Annotation of OpenXM_contrib/pari-2.2/src/basemath/base2.c, Revision 1.1
1.1 ! noro 1: /* $Id: base2.c,v 1.87 2001/10/01 12:11:28 karim Exp $
! 2:
! 3: Copyright (C) 2000 The PARI group.
! 4:
! 5: This file is part of the PARI/GP package.
! 6:
! 7: PARI/GP is free software; you can redistribute it and/or modify it under the
! 8: terms of the GNU General Public License as published by the Free Software
! 9: Foundation. It is distributed in the hope that it will be useful, but WITHOUT
! 10: ANY WARRANTY WHATSOEVER.
! 11:
! 12: Check the License for details. You should have received a copy of it, along
! 13: with the package; see the file 'COPYING'. If not, write to the Free Software
! 14: Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA. */
! 15:
! 16: /*******************************************************************/
! 17: /* */
! 18: /* MAXIMAL ORDERS */
! 19: /* */
! 20: /*******************************************************************/
! 21: #include "pari.h"
! 22:
! 23: extern GEN caractducos(GEN p, GEN x, int v);
! 24: extern GEN element_muli(GEN nf, GEN x, GEN y);
! 25: extern GEN element_mulid(GEN nf, GEN x, long i);
! 26: extern GEN eleval(GEN f,GEN h,GEN a);
! 27: extern GEN ideal_better_basis(GEN nf, GEN x, GEN M);
! 28: extern long int_elt_val(GEN nf, GEN x, GEN p, GEN bp, GEN *t, long v);
! 29: extern GEN mat_to_vecpol(GEN x, long v);
! 30: extern GEN nfidealdet1(GEN nf, GEN a, GEN b);
! 31: extern GEN nfsuppl(GEN nf, GEN x, long n, GEN prhall);
! 32: extern GEN pol_to_monic(GEN pol, GEN *lead);
! 33: extern GEN pol_to_vec(GEN x, long N);
! 34: extern GEN quicktrace(GEN x, GEN sym);
! 35: extern GEN respm(GEN f1,GEN f2,GEN pm);
! 36:
! 37: static void
! 38: allbase_check_args(GEN f, long code, GEN *y, GEN *ptw1, GEN *ptw2)
! 39: {
! 40: GEN w;
! 41: if (typ(f)!=t_POL) err(notpoler,"allbase");
! 42: if (degpol(f) <= 0) err(constpoler,"allbase");
! 43: if (DEBUGLEVEL) timer2();
! 44: switch(code)
! 45: {
! 46: case 0: case 1:
! 47: *y = ZX_disc(f);
! 48: if (!signe(*y)) err(talker,"reducible polynomial in allbase");
! 49: w = auxdecomp(absi(*y),1-code);
! 50: break;
! 51: default:
! 52: w = (GEN)code;
! 53: *y = factorback(w, NULL);
! 54: }
! 55: if (DEBUGLEVEL) msgtimer("disc. factorisation");
! 56: *ptw1 = (GEN)w[1];
! 57: *ptw2 = (GEN)w[2];
! 58: }
! 59:
! 60: /*******************************************************************/
! 61: /* */
! 62: /* ROUND 2 */
! 63: /* */
! 64: /*******************************************************************/
! 65: /* Normalized quotient and remainder ( -1/2 |y| < r = x-q*y <= 1/2 |y| ) */
! 66: static GEN
! 67: rquot(GEN x, GEN y)
! 68: {
! 69: long av=avma,av1;
! 70: GEN u,v,w,p;
! 71:
! 72: u=absi(y); v=shifti(x,1); w=shifti(y,1);
! 73: if (cmpii(u,v)>0) p=subii(v,u);
! 74: else p=addsi(-1,addii(u,v));
! 75: av1=avma; return gerepile(av,av1,divii(p,w));
! 76: }
! 77:
! 78: /* space needed lx + 2*ly */
! 79: static GEN
! 80: rrmdr(GEN x, GEN y)
! 81: {
! 82: long av=avma,tetpil,k;
! 83: GEN r,ys2;
! 84:
! 85: if (!signe(x)) return gzero;
! 86: r = resii(x,y); tetpil = avma;
! 87: ys2 = shifti(y,-1);
! 88: k = absi_cmp(r, ys2);
! 89: if (k>0 || (k==0 && signe(r)>0))
! 90: {
! 91: avma = tetpil;
! 92: if (signe(y) == signe(r)) r = subii(r,y); else r = addii(r,y);
! 93: return gerepile(av,tetpil,r);
! 94: }
! 95: avma = tetpil; return r;
! 96: }
! 97:
! 98: /* companion matrix of unitary polynomial x */
! 99: static GEN
! 100: companion(GEN x) /* cf assmat */
! 101: {
! 102: long i,j,l;
! 103: GEN y;
! 104:
! 105: l=degpol(x)+1; y=cgetg(l,t_MAT);
! 106: for (j=1; j<l; j++)
! 107: {
! 108: y[j] = lgetg(l,t_COL);
! 109: for (i=1; i<l-1; i++)
! 110: coeff(y,i,j)=(i+1==j)? un: zero;
! 111: coeff(y,i,j) = lneg((GEN)x[j+1]);
! 112: }
! 113: return y;
! 114: }
! 115:
! 116: /* assume x, y are square integer matrices of same dim. Multiply them */
! 117: static GEN
! 118: mulmati(GEN x, GEN y)
! 119: {
! 120: long n = lg(x),i,j,k,av;
! 121: GEN z = cgetg(n,t_MAT),p1,p2;
! 122:
! 123: for (j=1; j<n; j++)
! 124: {
! 125: z[j] = lgetg(n,t_COL);
! 126: for (i=1; i<n; i++)
! 127: {
! 128: p1=gzero; av=avma;
! 129: for (k=1; k<n; k++)
! 130: {
! 131: p2=mulii(gcoeff(x,i,k),gcoeff(y,k,j));
! 132: if (p2 != gzero) p1=addii(p1,p2);
! 133: }
! 134: coeff(z,i,j)=lpileupto(av,p1);
! 135: }
! 136: }
! 137: return z;
! 138: }
! 139:
! 140: static GEN
! 141: powmati(GEN x, long m)
! 142: {
! 143: long av=avma,j;
! 144: GEN y = x;
! 145:
! 146: j=1+bfffo(m); m<<=j; j = BITS_IN_LONG-j;
! 147: for (; j; m<<=1,j--)
! 148: {
! 149: y=mulmati(y,y);
! 150: if (m<0) y=mulmati(y,x);
! 151: }
! 152: return gerepileupto(av,y);
! 153: }
! 154:
! 155: static GEN
! 156: rtran(GEN v, GEN w, GEN q)
! 157: {
! 158: long av,tetpil;
! 159: GEN p1;
! 160:
! 161: if (signe(q))
! 162: {
! 163: av=avma; p1=gneg(gmul(q,w)); tetpil=avma;
! 164: return gerepile(av,tetpil,gadd(v,p1));
! 165: }
! 166: return v;
! 167: }
! 168:
! 169: /* return (v - qw) mod m (only compute entries k0,..,n)
! 170: * v and w are expected to have entries smaller than m */
! 171: static GEN
! 172: mtran(GEN v, GEN w, GEN q, GEN m, long k0)
! 173: {
! 174: long k,l;
! 175: GEN p1;
! 176:
! 177: if (signe(q))
! 178: {
! 179: l = lgefint(m) << 2;
! 180: for (k=lg(v)-1; k>= k0; k--)
! 181: {
! 182: long av = avma; (void)new_chunk(l);
! 183: p1 = subii((GEN)v[k], mulii(q,(GEN)w[k]));
! 184: avma = av; v[k]=(long)rrmdr(p1, m);
! 185: }
! 186: }
! 187: return v;
! 188: }
! 189:
! 190: /* entries of v and w are C small integers */
! 191: static GEN
! 192: mtran_long(GEN v, GEN w, long q, long m, long k0)
! 193: {
! 194: long k, p1;
! 195:
! 196: if (q)
! 197: {
! 198: for (k=lg(v)-1; k>= k0; k--)
! 199: {
! 200: p1 = v[k] - q * w[k];
! 201: v[k] = p1 % m;
! 202: }
! 203: }
! 204: return v;
! 205: }
! 206:
! 207: /* coeffs of a are C-long integers */
! 208: static void
! 209: rowred_long(GEN a, long rmod)
! 210: {
! 211: long q,j,k,pro, c = lg(a), r = lg(a[1]);
! 212:
! 213: for (j=1; j<r; j++)
! 214: {
! 215: for (k=j+1; k<c; k++)
! 216: while (coeff(a,j,k))
! 217: {
! 218: q = coeff(a,j,j) / coeff(a,j,k);
! 219: pro=(long)mtran_long((GEN)a[j],(GEN)a[k],q,rmod, j);
! 220: a[j]=a[k]; a[k]=pro;
! 221: }
! 222: if (coeff(a,j,j) < 0)
! 223: for (k=j; k<r; k++) coeff(a,k,j)=-coeff(a,k,j);
! 224: for (k=1; k<j; k++)
! 225: {
! 226: q = coeff(a,j,k) / coeff(a,j,j);
! 227: a[k]=(long)mtran_long((GEN)a[k],(GEN)a[j],q,rmod, k);
! 228: }
! 229: }
! 230: /* don't update the 0s in the last columns */
! 231: for (j=1; j<r; j++)
! 232: for (k=1; k<r; k++) coeff(a,j,k) = lstoi(coeff(a,j,k));
! 233: }
! 234:
! 235: static void
! 236: rowred(GEN a, GEN rmod)
! 237: {
! 238: long j,k,pro, c = lg(a), r = lg(a[1]);
! 239: long av=avma, lim=stack_lim(av,1);
! 240: GEN q;
! 241:
! 242: for (j=1; j<r; j++)
! 243: {
! 244: for (k=j+1; k<c; k++)
! 245: while (signe(gcoeff(a,j,k)))
! 246: {
! 247: q=rquot(gcoeff(a,j,j),gcoeff(a,j,k));
! 248: pro=(long)mtran((GEN)a[j],(GEN)a[k],q,rmod, j);
! 249: a[j]=a[k]; a[k]=pro;
! 250: }
! 251: if (signe(gcoeff(a,j,j)) < 0)
! 252: for (k=j; k<r; k++) coeff(a,k,j)=lnegi(gcoeff(a,k,j));
! 253: for (k=1; k<j; k++)
! 254: {
! 255: q=rquot(gcoeff(a,j,k),gcoeff(a,j,j));
! 256: a[k]=(long)mtran((GEN)a[k],(GEN)a[j],q,rmod, k);
! 257: }
! 258: if (low_stack(lim, stack_lim(av,1)))
! 259: {
! 260: long j1,k1;
! 261: GEN p1 = a;
! 262: if(DEBUGMEM>1) err(warnmem,"rowred j=%ld", j);
! 263: p1 = gerepilecopy(av,a);
! 264: for (j1=1; j1<r; j1++)
! 265: for (k1=1; k1<c; k1++) coeff(a,j1,k1) = coeff(p1,j1,k1);
! 266: }
! 267: }
! 268: }
! 269:
! 270: /* Calcule d/x ou d est entier et x matrice triangulaire inferieure
! 271: * entiere dont les coeff diagonaux divisent d (resultat entier).
! 272: */
! 273: static GEN
! 274: matinv(GEN x, GEN d, long n)
! 275: {
! 276: long i,j,k,av,av1;
! 277: GEN y,h;
! 278:
! 279: y=idmat(n);
! 280: for (i=1; i<=n; i++)
! 281: coeff(y,i,i)=ldivii(d,gcoeff(x,i,i));
! 282: av=avma;
! 283: for (i=2; i<=n; i++)
! 284: for (j=i-1; j; j--)
! 285: {
! 286: for (h=gzero,k=j+1; k<=i; k++)
! 287: {
! 288: GEN p1 = mulii(gcoeff(y,i,k),gcoeff(x,k,j));
! 289: if (p1 != gzero) h=addii(h,p1);
! 290: }
! 291: setsigne(h,-signe(h)); av1=avma;
! 292: coeff(y,i,j) = lpile(av,av1,divii(h,gcoeff(x,j,j)));
! 293: av = avma;
! 294: }
! 295: return y;
! 296: }
! 297:
! 298: static GEN
! 299: ordmax(GEN *cf, GEN p, long epsilon, GEN *ptdelta)
! 300: {
! 301: long sp,hard_case_exponent,i,n=lg(cf)-1,av=avma, av2,limit;
! 302: GEN T,T2,Tn,m,v,delta, *w;
! 303: const GEN pp = sqri(p);
! 304: const long pps = (2*expi(pp)+2<BITS_IN_LONG)? pp[2]: 0;
! 305:
! 306: if (cmpis(p,n) > 0)
! 307: {
! 308: hard_case_exponent = 0;
! 309: sp = 0; /* gcc -Wall */
! 310: }
! 311: else
! 312: {
! 313: long k;
! 314: k = sp = itos(p);
! 315: i=1; while (k < n) { k *= sp; i++; }
! 316: hard_case_exponent = i;
! 317: }
! 318: T=cgetg(n+1,t_MAT); for (i=1; i<=n; i++) T[i]=lgetg(n+1,t_COL);
! 319: T2=cgetg(2*n+1,t_MAT); for (i=1; i<=2*n; i++) T2[i]=lgetg(n+1,t_COL);
! 320: Tn=cgetg(n*n+1,t_MAT); for (i=1; i<=n*n; i++) Tn[i]=lgetg(n+1,t_COL);
! 321: v = new_chunk(n+1);
! 322: w = (GEN*)new_chunk(n+1);
! 323:
! 324: av2 = avma; limit = stack_lim(av2,1);
! 325: delta=gun; m=idmat(n);
! 326:
! 327: for(;;)
! 328: {
! 329: long j,k,h, av0 = avma;
! 330: GEN t,b,jp,hh,index,p1, dd = sqri(delta), ppdd = mulii(dd,pp);
! 331:
! 332: if (DEBUGLEVEL > 3)
! 333: fprintferr("ROUND2: epsilon = %ld\tavma = %ld\n",epsilon,avma);
! 334:
! 335: b=matinv(m,delta,n);
! 336: for (i=1; i<=n; i++)
! 337: {
! 338: for (j=1; j<=n; j++)
! 339: for (k=1; k<=n; k++)
! 340: {
! 341: p1 = j==k? gcoeff(m,i,1): gzero;
! 342: for (h=2; h<=n; h++)
! 343: {
! 344: GEN p2 = mulii(gcoeff(m,i,h),gcoeff(cf[h],j,k));
! 345: if (p2!=gzero) p1 = addii(p1,p2);
! 346: }
! 347: coeff(T,j,k) = (long)rrmdr(p1, ppdd);
! 348: }
! 349: p1 = mulmati(m, mulmati(T,b));
! 350: for (j=1; j<=n; j++)
! 351: for (k=1; k<=n; k++)
! 352: coeff(p1,j,k)=(long)rrmdr(divii(gcoeff(p1,j,k),dd),pp);
! 353: w[i] = p1;
! 354: }
! 355:
! 356: if (hard_case_exponent)
! 357: {
! 358: for (j=1; j<=n; j++)
! 359: {
! 360: for (i=1; i<=n; i++) coeff(T,i,j) = coeff(w[j],1,i);
! 361: /* ici la boucle en k calcule la puissance p mod p de w[j] */
! 362: for (k=1; k<sp; k++)
! 363: {
! 364: for (i=1; i<=n; i++)
! 365: {
! 366: p1 = gzero;
! 367: for (h=1; h<=n; h++)
! 368: {
! 369: GEN p2=mulii(gcoeff(T,h,j),gcoeff(w[j],h,i));
! 370: if (p2!=gzero) p1 = addii(p1,p2);
! 371: }
! 372: v[i] = lmodii(p1, p);
! 373: }
! 374: for (i=1; i<=n; i++) coeff(T,i,j)=v[i];
! 375: }
! 376: }
! 377: t = powmati(T, hard_case_exponent);
! 378: }
! 379: else
! 380: {
! 381: for (i=1; i<=n; i++)
! 382: for (j=1; j<=n; j++)
! 383: {
! 384: long av1 = avma;
! 385: p1 = gzero;
! 386: for (k=1; k<=n; k++)
! 387: for (h=1; h<=n; h++)
! 388: {
! 389: const GEN r=modii(gcoeff(w[i],k,h),p);
! 390: const GEN s=modii(gcoeff(w[j],h,k),p);
! 391: const GEN p2 = mulii(r,s);
! 392: if (p2!=gzero) p1 = addii(p1,p2);
! 393: }
! 394: coeff(T,i,j) = lpileupto(av1,p1);
! 395: }
! 396: t = T;
! 397: }
! 398:
! 399: if (pps)
! 400: {
! 401: long ps = p[2];
! 402: for (i=1; i<=n; i++)
! 403: for (j=1; j<=n; j++)
! 404: {
! 405: coeff(T2,j,i)=(i==j)? ps: 0;
! 406: coeff(T2,j,n+i)=smodis(gcoeff(t,i,j),ps);
! 407: }
! 408: rowred_long(T2,pps);
! 409: }
! 410: else
! 411: {
! 412: for (i=1; i<=n; i++)
! 413: for (j=1; j<=n; j++)
! 414: {
! 415: coeff(T2,j,i)=(i==j)? (long)p: zero;
! 416: coeff(T2,j,n+i)=lmodii(gcoeff(t,i,j),p);
! 417: }
! 418: rowred(T2,pp);
! 419: }
! 420: jp=matinv(T2,p,n);
! 421: if (pps)
! 422: {
! 423: for (k=1; k<=n; k++)
! 424: {
! 425: long av1=avma;
! 426: t = mulmati(mulmati(jp,w[k]), T2);
! 427: for (h=i=1; i<=n; i++)
! 428: for (j=1; j<=n; j++)
! 429: { coeff(Tn,k,h) = itos(divii(gcoeff(t,i,j), p)) % pps; h++; }
! 430: avma=av1;
! 431: }
! 432: avma = av0;
! 433: rowred_long(Tn,pps);
! 434: }
! 435: else
! 436: {
! 437: for (k=1; k<=n; k++)
! 438: {
! 439: t = mulmati(mulmati(jp,w[k]), T2);
! 440: for (h=i=1; i<=n; i++)
! 441: for (j=1; j<=n; j++)
! 442: { coeff(Tn,k,h) = ldivii(gcoeff(t,i,j), p); h++; }
! 443: }
! 444: rowred(Tn,pp);
! 445: }
! 446: for (index=gun,i=1; i<=n; i++)
! 447: index = mulii(index,gcoeff(Tn,i,i));
! 448: if (gcmp1(index)) break;
! 449:
! 450: m = mulmati(matinv(Tn,index,n), m);
! 451: hh = delta = mulii(index,delta);
! 452: for (i=1; i<=n; i++)
! 453: for (j=1; j<=n; j++)
! 454: hh = mppgcd(gcoeff(m,i,j),hh);
! 455: if (!is_pm1(hh))
! 456: {
! 457: m = gdiv(m,hh);
! 458: delta = divii(delta,hh);
! 459: }
! 460: epsilon -= 2 * ggval(index,p);
! 461: if (epsilon < 2) break;
! 462: if (low_stack(limit,stack_lim(av2,1)))
! 463: {
! 464: GEN *gptr[3]; gptr[0]=&m; gptr[1]=δ
! 465: if(DEBUGMEM>1) err(warnmem,"ordmax");
! 466: gerepilemany(av2, gptr,2);
! 467: }
! 468: }
! 469: {
! 470: GEN *gptr[2]; gptr[0]=&m; gptr[1]=δ
! 471: gerepilemany(av,gptr,2);
! 472: }
! 473: *ptdelta=delta; return m;
! 474: }
! 475:
! 476: /* Input:
! 477: * x normalized integral polynomial of degree n, defining K=Q(theta).
! 478: *
! 479: * code 0, 1 or (long)p if we want base, smallbase ou factoredbase (resp.).
! 480: * y is GEN *, which will receive the discriminant of K.
! 481: *
! 482: * Output
! 483: * 1) A t_COL whose n components are rationnal polynomials (with degree
! 484: * 0,1...n-1) : integral basis for K (putting x=theta).
! 485: * Rem: common denominator is in da.
! 486: *
! 487: * 2) discriminant of K (in *y).
