPE("idealpow",n);
2305 tx = idealtyp(&x,&ax);
2306 res = ax? cgetg(3,t_VEC): NULL;
2307 av = avma;
2308 x = gerepileupto(av, idealpow_aux(checknf(nf), x, tx, n));
2309 if (!ax) return x;
2310 ax = ext_pow(nf, ax, n);
2311 gel(res,1) = x;
2312 gel(res,2) = ax;
2313 return res;
2314 }
2315
2316 /* Return ideal^e in number field nf. e is a C integer. */
2317 GEN
idealpows(GEN nf,GEN ideal,long e)2318 idealpows(GEN nf, GEN ideal, long e)
2319 {
2320 long court[] = {evaltyp(t_INT) | _evallg(3),0,0};
2321 affsi(e,court); return idealpow(nf,ideal,court);
2322 }
2323
2324 static GEN
_idealmulred(GEN nf,GEN x,GEN y)2325 _idealmulred(GEN nf, GEN x, GEN y)
2326 { return idealred(nf,idealmul(nf,x,y)); }
2327 static GEN
_idealsqrred(GEN nf,GEN x)2328 _idealsqrred(GEN nf, GEN x)
2329 { return idealred(nf,idealsqr(nf,x)); }
2330 static GEN
_mul(void * data,GEN x,GEN y)2331 _mul(void *data, GEN x, GEN y) { return _idealmulred((GEN)data,x,y); }
2332 static GEN
_sqr(void * data,GEN x)2333 _sqr(void *data, GEN x) { return _idealsqrred((GEN)data, x); }
2334
2335 /* compute x^n (x ideal, n integer), reducing along the way */
2336 GEN
idealpowred(GEN nf,GEN x,GEN n)2337 idealpowred(GEN nf, GEN x, GEN n)
2338 {
2339 pari_sp av = avma, av2;
2340 long s;
2341 GEN y;
2342
2343 if (typ(n) != t_INT) pari_err_TYPE("idealpowred",n);
2344 s = signe(n); if (s == 0) return idealpow(nf,x,n);
2345 y = gen_pow_i(x, n, (void*)nf, &_sqr, &_mul);
2346 av2 = avma;
2347 if (s < 0) y = idealinv(nf,y);
2348 if (s < 0 || is_pm1(n)) y = idealred(nf,y);
2349 return avma == av2? gerepilecopy(av,y): gerepileupto(av,y);
2350 }
2351
2352 GEN
idealmulred(GEN nf,GEN x,GEN y)2353 idealmulred(GEN nf, GEN x, GEN y)
2354 {
2355 pari_sp av = avma;
2356 return gerepileupto(av, _idealmulred(nf,x,y));
2357 }
2358
2359 long
isideal(GEN nf,GEN x)2360 isideal(GEN nf,GEN x)
2361 {
2362 long N, i, j, lx, tx = typ(x);
2363 pari_sp av;
2364 GEN T, xZ;
2365
2366 nf = checknf(nf); T = nf_get_pol(nf); lx = lg(x);
2367 if (tx==t_VEC && lx==3) { x = gel(x,1); tx = typ(x); lx = lg(x); }
2368 switch(tx)
2369 {
2370 case t_INT: case t_FRAC: return 1;
2371 case t_POL: return varn(x) == varn(T);
2372 case t_POLMOD: return RgX_equal_var(T, gel(x,1));
2373 case t_VEC: return get_prid(x)? 1 : 0;
2374 case t_MAT: break;
2375 default: return 0;
2376 }
2377 N = degpol(T);
2378 if (lx-1 != N) return (lx == 1);
2379 if (nbrows(x) != N) return 0;
2380
2381 av = avma; x = Q_primpart(x);
2382 if (!ZM_ishnf(x)) return 0;
2383 xZ = gcoeff(x,1,1);
2384 for (j=2; j<=N; j++)
2385 if (!dvdii(xZ, gcoeff(x,j,j))) return gc_long(av,0);
2386 for (i=2; i<=N; i++)
2387 for (j=2; j<=N; j++)
2388 if (! hnf_invimage(x, zk_ei_mul(nf,gel(x,i),j))) return gc_long(av,0);
2389 return gc_long(av,1);
2390 }
2391
2392 GEN
idealdiv(GEN nf,GEN x,GEN y)2393 idealdiv(GEN nf, GEN x, GEN y)
2394 {
2395 pari_sp av = avma, tetpil;
2396 GEN z = idealinv(nf,y);
2397 tetpil = avma; return gerepile(av,tetpil, idealmul(nf,x,z));
2398 }
2399
2400 /* This routine computes the quotient x/y of two ideals in the number field nf.
2401 * It assumes that the quotient is an integral ideal. The idea is to find an
2402 * ideal z dividing y such that gcd(Nx/Nz, Nz) = 1. Then
2403 *
2404 * x + (Nx/Nz) x
2405 * ----------- = ---
2406 * y + (Ny/Nz) y
2407 *
2408 * Proof: we can assume x and y are integral. Let p be any prime ideal
2409 *
2410 * If p | Nz, then it divides neither Nx/Nz nor Ny/Nz (since Nx/Nz is the
2411 * product of the integers N(x/y) and N(y/z)). Both the numerator and the
2412 * denominator on the left will be coprime to p. So will x/y, since x/y is
2413 * assumed integral and its norm N(x/y) is coprime to p.
