Annotation of OpenXM_contrib/pari/src/basemath/base2.c, Revision 1.1
1.1 ! maekawa 1: /*******************************************************************/
! 2: /* */
! 3: /* MAXIMAL ORDERS */
! 4: /* */
! 5: /*******************************************************************/
! 6: /* $Id: base2.c,v 1.4 1999/09/21 19:17:40 karim Exp $ */
! 7: #include "pari.h"
! 8:
! 9: GEN caractducos(GEN p, GEN x, int v);
! 10: GEN element_muli(GEN nf, GEN x, GEN y);
! 11: GEN element_mulid(GEN nf, GEN x, long i);
! 12: GEN eleval(GEN f,GEN h,GEN a);
! 13: GEN ideal_better_basis(GEN nf, GEN x, GEN M);
! 14: long int_elt_val(GEN nf, GEN x, GEN p, GEN bp, long v);
! 15: GEN mat_to_vecpol(GEN x, long v);
! 16: GEN nfidealdet1(GEN nf, GEN a, GEN b);
! 17: GEN nfsuppl(GEN nf, GEN x, long n, GEN prhall);
! 18: GEN pol_to_monic(GEN pol, GEN *lead);
! 19: GEN pol_to_vec(GEN x, long N);
! 20: GEN quicktrace(GEN x, GEN sym);
! 21: GEN respm(GEN f1,GEN f2,GEN pm);
! 22:
! 23: static void
! 24: allbase_check_args(GEN f, long code, GEN *y, GEN *ptw1, GEN *ptw2)
! 25: {
! 26: GEN w,w1,w2,q;
! 27: long i,h;
! 28:
! 29: if (typ(f)!=t_POL) err(notpoler,"allbase");
! 30: if (lgef(f)<=3) err(constpoler,"allbase");
! 31: *y=discsr(f);
! 32: if (!signe(*y)) err(talker,"reducible polynomial in allbase");
! 33: if (DEBUGLEVEL) timer2();
! 34: switch(code)
! 35: {
! 36: case 0: case 1:
! 37: w=auxdecomp(absi(*y),1-code);
! 38: w1=(GEN)w[1]; w2=(GEN)w[2]; break;
! 39: default: w=(GEN)code;
! 40: if (typ(w)!=t_MAT || lg(w)!=3)
! 41: err(talker,"not a n x 2 matrix as factorization in factoredbase");
! 42: w1=(GEN)w[1]; w2=(GEN)w[2]; h=lg(w1); q=gun;
! 43: for (i=1; i<h; i++)
! 44: q=gmul(q,powgi((GEN)w1[i], (GEN)w2[i]));
! 45: if (gcmp(absi(q), absi(*y)))
! 46: err(talker,"incorrect factorization in factoredbase");
! 47: }
! 48: if (DEBUGLEVEL) msgtimer("disc. factorisation");
! 49: *ptw1=w1; *ptw2=w2;
! 50: }
! 51:
! 52: /*******************************************************************/
! 53: /* */
! 54: /* ROUND 2 */
! 55: /* */
! 56: /*******************************************************************/
! 57: /* Normalized quotient and remainder ( -1/2 |y| < r = x-q*y <= 1/2 |y| ) */
! 58: static GEN
! 59: rquot(GEN x, GEN y)
! 60: {
! 61: long av=avma,av1;
! 62: GEN u,v,w,p;
! 63:
! 64: u=absi(y); v=shifti(x,1); w=shifti(y,1);
! 65: if (cmpii(u,v)>0) p=subii(v,u);
! 66: else p=addsi(-1,addii(u,v));
! 67: av1=avma; return gerepile(av,av1,divii(p,w));