! 488: */
! 489: GEN
! 490: allbase(GEN f, long code, GEN *y)
! 491: {
! 492: GEN w1,w2,a,pro,at,bt,b,da,db,q, *cf,*gptr[2];
! 493: long av=avma,tetpil,n,h,j,i,k,r,s,t,v,mf;
! 494:
! 495: allbase_check_args(f,code,y, &w1,&w2);
! 496: v = varn(f); n = degpol(f); h = lg(w1)-1;
! 497: cf = (GEN*)cgetg(n+1,t_VEC);
! 498: cf[2]=companion(f);
! 499: for (i=3; i<=n; i++) cf[i]=mulmati(cf[2],cf[i-1]);
! 500:
! 501: a=idmat(n); da=gun;
! 502: for (i=1; i<=h; i++)
! 503: {
! 504: long av1 = avma;
! 505: mf=itos((GEN)w2[i]); if (mf==1) continue;
! 506: if (DEBUGLEVEL) fprintferr("Treating p^k = %Z^%ld\n",w1[i],mf);
! 507:
! 508: b=ordmax(cf,(GEN)w1[i],mf,&db);
! 509: a=gmul(db,a); b=gmul(da,b);
! 510: da=mulii(db,da);
! 511: at=gtrans(a); bt=gtrans(b);
! 512: for (r=n; r; r--)
! 513: for (s=r; s; s--)
! 514: while (signe(gcoeff(bt,s,r)))
! 515: {
! 516: q=rquot(gcoeff(at,s,s),gcoeff(bt,s,r));
! 517: pro=rtran((GEN)at[s],(GEN)bt[r],q);
! 518: for (t=s-1; t; t--)
! 519: {
! 520: q=rquot(gcoeff(at,t,s),gcoeff(at,t,t));
! 521: pro=rtran(pro,(GEN)at[t],q);
! 522: }
! 523: at[s]=bt[r]; bt[r]=(long)pro;
! 524: }
! 525: for (j=n; j; j--)
! 526: {
! 527: for (k=1; k<j; k++)
! 528: {
! 529: while (signe(gcoeff(at,j,k)))
! 530: {
! 531: q=rquot(gcoeff(at,j,j),gcoeff(at,j,k));
! 532: pro=rtran((GEN)at[j],(GEN)at[k],q);
! 533: at[j]=at[k]; at[k]=(long)pro;
! 534: }
! 535: }
! 536: if (signe(gcoeff(at,j,j))<0)
! 537: for (k=1; k<=j; k++) coeff(at,k,j)=lnegi(gcoeff(at,k,j));
! 538: for (k=j+1; k<=n; k++)
! 539: {
! 540: q=rquot(gcoeff(at,j,k),gcoeff(at,j,j));
! 541: at[k]=(long)rtran((GEN)at[k],(GEN)at[j],q);
! 542: }
! 543: }
! 544: for (j=2; j<=n; j++)
! 545: if (egalii(gcoeff(at,j,j), gcoeff(at,j-1,j-1)))
! 546: {
! 547: coeff(at,1,j)=zero;
! 548: for (k=2; k<=j; k++) coeff(at,k,j)=coeff(at,k-1,j-1);
! 549: }
! 550: tetpil=avma; a=gtrans(at);
! 551: {
! 552: GEN *gptr[2];
! 553: da = icopy(da); gptr[0]=&a; gptr[1]=&da;
! 554: gerepilemanysp(av1,tetpil,gptr,2);
! 555: }
! 556: }
! 557: for (j=1; j<=n; j++)
! 558: *y = divii(mulii(*y,sqri(gcoeff(a,j,j))), sqri(da));
! 559: tetpil=avma; *y=icopy(*y);
! 560: at=cgetg(n+1,t_VEC); v=varn(f);
! 561: for (k=1; k<=n; k++)
! 562: {
! 563: q=cgetg(k+2,t_POL); at[k]=(long)q;
! 564: q[1] = evalsigne(1) | evallgef(2+k) | evalvarn(v);
! 565: for (j=1; j<=k; j++) q[j+1]=ldiv(gcoeff(a,k,j),da);
! 566: }
! 567: gptr[0]=&at; gptr[1]=y;
! 568: gerepilemanysp(av,tetpil,gptr,2);
! 569: return at;
! 570: }
! 571:
! 572: GEN
! 573: base2(GEN x, GEN *y)
! 574: {
! 575: return allbase(x,0,y);
! 576: }
! 577:
! 578: GEN
! 579: discf2(GEN x)
! 580: {
! 581: GEN y;
! 582: long av=avma,tetpil;
! 583:
! 584: allbase(x,0,&y); tetpil=avma;
! 585: return gerepile(av,tetpil,icopy(y));
! 586: }
! 587:
! 588: /*******************************************************************/
! 589: /* */
! 590: /* ROUND 4 */
! 591: /* */
! 592: /*******************************************************************/
! 593:
! 594: GEN nilord(GEN p,GEN fx,long mf,GEN gx,long flag);
! 595: GEN Decomp(GEN p,GEN f,long mf,GEN theta,GEN chi,GEN nu,long r);
! 596: static GEN dbasis(GEN p, GEN f, long mf, GEN alpha, GEN U);
! 597: static GEN maxord(GEN p,GEN f,long mf);
! 598: static GEN nbasis(GEN ibas,GEN pd);
! 599: static GEN testb2(GEN p,GEN fa,long Fa,GEN theta,GEN pmf,long Ft,GEN ns);
! 600: static GEN testc2(GEN p,GEN fa,GEN pmr,GEN pmf,GEN alph2,
! 601: long Ea,GEN thet2,long Et,GEN ns);
! 602:
! 603: static int
! 604: fnz(GEN x,long j)
! 605: {
! 606: long i;
! 607: for (i=1; i<j; i++)
! 608: if (signe(x[i])) return 0;
! 609: return 1;
! 610: }
! 611:
! 612: /* retourne la base, dans y le discf et dans ptw la factorisation (peut
! 613: etre partielle) de discf */
! 614: GEN
! 615: allbase4(GEN f,long code, GEN *y, GEN *ptw)
! 616: {
! 617: GEN w,w1,w2,a,da,b,db,bas,q,p1,*gptr[3];
! 618: long v,n,mf,h,lfa,i,j,k,l,tetpil,av = avma;
! 619:
! 620: allbase_check_args(f,code,y, &w1,&w2);
! 621: v = varn(f); n = degpol(f); h = lg(w1)-1;
! 622: a = NULL; /* gcc -Wall */
! 623: da= NULL;
! 624: for (i=1; i<=h; i++)
! 625: {
! 626: mf=itos((GEN)w2[i]); if (mf == 1) continue;
! 627: if (DEBUGLEVEL) fprintferr("Treating p^k = %Z^%ld\n",w1[i],mf);
! 628:
! 629: b = maxord((GEN)w1[i],f,mf); db = gun;
! 630: for (j=1; j<=n; j++)
! 631: {
! 632: p1 = denom(gcoeff(b,j,j));
! 633: if (cmpii(p1,db) > 0) db = p1;
! 634: }
! 635: if (db != gun)
! 636: { /* db = denom(diag(b)), (da,db) = 1 */
! 637: b = gmul(b,db);
! 638: if (!da) { da=db; a=b; }
! 639: else
! 640: {
! 641: j=1; while (j<=n && fnz((GEN)a[j],j) && fnz((GEN)b[j],j)) j++;
! 642: b = gmul(da,b); a = gmul(db,a);
! 643: k=j-1; p1=cgetg(2*n-k+1,t_MAT);
! 644: for (j=1; j<=k; j++)
! 645: {
! 646: p1[j] = a[j];
! 647: coeff(p1,j,j) = lmppgcd(gcoeff(a,j,j),gcoeff(b,j,j));
! 648: }
! 649: for ( ; j<=n; j++) p1[j] = a[j];
! 650: for ( ; j<=2*n-k; j++) p1[j] = b[j+k-n];
! 651: da = mulii(da,db); a = hnfmodid(p1, da);
! 652: }
! 653: }
! 654: if (DEBUGLEVEL>5)
! 655: fprintferr("Result for prime %Z is:\n%Z\n",w1[i],b);
! 656: }
! 657: if (da)
! 658: {
! 659: for (j=1; j<=n; j++)
! 660: *y = mulii(divii(*y,sqri(da)),sqri(gcoeff(a,j,j)));
! 661: for (j=n-1; j; j--)
! 662: if (cmpis(gcoeff(a,j,j),2) > 0)
! 663: {
! 664: p1=shifti(gcoeff(a,j,j),-1);
! 665: for (k=j+1; k<=n; k++)
! 666: if (cmpii(gcoeff(a,j,k),p1) > 0)
! 667: for (l=1; l<=j; l++)
! 668: coeff(a,l,k)=lsubii(gcoeff(a,l,k),gcoeff(a,l,j));
! 669: }
! 670: }
! 671: lfa = 0;
! 672: if (ptw)
! 673: {
! 674: for (j=1; j<=h; j++)
! 675: {
! 676: k=ggval(*y,(GEN)w1[j]);
! 677: if (k) { lfa++; w1[lfa]=w1[j]; w2[lfa]=k; }
! 678: }
! 679: }
! 680: tetpil=avma; *y=icopy(*y);
! 681: bas=cgetg(n+1,t_VEC); v=varn(f);
! 682: for (k=1; k<=n; k++)
! 683: {
! 684: q=cgetg(k+2,t_POL); bas[k]=(long)q;
! 685: q[1] = evalsigne(1) | evallgef(k+2) | evalvarn(v);
! 686: if (da)
! 687: for (j=1; j<=k; j++) q[j+1]=ldiv(gcoeff(a,j,k),da);
! 688: else
! 689: {
! 690: for (j=2; j<=k; j++) q[j]=zero;
! 691: q[j]=un;
! 692: }
! 693: }
! 694: if (ptw)
! 695: {
! 696: *ptw=w=cgetg(3,t_MAT);
! 697: w[1]=lgetg(lfa+1,t_COL);
! 698: w[2]=lgetg(lfa+1,t_COL);
! 699: for (j=1; j<=lfa; j++)
! 700: {
! 701: coeff(w,j,1)=(long)icopy((GEN)w1[j]);
! 702: coeff(w,j,2)=lstoi(w2[j]);
! 703: }
! 704: gptr[2]=ptw;
! 705: }
! 706: gptr[0]=&bas; gptr[1]=y;
! 707: gerepilemanysp(av,tetpil,gptr, ptw?3:2);
! 708: return bas;
! 709: }
! 710:
! 711: extern GEN merge_factor_i(GEN f, GEN g);
! 712:
! 713: static GEN
! 714: update_fact(GEN x, GEN f)
! 715: {
! 716: GEN e,q,d = ZX_disc(x), g = cgetg(3, t_MAT), p = (GEN)f[1];
! 717: long iq,i,k,l;
! 718: if (typ(f)!=t_MAT || lg(f)!=3)
! 719: err(talker,"not a factorisation in nfbasis");
! 720: l = lg(p);
! 721: q = cgetg(l,t_COL); g[1]=(long)q;
! 722: e = cgetg(l,t_COL); g[2]=(long)e; iq = 1;
! 723: for (i=1; i<l; i++)
! 724: {
! 725: k = pvaluation(d, (GEN)p[i], &d);
! 726: if (k) { q[iq] = p[i]; e[iq] = lstoi(k); iq++; }
! 727: }
! 728: setlg(q,iq); setlg(e,iq);
! 729: return merge_factor_i(decomp(d), g);
! 730: }
! 731:
! 732: /* if y is non-NULL, it receives the discriminant
! 733: * return basis if (ret_basis != 0), discriminant otherwise
! 734: */
! 735: static GEN
! 736: nfbasis00(GEN x0, long flag, GEN p, long ret_basis, GEN *y)
! 737: {
! 738: GEN x, disc, basis, lead;
! 739: GEN *gptr[2];
! 740: long k, tetpil, av = avma, l = lgef(x0), smll;
! 741:
! 742: if (typ(x0)!=t_POL) err(typeer,"nfbasis00");
! 743: if (l<=3) err(zeropoler,"nfbasis00");
! 744: for (k=2; k<l; k++)
! 745: if (typ(x0[k])!=t_INT) err(talker,"polynomial not in Z[X] in nfbasis");
! 746:
! 747: x = pol_to_monic(x0,&lead);
! 748:
! 749: if (!p || gcmp0(p))
! 750: smll = (flag & 1); /* small basis */
! 751: else
! 752: {
! 753: if (lead) p = update_fact(x,p);
! 754: smll = (long) p; /* factored basis */
! 755: }
! 756:
! 757: if (flag & 2)
! 758: basis = allbase(x,smll,&disc); /* round 2 */
! 759: else
! 760: basis = allbase4(x,smll,&disc,NULL); /* round 4 */
! 761:
! 762: if (!ret_basis) return gerepilecopy(av,disc);
! 763:
! 764: tetpil=avma;
! 765: if (!lead) basis = gcopy(basis);
! 766: else
! 767: {
! 768: long v = varn(x);
! 769: GEN pol = gmul(polx[v],lead);
! 770:
! 771: tetpil = avma; basis = gsubst(basis,v,pol);
! 772: }
! 773: if (!y)
! 774: return gerepile(av,tetpil,basis);
! 775:
! 776: *y = gcopy(disc);
! 777: gptr[0]=&basis; gptr[1]=y;
! 778: gerepilemanysp(av,tetpil,gptr,2);
! 779: return basis;
! 780: }
! 781:
! 782: GEN
! 783: nfbasis(GEN x, GEN *y, long flag, GEN p)
! 784: {
! 785: return nfbasis00(x,flag,p,1,y);
! 786: }
! 787:
! 788: GEN
! 789: nfbasis0(GEN x, long flag, GEN p)
! 790: {
! 791: return nfbasis00(x,flag,p,1,NULL);
! 792: }
! 793:
! 794: GEN
! 795: nfdiscf0(GEN x, long flag, GEN p)
! 796: {
! 797: return nfbasis00(x,flag,p,0,&p);
! 798: }
! 799:
! 800: GEN
! 801: base(GEN x, GEN *y)
! 802: {
! 803: return allbase4(x,0,y,NULL);
! 804: }
! 805:
! 806: GEN
! 807: smallbase(GEN x, GEN *y)
! 808: {
! 809: return allbase4(x,1,y,NULL);
! 810: }
! 811:
! 812: GEN
! 813: factoredbase(GEN x, GEN p, GEN *y)
! 814: {
! 815: return allbase4(x,(long)p,y,NULL);
! 816: }
! 817:
! 818: GEN
! 819: discf(GEN x)
! 820: {
! 821: GEN y;
! 822: long av=avma,tetpil;
! 823:
! 824: allbase4(x,0,&y,NULL); tetpil=avma;
! 825: return gerepile(av,tetpil,icopy(y));
! 826: }
! 827:
! 828: GEN
! 829: smalldiscf(GEN x)
! 830: {
! 831: GEN y;
! 832: long av=avma,tetpil;
! 833:
! 834: allbase4(x,1,&y,NULL); tetpil=avma;
! 835: return gerepile(av,tetpil,icopy(y));
! 836: }
! 837:
! 838: GEN
! 839: factoreddiscf(GEN x, GEN p)
! 840: {
! 841: GEN y;
! 842: long av=avma,tetpil;
! 843:
! 844: allbase4(x,(long)p,&y,NULL); tetpil=avma;
! 845: return gerepile(av,tetpil,icopy(y));
! 846: }
! 847:
! 848: /* return U if Z[alpha] is not maximal or 2*dU < m-1; else return NULL */
! 849: static GEN
! 850: dedek(GEN f, long mf, GEN p,GEN g)
! 851: {
! 852: GEN k,h;
! 853: long dk;
! 854:
! 855: if (DEBUGLEVEL>=3)
! 856: {
! 857: fprintferr(" entering dedek ");
! 858: if (DEBUGLEVEL>5)
! 859: fprintferr("with parameters p=%Z,\n f=%Z",p,f);
! 860: fprintferr("\n");
! 861: }
! 862: h = FpX_div(f,g,p);
! 863: k = gdivexact(gadd(f, gneg_i(gmul(g,h))), p);
! 864: k = FpX_gcd(k, FpX_gcd(g,h, p), p);
! 865:
! 866: dk = degpol(k);
! 867: if (DEBUGLEVEL>=3) fprintferr(" gcd has degree %ld\n", dk);
! 868: if (2*dk >= mf-1) return FpX_div(f,k,p);
! 869: return dk? (GEN)NULL: f;
! 870: }
! 871:
! 872: /* p-maximal order of Af; mf = v_p(Disc(f)) */
! 873: static GEN
! 874: maxord(GEN p,GEN f,long mf)
! 875: {
! 876: long j,r, av = avma, flw = (cmpsi(degpol(f),p) < 0);
! 877: GEN w,g,h,res;
! 878:
! 879: if (flw)
! 880: {
! 881: h = NULL; r = 0; /* gcc -Wall */
! 882: g = FpX_div(f, FpX_gcd(f,derivpol(f), p), p);
! 883: }
! 884: else
! 885: {
! 886: w=(GEN)factmod(f,p)[1]; r=lg(w)-1;
! 887: g = h = lift_intern((GEN)w[r]); /* largest factor */
! 888: for (j=1; j<r; j++) g = FpX_red(gmul(g, lift_intern((GEN)w[j])), p);
! 889: }
! 890: res = dedek(f,mf,p,g);
! 891: if (res)
! 892: res = dbasis(p,f,mf,polx[varn(f)],res);
! 893: else
! 894: {
! 895: if (flw) { w=(GEN)factmod(f,p)[1]; r=lg(w)-1; h=lift_intern((GEN)w[r]); }
! 896: res = (r==1)? nilord(p,f,mf,h,0): Decomp(p,f,mf,polx[varn(f)],f,h,0);
! 897: }
! 898: return gerepileupto(av,res);
! 899: }
! 900:
! 901: /* do a centermod on integer or rational number */
! 902: static GEN
! 903: polmodiaux(GEN x, GEN y, GEN ys2)
! 904: {
! 905: if (typ(x)!=t_INT)
! 906: x = mulii((GEN)x[1], mpinvmod((GEN)x[2],y));
! 907: x = modii(x,y);
! 908: if (cmpii(x,ys2) > 0) x = subii(x,y);
! 909: return x;
! 910: }
! 911:
! 912: /* x polynomial with integer or rational coeff. Reduce them mod y IN PLACE */
! 913: GEN
! 914: polmodi(GEN x, GEN y)
! 915: {
! 916: long lx=lgef(x), i;
! 917: GEN ys2 = shifti(y,-1);
! 918: for (i=2; i<lx; i++) x[i]=(long)polmodiaux((GEN)x[i],y,ys2);
! 919: return normalizepol_i(x, lx);
! 920: }
! 921:
! 922: /* same but not in place */
! 923: GEN
! 924: polmodi_keep(GEN x, GEN y)
! 925: {
! 926: long lx=lgef(x), i;
! 927: GEN ys2 = shifti(y,-1);
! 928: GEN z = cgetg(lx,t_POL);
! 929: for (i=2; i<lx; i++) z[i]=(long)polmodiaux((GEN)x[i],y,ys2);
! 930: z[1]=x[1]; return normalizepol_i(z, lx);
! 931: }
! 932:
! 933: static GEN
! 934: dbasis(GEN p, GEN f, long mf, GEN alpha, GEN U)
! 935: {
! 936: long n=degpol(f),dU,c;
! 937: GEN b,ha,pd,pdp;
! 938:
! 939: if (n == 1) return gscalmat(gun, 1);
! 940: if (DEBUGLEVEL>=3)
! 941: {
! 942: fprintferr(" entering Dedekind Basis ");
! 943: if (DEBUGLEVEL>5)
! 944: {
! 945: fprintferr("with parameters p=%Z\n",p);
! 946: fprintferr(" f = %Z,\n alpha = %Z",f,alpha);
! 947: }
! 948: fprintferr("\n");
! 949: }
! 950: ha = pd = gpuigs(p,mf/2); pdp = mulii(pd,p);
! 951: dU = typ(U)==t_POL? degpol(U): 0;
! 952: b = cgetg(n,t_MAT); /* Z[a] + U/p Z[a] is maximal */
! 953: /* skip first column = gscalcol(pd,n) */
! 954: for (c=1; c<n; c++)
! 955: {
! 956: if (c == dU)
! 957: {
! 958: ha = gdiv(gmul(pd,eleval(f,U,alpha)),p);
! 959: ha = polmodi(ha,pdp);
! 960: }
! 961: else
! 962: {
! 963: GEN p2, mod;
! 964: ha = gmul(ha,alpha);
! 965: p2 = content(ha); /* to cancel denominator */
! 966: if (gcmp1(p2)) { p2 = NULL; mod = pdp; }
! 967: else
! 968: {
! 969: ha = gdiv(ha,p2);
! 970: if (typ(p2)==t_INT)
! 971: mod = divii(pdp, mppgcd(pdp,p2));
! 972: else
! 973: mod = mulii(pdp, (GEN)p2[2]); /* p2 = a / p^e */
! 974: }
! 975: ha = FpX_res(ha, f, mod);
! 976: if (p2) ha = gmul(ha,p2);
! 977: }
! 978: b[c] = (long)pol_to_vec(ha,n);
! 979: }
! 980: b = hnfmodid(b,pd);
! 981: if (DEBUGLEVEL>5) fprintferr(" new order: %Z\n",b);
! 982: return gdiv(b,pd);
! 983: }
! 984:
! 985: static GEN
! 986: get_partial_order_as_pols(GEN p, GEN f)
! 987: {
! 988: long i,j, n = degpol(f), vf = varn(f);