2414 *
2415 * If instead p does not divide Nz, then v_p (Nx/Nz) = v_p (Nx) >= v_p(x).
2416 * Hence v_p (x + Nx/Nz) = v_p(x). Likewise for the denominators. QED.
2417 *
2418 * Peter Montgomery. July, 1994. */
2419 static void
err_divexact(GEN x,GEN y)2420 err_divexact(GEN x, GEN y)
2421 { pari_err_DOMAIN("idealdivexact","denominator(x/y)", "!=",
2422 gen_1,mkvec2(x,y)); }
2423 GEN
idealdivexact(GEN nf,GEN x0,GEN y0)2424 idealdivexact(GEN nf, GEN x0, GEN y0)
2425 {
2426 pari_sp av = avma;
2427 GEN x, y, xZ, yZ, Nx, Ny, Nz, cy, q, r;
2428
2429 nf = checknf(nf);
2430 x = idealhnf_shallow(nf, x0);
2431 y = idealhnf_shallow(nf, y0);
2432 if (lg(y) == 1) pari_err_INV("idealdivexact", y0);
2433 if (lg(x) == 1) { set_avma(av); return cgetg(1, t_MAT); } /* numerator is zero */
2434 y = Q_primitive_part(y, &cy);
2435 if (cy) x = RgM_Rg_div(x,cy);
2436 xZ = gcoeff(x,1,1); if (typ(xZ) != t_INT) err_divexact(x,y);
2437 yZ = gcoeff(y,1,1); if (isint1(yZ)) return gerepilecopy(av, x);
2438 Nx = idealnorm(nf,x);
2439 Ny = idealnorm(nf,y);
2440 if (typ(Nx) != t_INT) err_divexact(x,y);
2441 q = dvmdii(Nx,Ny, &r);
2442 if (signe(r)) err_divexact(x,y);
2443 if (is_pm1(q)) { set_avma(av); return matid(nf_get_degree(nf)); }
2444 /* Find a norm Nz | Ny such that gcd(Nx/Nz, Nz) = 1 */
2445 for (Nz = Ny;;) /* q = Nx/Nz */
2446 {
2447 GEN p1 = gcdii(Nz, q);
2448 if (is_pm1(p1)) break;
2449 Nz = diviiexact(Nz,p1);
2450 q = mulii(q,p1);
2451 }
2452 xZ = gcoeff(x,1,1); q = gcdii(q, xZ);
2453 if (!equalii(xZ,q))
2454 { /* Replace x/y by x+(Nx/Nz) / y+(Ny/Nz) */
2455 x = ZM_hnfmodid(x, q);
2456 /* y reduced to unit ideal ? */
2457 if (Nz == Ny) return gerepileupto(av, x);
2458
2459 yZ = gcoeff(y,1,1); q = gcdii(diviiexact(Ny,Nz), yZ);
2460 y = ZM_hnfmodid(y, q);
2461 }
2462 yZ = gcoeff(y,1,1);
2463 y = idealHNF_mul(nf,x, idealHNF_inv_Z(nf,y));
2464 return gerepileupto(av, ZM_Z_divexact(y, yZ));
2465 }
2466
2467 GEN
idealintersect(GEN nf,GEN x,GEN y)2468 idealintersect(GEN nf, GEN x, GEN y)
2469 {
2470 pari_sp av = avma;
2471 long lz, lx, i;
2472 GEN z, dx, dy, xZ, yZ;;
2473
2474 nf = checknf(nf);
2475 x = idealhnf_shallow(nf,x);
2476 y = idealhnf_shallow(nf,y);
2477 if (lg(x) == 1 || lg(y) == 1) { set_avma(av); return cgetg(1,t_MAT); }
2478 x = Q_remove_denom(x, &dx);
2479 y = Q_remove_denom(y, &dy);
2480 if (dx) y = ZM_Z_mul(y, dx);
2481 if (dy) x = ZM_Z_mul(x, dy);
2482 xZ = gcoeff(x,1,1);
2483 yZ = gcoeff(y,1,1);
2484 dx = mul_denom(dx,dy);
2485 z = ZM_lll(shallowconcat(x,y), 0.99, LLL_KER); lz = lg(z);
2486 lx = lg(x);
2487 for (i=1; i<lz; i++) setlg(z[i], lx);
2488 z = ZM_hnfmodid(ZM_mul(x,z), lcmii(xZ, yZ));
2489 if (dx) z = RgM_Rg_div(z,dx);
2490 return gerepileupto(av,z);
2491 }
2492
2493 /*******************************************************************/
2494 /* */
2495 /* T2-IDEAL REDUCTION */
2496 /* */
2497 /*******************************************************************/
2498
2499 static GEN
chk_vdir(GEN nf,GEN vdir)2500 chk_vdir(GEN nf, GEN vdir)
2501 {
2502 long i, l = lg(vdir);
2503 GEN v;
2504 if (l != lg(nf_get_roots(nf))) pari_err_DIM("idealred");
2505 switch(typ(vdir))
2506 {
2507 case t_VECSMALL: return vdir;
2508 case t_VEC: break;
2509 default: pari_err_TYPE("idealred",vdir);
2510 }
2511 v = cgetg(l, t_VECSMALL);