! 68: }
! 69:
! 70: /* space needed lx + 2*ly */
! 71: static GEN
! 72: rrmdr(GEN x, GEN y)
! 73: {
! 74: long av=avma,tetpil,k;
! 75: GEN r,ys2;
! 76:
! 77: if (!signe(x)) return gzero;
! 78: r = resii(x,y); tetpil = avma;
! 79: ys2 = shifti(y,-1);
! 80: k = absi_cmp(r, ys2);
! 81: if (k>0 || (k==0 && signe(r)>0))
! 82: {
! 83: avma = tetpil;
! 84: if (signe(y) == signe(r)) r = subii(r,y); else r = addii(r,y);
! 85: return gerepile(av,tetpil,r);
! 86: }
! 87: avma = tetpil; return r;
! 88: }
! 89:
! 90: /* companion matrix of unitary polynomial x */
! 91: static GEN
! 92: companion(GEN x) /* cf assmat */
! 93: {
! 94: long i,j,l;
! 95: GEN y;
! 96:
! 97: l=lgef(x)-2; y=cgetg(l,t_MAT);
! 98: for (j=1; j<l; j++)
! 99: {
! 100: y[j] = lgetg(l,t_COL);
! 101: for (i=1; i<l-1; i++)
! 102: coeff(y,i,j)=(i+1==j)? un: zero;
! 103: coeff(y,i,j) = lneg((GEN)x[j+1]);
! 104: }
! 105: return y;
! 106: }
! 107:
! 108: /* assume x, y are square integer matrices of same dim. Multiply them */
! 109: static GEN
! 110: mulmati(GEN x, GEN y)
! 111: {
! 112: long n = lg(x),i,j,k,av;
! 113: GEN z = cgetg(n,t_MAT),p1,p2;
! 114:
! 115: for (j=1; j<n; j++)
! 116: {
! 117: z[j] = lgetg(n,t_COL);
! 118: for (i=1; i<n; i++)
! 119: {
! 120: p1=gzero; av=avma;
! 121: for (k=1; k<n; k++)
! 122: {
! 123: p2=mulii(gcoeff(x,i,k),gcoeff(y,k,j));
! 124: if (p2 != gzero) p1=addii(p1,p2);
! 125: }
! 126: coeff(z,i,j)=lpileupto(av,p1);
! 127: }
! 128: }
! 129: return z;
! 130: }
! 131:
! 132: static GEN
! 133: powmati(GEN x, long m)
! 134: {
! 135: long av=avma,j;
! 136: GEN y = x;
! 137:
! 138: j=1+bfffo(m); m<<=j; j = BITS_IN_LONG-j;
! 139: for (; j; m<<=1,j--)
! 140: {
! 141: y=mulmati(y,y);
! 142: if (m<0) y=mulmati(y,x);
! 143: }
! 144: return gerepileupto(av,y);
! 145: }
! 146:
! 147: static GEN
! 148: rtran(GEN v, GEN w, GEN q)
! 149: {
! 150: long av,tetpil;
! 151: GEN p1;
! 152:
! 153: if (signe(q))
! 154: {
! 155: av=avma; p1=gneg(gmul(q,w)); tetpil=avma;
! 156: return gerepile(av,tetpil,gadd(v,p1));
! 157: }
! 158: return v;
! 159: }
! 160:
! 161: /* return (v - qw) mod m (only compute entries k0,..,n)
! 162: * v and w are expected to have entries smaller than m */
! 163: static GEN
! 164: mtran(GEN v, GEN w, GEN q, GEN m, long k0)
! 165: {
! 166: long k,l;
! 167: GEN p1;
! 168:
! 169: if (signe(q))
! 170: {
! 171: l = lgefint(m) << 2;
! 172: for (k=lg(v)-1; k>= k0; k--)
! 173: {