! 989: GEN b,ib,h,col;
! 990:
! 991: b = maxord(p,f, ggval(ZX_disc(f),p));
! 992: ib = cgetg(n+1,t_VEC);
! 993: for (i=1; i<=n; i++)
! 994: {
! 995: h=cgetg(i+2,t_POL); ib[i]=(long)h; col=(GEN)b[i];
! 996: h[1]=evalsigne(1)|evallgef(i+2)|evalvarn(vf);
! 997: for (j=1;j<=i;j++) h[j+1]=col[j];
! 998: }
! 999: return ib;
! 1000: }
! 1001:
! 1002: /* if flag != 0, factorization to precision r (maximal order otherwise) */
! 1003: GEN
! 1004: Decomp(GEN p,GEN f,long mf,GEN theta,GEN chi,GEN nu,long flag)
! 1005: {
! 1006: GEN res,pr,pk,ph,pdr,unmodp,b1,b2,b3,a1,e,f1,f2;
! 1007:
! 1008: if (DEBUGLEVEL>2)
! 1009: {
! 1010: fprintferr(" entering Decomp ");
! 1011: if (DEBUGLEVEL>5)
! 1012: {
! 1013: fprintferr("with parameters: p=%Z, expo=%ld\n",p,mf);
! 1014: if (flag) fprintferr("precision = %ld\n",flag);
! 1015: fprintferr(" f=%Z",f);
! 1016: }
! 1017: fprintferr("\n");
! 1018: }
! 1019: unmodp = gmodulsg(1,p);
! 1020: b1=lift_intern(gmul(chi,unmodp));
! 1021: a1=gun; b2=gun;
! 1022: b3=lift_intern(gmul(nu,unmodp));
! 1023: while (degpol(b3) > 0)
! 1024: {
! 1025: GEN p1;
! 1026: b1 = FpX_div(b1,b3, p);
! 1027: b2 = FpX_red(gmul(b2,b3), p);
! 1028: b3 = FpX_extgcd(b2,b1, p, &a1,&p1); /* p1 = junk */
! 1029: p1 = leading_term(b3);
! 1030: if (!gcmp1(p1))
! 1031: { /* FpX_extgcd does not return normalized gcd */
! 1032: p1 = mpinvmod(p1,p);
! 1033: b3 = gmul(b3,p1);
! 1034: a1 = gmul(a1,p1);
! 1035: }
! 1036: }
! 1037: pdr = respm(f,derivpol(f),gpuigs(p,mf+1));
! 1038: e = eleval(f,FpX_red(gmul(a1,b2), p),theta);
! 1039: e = gdiv(polmodi(gmul(pdr,e), mulii(pdr,p)),pdr);
! 1040:
! 1041: pr = flag? gpowgs(p,flag): mulii(p,sqri(pdr));
! 1042: pk=p; ph=mulii(pdr,pr);
! 1043: /* E(t) - e(t) belongs to p^k Op, which is contained in p^(k-df)*Zp[xi] */
! 1044: while (cmpii(pk,ph) < 0)
! 1045: {
! 1046: e = gmul(gsqr(e), gsubsg(3,gmul2n(e,1)));
! 1047: e = gres(e,f); pk = sqri(pk);
! 1048: e = gdiv(polmodi(gmul(pdr,e), mulii(pdr,pk)), pdr);
! 1049: }
! 1050: f1 = gcdpm(f,gmul(pdr,gsubsg(1,e)), ph);
! 1051: f1 = FpX_res(f1,f, pr);
! 1052: f2 = FpX_res(FpX_div(f,f1, pr), f, pr);
! 1053:
! 1054: if (DEBUGLEVEL>2)
! 1055: {
! 1056: fprintferr(" leaving Decomp");
! 1057: if (DEBUGLEVEL>5)
! 1058: fprintferr(" with parameters: f1 = %Z\nf2 = %Z\ne = %Z\n", f1,f2,e);
! 1059: fprintferr("\n");
! 1060: }
! 1061:
! 1062: if (flag)
! 1063: {
! 1064: b1=factorpadic4(f1,p,flag);
! 1065: b2=factorpadic4(f2,p,flag); res=cgetg(3,t_MAT);
! 1066: res[1]=lconcat((GEN)b1[1],(GEN)b2[1]);
! 1067: res[2]=lconcat((GEN)b1[2],(GEN)b2[2]); return res;
! 1068: }
! 1069: else
! 1070: {
! 1071: GEN ib1,ib2;
! 1072: long n1,n2,i;
! 1073: ib1 = get_partial_order_as_pols(p,f1); n1=lg(ib1)-1;
! 1074: ib2 = get_partial_order_as_pols(p,f2); n2=lg(ib2)-1;
! 1075: res=cgetg(n1+n2+1,t_VEC);
! 1076: for (i=1; i<=n1; i++)
! 1077: res[i]=(long)polmodi(gmod(gmul(gmul(pdr,(GEN)ib1[i]),e),f), pdr);
! 1078: e=gsubsg(1,e); ib2 -= n1;
! 1079: for ( ; i<=n1+n2; i++)
! 1080: res[i]=(long)polmodi(gmod(gmul(gmul(pdr,(GEN)ib2[i]),e),f), pdr);
! 1081: return nbasis(res,pdr);
! 1082: }
! 1083: }
! 1084:
! 1085: /* minimum extension valuation: res[0]/res[1] (both are longs) */
! 1086: static long *
! 1087: vstar(GEN p,GEN h)
! 1088: {
! 1089: static long res[2];
! 1090: long m,first,j,k,v,w;
! 1091:
! 1092: m=degpol(h); first=1; k=1; v=0;
! 1093: for (j=1; j<=m; j++)
! 1094: if (! gcmp0((GEN)h[m-j+2]))
! 1095: {
! 1096: w = ggval((GEN)h[m-j+2],p);
! 1097: if (first || w*k < v*j) { v=w; k=j; }
! 1098: first=0;
! 1099: }
! 1100: m = cgcd(v,k);
! 1101: res[0]=v/m; res[1]=k/m; return res;
! 1102: }
! 1103:
! 1104: /* reduce the element elt modulo rd, taking first care of the denominators */
! 1105: static GEN
! 1106: redelt(GEN elt, GEN rd, GEN pd)
! 1107: {
! 1108: GEN den, nelt, nrd, relt;
! 1109:
! 1110: den = ggcd(denom(content(elt)), pd);
! 1111: nelt = gmul(den, elt);
! 1112: nrd = gmul(den, rd);
! 1113:
! 1114: if (typ(elt) == t_POL)
! 1115: relt = polmodi(nelt, nrd);
! 1116: else
! 1117: relt = centermod(nelt, nrd);
! 1118:
! 1119: return gdiv(relt, den);
! 1120: }
! 1121:
! 1122: /* compute the Newton sums of g(x) mod pp from its coefficients */
! 1123: GEN
! 1124: polsymmodpp(GEN g, GEN pp)
! 1125: {
! 1126: long av1, av2, d = degpol(g), i, k;
! 1127: GEN s , y;
! 1128:
! 1129: y = cgetg(d + 1, t_COL);
! 1130: y[1] = lstoi(d);
! 1131: for (k = 1; k < d; k++)
! 1132: {
! 1133: av1 = avma;
! 1134: s = centermod(gmulsg(k, polcoeff0(g,d-k,-1)), pp);
! 1135: for (i = 1; i < k; i++)
! 1136: s = gadd(s, gmul((GEN)y[k-i+1], polcoeff0(g,d-i,-1)));
! 1137: av2 = avma;
! 1138: y[k + 1] = lpile(av1, av2, centermod(gneg(s), pp));
! 1139: }
! 1140:
! 1141: return y;
! 1142: }
! 1143:
! 1144: /* no GC */
! 1145: static GEN
! 1146: manage_cache(GEN chi, GEN pp, GEN ns)
! 1147: {
! 1148: long j, n = degpol(chi);
! 1149: GEN ns2, npp = (GEN)ns[n+1];
! 1150:
! 1151: if (gcmp(pp, npp) > 0)
! 1152: {
! 1153: if (DEBUGLEVEL > 4)
! 1154: fprintferr("newtonsums: result too large to fit in cache\n");
! 1155: return polsymmodpp(chi, pp);
! 1156: }
! 1157:
! 1158: if (!signe((GEN)ns[1]))
! 1159: {
! 1160: ns2 = polsymmodpp(chi, pp);
! 1161: for (j = 1; j <= n; j++)
! 1162: affii((GEN)ns2[j], (GEN)ns[j]);
! 1163: }
! 1164:
! 1165: return ns;
! 1166: }
! 1167:
! 1168: /* compute the Newton sums modulo pp of the characteristic
! 1169: polynomial of a(x) mod g(x) */
! 1170: static GEN
! 1171: newtonsums(GEN a, GEN chi, GEN pp, GEN ns)
! 1172: {
! 1173: GEN va, pa, s, ns2;
! 1174: long j, k, n = degpol(chi), av2, lim;
! 1175:
! 1176: ns2 = manage_cache(chi, pp, ns);
! 1177:
! 1178: av2 = avma;
! 1179: lim = stack_lim(av2, 1);
! 1180:
! 1181: pa = gun;
! 1182: va = zerovec(n);
! 1183:
! 1184: for (j = 1; j <= n; j++)
! 1185: {
! 1186: pa = gmul(pa, a);
! 1187: if (pp) pa = polmodi(pa, pp);
! 1188: pa = gmod(pa, chi);
! 1189: if (pp) pa = polmodi(pa, pp);
! 1190:
! 1191: s = gzero;
! 1192:
! 1193: for (k = 0; k <= n-1; k++)
! 1194: s = addii(s, mulii(polcoeff0(pa, k, -1), (GEN)ns2[k+1]));
! 1195:
! 1196: if (pp) va[j] = (long)centermod(s, pp);
! 1197:
! 1198: if (low_stack(lim, stack_lim(av2, 1)))
! 1199: {
! 1200: GEN *gptr[2];
! 1201: gptr[0]=&pa; gptr[1]=&va;
! 1202: if(DEBUGMEM>1) err(warnmem, "newtonsums");
! 1203: gerepilemany(av2, gptr, 2);
! 1204: }
! 1205: }
! 1206:
! 1207: return va;
! 1208: }
! 1209:
! 1210: /* compute the characteristic polynomial of a mod g
! 1211: to a precision of pp using Newton sums */
! 1212: static GEN
! 1213: newtoncharpoly(GEN a, GEN chi, GEN pp, GEN ns)
! 1214: {
! 1215: GEN v, c, s, t;
! 1216: long n = degpol(chi), j, k, vn = varn(chi), av = avma, av2, lim;
! 1217:
! 1218: v = newtonsums(a, chi, pp, ns);
! 1219: av2 = avma;
! 1220: lim = stack_lim(av2, 1);
! 1221: c = cgetg(n + 2, t_VEC);
! 1222: c[1] = un;
! 1223: if (n%2) c[1] = lneg((GEN)c[1]);
! 1224: for (k = 2; k <= n+1; k++) c[k] = zero;
! 1225:
! 1226: for (k = 2; k <= n+1; k++)
! 1227: {
! 1228: s = gzero;
! 1229: for (j = 1; j < k; j++)
! 1230: {
! 1231: t = gmul((GEN)v[j], (GEN)c[k-j]);
! 1232: if (!(j%2)) t = gneg(t);
! 1233: s = gadd(s, t);
! 1234: }
! 1235: c[k] = ldiv(s, stoi(k - 1));
! 1236:
! 1237: if (low_stack(lim, stack_lim(av2, 1)))
! 1238: {
! 1239: if(DEBUGMEM>1) err(warnmem, "newtoncharpoly");
! 1240: c = gerepilecopy(av2, c);
! 1241: }
! 1242: }
! 1243:
! 1244: k = (n%2)? 1: 2;
! 1245: for ( ; k <= n+1; k += 2)
! 1246: c[k] = lneg((GEN)c[k]);
! 1247:
! 1248: return gerepileupto(av, gtopoly(c, vn));
! 1249: }
! 1250:
! 1251: static GEN
! 1252: mycaract(GEN f, GEN beta, GEN p, GEN pp, GEN ns)
! 1253: {
! 1254: GEN p1, p2, chi, chi2, npp;
! 1255: long j, a, v = varn(f), n = degpol(f);
! 1256:
! 1257: if (gcmp0(beta)) return zeropol(v);
! 1258:
! 1259: p1 = content(beta);
! 1260: if (gcmp1(p1)) p1 = NULL;
! 1261: else beta = gdiv(beta, p1);
! 1262:
! 1263: if (!pp)
! 1264: chi = caractducos(f, beta, v);
! 1265: else
! 1266: {
! 1267: a = 0;
! 1268: for (j = 1; j <= n; j++) /* compute the extra precision needed */
! 1269: a += ggval(stoi(j), p);
! 1270: npp = mulii(pp, gpowgs(p, a));
! 1271: if (p1) npp = gmul(npp, gpowgs(denom(p1), n));
! 1272:
! 1273: chi = newtoncharpoly(beta, f, npp, ns);
! 1274: }
! 1275:
! 1276: if (p1)
! 1277: {
! 1278: chi2 = cgetg(n+3, t_POL);
! 1279: chi2[1] = chi[1];
! 1280: p2 = gun;
! 1281: for (j = n+2; j >= 2; j--)
! 1282: {
! 1283: chi2[j] = lmul((GEN)chi[j], p2);
! 1284: p2 = gmul(p2, p1);
! 1285: }
! 1286: }
! 1287: else
! 1288: chi2 = chi;
! 1289:
! 1290: if (!pp) return chi2;
! 1291:
! 1292: /* this can happen only if gamma is incorrect (not an integer) */
! 1293: if (divise(denom(content(chi2)), p)) return NULL;
! 1294:
! 1295: return redelt(chi2, pp, pp);
! 1296: }
! 1297:
! 1298: /* Factor characteristic polynomial of beta mod p */
! 1299: static GEN
! 1300: factcp(GEN p, GEN f, GEN beta, GEN pp, GEN ns)
! 1301: {
! 1302: long av,l;
! 1303: GEN chi,nu, b = cgetg(4,t_VEC);
! 1304:
! 1305: chi=mycaract(f,beta,p,pp,ns);
! 1306: av=avma; nu=(GEN)factmod(chi,p)[1]; l=lg(nu)-1;
! 1307: nu=lift_intern((GEN)nu[1]);
! 1308: b[1]=(long)chi;
! 1309: b[2]=lpilecopy(av,nu);
! 1310: b[3]=lstoi(l); return b;
! 1311: }
! 1312:
! 1313: /* return the prime element in Zp[phi] */
! 1314: static GEN
! 1315: getprime(GEN p, GEN chi, GEN phi, GEN chip, GEN nup, long *Lp, long *Ep)
! 1316: {
! 1317: long v = varn(chi), L, E, r, s;
! 1318: GEN chin, pip, pp, vn;
! 1319:
! 1320: if (gegal(nup, polx[v]))
! 1321: chin = chip;
! 1322: else
! 1323: chin = mycaract(chip, nup, p, NULL, NULL);
! 1324:
! 1325: vn = vstar(p, chin);
! 1326: L = vn[0];
! 1327: E = vn[1];
! 1328:
! 1329: cbezout(L, -E, &r, &s);
! 1330:
! 1331: if (r <= 0)
! 1332: {
! 1333: long q = (-r) / E;
! 1334: q++;
! 1335: r += q*E;
! 1336: s += q*L;
! 1337: }
! 1338:
! 1339: pip = eleval(chi, nup, phi);
! 1340: pip = lift_intern(gpuigs(gmodulcp(pip, chi), r));
! 1341: pp = gpuigs(p, s);
! 1342:
! 1343: *Lp = L;
! 1344: *Ep = E;
! 1345: return gdiv(pip, pp);
! 1346: }
! 1347:
! 1348: static GEN
! 1349: update_alpha(GEN p, GEN fx, GEN alph, GEN chi, GEN pmr, GEN pmf, long mf,
! 1350: GEN ns)
! 1351: {
! 1352: long l, v = varn(fx);
! 1353: GEN nalph = NULL, nchi, w, nnu, pdr, npmr, rep;
! 1354:
! 1355: affii(gzero, (GEN)ns[1]); /* kill cache */
! 1356:
! 1357: if (!chi)
! 1358: nchi = mycaract(fx, alph, p, pmf, ns);
! 1359: else
! 1360: {
! 1361: nchi = chi;
! 1362: nalph = alph;
! 1363: }
! 1364:
! 1365: pdr = respm(nchi, derivpol(nchi), pmr);
! 1366: for (;;)
! 1367: {
! 1368: if (signe(pdr)) break;
! 1369: if (!nalph) nalph = gadd(alph, gmul(p, polx[v]));
! 1370: /* nchi was too reduced at this point */
! 1371: nchi = mycaract(fx, nalph, NULL, NULL, ns);
! 1372: pdr = respm(nchi, derivpol(nchi), pmf);
! 1373: if (signe(pdr)) break;
! 1374: if (DEBUGLEVEL >= 6)
! 1375: fprintferr(" non separable polynomial in update_alpha!\n");
! 1376: /* at this point, we assume that chi is not square-free */
! 1377: nalph = gadd(nalph, gmul(p, polx[v]));
! 1378: w = factcp(p, fx, nalph, NULL, ns);
! 1379: nchi = (GEN)w[1];
! 1380: nnu = (GEN)w[2];
! 1381: l = itos((GEN)w[3]);
! 1382: if (l > 1) return Decomp(p, fx, mf, nalph, nchi, nnu, 0);
! 1383: pdr = respm(nchi, derivpol(nchi), pmr);
! 1384: }
! 1385:
! 1386: if (is_pm1(pdr))
! 1387: npmr = gun;
! 1388: else
! 1389: {
! 1390: npmr = mulii(sqri(pdr), p);
! 1391: nchi = polmodi(nchi, npmr);
! 1392: if (!nalph) nalph = redelt(alph, npmr, pmf);
! 1393: else nalph = redelt(nalph, npmr, pmf);
! 1394: }
! 1395:
! 1396: affii(gzero, (GEN)ns[1]); /* kill cache again (contains data for fx) */
! 1397:
! 1398: rep = cgetg(5, t_VEC);
! 1399: rep[1] = (long)nalph;
! 1400: rep[2] = (long)nchi;
! 1401: rep[3] = (long)npmr;
! 1402: rep[4] = lmulii(p, pdr);
! 1403:
! 1404: return rep;
! 1405: }
! 1406:
! 1407: extern GEN Fp_factor_irred(GEN P,GEN l, GEN Q);
! 1408:
! 1409: /* flag != 0 iff we're looking for the p-adic factorization,
! 1410: in which case it is the p-adic precision we want */
! 1411: GEN
! 1412: nilord(GEN p, GEN fx, long mf, GEN gx, long flag)
! 1413: {
! 1414: long Fa, La, Ea, oE, Fg, eq, er, v = varn(fx), i, nv, Le, Ee, N, l, vn;
! 1415: GEN p1, alph, chi, nu, w, phi, pmf, pdr, pmr, kapp, pie, chib, ns;
! 1416: GEN gamm, chig, nug, delt, beta, eta, chie, nue, pia, vb, opa;
! 1417:
! 1418: if (DEBUGLEVEL >= 3)
! 1419: {
! 1420: if (flag)
! 1421: fprintferr(" entering Nilord2 (factorization)");
! 1422: else
! 1423: fprintferr(" entering Nilord2 (basis/discriminant)");
! 1424: if (DEBUGLEVEL >= 5)
! 1425: {
! 1426: fprintferr(" with parameters: p = %Z, expo = %ld\n", p, mf);
! 1427: fprintferr(" fx = %Z, gx = %Z", fx, gx);
! 1428: }
! 1429: fprintferr("\n");
! 1430: }
! 1431:
! 1432: pmf = gpowgs(p, mf + 1);
! 1433: pdr = respm(fx, derivpol(fx), pmf);
! 1434: pmr = mulii(sqri(pdr), p);
! 1435: pdr = mulii(p, pdr);
! 1436: chi = polmodi_keep(fx, pmr);
! 1437:
! 1438: alph = polx[v];
! 1439: nu = gx;
! 1440: N = degpol(fx);
! 1441: oE = 0;
! 1442: opa = NULL;
! 1443:
! 1444: /* used to cache the newton sums of chi */
! 1445: ns = cgetg(N + 2, t_COL);
! 1446: p1 = powgi(p, gceil(gdivsg(N, mulii(p, subis(p, 1))))); /* p^(N/(p(p-1))) */
! 1447: p1 = mulii(p1, mulii(pmf, gpowgs(pmr, N)));
! 1448: l = lg(p1); /* enough in general... */
! 1449: for (i = 1; i <= N + 1; i++) ns[i] = lgeti(l);
! 1450: ns[N+1] = (long)p1;
! 1451: affii(gzero, (GEN)ns[1]); /* zero means: need to be computed */
! 1452:
! 1453: for(;;)
! 1454: {
! 1455: /* kappa need to be recomputed */
! 1456: kapp = NULL;
! 1457: Fa = degpol(nu);
! 1458: /* the prime element in Zp[alpha] */
! 1459: pia = getprime(p, chi, polx[v], chi, nu, &La, &Ea);
! 1460: pia = redelt(pia, pmr, pmf);
! 1461:
! 1462: if (Ea < oE)
! 1463: {
! 1464: alph = gadd(alph, opa);