2512 for (i = 1; i < l; i++) v[i] = itos(gceil(gel(vdir,i)));
2513 return v;
2514 }
2515
2516 static void
twistG(GEN G,long r1,long i,long v)2517 twistG(GEN G, long r1, long i, long v)
2518 {
2519 long j, lG = lg(G);
2520 if (i <= r1) {
2521 for (j=1; j<lG; j++) gcoeff(G,i,j) = gmul2n(gcoeff(G,i,j), v);
2522 } else {
2523 long k = (i<<1) - r1;
2524 for (j=1; j<lG; j++)
2525 {
2526 gcoeff(G,k-1,j) = gmul2n(gcoeff(G,k-1,j), v);
2527 gcoeff(G,k ,j) = gmul2n(gcoeff(G,k ,j), v);
2528 }
2529 }
2530 }
2531
2532 GEN
nf_get_Gtwist(GEN nf,GEN vdir)2533 nf_get_Gtwist(GEN nf, GEN vdir)
2534 {
2535 long i, l, v, r1;
2536 GEN G;
2537
2538 if (!vdir) return nf_get_roundG(nf);
2539 if (typ(vdir) == t_MAT)
2540 {
2541 long N = nf_get_degree(nf);
2542 if (lg(vdir) != N+1 || lgcols(vdir) != N+1) pari_err_DIM("idealred");
2543 return vdir;
2544 }
2545 vdir = chk_vdir(nf, vdir);
2546 G = RgM_shallowcopy(nf_get_G(nf));
2547 r1 = nf_get_r1(nf);
2548 l = lg(vdir);
2549 for (i=1; i<l; i++)
2550 {
2551 v = vdir[i]; if (!v) continue;
2552 twistG(G, r1, i, v);
2553 }
2554 return RM_round_maxrank(G);
2555 }
2556 GEN
nf_get_Gtwist1(GEN nf,long i)2557 nf_get_Gtwist1(GEN nf, long i)
2558 {
2559 GEN G = RgM_shallowcopy( nf_get_G(nf) );
2560 long r1 = nf_get_r1(nf);
2561 twistG(G, r1, i, 10);
2562 return RM_round_maxrank(G);
2563 }
2564
2565 GEN
RM_round_maxrank(GEN G0)2566 RM_round_maxrank(GEN G0)
2567 {
2568 long e, r = lg(G0)-1;
2569 pari_sp av = avma;
2570 for (e = 4; ; e <<= 1, set_avma(av))
2571 {
2572 GEN G = gmul2n(G0, e), H = ground(G);
2573 if (ZM_rank(H) == r) return H; /* maximal rank ? */
2574 }
2575 }
2576
2577 GEN
idealred0(GEN nf,GEN I,GEN vdir)2578 idealred0(GEN nf, GEN I, GEN vdir)
2579 {
2580 pari_sp av = avma;
2581 GEN G, aI, IZ, J, y, yZ, my, c1 = NULL;
2582 long N;
2583
2584 nf = checknf(nf);
2585 N = nf_get_degree(nf);
2586 /* put first for sanity checks, unused when I obviously principal */
2587 G = nf_get_Gtwist(nf, vdir);
2588 switch (idealtyp(&I,&aI))
2589 {
2590 case id_PRIME:
2591 if (pr_is_inert(I)) {
2592 if (!aI) { set_avma(av); return matid(N); }
2593 c1 = gel(I,1); I = matid(N);
2594 goto END;
2595 }
2596 IZ = pr_get_p(I);
2597 J = pr_inv_p(I);
2598 I = idealhnf_two(nf,I);
2599 break;
2600 case id_MAT:
2601 I = Q_primitive_part(I, &c1);
2602 IZ = gcoeff(I,1,1);
2603 if (is_pm1(IZ))
2604 {
2605 if (!aI) { set_avma(av); return matid(N); }
2606 goto END;
2607 }
2608 J = idealHNF_inv_Z(nf, I);
2609 break;
2610 default: /* id_PRINCIPAL, silly case */
2611 if (gequal0(I)) I = cgetg(1,t_MAT); else { c1 = I; I = matid(N); }
2612 if (!aI) return I;
2613 goto END;
2614 }
2615 /* now I integral, HNF; and J = (I\cap Z) I^(-1), integral */
2616 y = idealpseudomin(J, G); /* small elt in (I\cap Z)I^(-1), integral */
2617 if (equalii(ZV_content(y), IZ))
2618 { /* already reduced */
2619 if (!aI) return gerepilecopy(av, I);
2620 goto END;
2621 }
2622
2623 my = zk_multable(nf, y);
2624 I = ZM_Z_divexact(ZM_mul(my, I), IZ); /* y I / (I\cap Z), integral */
2625 c1 = mul_content(c1, IZ);
2626 my = ZM_gauss(my, col_ei(N,1)); /* y^-1 */
2627 yZ = Q_denom(my); /* (y) \cap Z */
2628 I = hnfmodid(I, yZ);
2629 if (!aI) return gerepileupto(av, I);
2630 c1 = RgC_Rg_mul(my, c1);
2631 END:
2632 if (c1) aI = ext_mul(nf, aI,c1);