! 174: long av = avma; (void)new_chunk(l);
! 175: p1 = subii((GEN)v[k], mulii(q,(GEN)w[k]));
! 176: avma = av; v[k]=(long)rrmdr(p1, m);
! 177: }
! 178: }
! 179: return v;
! 180: }
! 181:
! 182: /* entries of v and w are C small integers */
! 183: static GEN
! 184: mtran_long(GEN v, GEN w, long q, long m, long k0)
! 185: {
! 186: long k, p1;
! 187:
! 188: if (q)
! 189: {
! 190: for (k=lg(v)-1; k>= k0; k--)
! 191: {
! 192: p1 = v[k] - q * w[k];
! 193: v[k] = p1 % m;
! 194: }
! 195: }
! 196: return v;
! 197: }
! 198:
! 199: /* coeffs of a are C-long integers */
! 200: static void
! 201: rowred_long(GEN a, long rmod)
! 202: {
! 203: long q,j,k,pro, c = lg(a), r = lg(a[1]);
! 204:
! 205: for (j=1; j<r; j++)
! 206: {
! 207: for (k=j+1; k<c; k++)
! 208: while (coeff(a,j,k))
! 209: {
! 210: q = coeff(a,j,j) / coeff(a,j,k);
! 211: pro=(long)mtran_long((GEN)a[j],(GEN)a[k],q,rmod, j);
! 212: a[j]=a[k]; a[k]=pro;
! 213: }
! 214: if (coeff(a,j,j) < 0)
! 215: for (k=j; k<r; k++) coeff(a,k,j)=-coeff(a,k,j);
! 216: for (k=1; k<j; k++)
! 217: {
! 218: q = coeff(a,j,k) / coeff(a,j,j);
! 219: a[k]=(long)mtran_long((GEN)a[k],(GEN)a[j],q,rmod, k);
! 220: }
! 221: }
! 222: /* don't update the 0s in the last columns */
! 223: for (j=1; j<r; j++)
! 224: for (k=1; k<r; k++) coeff(a,j,k) = lstoi(coeff(a,j,k));
! 225: }
! 226:
! 227: static void
! 228: rowred(GEN a, GEN rmod)
! 229: {
! 230: long j,k,pro, c = lg(a), r = lg(a[1]);
! 231: long av=avma, lim=stack_lim(av,1);
! 232: GEN q;
! 233:
! 234: for (j=1; j<r; j++)
! 235: {
! 236: for (k=j+1; k<c; k++)
! 237: while (signe(gcoeff(a,j,k)))
! 238: {
! 239: q=rquot(gcoeff(a,j,j),gcoeff(a,j,k));
! 240: pro=(long)mtran((GEN)a[j],(GEN)a[k],q,rmod, j);
! 241: a[j]=a[k]; a[k]=pro;
! 242: }
! 243: if (signe(gcoeff(a,j,j)) < 0)
! 244: for (k=j; k<r; k++) coeff(a,k,j)=lnegi(gcoeff(a,k,j));
! 245: for (k=1; k<j; k++)
! 246: {
! 247: q=rquot(gcoeff(a,j,k),gcoeff(a,j,j));
! 248: a[k]=(long)mtran((GEN)a[k],(GEN)a[j],q,rmod, k);
! 249: }
! 250: if (low_stack(lim, stack_lim(av,1)))
! 251: {
! 252: long j1,k1, tetpil = avma;
! 253: GEN p1 = a;
! 254: if(DEBUGMEM>1) err(warnmem,"rowred j=%ld", j);
! 255: p1 = gerepile(av,tetpil,gcopy(a));
! 256: for (j1=1; j1<r; j1++)
! 257: for (k1=1; k1<c; k1++) coeff(a,j1,k1) = coeff(p1,j1,k1);
! 258: }
! 259: }
! 260: }
! 261:
! 262: /* Calcule d/x ou d est entier et x matrice triangulaire inferieure
! 263: * entiere dont les coeff diagonaux divisent d (resultat entier).