! 1465: w = update_alpha(p, fx, alph, NULL, pmr, pmf, mf, ns);
! 1466: alph = (GEN)w[1];
! 1467: chi = (GEN)w[2];
! 1468: pmr = (GEN)w[3];
! 1469: pdr = (GEN)w[4];
! 1470: kapp = NULL;
! 1471: pia = getprime(p, chi, polx[v], chi, nu, &La, &Ea);
! 1472: pia = redelt(pia, pmr, pmf);
! 1473: }
! 1474:
! 1475: oE = Ea; opa = eleval(fx, pia, alph);
! 1476:
! 1477: if (DEBUGLEVEL >= 5)
! 1478: fprintferr(" Fa = %ld and Ea = %ld \n", Fa, Ea);
! 1479:
! 1480: /* we change alpha such that nu = pia */
! 1481: if (La > 1)
! 1482: {
! 1483: alph = gadd(alph, eleval(fx, pia, alph));
! 1484:
! 1485: w = update_alpha(p, fx, alph, NULL, pmr, pmf, mf, ns);
! 1486: alph = (GEN)w[1];
! 1487: chi = (GEN)w[2];
! 1488: pmr = (GEN)w[3];
! 1489: pdr = (GEN)w[4];
! 1490: }
! 1491:
! 1492: /* if Ea*Fa == N then O = Zp[alpha] */
! 1493: if (Ea*Fa == N)
! 1494: {
! 1495: if (flag) return NULL;
! 1496:
! 1497: alph = redelt(alph, sqri(p), pmf);
! 1498: return dbasis(p, fx, mf, alph, p);
! 1499: }
! 1500:
! 1501: /* during the process beta tends to a factor of chi */
! 1502: beta = lift_intern(gpowgs(gmodulcp(nu, chi), Ea));
! 1503:
! 1504: for (;;)
! 1505: {
! 1506: if (DEBUGLEVEL >= 5)
! 1507: fprintferr(" beta = %Z\n", beta);
! 1508:
! 1509: /* if pmf divides norm(beta) then it's useless */
! 1510: p1 = gmod(gnorm(gmodulcp(beta, chi)), pmf);
! 1511: if (signe(p1))
! 1512: {
! 1513: chib = NULL;
! 1514: vn = ggval(p1, p);
! 1515: eq = (long)(vn / N);
! 1516: er = (long)(vn*Ea/N - eq*Ea);
! 1517: }
! 1518: else
! 1519: {
! 1520: chib = mycaract(chi, beta, NULL, NULL, ns);
! 1521: vb = vstar(p, chib);
! 1522: eq = (long)(vb[0] / vb[1]);
! 1523: er = (long)(vb[0]*Ea / vb[1] - eq*Ea);
! 1524: }
! 1525:
! 1526: /* eq and er are such that gamma = beta.p^-eq.nu^-er is a unit */
! 1527: if (eq) gamm = gdiv(beta, gpowgs(p, eq));
! 1528: else gamm = beta;
! 1529:
! 1530: if (er)
! 1531: {
! 1532: /* kappa = nu^-1 in Zp[alpha] */
! 1533: if (!kapp)
! 1534: {
! 1535: kapp = ginvmod(nu, chi);
! 1536: kapp = redelt(kapp, pmr, pmf);
! 1537: kapp = gmodulcp(kapp, chi);
! 1538: }
! 1539: gamm = lift(gmul(gamm, gpowgs(kapp, er)));
! 1540: gamm = redelt(gamm, p, pmf);
! 1541: }
! 1542:
! 1543: if (DEBUGLEVEL >= 6)
! 1544: fprintferr(" gamma = %Z\n", gamm);
! 1545:
! 1546: if (er || !chib)
! 1547: chig = mycaract(chi, gamm, p, pmf, ns);
! 1548: else
! 1549: {
! 1550: chig = poleval(chib, gmul(polx[v], gpowgs(p, eq)));
! 1551: chig = gdiv(chig, gpowgs(p, N*eq));
! 1552: chig = polmodi(chig, pmf);
! 1553: }
! 1554:
! 1555: if (!chig || !gcmp1(denom(content(chig))))
! 1556: {
! 1557: /* the valuation of beta was wrong... This also means
! 1558: that chi_gamma has more than one factor modulo p */
! 1559: if (!chig) chig = mycaract(chi, gamm, p, NULL, NULL);
! 1560:
! 1561: vb = vstar(p, chig);
! 1562: eq = (long)(-vb[0] / vb[1]);
! 1563: er = (long)(-vb[0]*Ea / vb[1] - eq*Ea);
! 1564: if (eq) gamm = gmul(gamm, gpowgs(p, eq));
! 1565: if (er)
! 1566: {
! 1567: gamm = gmul(gamm, gpowgs(nu, er));
! 1568: gamm = gmod(gamm, chi);
! 1569: gamm = redelt(gamm, p, pmr);
! 1570: }
! 1571: if (eq || er) chig = mycaract(chi, gamm, p, pmf, ns);
! 1572: }
! 1573:
! 1574: nug = (GEN)factmod(chig, p)[1];
! 1575: l = lg(nug) - 1;
! 1576: nug = lift((GEN)nug[l]);
! 1577:
! 1578: if (l > 1)
! 1579: {
! 1580: /* there are at least 2 factors mod. p => chi can be split */
! 1581: phi = eleval(fx, gamm, alph);
! 1582: phi = redelt(phi, p, pmf);
! 1583: if (flag) mf += 3;
! 1584: return Decomp(p, fx, mf, phi, chig, nug, flag);
! 1585: }
! 1586:
! 1587: Fg = degpol(nug);
! 1588: if (Fa%Fg)
! 1589: {
! 1590: if (DEBUGLEVEL >= 5)
! 1591: fprintferr(" Increasing Fa\n");
! 1592: /* we compute a new element such F = lcm(Fa, Fg) */
! 1593: w = testb2(p, chi, Fa, gamm, pmf, Fg, ns);
! 1594: if (gcmp1((GEN)w[1]))
! 1595: {
! 1596: /* there are at least 2 factors mod. p => chi can be split */
! 1597: phi = eleval(fx, (GEN)w[2], alph);
! 1598: phi = redelt(phi, p, pmf);
! 1599: if (flag) mf += 3;
! 1600: return Decomp(p, fx, mf, phi, (GEN)w[3], (GEN)w[4], flag);
! 1601: }
! 1602: break;
! 1603: }
! 1604:
! 1605: /* we look for a root delta of nug in Fp[alpha] such that
! 1606: vp(gamma - delta) > 0. This root can then be used to
! 1607: improved the approximation given by beta */
! 1608: nv = fetch_var();
! 1609: w = Fp_factor_irred(nug, p, gsubst(nu, varn(nu), polx[nv]));
! 1610: if (degpol(w[1]) != 1)
! 1611: err(talker,"bug in nilord (no root). Is p a prime ?");
! 1612:
! 1613: for (i = 1;; i++)
! 1614: {
! 1615: if (i >= lg(w))
! 1616: err(talker, "bug in nilord (no suitable root), is p = %Z a prime?",
! 1617: (long)p);
! 1618: delt = gneg_i(gsubst(gcoeff(w, 2, i), nv, polx[v]));
! 1619: eta = gsub(gamm, delt);
! 1620: if (typ(delt) == t_INT)
! 1621: {
! 1622: chie = poleval(chig, gadd(polx[v], delt));
! 1623: nue = (GEN)factmod(chie, p)[1];
! 1624: l = lg(nue) - 1;
! 1625: nue = lift((GEN)nue[l]);
! 1626: }
! 1627: else
! 1628: {
! 1629: p1 = factcp(p, chi, eta, pmf, ns);
! 1630: chie = (GEN)p1[1];
! 1631: nue = (GEN)p1[2];
! 1632: l = itos((GEN)p1[3]);
! 1633: }
! 1634: if (l > 1)
! 1635: {
! 1636: /* there are at least 2 factors mod. p => chi can be split */
! 1637: delete_var();
! 1638: phi = eleval(fx, eta, alph);
! 1639: phi = redelt(phi, p, pmf);
! 1640: if (flag) mf += 3;
! 1641: return Decomp(p, fx, mf, phi, chie, nue, flag);
! 1642: }
! 1643:
! 1644: /* if vp(eta) = vp(gamma - delta) > 0 */
! 1645: if (gegal(nue, polx[v])) break;
! 1646: }
! 1647: delete_var();
! 1648:
! 1649: if (!signe(modii((GEN)chie[2], pmr)))
! 1650: chie = mycaract(chi, eta, p, pmf, ns);
! 1651:
! 1652: pie = getprime(p, chi, eta, chie, nue, &Le, &Ee);
! 1653: if (Ea%Ee)
! 1654: {
! 1655: if (DEBUGLEVEL >= 5)
! 1656: fprintferr(" Increasing Ea\n");
! 1657: pie = redelt(pie, p, pmf);
! 1658: /* we compute a new element such E = lcm(Ea, Ee) */
! 1659: w = testc2(p, chi, pmr, pmf, nu, Ea, pie, Ee, ns);
! 1660: if (gcmp1((GEN)w[1]))
! 1661: {
! 1662: /* there are at least 2 factors mod. p => chi can be split */
! 1663: phi = eleval(fx, (GEN)w[2], alph);
! 1664: phi = redelt(phi, p, pmf);
! 1665: if (flag) mf += 3;
! 1666: return Decomp(p, fx, mf, phi, (GEN)w[3], (GEN)w[4], flag);
! 1667: }
! 1668: break;
! 1669: }
! 1670:
! 1671: if (eq) delt = gmul(delt, gpowgs(p, eq));
! 1672: if (er) delt = gmul(delt, gpowgs(nu, er));
! 1673: beta = gsub(beta, delt);
! 1674: }
! 1675:
! 1676: /* we replace alpha by a new alpha with a larger F or E */
! 1677: alph = eleval(fx, (GEN)w[2], alph);
! 1678: chi = (GEN)w[3];
! 1679: nu = (GEN)w[4];
! 1680:
! 1681: w = update_alpha(p, fx, alph, chi, pmr, pmf, mf, ns);
! 1682: alph = (GEN)w[1];
! 1683: chi = (GEN)w[2];
! 1684: pmr = (GEN)w[3];
! 1685: pdr = (GEN)w[4];
! 1686:
! 1687: /* that can happen if p does not divide the field discriminant! */
! 1688: if (is_pm1(pmr))
! 1689: {
! 1690: if (flag)
! 1691: {
! 1692: p1 = lift((GEN)factmod(chi, p)[1]);
! 1693: l = lg(p1) - 1;
! 1694: if (l == 1) return NULL;
! 1695: phi = redelt(alph, p, pmf);
! 1696: return Decomp(p, fx, ggval(pmf, p), phi, chi, (GEN)p1[l], flag);
! 1697: }
! 1698: else
! 1699: return dbasis(p, fx, mf, alph, chi);
! 1700: }
! 1701: }
! 1702: }
! 1703:
! 1704: /* Returns [1,phi,chi,nu] if phi non-primary
! 1705: * [2,phi,chi,nu] with F_phi = lcm (F_alpha, F_theta)
! 1706: * and E_phi = E_alpha
! 1707: */
! 1708: static GEN
! 1709: testb2(GEN p, GEN fa, long Fa, GEN theta, GEN pmf, long Ft, GEN ns)
! 1710: {
! 1711: long m, Dat, t, v = varn(fa);
! 1712: GEN b, w, phi, h;
! 1713:
! 1714: Dat = clcm(Fa, Ft) + 3;
! 1715: b = cgetg(5, t_VEC);
! 1716: m = p[2];
! 1717: if (degpol(p) > 0 || m < 0) m = 0;
! 1718:
! 1719: for (t = 1;; t++)
! 1720: {
! 1721: h = m? stopoly(t, m, v): scalarpol(stoi(t), v);
! 1722: phi = gadd(theta, gmod(h, fa));
! 1723: w = factcp(p, fa, phi, pmf, ns);
! 1724: h = (GEN)w[3];
! 1725: if (h[2] > 1) { b[1] = un; break; }
! 1726: if (lgef(w[2]) == Dat) { b[1] = deux; break; }
! 1727: }
! 1728:
! 1729: b[2] = (long)phi;
! 1730: b[3] = w[1];
! 1731: b[4] = w[2];
! 1732: return b;
! 1733: }
! 1734:
! 1735: /* Returns [1, phi, chi, nu] if phi non-primary
! 1736: * [2, phi, chi, nu] if E_phi = lcm (E_alpha, E_theta)
! 1737: */
! 1738: static GEN
! 1739: testc2(GEN p, GEN fa, GEN pmr, GEN pmf, GEN alph2, long Ea, GEN thet2,
! 1740: long Et, GEN ns)
! 1741: {
! 1742: GEN b, c1, c2, c3, psi, phi, w, h;
! 1743: long r, s, t, v = varn(fa);
! 1744:
! 1745: b=cgetg(5, t_VEC);
! 1746:
! 1747: cbezout(Ea, Et, &r, &s); t = 0;
! 1748: while (r < 0) { r = r + Et; t++; }
! 1749: while (s < 0) { s = s + Ea; t++; }
! 1750:
! 1751: c1 = lift_intern(gpuigs(gmodulcp(alph2, fa), s));
! 1752: c2 = lift_intern(gpuigs(gmodulcp(thet2, fa), r));
! 1753: c3 = gdiv(gmod(gmul(c1, c2), fa), gpuigs(p, t));
! 1754:
! 1755: psi = redelt(c3, pmr, pmr);
! 1756: phi = gadd(polx[v], psi);
! 1757:
! 1758: w = factcp(p, fa, phi, pmf, ns);
! 1759: h = (GEN)w[3];
! 1760: b[1] = (h[2] > 1)? un: deux;
! 1761: b[2] = (long)phi;
! 1762: b[3] = w[1];
! 1763: b[4] = w[2];
! 1764: return b;
! 1765: }
! 1766:
! 1767: /* evaluate h(a) mod f */
! 1768: GEN
! 1769: eleval(GEN f,GEN h,GEN a)
! 1770: {
! 1771: long n,av,tetpil;
! 1772: GEN y;
! 1773:
! 1774: if (typ(h) != t_POL) return gcopy(h);
! 1775: av = tetpil = avma;
! 1776: n=lgef(h)-1; y=(GEN)h[n];
! 1777: for (n--; n>=2; n--)
! 1778: {
! 1779: y = gadd(gmul(y,a),(GEN)h[n]);
! 1780: tetpil=avma; y = gmod(y,f);
! 1781: }
! 1782: return gerepile(av,tetpil,y);
! 1783: }
! 1784:
! 1785: GEN addshiftw(GEN x, GEN y, long d);
! 1786:
! 1787: static GEN
! 1788: shiftpol(GEN x, long v)
! 1789: {
! 1790: x = addshiftw(x, zeropol(v), 1);
! 1791: setvarn(x,v); return x;
! 1792: }
! 1793:
! 1794: /* Sylvester's matrix, mod p^m (assumes f1 monic) */
! 1795: static GEN
! 1796: sylpm(GEN f1,GEN f2,GEN pm)
! 1797: {
! 1798: long n,j,v=varn(f1);
! 1799: GEN a,h;
! 1800:
! 1801: n=degpol(f1); a=cgetg(n+1,t_MAT);
! 1802: h = FpX_res(f2,f1,pm);
! 1803: for (j=1;; j++)
! 1804: {
! 1805: a[j] = (long)pol_to_vec(h,n);
! 1806: if (j == n) break;
! 1807: h = FpX_res(shiftpol(h,v),f1,pm);
! 1808: }
! 1809: return hnfmodid(a,pm);
! 1810: }
! 1811:
! 1812: /* polynomial gcd mod p^m (assumes f1 monic) */
! 1813: GEN
! 1814: gcdpm(GEN f1,GEN f2,GEN pm)
! 1815: {
! 1816: long n,c,v=varn(f1),av=avma,tetpil;
! 1817: GEN a,col;
! 1818:
! 1819: n=degpol(f1); a=sylpm(f1,f2,pm);
! 1820: for (c=1; c<=n; c++)
! 1821: if (signe(resii(gcoeff(a,c,c),pm))) break;
! 1822: if (c > n) { avma=av; return zeropol(v); }
! 1823:
! 1824: col = gdiv((GEN)a[c], gcoeff(a,c,c)); tetpil=avma;
! 1825: return gerepile(av,tetpil, gtopolyrev(col,v));
! 1826: }
! 1827:
! 1828: /* reduced resultant mod p^m (assumes x monic) */
! 1829: GEN
! 1830: respm(GEN x,GEN y,GEN pm)
! 1831: {
! 1832: long av = avma;
! 1833: GEN p1 = sylpm(x,y,pm);
! 1834:
! 1835: p1 = gcoeff(p1,1,1);
! 1836: if (egalii(p1,pm)) { avma = av; return gzero; }
! 1837: return gerepileuptoint(av, icopy(p1));
! 1838: }
! 1839:
! 1840: /* Normalized integral basis */
! 1841: static GEN
! 1842: nbasis(GEN ibas,GEN pd)
! 1843: {
! 1844: long k, n = lg(ibas)-1;
! 1845: GEN a = cgetg(n+1,t_MAT);
! 1846: for (k=1; k<=n; k++) a[k] = (long)pol_to_vec((GEN)ibas[k],n);
! 1847: return gdiv(hnfmodid(a,pd), pd);
! 1848: }
! 1849:
! 1850: /*******************************************************************/
! 1851: /* */
! 1852: /* BUCHMANN-LENSTRA ALGORITHM */
! 1853: /* */
! 1854: /*******************************************************************/
! 1855: static GEN lens(GEN nf,GEN p,GEN a);
! 1856: GEN element_powid_mod_p(GEN nf, long I, GEN n, GEN p);
! 1857:
! 1858: /* return a Z basis of Z_K's p-radical, modfrob = x--> x^p-x */
! 1859: static GEN
! 1860: pradical(GEN nf, GEN p, GEN *modfrob)
! 1861: {
! 1862: long i,N = degpol(nf[1]);
! 1863: GEN p1,m,frob,rad;
! 1864:
! 1865: frob = cgetg(N+1,t_MAT);
! 1866: for (i=1; i<=N; i++)
! 1867: frob[i] = (long) element_powid_mod_p(nf,i,p, p);
! 1868:
! 1869: /* p1 = smallest power of p st p^k >= N */
! 1870: p1=p; while (cmpis(p1,N)<0) p1=mulii(p1,p);
! 1871: if (p1==p) m = frob;
! 1872: else
! 1873: {
! 1874: m=cgetg(N+1,t_MAT); p1 = divii(p1,p);
! 1875: for (i=1; i<=N; i++)
! 1876: m[i]=(long)element_pow_mod_p(nf,(GEN)frob[i],p1, p);
! 1877: }
! 1878: rad = FpM_ker(m, p);
! 1879: for (i=1; i<=N; i++)
! 1880: coeff(frob,i,i) = lsubis(gcoeff(frob,i,i), 1);
! 1881: *modfrob = frob; return rad;
! 1882: }
! 1883:
! 1884: static GEN
! 1885: project(GEN algebre, GEN x, long k, long kbar)
! 1886: {
! 1887: x = inverseimage(algebre,x);
! 1888: x += k; x[0] = evaltyp(t_COL) | evallg(kbar+1);
! 1889: return x;
! 1890: }
! 1891:
! 1892: /* Calcule le polynome minimal de alpha dans algebre (coeffs dans Z) */
! 1893: static GEN
! 1894: pol_min(GEN alpha,GEN nf,GEN algebre,long kbar,GEN p)
! 1895: {
! 1896: long av=avma,tetpil,i,N,k;
! 1897: GEN p1,puiss;
! 1898:
! 1899: N = lg(nf[1])-3; puiss=cgetg(N+2,t_MAT);
! 1900: k = N-kbar; p1=alpha;
! 1901: puiss[1] = (long)gscalcol_i(gun,kbar);
! 1902: for (i=2; i<=N+1; i++)
! 1903: {
! 1904: if (i>2) p1 = element_mul(nf,p1,alpha);
! 1905: puiss[i] = (long) project(algebre,p1,k,kbar);
! 1906: }
! 1907: puiss = lift_intern(puiss);
! 1908: p1 = (GEN)FpM_ker(puiss, p)[1]; tetpil=avma;
! 1909: return gerepile(av,tetpil,gtopolyrev(p1,0));
! 1910: }
! 1911:
! 1912: /* Evalue le polynome pol en alpha,element de nf */
! 1913: static GEN
! 1914: eval_pol(GEN nf,GEN pol,GEN alpha,GEN algebre,GEN algebre1)
! 1915: {
! 1916: long av=avma,tetpil,i,kbar,k, lx = lgef(pol)-1, N = degpol(nf[1]);
! 1917: GEN res;
! 1918:
! 1919: kbar = lg(algebre1)-1; k = N-kbar;
! 1920: res = gscalcol_i((GEN)pol[lx], N);
! 1921: for (i=2; i<lx; i++)
! 1922: {
! 1923: res = element_mul(nf,alpha,res);