2633 return gerepilecopy(av, mkvec2(I, aI));
2634 }
2635
2636 GEN
idealmin(GEN nf,GEN x,GEN vdir)2637 idealmin(GEN nf, GEN x, GEN vdir)
2638 {
2639 pari_sp av = avma;
2640 GEN y, dx;
2641 nf = checknf(nf);
2642 switch( idealtyp(&x,&y) )
2643 {
2644 case id_PRINCIPAL: return gcopy(x);
2645 case id_PRIME: x = pr_hnf(nf,x); break;
2646 case id_MAT: if (lg(x) == 1) return gen_0;
2647 }
2648 x = Q_remove_denom(x, &dx);
2649 y = idealpseudomin(x, nf_get_Gtwist(nf,vdir));
2650 if (dx) y = RgC_Rg_div(y, dx);
2651 return gerepileupto(av, y);
2652 }
2653
2654 /*******************************************************************/
2655 /* */
2656 /* APPROXIMATION THEOREM */
2657 /* */
2658 /*******************************************************************/
2659 /* a = ppi(a,b) ppo(a,b), where ppi regroups primes common to a and b
2660 * and ppo(a,b) = Z_ppo(a,b) */
2661 /* return gcd(a,b),ppi(a,b),ppo(a,b) */
2662 GEN
Z_ppio(GEN a,GEN b)2663 Z_ppio(GEN a, GEN b)
2664 {
2665 GEN x, y, d = gcdii(a,b);
2666 if (is_pm1(d)) return mkvec3(gen_1, gen_1, a);
2667 x = d; y = diviiexact(a,d);
2668 for(;;)
2669 {
2670 GEN g = gcdii(x,y);
2671 if (is_pm1(g)) return mkvec3(d, x, y);
2672 x = mulii(x,g); y = diviiexact(y,g);
2673 }
2674 }
2675 /* a = ppg(a,b)pple(a,b), where ppg regroups primes such that v(a) > v(b)
2676 * and pple all others */
2677 /* return gcd(a,b),ppg(a,b),pple(a,b) */
2678 GEN
Z_ppgle(GEN a,GEN b)2679 Z_ppgle(GEN a, GEN b)
2680 {
2681 GEN x, y, g, d = gcdii(a,b);
2682 if (equalii(a, d)) return mkvec3(a, gen_1, a);
2683 x = diviiexact(a,d); y = d;
2684 for(;;)
2685 {
2686 g = gcdii(x,y);
2687 if (is_pm1(g)) return mkvec3(d, x, y);
2688 x = mulii(x,g); y = diviiexact(y,g);
2689 }
2690 }
2691 static void
Z_dcba_rec(GEN L,GEN a,GEN b)2692 Z_dcba_rec(GEN L, GEN a, GEN b)
2693 {
2694 GEN x, r, v, g, h, c, c0;
2695 long n;
2696 if (is_pm1(b)) {
2697 if (!is_pm1(a)) vectrunc_append(L, a);
2698 return;
2699 }
2700 v = Z_ppio(a,b);
2701 a = gel(v,2);
2702 r = gel(v,3);
2703 if (!is_pm1(r)) vectrunc_append(L, r);
2704 v = Z_ppgle(a,b);
2705 g = gel(v,1);
2706 h = gel(v,2);
2707 x = c0 = gel(v,3);
2708 for (n = 1; !is_pm1(h); n++)
2709 {
2710 GEN d, y;
2711 long i;
2712 v = Z_ppgle(h,sqri(g));
2713 g = gel(v,1);
2714 h = gel(v,2);
2715 c = gel(v,3); if (is_pm1(c)) continue;
2716 d = gcdii(c,b);
2717 x = mulii(x,d);
2718 y = d; for (i=1; i < n; i++) y = sqri(y);
2719 Z_dcba_rec(L, diviiexact(c,y), d);
2720 }
2721 Z_dcba_rec(L,diviiexact(b,x), c0);
2722 }
2723 static GEN
Z_cba_rec(GEN L,GEN a,GEN b)2724 Z_cba_rec(GEN L, GEN a, GEN b)
2725 {
2726 GEN g;
2727 if (lg(L) > 10)
2728 { /* a few naive steps before switching to dcba */
2729 Z_dcba_rec(L, a, b);
2730 return gel(L, lg(L)-1);
2731 }
2732 if (is_pm1(a)) return b;
2733 g = gcdii(a,b);
2734 if (is_pm1(g)) { vectrunc_append(L, a); return b; }
2735 a = diviiexact(a,g);
2736 b = diviiexact(b,g);
2737 return Z_cba_rec(L, Z_cba_rec(L, a, g), b);
2738 }
2739 GEN
Z_cba(GEN a,GEN b)2740 Z_cba(GEN a, GEN b)
2741 {
2742 GEN L = vectrunc_init(expi(a) + expi(b) + 2);
2743 GEN t = Z_cba_rec(L, a, b);
2744 if (!is_pm1(t)) vectrunc_append(L, t);
2745 return L;
2746 }
2747 /* P = coprime base, extend it by b; TODO: quadratic for now */
2748 GEN
ZV_cba_extend(GEN P,GEN b)2749 ZV_cba_extend(GEN P, GEN b)
2750 {