! 1924: res[1] = ladd((GEN)res[1],(GEN)pol[i]);
! 1925: }
! 1926: res = project(algebre,res,k,kbar); tetpil=avma;
! 1927: return gerepile(av,tetpil,gmul(algebre1,res));
! 1928: }
! 1929:
! 1930: static GEN
! 1931: kerlens2(GEN x, GEN p)
! 1932: {
! 1933: long i,j,k,t,nbc,nbl,av,av1;
! 1934: GEN a,c,l,d,y,q;
! 1935:
! 1936: av=avma; a=gmul(x,gmodulsg(1,p));
! 1937: nbl=nbc=lg(x)-1;
! 1938: c=new_chunk(nbl+1); for (i=1; i<=nbl; i++) c[i]=0;
! 1939: l=new_chunk(nbc+1);
! 1940: d=new_chunk(nbc+1);
! 1941: k = t = 1;
! 1942: while (t<=nbl && k<=nbc)
! 1943: {
! 1944: for (j=1; j<k; j++)
! 1945: for (i=1; i<=nbl; i++)
! 1946: if (i!=l[j])
! 1947: coeff(a,i,k) = lsub(gmul((GEN)d[j],gcoeff(a,i,k)),
! 1948: gmul(gcoeff(a,l[j],k),gcoeff(a,i,j)));
! 1949: t=1; while (t<=nbl && (c[t] || gcmp0(gcoeff(a,t,k)))) t++;
! 1950: if (t<=nbl) { d[k]=coeff(a,t,k); c[t]=k; l[k]=t; k++; }
! 1951: }
! 1952: if (k>nbc) err(bugparier,"kerlens2");
! 1953: y=cgetg(nbc+1,t_COL);
! 1954: y[1]=(k>1)?coeff(a,l[1],k):un;
! 1955: for (q=gun,j=2; j<k; j++)
! 1956: {
! 1957: q=gmul(q,(GEN)d[j-1]);
! 1958: y[j]=lmul(gcoeff(a,l[j],k),q);
! 1959: }
! 1960: if (k>1) y[k]=lneg(gmul(q,(GEN)d[k-1]));
! 1961: for (j=k+1; j<=nbc; j++) y[j]=zero;
! 1962: av1=avma; return gerepile(av,av1,lift(y));
! 1963: }
! 1964:
! 1965: static GEN
! 1966: kerlens(GEN x, GEN pgen)
! 1967: {
! 1968: long av = avma, i,j,k,t,nbc,nbl,p,q,*c,*l,*d,**a;
! 1969: GEN y;
! 1970:
! 1971: if (cmpis(pgen, MAXHALFULONG>>1) > 0)
! 1972: return kerlens2(x,pgen);
! 1973: /* ici p <= (MAXHALFULONG>>1) ==> long du C */
! 1974: p=itos(pgen); nbl=nbc=lg(x)-1;
! 1975: a=(long**)new_chunk(nbc+1);
! 1976: for (j=1; j<=nbc; j++)
! 1977: {
! 1978: c=a[j]=new_chunk(nbl+1);
! 1979: for (i=1; i<=nbl; i++) c[i]=smodis(gcoeff(x,i,j),p);
! 1980: }
! 1981: c=new_chunk(nbl+1); for (i=1; i<=nbl; i++) c[i]=0;
! 1982: l=new_chunk(nbc+1);
! 1983: d=new_chunk(nbc+1);
! 1984: k = t = 1;
! 1985: while (t<=nbl && k<=nbc)
! 1986: {
! 1987: for (j=1; j<k; j++)
! 1988: for (i=1; i<=nbl; i++)
! 1989: if (i!=l[j])
! 1990: a[k][i] = (d[j]*a[k][i] - a[j][i]*a[k][l[j]]) % p;
! 1991: t=1; while (t<=nbl && (c[t] || !a[k][t])) t++;
! 1992: if (t<=nbl) { d[k]=a[k][t]; c[t]=k; l[k++]=t; }
! 1993: }
! 1994: if (k>nbc) err(bugparier,"kerlens");
! 1995: avma=av; y=cgetg(nbc+1,t_COL);
! 1996: t=(k>1) ? a[k][l[1]]:1;
! 1997: y[1]=(t>0)? lstoi(t):lstoi(t+p);
! 1998: for (q=1,j=2; j<k; j++)
! 1999: {
! 2000: q = (q*d[j-1]) % p;
! 2001: t = (a[k][l[j]]*q) % p;
! 2002: y[j] = (t>0) ? lstoi(t) : lstoi(t+p);
! 2003: }
! 2004: if (k>1)
! 2005: {
! 2006: t = (q*d[k-1]) % p;
! 2007: y[k] = (t>0) ? lstoi(p-t) : lstoi(-t);
! 2008: }
! 2009: for (j=k+1; j<=nbc; j++) y[j]=zero;
! 2010: return y;
! 2011: }
! 2012:
! 2013: /* Calcule la constante de lenstra de l'ideal p.Z_K+a.Z_K ou a est un
! 2014: vecteur sur la base d'entiers */
! 2015: static GEN
! 2016: lens(GEN nf, GEN p, GEN a)
! 2017: {
! 2018: long av=avma,tetpil,N=degpol(nf[1]),j;
! 2019: GEN mat=cgetg(N+1,t_MAT);
! 2020: for (j=1; j<=N; j++) mat[j]=(long)element_mulid(nf,a,j);
! 2021: tetpil=avma; return gerepile(av,tetpil,kerlens(mat,p));
! 2022: }
! 2023:
! 2024: extern GEN det_mod_P_n(GEN a, GEN N, GEN P);
! 2025: GEN sylvestermatrix_i(GEN x, GEN y);
! 2026:
! 2027: /* check if p^va doesnt divide norm x (or norm(x+p)) */
! 2028: #if 0
! 2029: /* compute norm mod p^whatneeded using Sylvester's matrix */
! 2030: /* looks slower than the new subresultant. Have to re-check this */
! 2031: static GEN
! 2032: prime_check_elt(GEN a, GEN pol, GEN p, GEN pf)
! 2033: {
! 2034: GEN M,mod,x, c = denom(content(a));
! 2035: long v = pvaluation(c, p, &x); /* x is junk */
! 2036:
! 2037: mod = mulii(pf, gpowgs(p, degpol(pol)*v + 1));
! 2038:
! 2039: x = FpX_red(gmul(a,c), mod);
! 2040: M = sylvestermatrix_i(pol,x);
! 2041: if (det_mod_P_n(M,mod,p) == gzero)
! 2042: {
! 2043: x[2] = ladd((GEN)x[2], mulii(p,c));
! 2044: M = sylvestermatrix_i(pol,x);
! 2045: if (det_mod_P_n(M,mod,p) == gzero) return NULL;
! 2046: a[2] = ladd((GEN)a[2], p);
! 2047: }
! 2048: return a;
! 2049: }
! 2050: #else
! 2051: /* use subres to compute norm */
! 2052: static GEN
! 2053: prime_check_elt(GEN a, GEN pol, GEN p, GEN pf)
! 2054: {
! 2055: GEN norme=subres(pol,a);
! 2056: if (resii(divii(norme,pf),p) != gzero) return a;
! 2057: /* Note: a+p can't succeed if e > 1, can we know this at this point ? */
! 2058: a=gadd(a,p); norme=subres(pol,a);
! 2059: if (resii(divii(norme,pf),p) != gzero) return a;
! 2060: return NULL;
! 2061: }
! 2062: #endif
! 2063:
! 2064: #if 0
! 2065: static GEN
! 2066: prime_two_elt_loop(GEN beta, GEN pol, GEN p, GEN pf)
! 2067: {
! 2068: long av, m = lg(beta)-1;
! 2069: int i,j,K, *x = (int*)new_chunk(m+1);
! 2070: GEN a;
! 2071:
! 2072: K = 1; av = avma;
! 2073: for(;;)
! 2074: { /* x runs through strictly increasing sequences of length K,
! 2075: * 1 <= x[i] <= m */
! 2076: nextK:
! 2077: if (DEBUGLEVEL) fprintferr("K = %d\n", K);
! 2078: for (i=1; i<=K; i++) x[i] = i;
! 2079: for(;;)
! 2080: {
! 2081: if (DEBUGLEVEL > 1)
! 2082: {
! 2083: for (i=1; i<=K; i++) fprintferr("%d ",x[i]);
! 2084: fprintferr("\n"); flusherr();
! 2085: }
! 2086: a = (GEN)beta[x[1]];
! 2087: for (i=2; i<=K; i++) a = gadd(a, (GEN)beta[x[i]]);
! 2088: if ((a = prime_check_elt(a,pol,p,pf))) return a;
! 2089: avma = av;
! 2090:
! 2091: /* start: i = K+1; */
! 2092: do
! 2093: {
! 2094: if (--i == 0)
! 2095: {
! 2096: if (++K > m) return NULL; /* fail */
! 2097: goto nextK;
! 2098: }
! 2099: x[i]++;
! 2100: } while (x[i] > m - K + i);
! 2101: for (j=i; j<K; j++) x[j+1] = x[j]+1;
! 2102: }
! 2103: }
! 2104: }
! 2105: #endif
! 2106:
! 2107: static GEN
! 2108: random_prime_two_elt_loop(GEN beta, GEN pol, GEN p, GEN pf)
! 2109: {
! 2110: long av = avma, z,i, m = lg(beta)-1;
! 2111: long keep = getrand();
! 2112: int c = 0;
! 2113: GEN a;
! 2114:
! 2115: for(i=1; i<=m; i++)
! 2116: if ((a = prime_check_elt((GEN)beta[i],pol,p,pf))) return a;
! 2117: (void)setrand(1);
! 2118: if (DEBUGLEVEL) fprintferr("prime_two_elt_loop, hard case: ");
! 2119: for(;;avma=av)
! 2120: {
! 2121: if (DEBUGLEVEL) fprintferr("%d ", ++c);
! 2122: a = gzero;
! 2123: for (i=1; i<=m; i++)
! 2124: {
! 2125: z = mymyrand() >> (BITS_IN_RANDOM-5); /* in [0,15] */
! 2126: if (z >= 9) z -= 7;
! 2127: a = gadd(a,gmulsg(z,(GEN)beta[i]));
! 2128: }
! 2129: if ((a = prime_check_elt(a,pol,p,pf)))
! 2130: {
! 2131: if (DEBUGLEVEL) fprintferr("\n");
! 2132: (void)setrand(keep); return a;
! 2133: }
! 2134: }
! 2135: }
! 2136:
! 2137: /* Input: an ideal mod p (!= Z_K)
! 2138: * Output: a 2-elt representation [p, x] */
! 2139: static GEN
! 2140: prime_two_elt(GEN nf, GEN p, GEN ideal)
! 2141: {
! 2142: GEN beta,a,pf, pol = (GEN)nf[1];
! 2143: long f, N=degpol(pol), m=lg(ideal)-1;
! 2144: ulong av;
! 2145:
! 2146: if (!m) return gscalcol_i(p,N);
! 2147:
! 2148: /* we want v_p(Norm(beta)) = p^f, f = N-m */
! 2149: av = avma; f = N-m; pf = gpuigs(p,f);
! 2150: ideal = centerlift(ideal);
! 2151: ideal = concatsp(gscalcol(p,N), ideal);
! 2152: ideal = ideal_better_basis(nf, ideal, p);
! 2153: beta = gmul((GEN)nf[7], ideal);
! 2154:
! 2155: #if 0
! 2156: a = prime_two_elt_loop(beta,pol,p,pf);
! 2157: if (!a) err(bugparier, "prime_two_elt (failed)");
! 2158: #else
! 2159: a = random_prime_two_elt_loop(beta,pol,p,pf);
! 2160: #endif
! 2161:
! 2162: a = centermod(algtobasis_intern(nf,a), p);
! 2163: if (resii(divii(subres(gmul((GEN)nf[7],a),pol),pf),p) == gzero)
! 2164: a[1] = laddii((GEN)a[1],p);
! 2165: return gerepilecopy(av,a);
! 2166: }
! 2167:
! 2168: static GEN
! 2169: apply_kummer(GEN nf,GEN pol,GEN e,GEN p,long N,GEN *beta)
! 2170: {
! 2171: GEN T,p1, res = cgetg(6,t_VEC);
! 2172: long f = degpol(pol);
! 2173:
! 2174: res[1]=(long)p;
! 2175: res[3]=(long)e;
! 2176: res[4]=lstoi(f);
! 2177: if (f == N) /* inert */
! 2178: {
! 2179: res[2]=(long)gscalcol_i(p,N);
! 2180: res[5]=(long)gscalcol_i(gun,N);
! 2181: }
! 2182: else
! 2183: {
! 2184: T = (GEN) nf[1];
! 2185: if (ggval(subres(pol,T),p) > f)
! 2186: pol[2] = laddii((GEN)pol[2],p);
! 2187: res[2] = (long) algtobasis_intern(nf,pol);
! 2188:
! 2189: p1 = FpX_div(T,pol,p);
! 2190: res[5] = (long) centermod(algtobasis_intern(nf,p1), p);
! 2191:
! 2192: if (beta)
! 2193: *beta = *beta? FpX_div(*beta,pol,p): p1;
! 2194: }
! 2195: return res;
! 2196: }
! 2197:
! 2198: /* prime ideal decomposition of p sorted by increasing residual degree */
! 2199: GEN
! 2200: primedec(GEN nf, GEN p)
! 2201: {
! 2202: long av=avma,tetpil,i,j,k,kbar,np,c,indice,N,lp;
! 2203: GEN ex,f,list,ip,elth,h;
! 2204: GEN modfrob,algebre,algebre1,b,mat1,T;
! 2205: GEN alpha,beta,p1,p2,unmodp,zmodp,idmodp;
! 2206:
! 2207: if (DEBUGLEVEL>=3) timer2();
! 2208: nf=checknf(nf); T=(GEN)nf[1]; N=degpol(T);
! 2209: f=factmod(T,p); ex=(GEN)f[2];
! 2210: f=centerlift((GEN)f[1]); np=lg(f);
! 2211: if (DEBUGLEVEL>=6) msgtimer("factmod");
! 2212:
! 2213: if (signe(modii((GEN)nf[4],p))) /* p doesn't divide index */
! 2214: {
! 2215: list=cgetg(np,t_VEC);
! 2216: for (i=1; i<np; i++)
! 2217: list[i]=(long)apply_kummer(nf,(GEN)f[i],(GEN)ex[i],p,N, NULL);
! 2218: if (DEBUGLEVEL>=6) msgtimer("simple primedec");
! 2219: p1=stoi(4); tetpil=avma;
! 2220: return gerepile(av,tetpil,vecsort(list,p1));
! 2221: }
! 2222:
! 2223: p1 = (GEN)f[1];
! 2224: for (i=2; i<np; i++)
! 2225: p1 = FpX_red(gmul(p1, (GEN)f[i]), p);
! 2226: p1 = FpX_red(gdiv(gadd(gmul(p1, FpX_div(T,p1,p)), gneg(T)), p), p);
! 2227: list = cgetg(N+1,t_VEC);
! 2228: indice=1; beta=NULL;
! 2229: for (i=1; i<np; i++) /* e = 1 or f[i] does not divide p1 (mod p) */
! 2230: if (is_pm1(ex[i]) || signe(FpX_res(p1,(GEN)f[i],p)))
! 2231: list[indice++] = (long)apply_kummer(nf,(GEN)f[i],(GEN)ex[i],p,N,&beta);
! 2232: if (DEBUGLEVEL>=3) msgtimer("unramified factors");
! 2233:
! 2234: /* modfrob = modified Frobenius: x -> x^p - x mod p */
! 2235: ip = pradical(nf,p,&modfrob);
! 2236: if (DEBUGLEVEL>=3) msgtimer("pradical");
! 2237:
! 2238: if (beta)
! 2239: {
! 2240: beta = algtobasis_intern(nf,beta);
! 2241: lp=lg(ip)-1; p1=cgetg(2*lp+N+1,t_MAT);
! 2242: for (i=1; i<=N; i++) p1[i]=(long)element_mulid(nf,beta,i);
! 2243: for ( ; i<=N+lp; i++)
! 2244: {
! 2245: p2 = (GEN) ip[i-N];
! 2246: p1[i+lp] = (long) p2;
! 2247: p1[i] = ldiv(element_mul(nf,lift(p2),beta),p);
! 2248: }
! 2249: ip = FpM_image(p1, p);
! 2250: if (lg(ip)>N) err(bugparier,"primedec (bad pradical)");
! 2251: }
! 2252: unmodp=gmodulsg(1,p); zmodp=gmodulsg(0,p);
! 2253: idmodp = idmat_intern(N,unmodp,zmodp);
! 2254: ip = gmul(ip, unmodp);
! 2255:
! 2256: h=cgetg(N+1,t_VEC); h[1]=(long)ip;
! 2257: for (c=1; c; c--)
! 2258: {
! 2259: elth=(GEN)h[c]; k=lg(elth)-1; kbar=N-k;
! 2260: p1 = concatsp(elth,(GEN)idmodp[1]);
! 2261: algebre = suppl_intern(p1,idmodp);
! 2262: algebre1 = cgetg(kbar+1,t_MAT);
! 2263: for (i=1; i<=kbar; i++) algebre1[i]=algebre[i+k];
! 2264: b = gmul(modfrob,algebre1);
! 2265: for (i=1;i<=kbar;i++)
! 2266: b[i] = (long) project(algebre,(GEN) b[i],k,kbar);
! 2267: mat1 = FpM_ker(lift_intern(b), p);
! 2268: if (lg(mat1)>2)
! 2269: {
! 2270: GEN mat2 = cgetg(k+N+1,t_MAT);
! 2271: for (i=1; i<=k; i++) mat2[i]=elth[i];
! 2272: alpha=gmul(algebre1,(GEN)mat1[2]);
! 2273: p1 = pol_min(alpha,nf,algebre,kbar,p);
! 2274: p1 = (GEN)factmod(p1,p)[1];
! 2275: for (i=1; i<lg(p1); i++)
! 2276: {
! 2277: beta = eval_pol(nf,(GEN)p1[i],alpha,algebre,algebre1);
! 2278: beta = lift_intern(beta);
! 2279: for (j=1; j<=N; j++)
! 2280: mat2[k+j] = (long)FpV(element_mulid(nf,beta,j), p);
! 2281: h[c] = (long) image(mat2); c++;
! 2282: }
! 2283: }
! 2284: else
! 2285: {
! 2286: long av1; p1 = cgetg(6,t_VEC);
! 2287: list[indice++] = (long)p1;
! 2288: p1[1]=(long)p; p1[4]=lstoi(kbar);
! 2289: p1[2]=(long)prime_two_elt(nf,p,elth);
! 2290: p1[5]=(long)lens(nf,p,(GEN)p1[2]);
! 2291: av1=avma;
! 2292: i = int_elt_val(nf,(GEN)p1[5],p,(GEN)p1[5],NULL,N-1);
! 2293: avma=av1;
! 2294: p1[3]=lstoi(i+1);
! 2295: }
! 2296: if (DEBUGLEVEL>=3) msgtimer("h[%ld]",c);
! 2297: }
! 2298: setlg(list, indice); tetpil=avma;
! 2299: return gerepile(av,tetpil,gen_sort(list,0,cmp_prime_over_p));
! 2300: }
! 2301:
! 2302: /* REDUCTION Modulo a prime ideal */
! 2303:
! 2304: /* x integral, reduce mod prh in HNF */
! 2305: GEN
! 2306: nfreducemodpr_i(GEN x, GEN prh)
! 2307: {
! 2308: GEN p = gcoeff(prh,1,1);
! 2309: long i,j;
! 2310:
! 2311: x = dummycopy(x);
! 2312: for (i=lg(x)-1; i>=2; i--)
! 2313: {
! 2314: GEN t = (GEN)prh[i], p1 = resii((GEN)x[i], p);
! 2315: x[i] = (long)p1;
! 2316: if (signe(p1) && is_pm1(t[i]))
! 2317: {
! 2318: for (j=1; j<i; j++)
! 2319: x[j] = lsubii((GEN)x[j], mulii(p1, (GEN)t[j]));
! 2320: x[i] = zero;
! 2321: }
! 2322: }
! 2323: x[1] = lresii((GEN)x[1], p); return x;
! 2324: }
! 2325:
! 2326: /* for internal use */
! 2327: GEN
! 2328: nfreducemodpr(GEN nf, GEN x, GEN prhall)
! 2329: {
! 2330: long i,v;
! 2331: GEN p,prh,den;
! 2332:
! 2333: for (i=lg(x)-1; i>0; i--)
! 2334: if (typ(x[i]) == t_INTMOD) { x=lift_intern(x); break; }
! 2335: prh=(GEN)prhall[1]; p=gcoeff(prh,1,1);
! 2336: den=denom(x);
! 2337: if (!gcmp1(den))
! 2338: {
! 2339: v=ggval(den,p);
! 2340: if (v) x=element_mul(nf,x,element_pow(nf,(GEN)prhall[2],stoi(v)));
! 2341: x = gmod(x,p);
! 2342: }
! 2343: return FpV(nfreducemodpr_i(x, prh), p);
! 2344: }
! 2345:
! 2346: /* public function */
! 2347: GEN
! 2348: nfreducemodpr2(GEN nf, GEN x, GEN prhall)
! 2349: {
! 2350: long av = avma; checkprhall(prhall);
! 2351: if (typ(x) != t_COL) x = algtobasis(nf,x);
! 2352: return gerepileupto(av, nfreducemodpr(nf,x,prhall));
! 2353: }
! 2354:
! 2355: /* relative ROUND 2
! 2356: *
! 2357: * input: nf = base field K
! 2358: * x monic polynomial, coefficients in Z_K, degree n defining a relative
! 2359: * extension L=K(theta).
! 2360: * One MUST have varn(x) < varn(nf[1]).
! 2361: * output: a pseudo-basis [A,I] of Z_L, where A is in M_n(K) in HNF form and
! 2362: * I a vector of n ideals.
! 2363: */
! 2364:
! 2365: /* given MODULES x and y by their pseudo-bases in HNF, gives a
! 2366: * pseudo-basis of the module generated by x and y. For internal use.
! 2367: */
! 2368: static GEN
! 2369: rnfjoinmodules(GEN nf, GEN x, GEN y)
! 2370: {
! 2371: long i,lx,ly;
! 2372: GEN p1,p2,z,Hx,Hy,Ix,Iy;
! 2373:
! 2374: if (x == NULL) return y;
! 2375: Hx=(GEN)x[1]; lx=lg(Hx); Ix=(GEN)x[2];
! 2376: Hy=(GEN)y[1]; ly=lg(Hy); Iy=(GEN)y[2];
! 2377: i = lx+ly-1;
! 2378: z = (GEN)gpmalloc(sizeof(long*)*(3+2*i));
! 2379: *z = evaltyp(t_VEC)|evallg(3);
! 2380: p1 = z+3; z[1]=(long)p1; *p1 = evaltyp(t_MAT)|evallg(i);
! 2381: p2 = p1+i; z[2]=(long)p2; *p2 = evaltyp(t_VEC)|evallg(i);
! 2382:
! 2383: for (i=1; i<lx; i++) { p1[i]=Hx[i]; p2[i]=Ix[i]; }
! 2384: for ( ; i<lx+ly-1; i++) { p1[i]=Hy[i-lx+1]; p2[i]=Iy[i-lx+1]; }
! 2385: x = nfhermite(nf,z); free(z); return x;
! 2386: }
! 2387:
! 2388: /* a usage interne, pas de gestion de pile : x et y sont des vecteurs dont
! 2389: * les coefficients sont les composantes sur nf[7]; avec reduction mod pr sauf
! 2390: * si prhall=NULL
! 2391: */
! 2392: static GEN
! 2393: rnfelement_mulidmod(GEN nf, GEN multab, GEN unnf, GEN x, long h, GEN prhall)
! 2394: {
! 2395: long j,k,N;
! 2396: GEN p1,c,v,s,znf;
! 2397:
! 2398: if (h==1) return gcopy(x);
! 2399: N = lg(x)-1; multab += (h-1)*N;
! 2400: x = lift(x); v = cgetg(N+1,t_COL);
! 2401: znf = gscalcol_i(gzero,lg(unnf)-1);
! 2402: for (k=1; k<=N; k++)
! 2403: {
! 2404: s = gzero;
! 2405: for (j=1; j<=N; j++)
! 2406: if (!gcmp0(p1 = (GEN)x[j]) && !gcmp0(c = gcoeff(multab,k,j)))
! 2407: {
! 2408: if (!gegal(c,unnf)) p1 = element_mul(nf,p1,c);
! 2409: s = gadd(s,p1);
! 2410: }
! 2411: if (s == gzero) s = znf;
! 2412: else
! 2413: if (prhall) s = nfreducemodpr(nf,s,prhall);
! 2414: v[k] = (long)s;
! 2415: }
! 2416: return v;
! 2417: }
! 2418:
! 2419: /* a usage interne, pas de gestion de pile : x est un vecteur dont
! 2420: * les coefficients sont les composantes sur nf[7]
! 2421: */
! 2422: static GEN
! 2423: rnfelement_sqrmod(GEN nf, GEN multab, GEN unnf, GEN x, GEN prhall)
! 2424: {
! 2425: long i,j,k,n;
! 2426: GEN p1,c,z,s;
! 2427:
! 2428: n=lg(x)-1; x=lift(x); z=cgetg(n+1,t_COL);
! 2429: for (k=1; k<=n; k++)
! 2430: {
! 2431: if (k == 1)
! 2432: s = element_sqr(nf,(GEN)x[1]);
! 2433: else
! 2434: s = gmul2n(element_mul(nf,(GEN)x[1],(GEN)x[k]), 1);
! 2435: for (i=2; i<=n; i++)
! 2436: {
! 2437: c = gcoeff(multab,k,(i-1)*n+i);
! 2438: if (!gcmp0(c))
! 2439: {
! 2440: p1=element_sqr(nf,(GEN)x[i]);
! 2441: if (!gegal(c,unnf)) p1 = element_mul(nf,p1,c);
! 2442: s = gadd(s,p1);
! 2443: }
! 2444: for (j=i+1; j<=n; j++)
! 2445: {
! 2446: c = gcoeff(multab,k,(i-1)*n+j);
! 2447: if (!gcmp0(c))
! 2448: {
! 2449: p1=gmul2n(element_mul(nf,(GEN)x[i],(GEN)x[j]),1);
! 2450: if (!gegal(c,unnf)) p1 = element_mul(nf,p1,c);
! 2451: s = gadd(s,p1);
! 2452: }
! 2453: }
! 2454: }
! 2455: if (prhall) s = nfreducemodpr(nf,s,prhall);
! 2456: z[k]=(long)s;
! 2457: }
! 2458: return z;
! 2459: }
! 2460:
! 2461: /* Compute x^n mod pr in the extension, assume n >= 0 [cf puissii()]*/
! 2462: static GEN
! 2463: rnfelementid_powmod(GEN nf, GEN multab, GEN matId, long h, GEN n, GEN prhall)
! 2464: {
! 2465: ulong av = avma;
! 2466: long i,k,m;
! 2467: GEN y,p1, unrnf=(GEN)matId[1], unnf=(GEN)unrnf[1];
! 2468:
! 2469: if (!signe(n)) return unrnf;
! 2470: y = (GEN)matId[h]; p1 = n+2; m = *p1;
! 2471: k = 1+bfffo(m); m<<=k; k = BITS_IN_LONG-k;
! 2472: for (i=lgefint(n)-2;;)
! 2473: {
! 2474: for (; k; m<<=1,k--)
! 2475: {
! 2476: y = rnfelement_sqrmod(nf,multab,unnf,y,prhall);
! 2477: if (m < 0) y = rnfelement_mulidmod(nf,multab,unnf,y,h,prhall);
! 2478: }
! 2479: if (--i == 0) break;
! 2480: m = *++p1; k = BITS_IN_LONG;
! 2481: }
! 2482: return gerepilecopy(av, y);
! 2483: }
! 2484:
! 2485: GEN
! 2486: mymod(GEN x, GEN p)
! 2487: {
! 2488: long i,lx, tx = typ(x);
! 2489: GEN y,p1;
! 2490:
! 2491: if (tx == t_INT) return resii(x,p);
! 2492: if (tx == t_FRAC)
! 2493: {
! 2494: p1 = resii((GEN)x[2], p);
! 2495: if (p1 != gzero) { cgiv(p1); return gmod(x,p); }
! 2496: return x;
! 2497: }
! 2498: if (!is_matvec_t(tx))
! 2499: err(bugparier, "mymod (missing type)");
! 2500: lx = lg(x); y = cgetg(lx,tx);
! 2501: for (i=1; i<lx; i++) y[i] = (long)mymod((GEN)x[i],p);
! 2502: return y;
! 2503: }
! 2504:
! 2505: static GEN
! 2506: rnfordmax(GEN nf, GEN pol, GEN pr, GEN unnf, GEN id, GEN matId)
! 2507: {
! 2508: long av=avma,tetpil,av1,lim,i,j,k,n,v1,v2,vpol,m,cmpt,sep;
! 2509: GEN p,q,q1,prhall,A,Aa,Aaa,A1,I,R,p1,p2,p3,multab,multabmod,Aainv;
! 2510: GEN pip,baseIp,baseOp,alpha,matprod,alphainv,matC,matG,vecpro,matH;
! 2511: GEN neworder,H,Hid,alphalistinv,alphalist,prhinv;
! 2512:
! 2513: if (DEBUGLEVEL>1) fprintferr(" treating %Z\n",pr);
! 2514: prhall=nfmodprinit(nf,pr);
! 2515: q=cgetg(3,t_VEC); q[1]=(long)pr;q[2]=(long)prhall;
! 2516: p1=rnfdedekind(nf,pol,q);
! 2517: if (gcmp1((GEN)p1[1])) return gerepilecopy(av,(GEN)p1[2]);
! 2518:
! 2519: sep=itos((GEN)p1[3]);
! 2520: A=gmael(p1,2,1);
! 2521: I=gmael(p1,2,2);
! 2522:
! 2523: n=degpol(pol); vpol=varn(pol);
! 2524: p=(GEN)pr[1]; q=powgi(p,(GEN)pr[4]); pip=(GEN)pr[2];
! 2525: q1=q; while (cmpis(q1,n)<0) q1=mulii(q1,q);
! 2526:
! 2527: multab=cgetg(n*n+1,t_MAT);
! 2528: for (j=1; j<=n*n; j++) multab[j]=lgetg(n+1,t_COL);
! 2529: prhinv = idealinv(nf,(GEN)prhall[1]);
! 2530: alphalistinv=cgetg(n+1,t_VEC);
! 2531: alphalist=cgetg(n+1,t_VEC);
! 2532: A1=cgetg(n+1,t_MAT);
! 2533: av1=avma; lim=stack_lim(av1,1);
! 2534: for(cmpt=1; ; cmpt++)
! 2535: {
! 2536: if (DEBUGLEVEL>1)
! 2537: {
! 2538: fprintferr(" %ld%s pass\n",cmpt,eng_ord(cmpt));
! 2539: flusherr();
! 2540: }
! 2541: for (i=1; i<=n; i++)
! 2542: {
! 2543: if (gegal((GEN)I[i],id)) alphalist[i]=alphalistinv[i]=(long)unnf;
! 2544: else
! 2545: {
! 2546: p1=ideal_two_elt(nf,(GEN)I[i]);
! 2547: v1=gcmp0((GEN)p1[1])? EXP220
! 2548: : element_val(nf,(GEN)p1[1],pr);
! 2549: v2=element_val(nf,(GEN)p1[2],pr);
! 2550: if (v1>v2) p2=(GEN)p1[2]; else p2=(GEN)p1[1];
! 2551: alphalist[i]=(long)p2;
! 2552: alphalistinv[i]=(long)element_inv(nf,p2);
! 2553: }
! 2554: }
! 2555: for (j=1; j<=n; j++)
! 2556: {
! 2557: p1=cgetg(n+1,t_COL); A1[j]=(long)p1;
! 2558: for (i=1; i<=n; i++)
! 2559: p1[i] = (long)element_mul(nf,gcoeff(A,i,j),(GEN)alphalist[j]);
! 2560: }
! 2561: Aa=basistoalg(nf,A1); Aainv=lift_intern(ginv(Aa));
! 2562: Aaa=mat_to_vecpol(Aa,vpol);
! 2563: for (i=1; i<=n; i++) for (j=i; j<=n; j++)
! 2564: {
! 2565: long tp;
! 2566: p1 = lift_intern(gres(gmul((GEN)Aaa[i],(GEN)Aaa[j]), pol));
! 2567: tp = typ(p1);
! 2568: if (is_scalar_t(tp) || (tp==t_POL && varn(p1)>vpol))
! 2569: p2 = gmul(p1, (GEN)Aainv[1]);
! 2570: else
! 2571: p2 = gmul(Aainv, pol_to_vec(p1, n));
! 2572:
! 2573: p3 = algtobasis(nf,p2);
! 2574: for (k=1; k<=n; k++)
! 2575: {
! 2576: coeff(multab,k,(i-1)*n+j) = p3[k];
! 2577: coeff(multab,k,(j-1)*n+i) = p3[k];
! 2578: }
! 2579: }
! 2580: R=cgetg(n+1,t_MAT); multabmod = mymod(multab,p);
! 2581: R[1] = matId[1];
! 2582: for (j=2; j<=n; j++)
! 2583: R[j] = (long) rnfelementid_powmod(nf,multabmod,matId, j,q1,prhall);
! 2584: baseIp = nfkermodpr(nf,R,prhall);
! 2585: baseOp = lift_intern(nfsuppl(nf,baseIp,n,prhall));
! 2586: alpha=cgetg(n+1,t_MAT);
! 2587: for (j=1; j<lg(baseIp); j++) alpha[j]=baseOp[j];
! 2588: for ( ; j<=n; j++)
! 2589: {
! 2590: p1=cgetg(n+1,t_COL); alpha[j]=(long)p1;
! 2591: for (i=1; i<=n; i++)
! 2592: p1[i]=(long)element_mul(nf,pip,gcoeff(baseOp,i,j));
! 2593: }
! 2594: matprod=cgetg(n+1,t_MAT);
! 2595: for (j=1; j<=n; j++)
! 2596: {
! 2597: p1=cgetg(n+1,t_COL); matprod[j]=(long)p1;
! 2598: for (i=1; i<=n; i++)
! 2599: {
! 2600: p2 = rnfelement_mulidmod(nf,multab,unnf, (GEN)alpha[i],j, NULL);
! 2601: for (k=1; k<=n; k++)
! 2602: p2[k] = lmul((GEN)nf[7], (GEN)p2[k]);
! 2603: p1[i] = (long)p2;
! 2604: }
! 2605: }
! 2606: alphainv = lift_intern(ginv(basistoalg(nf,alpha)));
! 2607: matC = cgetg(n+1,t_MAT);
! 2608: for (j=1; j<=n; j++)
! 2609: {
! 2610: p1=cgetg(n*n+1,t_COL); matC[j]=(long)p1;
! 2611: for (i=1; i<=n; i++)
! 2612: {
! 2613: p2 = gmul(alphainv, gcoeff(matprod,i,j));
! 2614: for (k=1; k<=n; k++)
! 2615: p1[(i-1)*n+k]=(long)nfreducemodpr(nf,algtobasis(nf,(GEN)p2[k]),prhall);
! 2616: }
! 2617: }
! 2618: matG=nfkermodpr(nf,matC,prhall); m=lg(matG)-1;
! 2619: vecpro=cgetg(3,t_VEC);
! 2620: p1=cgetg(n+m+1,t_MAT); vecpro[1]=(long)p1;
! 2621: p2=cgetg(n+m+1,t_VEC); vecpro[2]=(long)p2;
! 2622: for (j=1; j<=m; j++)
! 2623: {
! 2624: p1[j] = llift((GEN)matG[j]);
! 2625: p2[j] = (long)prhinv;
! 2626: }
! 2627: p1 += m;
! 2628: p2 += m;
! 2629: for (j=1; j<=n; j++)
! 2630: {
! 2631: p1[j] = matId[j];
! 2632: p2[j] = (long)idealmul(nf,(GEN)I[j],(GEN)alphalistinv[j]);
! 2633: }
! 2634: matH=nfhermite(nf,vecpro);
! 2635: p1=algtobasis(nf,gmul(Aa,basistoalg(nf,(GEN)matH[1])));
! 2636: p2=(GEN)matH[2];
! 2637:
! 2638: tetpil=avma; neworder=cgetg(3,t_VEC);
! 2639: H=cgetg(n+1,t_MAT); Hid=cgetg(n+1,t_VEC);
! 2640: for (j=1; j<=n; j++)
! 2641: {
! 2642: p3=cgetg(n+1,t_COL); H[j]=(long)p3;
! 2643: for (i=1; i<=n; i++)
! 2644: p3[i]=(long)element_mul(nf,gcoeff(p1,i,j),(GEN)alphalistinv[j]);
! 2645: Hid[j]=(long)idealmul(nf,(GEN)p2[j],(GEN)alphalist[j]);
! 2646: }
! 2647: if (DEBUGLEVEL>3)
! 2648: { fprintferr(" new order:\n"); outerr(H); outerr(Hid); }
! 2649: if (sep == 2 || gegal(I,Hid))
! 2650: {
! 2651: neworder[1]=(long)H; neworder[2]=(long)Hid;
! 2652: return gerepile(av,tetpil,neworder);
! 2653: }
! 2654:
! 2655: A=H; I=Hid;
! 2656: if (low_stack(lim, stack_lim(av1,1)) || (cmpt & 3) == 0)
! 2657: {
! 2658: GEN *gptr[2]; gptr[0]=&A; gptr[1]=&I;
! 2659: if(DEBUGMEM>1) err(warnmem,"rnfordmax");
! 2660: gerepilemany(av1,gptr,2);
! 2661: }
! 2662: }
! 2663: }
! 2664:
! 2665: static void
! 2666: check_pol(GEN x, long v)
! 2667: {
! 2668: long i,tx, lx = lg(x);
! 2669: if (varn(x) != v)
! 2670: err(talker,"incorrect variable in rnf function");
! 2671: for (i=2; i<lx; i++)
! 2672: {
! 2673: tx = typ(x[i]);
! 2674: if (!is_scalar_t(tx) || tx == t_POLMOD)
! 2675: err(talker,"incorrect polcoeff in rnf function");
! 2676: }
! 2677: }
! 2678:
! 2679: GEN
! 2680: fix_relative_pol(GEN nf, GEN x, int chk_lead)
! 2681: {
! 2682: GEN xnf = (typ(nf) == t_POL)? nf: (GEN)nf[1];
! 2683: long i, vnf = varn(xnf), lx = lg(x);
! 2684: if (typ(x) != t_POL || varn(x) >= vnf)
! 2685: err(talker,"incorrect polynomial in rnf function");
! 2686: x = dummycopy(x);
! 2687: for (i=2; i<lx; i++)
! 2688: switch(typ(x[i]))
! 2689: {
! 2690: case t_POL:
! 2691: check_pol((GEN)x[i], vnf);
! 2692: x[i] = lmodulcp((GEN)x[i], xnf); break;
! 2693: case t_POLMOD:
! 2694: if (!gegal(gmael(x,i,1), xnf)) err(consister,"rnf function");
! 2695: break;
! 2696: }
! 2697:
! 2698: if (chk_lead && !gcmp1(leading_term(x)))
! 2699: err(impl,"non-monic relative polynomials");
! 2700: return x;
! 2701: }
! 2702:
! 2703: static GEN
! 2704: rnfround2all(GEN nf, GEN pol, long all)
! 2705: {
! 2706: long av=avma,tetpil,i,j,n,N,nbidp,vpol,*ep;
! 2707: GEN p1,p2,p3,p4,polnf,list,unnf,id,matId,I,W,pseudo,y,discpol,d,D,sym;
! 2708:
! 2709: nf=checknf(nf); polnf=(GEN)nf[1]; vpol=varn(pol);
! 2710: pol = fix_relative_pol(nf,pol,1);
! 2711: N=degpol(polnf); n=degpol(pol); discpol=discsr(pol);
! 2712: list=idealfactor(nf,discpol); ep=(long*)list[2]; list=(GEN)list[1];
! 2713: nbidp=lg(list)-1; for(i=1;i<=nbidp;i++) ep[i]=itos((GEN)ep[i]);
! 2714: if (DEBUGLEVEL>1)
! 2715: {
! 2716: fprintferr("Ideals to consider:\n");
! 2717: for (i=1; i<=nbidp; i++)
! 2718: if (ep[i]>1) fprintferr("%Z^%ld\n",list[i],ep[i]);
! 2719: flusherr();
! 2720: }
! 2721: id=idmat(N); unnf=gscalcol_i(gun,N);
! 2722: matId=idmat_intern(n,unnf, gscalcol_i(gzero,N));
! 2723: pseudo = NULL;
! 2724: for (i=1; i<=nbidp; i++)
! 2725: if (ep[i] > 1)
! 2726: {
! 2727: y=rnfordmax(nf,pol,(GEN)list[i],unnf,id,matId);
! 2728: pseudo = rnfjoinmodules(nf,pseudo,y);
! 2729: }
! 2730: if (!pseudo)
! 2731: {
! 2732: I=cgetg(n+1,t_VEC); for (i=1; i<=n; i++) I[i]=(long)id;
! 2733: pseudo=cgetg(3,t_VEC); pseudo[1]=(long)matId; pseudo[2]=(long)I;
! 2734: }
! 2735: W=gmodulcp(mat_to_vecpol(basistoalg(nf,(GEN)pseudo[1]),vpol),pol);
! 2736: p2=cgetg(n+1,t_MAT); for (j=1; j<=n; j++) p2[j]=lgetg(n+1,t_COL);
! 2737: sym=polsym(pol,n-1);
! 2738: for (j=1; j<=n; j++)
! 2739: for (i=j; i<=n; i++)
! 2740: {
! 2741: p1 = lift_intern(gmul((GEN)W[i],(GEN)W[j]));
! 2742: coeff(p2,j,i)=coeff(p2,i,j)=(long)quicktrace(p1,sym);
! 2743: }
! 2744: d = algtobasis_intern(nf,det(p2));
! 2745:
! 2746: I=(GEN)pseudo[2];
! 2747: i=1; while (i<=n && gegal((GEN)I[i],id)) i++;
! 2748: if (i>n) D=id;
! 2749: else
! 2750: {
! 2751: D = (GEN)I[i];
! 2752: for (i++; i<=n; i++)
! 2753: if (!gegal((GEN)I[i],id)) D = idealmul(nf,D,(GEN)I[i]);
! 2754: D = idealpow(nf,D,gdeux);
! 2755: }
! 2756: p4=gun; p3=auxdecomp(content(d),0);
! 2757: for (i=1; i<lg(p3[1]); i++)
! 2758: p4 = gmul(p4, gpuigs(gcoeff(p3,i,1), itos(gcoeff(p3,i,2))>>1));
! 2759: p4 = gsqr(p4); tetpil=avma;
! 2760: i = all? 2: 0;
! 2761: p1=cgetg(3 + i,t_VEC);
! 2762: if (i) { p1[1]=lcopy((GEN)pseudo[1]); p1[2]=lcopy(I); }
! 2763: p1[1+i] = (long)idealmul(nf,D,d);
! 2764: p1[2+i] = ldiv(d,p4);
! 2765: return gerepile(av,tetpil,p1);
! 2766: }
! 2767:
! 2768: GEN
! 2769: rnfpseudobasis(GEN nf, GEN pol)
! 2770: {
! 2771: return rnfround2all(nf,pol,1);
! 2772: }
! 2773:
! 2774: GEN
! 2775: rnfdiscf(GEN nf, GEN pol)
! 2776: {
! 2777: return rnfround2all(nf,pol,0);
! 2778: }
! 2779:
! 2780: /* given bnf as output by buchinit and a pseudo-basis of an order
! 2781: * in HNF [A,I] (or [A,I,D,d] it does not matter), tries to simplify the
! 2782: * HNF as much as possible. The resulting matrix will be upper triangular
! 2783: * but the diagonal coefficients will not be equal to 1. The ideals
! 2784: * are guaranteed to be integral and primitive.