2751 long i, l = lg(P);
2752 GEN w = cgetg(l+1, t_VEC);
2753 for (i = 1; i < l; i++)
2754 {
2755 GEN v = Z_cba(gel(P,i), b);
2756 long nv = lg(v)-1;
2757 gel(w,i) = vecslice(v, 1, nv-1); /* those divide P[i] but not b */
2758 b = gel(v,nv);
2759 }
2760 gel(w,l) = b; return shallowconcat1(w);
2761 }
2762 GEN
ZV_cba(GEN v)2763 ZV_cba(GEN v)
2764 {
2765 long i, l = lg(v);
2766 GEN P;
2767 if (l <= 2) return v;
2768 P = Z_cba(gel(v,1), gel(v,2));
2769 for (i = 3; i < l; i++) P = ZV_cba_extend(P, gel(v,i));
2770 return P;
2771 }
2772
2773 /* write x = x1 x2, x2 maximal s.t. (x2,f) = 1, return x2 */
2774 GEN
Z_ppo(GEN x,GEN f)2775 Z_ppo(GEN x, GEN f)
2776 {
2777 for (;;)
2778 {
2779 f = gcdii(x, f); if (is_pm1(f)) break;
2780 x = diviiexact(x, f);
2781 }
2782 return x;
2783 }
2784 /* write x = x1 x2, x2 maximal s.t. (x2,f) = 1, return x2 */
2785 ulong
u_ppo(ulong x,ulong f)2786 u_ppo(ulong x, ulong f)
2787 {
2788 for (;;)
2789 {
2790 f = ugcd(x, f); if (f == 1) break;
2791 x /= f;
2792 }
2793 return x;
2794 }
2795
2796 /* result known to be representable as an ulong */
2797 static ulong
lcmuu(ulong a,ulong b)2798 lcmuu(ulong a, ulong b) { ulong d = ugcd(a,b); return (a/d) * b; }
2799
2800 /* assume 0 < x < N; return u in (Z/NZ)^* such that u x = gcd(x,N) (mod N);
2801 * set *pd = gcd(x,N) */
2802 ulong
Fl_invgen(ulong x,ulong N,ulong * pd)2803 Fl_invgen(ulong x, ulong N, ulong *pd)
2804 {
2805 ulong d, d0, e, v, v1;
2806 long s;
2807 *pd = d = xgcduu(N, x, 0, &v, &v1, &s);
2808 if (s > 0) v = N - v;
2809 if (d == 1) return v;
2810 /* vx = gcd(x,N) (mod N), v coprime to N/d but need not be coprime to N */
2811 e = N / d;
2812 d0 = u_ppo(d, e); /* d = d0 d1, d0 coprime to N/d, rad(d1) | N/d */
2813 if (d0 == 1) return v;
2814 e = lcmuu(e, d / d0);
2815 return u_chinese_coprime(v, 1, e, d0, e*d0);
2816 }
2817
2818 /* x t_INT, f ideal. Write x = x1 x2, sqf(x1) | f, (x2,f) = 1. Return x2 */
2819 static GEN
nf_coprime_part(GEN nf,GEN x,GEN listpr)2820 nf_coprime_part(GEN nf, GEN x, GEN listpr)
2821 {
2822 long v, j, lp = lg(listpr), N = nf_get_degree(nf);
2823 GEN x1, x2, ex;
2824
2825 #if 0 /*1) via many gcds. Expensive ! */
2826 GEN f = idealprodprime(nf, listpr);
2827 f = ZM_hnfmodid(f, x); /* first gcd is less expensive since x in Z */
2828 x = scalarmat(x, N);
2829 for (;;)
2830 {
2831 if (gequal1(gcoeff(f,1,1))) break;
2832 x = idealdivexact(nf, x, f);
2833 f = ZM_hnfmodid(shallowconcat(f,x), gcoeff(x,1,1)); /* gcd(f,x) */
2834 }
2835 x2 = x;
2836 #else /*2) from prime decomposition */
2837 x1 = NULL;
2838 for (j=1; j<lp; j++)
2839 {
2840 GEN pr = gel(listpr,j);
2841 v = Z_pval(x, pr_get_p(pr)); if (!v) continue;
2842
2843 ex = muluu(v, pr_get_e(pr)); /* = v_pr(x) > 0 */
2844 x1 = x1? idealmulpowprime(nf, x1, pr, ex)
2845 : idealpow(nf, pr, ex);
2846 }
2847 x = scalarmat(x, N);
2848 x2 = x1? idealdivexact(nf, x, x1): x;
2849 #endif
2850 return x2;
2851 }
2852
2853 /* L0 in K^*, assume (L0,f) = 1. Return L integral, L0 = L mod f */
2854 GEN
make_integral(GEN nf,GEN L0,GEN f,GEN listpr)2855 make_integral(GEN nf, GEN L0, GEN f, GEN listpr)
2856 {
2857 GEN fZ, t, L, D2, d1, d2, d;
2858
2859 L = Q_remove_denom(L0, &d);
2860 if (!d) return L0;
2861
2862 /* L0 = L / d, L integral */
2863 fZ = gcoeff(f,1,1);