! 2785: */
! 2786: GEN
! 2787: rnfsimplifybasis(GEN bnf, GEN order)
! 2788: {
! 2789: long av=avma,tetpil,j,N,n;
! 2790: GEN p1,id,Az,Iz,nf,A,I;
! 2791:
! 2792: bnf = checkbnf(bnf);
! 2793: if (typ(order)!=t_VEC || lg(order)<3)
! 2794: err(talker,"not a pseudo-basis in nfsimplifybasis");
! 2795: A=(GEN)order[1]; I=(GEN)order[2]; n=lg(A)-1; nf=(GEN)bnf[7];
! 2796: N=degpol(nf[1]); id=idmat(N); Iz=cgetg(n+1,t_VEC); Az=cgetg(n+1,t_MAT);
! 2797: for (j=1; j<=n; j++)
! 2798: {
! 2799: if (gegal((GEN)I[j],id)) { Iz[j]=(long)id; Az[j]=A[j]; }
! 2800: else
! 2801: {
! 2802: p1=content((GEN)I[j]);
! 2803: if (!gcmp1(p1))
! 2804: {
! 2805: Iz[j]=(long)gdiv((GEN)I[j],p1); Az[j]=lmul((GEN)A[j],p1);
! 2806: }
! 2807: else Az[j]=A[j];
! 2808: if (!gegal((GEN)Iz[j],id))
! 2809: {
! 2810: p1=isprincipalgen(bnf,(GEN)Iz[j]);
! 2811: if (gcmp0((GEN)p1[1]))
! 2812: {
! 2813: p1=(GEN)p1[2]; Iz[j]=(long)id;
! 2814: Az[j]=(long)element_mulvec(nf,p1,(GEN)Az[j]);
! 2815: }
! 2816: }
! 2817: }
! 2818: }
! 2819: tetpil=avma; p1=cgetg(lg(order),t_VEC); p1[1]=lcopy(Az); p1[2]=lcopy(Iz);
! 2820: for (j=3; j<lg(order); j++) p1[j]=lcopy((GEN)order[j]);
! 2821: return gerepile(av,tetpil,p1);
! 2822: }
! 2823:
! 2824: GEN
! 2825: rnfdet2(GEN nf, GEN A, GEN I)
! 2826: {
! 2827: long av,tetpil,i;
! 2828: GEN p1;
! 2829:
! 2830: nf=checknf(nf); av = tetpil = avma;
! 2831: p1=idealhermite(nf,det(matbasistoalg(nf,A)));
! 2832: for(i=1;i<lg(I);i++) { tetpil=avma; p1=idealmul(nf,p1,(GEN)I[i]); }
! 2833: tetpil=avma; return gerepile(av,tetpil,p1);
! 2834: }
! 2835:
! 2836: GEN
! 2837: rnfdet(GEN nf, GEN order)
! 2838: {
! 2839: if (typ(order)!=t_VEC || lg(order)<3)
! 2840: err(talker,"not a pseudo-matrix in rnfdet");
! 2841: return rnfdet2(nf,(GEN)order[1],(GEN)order[2]);
! 2842: }
! 2843:
! 2844: GEN
! 2845: rnfdet0(GEN nf, GEN x, GEN y)
! 2846: {
! 2847: return y? rnfdet2(nf,x,y): rnfdet(nf,x);
! 2848: }
! 2849:
! 2850: /* given a pseudo-basis of an order in HNF [A,I] (or [A,I,D,d] it does
! 2851: * not matter), gives an nxn matrix (not in HNF) of a pseudo-basis and
! 2852: * an ideal vector [id,id,...,id,I] such that order=nf[7]^(n-1)xI.
! 2853: * Since it uses the approximation theorem, can be long.
! 2854: */
! 2855: GEN
! 2856: rnfsteinitz(GEN nf, GEN order)
! 2857: {
! 2858: long av=avma,tetpil,i,n;
! 2859: GEN Id,A,I,p1,a,b;
! 2860:
! 2861: nf = checknf(nf);
! 2862: Id = idmat(degpol(nf[1]));
! 2863: if (typ(order)==t_POL) order=rnfpseudobasis(nf,order);
! 2864: if (typ(order)!=t_VEC || lg(order)<3)
! 2865: err(talker,"not a pseudo-matrix in rnfsteinitz");
! 2866: A=dummycopy((GEN)order[1]);
! 2867: I=dummycopy((GEN)order[2]); n=lg(A)-1;
! 2868: if (typ(A) != t_MAT || typ(I) != t_VEC || lg(I) != n+1)
! 2869: err(typeer,"rnfsteinitz");
! 2870: for (i=1; i<n; i++)
! 2871: {
! 2872: a = (GEN)I[i];
! 2873: if (!gegal(a,Id))
! 2874: {
! 2875: GEN c1 = (GEN)A[i];
! 2876: GEN c2 = (GEN)A[i+1];
! 2877: b = (GEN)I[i+1];
! 2878: if (gegal(b,Id))
! 2879: {
! 2880: A[i] = (long)c2;
! 2881: A[i+1]= lneg(c1);
! 2882: I[i] = (long)b;
! 2883: I[i+1]= (long)a;
! 2884: }
! 2885: else
! 2886: {
! 2887: p1 = nfidealdet1(nf,a,b);
! 2888: A[i] = ladd(element_mulvec(nf,(GEN)p1[1], c1),
! 2889: element_mulvec(nf,(GEN)p1[2], c2));
! 2890: A[i+1]= ladd(element_mulvec(nf,(GEN)p1[3], c1),
! 2891: element_mulvec(nf,(GEN)p1[4], c2));
! 2892: I[i] =(long)Id;
! 2893: I[i+1]=(long)idealmul(nf,a,b); p1 = content((GEN)I[i+1]);
! 2894: if (!gcmp1(p1))
! 2895: {
! 2896: I[i+1] = ldiv((GEN)I[i+1],p1);
! 2897: A[i+1] = lmul(p1,(GEN)A[i+1]);
! 2898: }
! 2899: }
! 2900: }
! 2901: }
! 2902: tetpil=avma; p1=cgetg(lg(order),t_VEC);
! 2903: p1[1]=lcopy(A); p1[2]=lcopy(I);
! 2904: for (i=3; i<lg(order); i++) p1[i]=lcopy((GEN)order[i]);
! 2905: return gerepile(av,tetpil,p1);
! 2906: }
! 2907:
! 2908: /* Given bnf as output by buchinit and either an order as output by
! 2909: * rnfpseudobasis or a polynomial, and outputs a basis if it is free,
! 2910: * an n+1-generating set if it is not
! 2911: */
! 2912: GEN
! 2913: rnfbasis(GEN bnf, GEN order)
! 2914: {
! 2915: ulong av = avma;
! 2916: long j,N,n;
! 2917: GEN nf,A,I,classe,p1,p2,id;
! 2918:
! 2919: bnf = checkbnf(bnf);
! 2920: nf=(GEN)bnf[7]; N=degpol(nf[1]); id=idmat(N);
! 2921: if (typ(order)==t_POL) order=rnfpseudobasis(nf,order);
! 2922: if (typ(order)!=t_VEC || lg(order)<3)
! 2923: err(talker,"not a pseudo-matrix in rnfbasis");
! 2924: A=(GEN)order[1]; I=(GEN)order[2]; n=lg(A)-1;
! 2925: j=1; while (j<n && gegal((GEN)I[j],id)) j++;
! 2926: if (j<n) order=rnfsteinitz(nf,order);
! 2927: A=(GEN)order[1]; I=(GEN)order[2]; classe=(GEN)I[n];
! 2928: p1=isprincipalgen(bnf,classe);
! 2929: if (gcmp0((GEN)p1[1]))
! 2930: {
! 2931: p2=cgetg(n+1,t_MAT);
! 2932: p2[n]=(long)element_mulvec(nf,(GEN)p1[2],(GEN)A[n]);
! 2933: }
! 2934: else
! 2935: {
! 2936: p1=ideal_two_elt(nf,classe);
! 2937: p2=cgetg(n+2,t_MAT);
! 2938: p2[n]=lmul((GEN)p1[1],(GEN)A[n]);
! 2939: p2[n+1]=(long)element_mulvec(nf,(GEN)p1[2],(GEN)A[n]);
! 2940: }
! 2941: for (j=1; j<n; j++) p2[j]=A[j];
! 2942: return gerepilecopy(av,p2);
! 2943: }
! 2944:
! 2945: /* Given bnf as output by buchinit and either an order as output by
! 2946: * rnfpseudobasis or a polynomial, and outputs a basis (not pseudo)
! 2947: * in Hermite Normal Form if it exists, zero if not
! 2948: */
! 2949: GEN
! 2950: rnfhermitebasis(GEN bnf, GEN order)
! 2951: {
! 2952: ulong av = avma;
! 2953: long j,N,n;
! 2954: GEN nf,A,I,p1,id;
! 2955:
! 2956: bnf = checkbnf(bnf); nf=(GEN)bnf[7];
! 2957: N=degpol(nf[1]); id=idmat(N);
! 2958: if (typ(order)==t_POL)
! 2959: {
! 2960: order=rnfpseudobasis(nf,order);
! 2961: A=(GEN)order[1];
! 2962: }
! 2963: else
! 2964: {
! 2965: if (typ(order)!=t_VEC || lg(order)<3)
! 2966: err(talker,"not a pseudo-matrix in rnfbasis");
! 2967: A=gcopy((GEN)order[1]);
! 2968: }
! 2969: I=(GEN)order[2]; n=lg(A)-1;
! 2970: for (j=1; j<=n; j++)
! 2971: {
! 2972: if (!gegal((GEN)I[j],id))
! 2973: {
! 2974: p1=isprincipalgen(bnf,(GEN)I[j]);
! 2975: if (gcmp0((GEN)p1[1]))
! 2976: A[j]=(long)element_mulvec(nf,(GEN)p1[2],(GEN)A[j]);
! 2977: else { avma=av; return gzero; }
! 2978: }
! 2979: }
! 2980: return gerepilecopy(av,A);
! 2981: }
! 2982:
! 2983: long
! 2984: rnfisfree(GEN bnf, GEN order)
! 2985: {
! 2986: long av=avma,n,N,j;
! 2987: GEN nf,p1,id,I;
! 2988:
! 2989: bnf = checkbnf(bnf);
! 2990: if (gcmp1(gmael3(bnf,8,1,1))) return 1;
! 2991:
! 2992: nf=(GEN)bnf[7]; N=degpol(nf[1]); id=idmat(N);
! 2993: if (typ(order)==t_POL) order=rnfpseudobasis(nf,order);
! 2994: if (typ(order)!=t_VEC || lg(order)<3)
! 2995: err(talker,"not a pseudo-matrix in rnfisfree");
! 2996:
! 2997: I=(GEN)order[2]; n=lg(I)-1;
! 2998: j=1; while (j<=n && gegal((GEN)I[j],id)) j++;
! 2999: if (j>n) { avma=av; return 1; }
! 3000:
! 3001: p1=(GEN)I[j];
! 3002: for (j++; j<=n; j++)
! 3003: if (!gegal((GEN)I[j],id)) p1=idealmul(nf,p1,(GEN)I[j]);
! 3004: j = gcmp0(isprincipal(bnf,p1));
! 3005: avma=av; return j;
! 3006: }
! 3007:
! 3008: /**********************************************************************/
! 3009: /** **/
! 3010: /** COMPOSITUM OF TWO NUMBER FIELDS **/
! 3011: /** **/
! 3012: /**********************************************************************/
! 3013: extern GEN ZY_ZXY_resultant_all(GEN A, GEN B0, long *lambda, GEN *LPRS);
! 3014: extern GEN squff2(GEN x, long klim, long hint);
! 3015: extern GEN to_polmod(GEN x, GEN mod);
! 3016:
! 3017: /* modular version. TODO: check that compositum2 is not slower */
! 3018: GEN
! 3019: polcompositum0(GEN A, GEN B, long flall)
! 3020: {
! 3021: ulong av = avma;
! 3022: long v,k;
! 3023: GEN C, LPRS;
! 3024:
! 3025: if (typ(A)!=t_POL || typ(B)!=t_POL) err(typeer,"polcompositum0");
! 3026: if (degpol(A)<=0 || degpol(B)<=0) err(constpoler,"compositum");
! 3027: v = varn(A);
! 3028: if (varn(B) != v) err(talker,"not the same variable in compositum");
! 3029: C = content(A); if (!gcmp1(C)) A = gdiv(A, C);
! 3030: C = content(B); if (!gcmp1(C)) B = gdiv(B, C);
! 3031: check_pol_int(A,"compositum");
! 3032: check_pol_int(B,"compositum");
! 3033: if (!ZX_is_squarefree(A)) err(talker,"compositum: %Z not separable", A);
! 3034: if (!ZX_is_squarefree(B)) err(talker,"compositum: %Z not separable", B);
! 3035:
! 3036: k = 1; C = ZY_ZXY_resultant_all(A, B, &k, flall? &LPRS: NULL);
! 3037: C = squff2(C,0,0); /* C = Res_Y (A, B(X + kY)) guaranteed squarefree */
! 3038: if (flall)
! 3039: {
! 3040: long i,l = lg(C);
! 3041: GEN w,a,b; /* a,b,c root of A,B,C = compositum, c = b - k a */
! 3042: for (i=1; i<l; i++)
! 3043: { /* invmod possibly very costly */
! 3044: a = gmul((GEN)LPRS[1], ZX_invmod((GEN)LPRS[2], (GEN)C[i]));
! 3045: a = gneg_i(gmod(a, (GEN)C[i]));
! 3046: b = gadd(polx[v], gmulsg(k,a));
! 3047: w = cgetg(5,t_VEC); /* [C, a, b, n ] */
! 3048: w[1] = C[i];
! 3049: w[2] = (long)to_polmod(a, (GEN)w[1]);
! 3050: w[3] = (long)to_polmod(b, (GEN)w[1]);
! 3051: w[4] = lstoi(-k); C[i] = (long)w;
! 3052: }
! 3053: }
! 3054: settyp(C, t_VEC); return gerepilecopy(av, C);
! 3055: }
! 3056:
! 3057: GEN
! 3058: compositum(GEN pol1,GEN pol2)
! 3059: {
! 3060: return polcompositum0(pol1,pol2,0);
! 3061: }
! 3062:
! 3063: GEN
! 3064: compositum2(GEN pol1,GEN pol2)
! 3065: {
! 3066: return polcompositum0(pol1,pol2,1);
! 3067: }
! 3068:
! 3069: extern int isrational(GEN x);
! 3070: extern GEN nfgcd(GEN P, GEN Q, GEN nf, GEN den);
! 3071:
! 3072: int
! 3073: nfissquarefree(GEN nf, GEN x)
! 3074: {
! 3075: ulong av = avma;
! 3076: GEN g, y = derivpol(x);
! 3077: if (isrational(x))
! 3078: g = modulargcd(x, y);
! 3079: else
! 3080: g = nfgcd(x, y, nf, NULL);
! 3081: avma = av; return (degpol(g) == 0);
! 3082: }
! 3083:
! 3084: GEN
! 3085: rnfequation0(GEN nf, GEN B, long flall)
! 3086: {
! 3087: ulong av = avma;
! 3088: long v,vpol,k,lA,lB;
! 3089: GEN cC,A,C,LPRS;
! 3090:
! 3091: if (typ(nf)==t_POL) A=nf; else { nf=checknf(nf); A=(GEN)nf[1]; }
! 3092: B = fix_relative_pol(nf,B,1);
! 3093: v = varn(A); lA = lgef(A);
! 3094: vpol= varn(B); lB = lgef(B);
! 3095: if (lA<=3 || lB<=3) err(constpoler,"rnfequation");
! 3096:
! 3097: check_pol_int(A,"rnfequation");
! 3098: B = lift_intern(B); B = gdiv(B, content(B));
! 3099: for (k=2; k<lB; k++)
! 3100: if (lgef(B[k]) >= lA) B[k] = lres((GEN)B[k],A);
! 3101:
! 3102: if (!nfissquarefree(A,B))
! 3103: err(talker,"not k separable relative equation in rnfequation");
! 3104:
! 3105: k = 0; C = ZY_ZXY_resultant_all(A, B, &k, flall? &LPRS: NULL);
! 3106: if (gsigne(leadingcoeff(C)) < 0) C = gneg_i(C);
! 3107: C = primitive_part(C, &cC);
! 3108: if (flall)
! 3109: {
! 3110: GEN w,a,b; /* a,b,c root of A,B,C = compositum, c = b - k a */
! 3111: /* invmod possibly very costly */
! 3112: a = gmul((GEN)LPRS[1], ZX_invmod((GEN)LPRS[2], C));
! 3113: a = gneg_i(gmod(a, C));
! 3114: b = gadd(polx[v], gmulsg(k,a));
! 3115: w = cgetg(4,t_VEC); /* [C, a, n ] */
! 3116: w[1] = (long)C;
! 3117: w[2] = (long)to_polmod(a, (GEN)w[1]);
! 3118: w[3] = lstoi(-k); C = w;
! 3119: }
! 3120: return gerepilecopy(av, C);
! 3121: }
! 3122:
! 3123: GEN
! 3124: rnfequation(GEN nf,GEN pol2)
! 3125: {
! 3126: return rnfequation0(nf,pol2,0);
! 3127: }
! 3128:
! 3129: GEN
! 3130: rnfequation2(GEN nf,GEN pol2)
! 3131: {
! 3132: return rnfequation0(nf,pol2,1);
! 3133: }
! 3134:
! 3135: static GEN
! 3136: nftau(long r1, GEN x)
! 3137: {
! 3138: long i, ru = lg(x);
! 3139: GEN s;
! 3140:
! 3141: s = r1 ? (GEN)x[1] : gmul2n(greal((GEN)x[1]),1);
! 3142: for (i=2; i<=r1; i++) s=gadd(s,(GEN)x[i]);
! 3143: for ( ; i<ru; i++) s=gadd(s,gmul2n(greal((GEN)x[i]),1));
! 3144: return s;
! 3145: }
! 3146:
! 3147: static GEN
! 3148: nftocomplex(GEN nf, GEN x)
! 3149: {
! 3150: long ru,vnf,k;
! 3151: GEN p2,p3,ronf;
! 3152:
! 3153: p2 = (typ(x)==t_POLMOD)? (GEN)x[2]: gmul((GEN)nf[7],x);
! 3154: vnf=varn(nf[1]);
! 3155: ronf=(GEN)nf[6]; ru=lg(ronf); p3=cgetg(ru,t_COL);
! 3156: for (k=1; k<ru; k++) p3[k]=lsubst(p2,vnf,(GEN)ronf[k]);
! 3157: return p3;
! 3158: }
! 3159:
! 3160: static GEN
! 3161: rnfscal(GEN mth, GEN xth, GEN yth)
! 3162: {
! 3163: long n,ru,i,j,kk;
! 3164: GEN x,y,m,res,p1,p2;
! 3165:
! 3166: n=lg(mth)-1; ru=lg(gcoeff(mth,1,1));
! 3167: res=cgetg(ru,t_COL);
! 3168: for (kk=1; kk<ru; kk++)
! 3169: {
! 3170: m=cgetg(n+1,t_MAT);
! 3171: for (j=1; j<=n; j++)
! 3172: {
! 3173: p1=cgetg(n+1,t_COL); m[j]=(long)p1;
! 3174: for (i=1; i<=n; i++) { p2=gcoeff(mth,i,j); p1[i]=p2[kk]; }
! 3175: }
! 3176: x=cgetg(n+1,t_VEC);
! 3177: for (j=1; j<=n; j++) x[j]=(long)gconj((GEN)((GEN)xth[j])[kk]);
! 3178: y=cgetg(n+1,t_COL);
! 3179: for (j=1; j<=n; j++) y[j]=((GEN)yth[j])[kk];
! 3180: res[kk]=(long)gmul(x,gmul(m,y));
! 3181: }
! 3182: return res;
! 3183: }
! 3184:
! 3185: static GEN
! 3186: rnfdiv(GEN x, GEN y)
! 3187: {
! 3188: long i, ru = lg(x);
! 3189: GEN z;
! 3190:
! 3191: z=cgetg(ru,t_COL);
! 3192: for (i=1; i<ru; i++) z[i]=(long)gdiv((GEN)x[i],(GEN)y[i]);
! 3193: return z;
! 3194: }
! 3195:
! 3196: static GEN
! 3197: rnfmul(GEN x, GEN y)
! 3198: {
! 3199: long i, ru = lg(x);
! 3200: GEN z;
! 3201:
! 3202: z=cgetg(ru,t_COL);
! 3203: for (i=1; i<ru; i++) z[i]=(long)gmul((GEN)x[i],(GEN)y[i]);
! 3204: return z;
! 3205: }
! 3206:
! 3207: static GEN
! 3208: rnfvecmul(GEN x, GEN v)
! 3209: {
! 3210: long i, lx = lg(v);
! 3211: GEN y;
! 3212:
! 3213: y=cgetg(lx,typ(v));
! 3214: for (i=1; i<lx; i++) y[i]=(long)rnfmul(x,(GEN)v[i]);
! 3215: return y;
! 3216: }
! 3217:
! 3218: static GEN
! 3219: allonge(GEN v, long N)
! 3220: {
! 3221: long r,r2,i;
! 3222: GEN y;
! 3223:
! 3224: r=lg(v)-1; r2=N-r;
! 3225: y=cgetg(N+1,t_COL);
! 3226: for (i=1; i<=r; i++) y[i]=v[i];
! 3227: for ( ; i<=N; i++) y[i]=(long)gconj((GEN)v[i-r2]);
! 3228: return y;
! 3229: }
! 3230:
! 3231: static GEN
! 3232: findmin(GEN nf, GEN ideal, GEN muf,long prec)
! 3233: {
! 3234: long av=avma,N,tetpil,i;
! 3235: GEN m,y;
! 3236:
! 3237: m = qf_base_change(gmael(nf,5,3), ideal, 0); /* nf[5][3] = T2 */
! 3238: m = lllgramintern(m,4,1,prec);
! 3239: if (!m)
! 3240: {
! 3241: m = lllint(ideal);
! 3242: m = qf_base_change(gmael(nf,5,3), gmul(ideal,m), 0);
! 3243: m = lllgramintern(m,4,1,prec);
! 3244: if (!m) err(precer,"rnflllgram");
! 3245: }
! 3246: ideal=gmul(ideal,m);
! 3247: N=lg(ideal)-1; y=cgetg(N+1,t_MAT);
! 3248: for (i=1; i<=N; i++)
! 3249: y[i] = (long) allonge(nftocomplex(nf,(GEN)ideal[i]),N);
! 3250: m=ground(greal(gauss(y,allonge(muf,N))));
! 3251: tetpil=avma; return gerepile(av,tetpil,gmul(ideal,m));
! 3252: }
! 3253:
! 3254: #define swap(x,y) { long _t=x; x=y; y=_t; }
! 3255:
! 3256: /* given a base field nf (e.g main variable y), a polynomial pol with
! 3257: * coefficients in nf (e.g main variable x), and an order as output
! 3258: * by rnfpseudobasis, outputs a reduced order.
! 3259: */
! 3260: GEN
! 3261: rnflllgram(GEN nf, GEN pol, GEN order,long prec)
! 3262: {
! 3263: long av=avma,tetpil,i,j,k,l,kk,kmax,r1,ru,lx,vnf;
! 3264: GEN p1,p2,M,I,U,ronf,poll,unro,roorder,powreorder,mth,s,MC,MPOL,MCS;
! 3265: GEN B,mu,Bf,temp,ideal,x,xc,xpol,muf,mufc,muno,y,z,Ikk_inv;
! 3266:
! 3267: /* Initializations and verifications */
! 3268:
! 3269: nf=checknf(nf);
! 3270: if (typ(order)!=t_VEC || lg(order)<3)
! 3271: err(talker,"not a pseudo-matrix in rnflllgram");
! 3272: M=(GEN)order[1]; I=(GEN)order[2]; lx=lg(I);
! 3273: if (lx < 3) return gcopy(order);
! 3274: if (lx-1 != degpol(pol)) err(consister,"rnflllgram");
! 3275: U=idmat(lx-1); I = dummycopy(I);
! 3276:
! 3277: /* Compute the relative T2 matrix of powers of theta */
! 3278:
! 3279: vnf=varn(nf[1]); ronf=(GEN)nf[6]; ru=lg(ronf); poll=lift(pol);
! 3280: r1 = nf_get_r1(nf);
! 3281: unro=cgetg(lx,t_COL); for (i=1; i<lx; i++) unro[i]=un;
! 3282: roorder=cgetg(ru,t_VEC);
! 3283: for (i=1; i<ru; i++)
! 3284: roorder[i]=lroots(gsubst(poll,vnf,(GEN)ronf[i]),prec);
! 3285: powreorder=cgetg(lx,t_MAT);
! 3286: p1=cgetg(ru,t_COL); powreorder[1]=(long)p1;
! 3287: for (i=1; i<ru; i++) p1[i]=(long)unro;
! 3288: for (k=2; k<lx; k++)
! 3289: {
! 3290: p1=cgetg(ru,t_COL); powreorder[k]=(long)p1;
! 3291: for (i=1; i<ru; i++)
! 3292: {
! 3293: p2=cgetg(lx,t_COL); p1[i]=(long)p2;
! 3294: for (j=1; j<lx; j++)
! 3295: p2[j] = lmul(gmael(roorder,i,j),gmael3(powreorder,k-1,i,j));
! 3296: }
! 3297: }
! 3298: mth=cgetg(lx,t_MAT);
! 3299: for (l=1; l<lx; l++)
! 3300: {
! 3301: p1=cgetg(lx,t_COL); mth[l]=(long)p1;
! 3302: for (k=1; k<lx; k++)
! 3303: {
! 3304: p2=cgetg(ru,t_COL); p1[k]=(long)p2;
! 3305: for (i=1; i<ru; i++)
! 3306: {
! 3307: s=gzero;
! 3308: for (j=1; j<lx; j++)
! 3309: s = gadd(s,gmul(gconj(gmael3(powreorder,k,i,j)),
! 3310: gmael3(powreorder,l,i,j)));
! 3311: p2[i]=(long)s;
! 3312: }
! 3313: }
! 3314: }
! 3315:
! 3316: /* Transform the matrix M into a matrix with coefficients in K and also
! 3317: with coefficients polymod */
! 3318:
! 3319: MC=cgetg(lx,t_MAT); MPOL=cgetg(lx,t_MAT);
! 3320: for (j=1; j<lx; j++)
! 3321: {
! 3322: p1=cgetg(lx,t_COL); MC[j]=(long)p1;
! 3323: p2=cgetg(lx,t_COL); MPOL[j]=(long)p2;
! 3324: for (i=1; i<lx; i++)
! 3325: {
! 3326: p2[i]=(long)basistoalg(nf,gcoeff(M,i,j));
! 3327: p1[i]=(long)nftocomplex(nf,(GEN)p2[i]);
! 3328: }
! 3329: }
! 3330: MCS=cgetg(lx,t_MAT);
! 3331:
! 3332: /* Start LLL algorithm */
! 3333:
! 3334: mu=cgetg(lx,t_MAT); B=cgetg(lx,t_COL);
! 3335: for (j=1; j<lx; j++)
! 3336: {
! 3337: p1=cgetg(lx,t_COL); mu[j]=(long)p1; for (i=1; i<lx; i++) p1[i]=zero;
! 3338: B[j]=zero;
! 3339: }
! 3340: kk=2; if (DEBUGLEVEL) fprintferr("kk = %ld ",kk);
! 3341: kmax=1; B[1]=lreal(rnfscal(mth,(GEN)MC[1],(GEN)MC[1]));
! 3342: MCS[1]=lcopy((GEN)MC[1]);
! 3343: do
! 3344: {
! 3345: if (kk>kmax)
! 3346: {
! 3347: /* Incremental Gram-Schmidt */
! 3348: kmax=kk; MCS[kk]=lcopy((GEN)MC[kk]);
! 3349: for (j=1; j<kk; j++)
! 3350: {
! 3351: coeff(mu,kk,j) = (long) rnfdiv(rnfscal(mth,(GEN)MCS[j],(GEN)MC[kk]),
! 3352: (GEN) B[j]);
! 3353: MCS[kk] = lsub((GEN) MCS[kk], rnfvecmul(gcoeff(mu,kk,j),(GEN)MCS[j]));
! 3354: }
! 3355: B[kk] = lreal(rnfscal(mth,(GEN)MCS[kk],(GEN)MCS[kk]));
! 3356: if (gcmp0((GEN)B[kk])) err(lllger3);
! 3357: }
! 3358:
! 3359: /* RED(k,k-1) */
! 3360: l=kk-1; Ikk_inv=idealinv(nf, (GEN)I[kk]);
! 3361: ideal=idealmul(nf,(GEN)I[l],Ikk_inv);
! 3362: x=findmin(nf,ideal,gcoeff(mu,kk,l),2*prec-2);
! 3363: if (!gcmp0(x))
! 3364: {
! 3365: xpol=basistoalg(nf,x); xc=nftocomplex(nf,xpol);
! 3366: MC[kk]=lsub((GEN)MC[kk],rnfvecmul(xc,(GEN)MC[l]));
! 3367: U[kk]=lsub((GEN)U[kk],gmul(xpol,(GEN)U[l]));
! 3368: coeff(mu,kk,l)=lsub(gcoeff(mu,kk,l),xc);
! 3369: for (i=1; i<l; i++)
! 3370: coeff(mu,kk,i)=lsub(gcoeff(mu,kk,i),rnfmul(xc,gcoeff(mu,l,i)));
! 3371: }
! 3372: /* Test LLL condition */
! 3373: p1=nftau(r1,gadd((GEN) B[kk],
! 3374: gmul(gnorml2(gcoeff(mu,kk,kk-1)),(GEN)B[kk-1])));
! 3375: p2=gdivgs(gmulsg(9,nftau(r1,(GEN)B[kk-1])),10);
! 3376: if (gcmp(p1,p2)<=0)
! 3377: {
! 3378: /* Execute SWAP(k) */
! 3379: k=kk;
! 3380: swap(MC[k-1],MC[k]);
! 3381: swap(U[k-1],U[k]);
! 3382: swap(I[k-1],I[k]);
! 3383: for (j=1; j<=k-2; j++) swap(coeff(mu,k-1,j),coeff(mu,k,j));
! 3384: muf=gcoeff(mu,k,k-1);
! 3385: mufc=gconj(muf); muno=greal(rnfmul(muf,mufc));
! 3386: Bf=gadd((GEN)B[k],rnfmul(muno,(GEN)B[k-1]));
! 3387: p1=rnfdiv((GEN)B[k-1],Bf);
! 3388: coeff(mu,k,k-1)=(long)rnfmul(mufc,p1);
! 3389: temp=(GEN)MCS[k-1];
! 3390: MCS[k-1]=ladd((GEN)MCS[k],rnfvecmul(muf,(GEN)MCS[k-1]));
! 3391: MCS[k]=lsub(rnfvecmul(rnfdiv((GEN)B[k],Bf),temp),
! 3392: rnfvecmul(gcoeff(mu,k,k-1),(GEN)MCS[k]));
! 3393: B[k]=(long)rnfmul((GEN)B[k],p1); B[k-1]=(long)Bf;
! 3394: for (i=k+1; i<=kmax; i++)
! 3395: {
! 3396: temp=gcoeff(mu,i,k);
! 3397: coeff(mu,i,k)=lsub(gcoeff(mu,i,k-1),rnfmul(muf,gcoeff(mu,i,k)));
! 3398: coeff(mu,i,k-1) = ladd(temp, rnfmul(gcoeff(mu,k,k-1),gcoeff(mu,i,k)));
! 3399: }
! 3400: if (kk>2) { kk--; if (DEBUGLEVEL) fprintferr("%ld ",kk); }
! 3401: }
! 3402: else
! 3403: {
! 3404: for (l=kk-2; l; l--)
! 3405: {
! 3406: /* RED(k,l) */
! 3407: ideal=idealmul(nf,(GEN)I[l],Ikk_inv);
! 3408: x=findmin(nf,ideal,gcoeff(mu,kk,l),2*prec-2);
! 3409: if (!gcmp0(x))
! 3410: {
! 3411: xpol=basistoalg(nf,x); xc=nftocomplex(nf,xpol);
! 3412: MC[kk]=(long)gsub((GEN)MC[kk],rnfvecmul(xc,(GEN)MC[l]));
! 3413: U[kk]=(long)gsub((GEN)U[kk],gmul(xpol,(GEN)U[l]));
! 3414: coeff(mu,kk,l)=lsub(gcoeff(mu,kk,l),xc);
! 3415: for (i=1; i<l; i++)
! 3416: coeff(mu,kk,i) = lsub(gcoeff(mu,kk,i), rnfmul(xc,gcoeff(mu,l,i)));
! 3417: }
! 3418: }
! 3419: kk++; if (DEBUGLEVEL) fprintferr("%ld ",kk);
! 3420: }
! 3421: }
! 3422: while (kk<lx);
! 3423: if (DEBUGLEVEL) fprintferr("\n");
! 3424: p1=gmul(MPOL,U); tetpil=avma;
! 3425: y=cgetg(3,t_VEC); z=cgetg(3,t_VEC); y[1]=(long)z;
! 3426: z[2]=lcopy(I); z[1]=(long)algtobasis(nf,p1);
! 3427: y[2]=(long)algtobasis(nf,U);
! 3428: return gerepile(av,tetpil,y);
! 3429: }
! 3430:
! 3431: GEN
! 3432: rnfpolred(GEN nf, GEN pol, long prec)
! 3433: {
! 3434: ulong av = avma;
! 3435: long i,j,k,n,N, vpol = varn(pol);
! 3436: GEN id,id2,newid,newor,p1,p2,al,newpol,w,z;
! 3437: GEN bnf,zk,newideals,ideals,order,neworder;
! 3438:
! 3439: if (typ(pol)!=t_POL) err(typeer,"rnfpolred");
! 3440: if (typ(nf)!=t_VEC) err(idealer1);
! 3441: switch(lg(nf))
! 3442: {
! 3443: case 10: bnf = NULL; break;
! 3444: case 11: bnf = nf; nf = checknf((GEN)nf[7]); break;
! 3445: default: err(idealer1);
! 3446: return NULL; /* not reached */
! 3447: }
! 3448: if (degpol(pol) <= 1)
! 3449: {
! 3450: w=cgetg(2,t_VEC);
! 3451: w[1]=lpolx[vpol]; return w;
! 3452: }
! 3453: id=rnfpseudobasis(nf,pol); N=degpol(nf[1]);
! 3454: if (bnf && gcmp1(gmael3(bnf,8,1,1))) /* if bnf is principal */
! 3455: {
! 3456: ideals=(GEN)id[2]; n=lg(ideals)-1; order=(GEN)id[1];
! 3457: newideals=cgetg(n+1,t_VEC); neworder=cgetg(n+1,t_MAT);
! 3458: zk=idmat(N);
! 3459: for (j=1; j<=n; j++)
! 3460: {
! 3461: newideals[j]=(long)zk; p1=cgetg(n+1,t_COL); neworder[j]=(long)p1;
! 3462: p2=(GEN)order[j];
! 3463: al=(GEN)isprincipalgen(bnf,(GEN)ideals[j])[2];
! 3464: for (k=1; k<=n; k++)
! 3465: p1[k]=(long)element_mul(nf,(GEN)p2[k],al);
! 3466: }
! 3467: id=cgetg(3,t_VEC); id[1]=(long)neworder; id[2]=(long)newideals;
! 3468: }
! 3469: id2=rnflllgram(nf,pol,id,prec);
! 3470: z=(GEN)id2[1]; newid=(GEN)z[2]; newor=(GEN)z[1];
! 3471: n=lg(newor)-1; w=cgetg(n+1,t_VEC);
! 3472: for (j=1; j<=n; j++)
! 3473: {
! 3474: p1=(GEN)newid[j]; al=gmul(gcoeff(p1,1,1),(GEN)newor[j]);
! 3475: p1=basistoalg(nf,(GEN)al[n]);
! 3476: for (i=n-1; i; i--)
! 3477: p1=gadd(basistoalg(nf,(GEN)al[i]),gmul(polx[vpol],p1));
! 3478: newpol=gtopoly(gmodulcp(gtovec(caract2(lift(pol),lift(p1),vpol)),
! 3479: (GEN) nf[1]), vpol);
! 3480: p1 = ggcd(newpol, derivpol(newpol));
! 3481: if (degpol(p1)>0)
! 3482: {
! 3483: newpol=gdiv(newpol,p1);
! 3484: newpol=gdiv(newpol,leading_term(newpol));
! 3485: }
! 3486: w[j]=(long)newpol;
! 3487: if (DEBUGLEVEL>=4) outerr(newpol);
! 3488: }
! 3489: return gerepilecopy(av,w);
! 3490: }
! 3491:
! 3492: extern GEN vecpol_to_mat(GEN v, long n);
! 3493:
! 3494: /* given a relative polynomial pol over nf, compute a pseudo-basis for the
! 3495: * extension, then an absolute basis */
! 3496: GEN
! 3497: makebasis(GEN nf,GEN pol)
! 3498: {
! 3499: GEN elts,ids,polabs,plg,B,bs,p1,p2,a,den,vbs,vbspro,vpro,rnf;
! 3500: long av=avma,n,N,m,i,j, v = varn(pol);
! 3501:
! 3502: p1 = rnfequation2(nf,pol);
! 3503: polabs= (GEN)p1[1];
! 3504: plg = (GEN)p1[2];
! 3505: a = (GEN)p1[3];
! 3506: rnf = cgetg(12,t_VEC);
! 3507: for (i=2;i<=9;i++) rnf[i]=zero;
! 3508: rnf[1] =(long)pol;
! 3509: rnf[10]=(long)nf; p2=cgetg(4,t_VEC);
! 3510: rnf[11]=(long)p2; p2[1]=p2[2]=zero; p2[3]=(long)a;
! 3511: if (signe(a))
! 3512: pol = gsubst(pol,v,gsub(polx[v],
! 3513: gmul(a,gmodulcp(polx[varn(nf[1])],(GEN)nf[1]))));
! 3514: p1=rnfpseudobasis(nf,pol);
! 3515: elts= (GEN)p1[1];
! 3516: ids = (GEN)p1[2];
! 3517: if (DEBUGLEVEL>1) { fprintferr("relative basis computed\n"); flusherr(); }
! 3518: N=degpol(pol); n=degpol((GEN)nf[1]); m=n*N;
! 3519: den = denom(content(lift(plg)));
! 3520: vbs = cgetg(n+1,t_VEC);
! 3521: vbs[1] = un;
! 3522: vbs[2] = (long)plg; vbspro = gmul(den,plg);
! 3523: for(i=3;i<=n;i++)
! 3524: vbs[i] = ldiv(gmul((GEN)vbs[i-1],vbspro),den);
! 3525: bs = gmul(vbs, vecpol_to_mat((GEN)nf[7],n));
! 3526:
! 3527: vpro=cgetg(N+1,t_VEC);
! 3528: for (i=1;i<=N;i++)
! 3529: {
! 3530: p1=cgetg(3,t_POLMOD);
! 3531: p1[1]=(long)polabs;
! 3532: p1[2]=lpuigs(polx[v],i-1); vpro[i]=(long)p1;
! 3533: }
! 3534: vpro=gmul(vpro,elts); B = cgetg(m+1, t_MAT);
! 3535: for(i=1;i<=N;i++)
! 3536: for(j=1;j<=n;j++)
! 3537: {
! 3538: p1 = gmul(bs, element_mul(nf,(GEN)vpro[i],gmael(ids,i,j)));
! 3539: B[(i-1)*n+j] = (long)pol_to_vec(lift_intern(p1), m);
! 3540: }
! 3541: p1 = denom(B); B = gmul(B,p1);
! 3542: B = hnfmodid(B, p1); B = gdiv(B,p1);
! 3543: p1=cgetg(4,t_VEC);
! 3544: p1[1]=(long)polabs;
! 3545: p1[2]=(long)B;
! 3546: p1[3]=(long)rnf; return gerepilecopy(av, p1);
! 3547: }
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