2864 if (typ(L) == t_INT) return Fp_mul(L, Fp_inv(d, fZ), fZ);
2865 /* Kill denom part coprime to fZ */
2866 d2 = Z_ppo(d, fZ);
2867 t = Fp_inv(d2, fZ); if (!is_pm1(t)) L = ZC_Z_mul(L,t);
2868 if (equalii(d, d2)) return L;
2869
2870 d1 = diviiexact(d, d2);
2871 /* L0 = (L / d1) mod f. d1 not coprime to f
2872 * write (d1) = D1 D2, D2 minimal, (D2,f) = 1. */
2873 D2 = nf_coprime_part(nf, d1, listpr);
2874 t = idealaddtoone_i(nf, D2, f); /* in D2, 1 mod f */
2875 L = nfmuli(nf,t,L);
2876
2877 /* if (L0, f) = 1, then L in D1 ==> in D1 D2 = (d1) */
2878 return Q_div_to_int(L, d1); /* exact division */
2879 }
2880
2881 /* assume L is a list of prime ideals. Return the product */
2882 GEN
idealprodprime(GEN nf,GEN L)2883 idealprodprime(GEN nf, GEN L)
2884 {
2885 long l = lg(L), i;
2886 GEN z;
2887 if (l == 1) return matid(nf_get_degree(nf));
2888 z = pr_hnf(nf, gel(L,1));
2889 for (i=2; i<l; i++) z = idealHNF_mul_two(nf,z, gel(L,i));
2890 return z;
2891 }
2892
2893 /* optimize for the frequent case I = nfhnf()[2]: lots of them are 1 */
2894 GEN
idealprod(GEN nf,GEN I)2895 idealprod(GEN nf, GEN I)
2896 {
2897 long i, l = lg(I);
2898 GEN z;
2899 for (i = 1; i < l; i++)
2900 if (!equali1(gel(I,i))) break;
2901 if (i == l) return gen_1;
2902 z = gel(I,i);
2903 for (i++; i<l; i++) z = idealmul(nf, z, gel(I,i));
2904 return z;
2905 }
2906
2907 /* v_pr(idealprod(nf,I)) */
2908 long
idealprodval(GEN nf,GEN I,GEN pr)2909 idealprodval(GEN nf, GEN I, GEN pr)
2910 {
2911 long i, l = lg(I), v = 0;
2912 for (i = 1; i < l; i++)
2913 if (!equali1(gel(I,i))) v += idealval(nf, gel(I,i), pr);
2914 return v;
2915 }
2916
2917 /* assume L is a list of prime ideals. Return prod L[i]^e[i] */
2918 GEN
factorbackprime(GEN nf,GEN L,GEN e)2919 factorbackprime(GEN nf, GEN L, GEN e)
2920 {
2921 long l = lg(L), i;
2922 GEN z;
2923
2924 if (l == 1) return matid(nf_get_degree(nf));
2925 z = idealpow(nf, gel(L,1), gel(e,1));
2926 for (i=2; i<l; i++)
2927 if (signe(gel(e,i))) z = idealmulpowprime(nf,z, gel(L,i),gel(e,i));
2928 return z;
2929 }
2930
2931 /* F in Z, divisible exactly by pr.p. Return F-uniformizer for pr, i.e.
2932 * a t in Z_K such that v_pr(t) = 1 and (t, F/pr) = 1 */
2933 GEN
pr_uniformizer(GEN pr,GEN F)2934 pr_uniformizer(GEN pr, GEN F)
2935 {
2936 GEN p = pr_get_p(pr), t = pr_get_gen(pr);
2937 if (!equalii(F, p))
2938 {
2939 long e = pr_get_e(pr);
2940 GEN u, v, q = (e == 1)? sqri(p): p;
2941 u = mulii(q, Fp_inv(q, diviiexact(F,p))); /* 1 mod F/p, 0 mod q */
2942 v = subui(1UL, u); /* 0 mod F/p, 1 mod q */
2943 if (pr_is_inert(pr))
2944 t = addii(mulii(p, v), u);
2945 else
2946 {
2947 t = ZC_Z_mul(t, v);
2948 gel(t,1) = addii(gel(t,1), u); /* return u + vt */
2949 }
2950 }
2951 return t;
2952 }
2953 /* L = list of prime ideals, return lcm_i (L[i] \cap \ZM) */
2954 GEN
prV_lcm_capZ(GEN L)2955 prV_lcm_capZ(GEN L)
2956 {
2957 long i, r = lg(L);
2958 GEN F;
2959 if (r == 1) return gen_1;
2960 F = pr_get_p(gel(L,1));
2961 for (i = 2; i < r; i++)
2962 {
2963 GEN pr = gel(L,i), p = pr_get_p(pr);
2964 if (!dvdii(F, p)) F = mulii(F,p);
2965 }
2966 return F;
2967 }
2968
2969 /* Given a prime ideal factorization with possibly zero or negative
2970 * exponents, gives b such that v_p(b) = v_p(x) for all prime ideals pr | x
2971 * and v_pr(b) >= 0 for all other pr.
2972 * For optimal performance, all [anti-]uniformizers should be precomputed,
2973 * but no support for this yet.
2974 *
2975 * If nored, do not reduce result.
2976 * No garbage collecting */
2977 static GEN
idealapprfact_i(GEN nf,GEN x,int nored)2978 idealapprfact_i(GEN nf, GEN x, int nored)
2979 {
2980 GEN z, d, L, e, e2, F;
2981 long i, r;
2982 int flagden;
2983
2984 nf = checknf(nf);
2985 L = gel(x,1);
2986 e = gel(x,2);
2987 F = prV_lcm_capZ(L);
2988 flagden = 0;
2989 z = NULL; r = lg(e);
2990 for (i = 1; i < r; i++)
2991 {
2992 long s = signe(gel(e,i));
2993 GEN pi, q;
2994 if (!s) continue;
2995 if (s < 0) flagden = 1;
2996 pi = pr_uniformizer(gel(L,i), F);
2997 q = nfpow(nf, pi, gel(e,i));
2998 z = z? nfmul(nf, z, q): q;
2999 }
3000 if (!z) return gen_1;
3001 if (nored || typ(z) != t_COL) return z;
3002 e2 = cgetg(r, t_VEC);
3003 for (i=1; i<r; i++) gel(e2,i) = addiu(gel(e,i), 1);
3004 x = factorbackprime(nf, L,e2);
3005 if (flagden) /* denominator */
3006 {
3007 z = Q_remove_denom(z, &d);
3008 d = diviiexact(d, Z_ppo(d, F));
3009 x = RgM_Rg_mul(x, d);
3010 }
3011 else
3012 d = NULL;
3013 z = ZC_reducemodlll(z, x);
3014 return d? RgC_Rg_div(z,d): z;
3015 }
3016
3017 GEN
idealapprfact(GEN nf,GEN x)3018 idealapprfact(GEN nf, GEN x) {
3019 pari_sp av = avma;
3020 return gerepileupto(av, idealapprfact_i(nf, x, 0));
3021 }
3022 GEN
idealappr(GEN nf,GEN x)3023 idealappr(GEN nf, GEN x) {
3024 pari_sp av = avma;
3025 if (!is_nf_extfactor(x)) x = idealfactor(nf, x);
3026 return gerepileupto(av, idealapprfact_i(nf, x, 0));
3027 }
3028
3029 /* OBSOLETE */
3030 GEN
idealappr0(GEN nf,GEN x,long fl)3031 idealappr0(GEN nf, GEN x, long fl) { (void)fl; return idealappr(nf, x); }
3032
3033 static GEN
mat_ideal_two_elt2(GEN nf,GEN x,GEN a)3034 mat_ideal_two_elt2(GEN nf, GEN x, GEN a)
3035 {
3036 GEN F = idealfactor(nf,a), P = gel(F,1), E = gel(F,2);
3037 long i, r = lg(E);
3038 for (i=1; i<r; i++) gel(E,i) = stoi( idealval(nf,x,gel(P,i)) );
3039 return idealapprfact_i(nf,F,1);
3040 }
3041
3042 static void
not_in_ideal(GEN a)3043 not_in_ideal(GEN a) {
3044 pari_err_DOMAIN("idealtwoelt2","element mod ideal", "!=", gen_0, a);
3045 }
3046 /* x integral in HNF, a an 'nf' */
3047 static int
in_ideal(GEN x,GEN a)3048 in_ideal(GEN x, GEN a)
3049 {
3050 switch(typ(a))
3051 {
3052 case t_INT: return dvdii(a, gcoeff(x,1,1));
3053 case t_COL: return RgV_is_ZV(a) && !!hnf_invimage(x, a);
3054 default: return 0;
3055 }
3056 }
3057
3058 /* Given an integral ideal x and a in x, gives a b such that
3059 * x = aZ_K + bZ_K using the approximation theorem */
3060 GEN
idealtwoelt2(GEN nf,GEN x,GEN a)3061 idealtwoelt2(GEN nf, GEN x, GEN a)
3062 {
3063 pari_sp av = avma;
3064 GEN cx, b;
3065
3066 nf = checknf(nf);
3067 a = nf_to_scalar_or_basis(nf, a);
3068 x = idealhnf_shallow(nf,x);
3069 if (lg(x) == 1)
3070 {
3071 if (!isintzero(a)) not_in_ideal(a);
3072 set_avma(av); return gen_0;
3073 }
3074 x = Q_primitive_part(x, &cx);
3075 if (cx) a = gdiv(a, cx);
3076 if (!in_ideal(x, a)) not_in_ideal(a);
3077 b = mat_ideal_two_elt2(nf, x, a);
3078 if (typ(b) == t_COL)
3079 {
3080 GEN mod = idealhnf_principal(nf,a);
3081 b = ZC_hnfrem(b,mod);
3082 if (ZV_isscalar(b)) b = gel(b,1);
3083 }
3084 else
3085 {
3086 GEN aZ = typ(a) == t_COL? Q_denom(zk_inv(nf,a)): a; /* (a) \cap Z */
3087 b = centermodii(b, aZ, shifti(aZ,-1));
3088 }
3089 b = cx? gmul(b,cx): gcopy(b);
3090 return gerepileupto(av, b);
3091 }
3092
3093 /* Given 2 integral ideals x and y in nf, returns a beta in nf such that
3094 * beta * x is an integral ideal coprime to y */
3095 GEN
idealcoprimefact(GEN nf,GEN x,GEN fy)3096 idealcoprimefact(GEN nf, GEN x, GEN fy)
3097 {
3098 GEN L = gel(fy,1), e;
3099 long i, r = lg(L);
3100
3101 e = cgetg(r, t_COL);
3102 for (i=1; i<r; i++) gel(e,i) = stoi( -idealval(nf,x,gel(L,i)) );
3103 return idealapprfact_i(nf, mkmat2(L,e), 0);
3104 }
3105 GEN
idealcoprime(GEN nf,GEN x,GEN y)3106 idealcoprime(GEN nf, GEN x, GEN y)
3107 {
3108 pari_sp av = avma;
3109 return gerepileupto(av, idealcoprimefact(nf, x, idealfactor(nf,y)));
3110 }
3111
3112 GEN
nfmulmodpr(GEN nf,GEN x,GEN y,GEN modpr)3113 nfmulmodpr(GEN nf, GEN x, GEN y, GEN modpr)
3114 {
3115 pari_sp av = avma;
3116 GEN z, p, pr = modpr, T;
3117
3118 nf = checknf(nf); modpr = nf_to_Fq_init(nf,&pr,&T,&p);
3119 x = nf_to_Fq(nf,x,modpr);
3120 y = nf_to_Fq(nf,y,modpr);
3121 z = Fq_mul(x,y,T,p);
3122 return gerepileupto(av, algtobasis(nf, Fq_to_nf(z,modpr)));
3123 }
3124
3125 GEN
nfdivmodpr(GEN nf,GEN x,GEN y,GEN modpr)3126 nfdivmodpr(GEN nf, GEN x, GEN y, GEN modpr)
3127 {
3128 pari_sp av = avma;
3129 nf = checknf(nf);
3130 return gerepileupto(av, nfreducemodpr(nf, nfdiv(nf,x,y), modpr));
3131 }
3132
3133 GEN
nfpowmodpr(GEN nf,GEN x,GEN k,GEN modpr)3134 nfpowmodpr(GEN nf, GEN x, GEN k, GEN modpr)
3135 {
3136 pari_sp av=avma;
3137 GEN z, T, p, pr = modpr;
3138
3139 nf = checknf(nf); modpr = nf_to_Fq_init(nf,&pr,&T,&p);
3140 z = nf_to_Fq(nf,x,modpr);
3141 z = Fq_pow(z,k,T,p);
3142 return gerepileupto(av, algtobasis(nf, Fq_to_nf(z,modpr)));
3143 }
3144
3145 GEN
nfkermodpr(GEN nf,GEN x,GEN modpr)3146 nfkermodpr(GEN nf, GEN x, GEN modpr)
3147 {
3148 pari_sp av = avma;
3149 GEN T, p, pr = modpr;
3150
3151 nf = checknf(nf); modpr = nf_to_Fq_init(nf, &pr,&T,&p);
3152 if (typ(x)!=t_MAT) pari_err_TYPE("nfkermodpr",x);
3153 x = nfM_to_FqM(x, nf, modpr);
3154 return gerepilecopy(av, FqM_to_nfM(FqM_ker(x,T,p), modpr));
3155 }
3156
3157 GEN
nfsolvemodpr(GEN nf,GEN a,GEN b,GEN pr)3158 nfsolvemodpr(GEN nf, GEN a, GEN b, GEN pr)
3159 {
3160 const char *f = "nfsolvemodpr";
3161 pari_sp av = avma;
3162 GEN T, p, modpr;
3163
3164 nf = checknf(nf);
3165 modpr = nf_to_Fq_init(nf, &pr,&T,&p);
3166 if (typ(a)!=t_MAT) pari_err_TYPE(f,a);
3167 a = nfM_to_FqM(a, nf, modpr);
3168 switch(typ(b))
3169 {
3170 case t_MAT:
3171 b = nfM_to_FqM(b, nf, modpr);
3172 b = FqM_gauss(a,b,T,p);
3173 if (!b) pari_err_INV(f,a);
3174 a = FqM_to_nfM(b, modpr);
3175 break;
3176 case t_COL:
3177 b = nfV_to_FqV(b, nf, modpr);
3178 b = FqM_FqC_gauss(a,b,T,p);
3179 if (!b) pari_err_INV(f,a);
3180 a = FqV_to_nfV(b, modpr);
3181 break;
3182 default: pari_err_TYPE(f,b);
3183 }
3184 return gerepilecopy(av, a);
3185 }
3186