! 264: */
! 265: static GEN
! 266: matinv(GEN x, GEN d, long n)
! 267: {
! 268: long i,j,k,av,av1;
! 269: GEN y,h;
! 270:
! 271: y=idmat(n);
! 272: for (i=1; i<=n; i++)
! 273: coeff(y,i,i)=ldivii(d,gcoeff(x,i,i));
! 274: av=avma;
! 275: for (i=2; i<=n; i++)
! 276: for (j=i-1; j; j--)
! 277: {
! 278: for (h=gzero,k=j+1; k<=i; k++)
! 279: {
! 280: GEN p1 = mulii(gcoeff(y,i,k),gcoeff(x,k,j));
! 281: if (p1 != gzero) h=addii(h,p1);
! 282: }
! 283: setsigne(h,-signe(h)); av1=avma;
! 284: coeff(y,i,j) = lpile(av,av1,divii(h,gcoeff(x,j,j)));
! 285: av = avma;
! 286: }
! 287: return y;
! 288: }
! 289:
! 290: static GEN
! 291: ordmax(GEN *cf, GEN p, long epsilon, GEN *ptdelta)
! 292: {
! 293: long sp,hard_case_exponent,i,n=lg(cf)-1,av=avma, av2,limit;
! 294: GEN T,T2,Tn,m,v,delta, *w;
! 295: const GEN pp = sqri(p);
! 296: const long pps = (2*expi(pp)+2<BITS_IN_LONG)? pp[2]: 0;
! 297:
! 298: if (cmpis(p,n) > 0) hard_case_exponent = 0;
! 299: else
! 300: {
! 301: long k;
! 302: k = sp = itos(p);
! 303: i=1; while (k < n) { k *= sp; i++; }
! 304: hard_case_exponent = i;
! 305: }
! 306: T=cgetg(n+1,t_MAT); for (i=1; i<=n; i++) T[i]=lgetg(n+1,t_COL);
! 307: T2=cgetg(2*n+1,t_MAT); for (i=1; i<=2*n; i++) T2[i]=lgetg(n+1,t_COL);
! 308: Tn=cgetg(n*n+1,t_MAT); for (i=1; i<=n*n; i++) Tn[i]=lgetg(n+1,t_COL);
! 309: v = new_chunk(n+1);
! 310: w = (GEN*)new_chunk(n+1);
! 311:
! 312: av2 = avma; limit = stack_lim(av2,1);
! 313: delta=gun; m=idmat(n);
! 314:
! 315: for(;;)
! 316: {
! 317: long j,k,h, av0 = avma;
! 318: GEN t,b,jp,hh,index,p1, dd = sqri(delta), ppdd = mulii(dd,pp);
! 319:
! 320: if (DEBUGLEVEL > 3)
! 321: fprintferr("ROUND2: epsilon = %ld\tavma = %ld\n",epsilon,avma);
! 322:
! 323: b=matinv(m,delta,n);
! 324: for (i=1; i<=n; i++)
! 325: {
! 326: for (j=1; j<=n; j++)
! 327: for (k=1; k<=n; k++)
! 328: {
! 329: p1 = j==k? gcoeff(m,i,1): gzero;
! 330: for (h=2; h<=n; h++)
! 331: {
! 332: GEN p2 = mulii(gcoeff(m,i,h),gcoeff(cf[h],j,k));
! 333: if (p2!=gzero) p1 = addii(p1,p2);
! 334: }
! 335: coeff(T,j,k) = (long)rrmdr(p1, ppdd);
! 336: }
! 337: p1 = mulmati(m, mulmati(T,b));
! 338: for (j=1; j<=n; j++)
! 339: for (k=1; k<=n; k++)
! 340: coeff(p1,j,k)=(long)rrmdr(divii(gcoeff(p1,j,k),dd),pp);
! 341: w[i] = p1;
! 342: }
! 343:
! 344: if (hard_case_exponent)
! 345: {
! 346: for (j=1; j<=n; j++)
! 347: {
! 348: for (i=1; i<=n; i++) coeff(T,i,j) = coeff(w[j],1,i);
! 349: /* ici la boucle en k calcule la puissance p mod p de w[j] */
! 350: for (k=1; k<sp; k++)
! 351: {
! 352: for (i=1; i<=n; i++)
! 353: {
! 354: p1 = gzero;
! 355: for (h=1; h<=n; h++)
! 356: {
! 357: GEN p2=mulii(gcoeff(T,h,j),gcoeff(w[j],h,i));
! 358: if (p2!=gzero) p1 = addii(p1,p2);
! 359: }
! 360: v[i] = lmodii(p1, p);
! 361: }
! 362: for (i=1; i<=n; i++) coeff(T,i,j)=v[i];
! 363: }
! 364: }
! 365: t = powmati(T, hard_case_exponent);
! 366: }
! 367: else
! 368: {
! 369: for (i=1; i<=n; i++)
! 370: for (j=1; j<=n; j++)
! 371: {
! 372: long av1 = avma;
! 373: p1 = gzero;
! 374: for (k=1; k<=n; k++)
! 375: for (h=1; h<=n; h++)
! 376: {
! 377: const GEN r=modii(gcoeff(w[i],k,h),p);
! 378: const GEN s=modii(gcoeff(w[j],h,k),p);
! 379: const GEN p2 = mulii(r,s);
! 380: if (p2!=gzero) p1 = addii(p1,p2);
! 381: }
! 382: coeff(T,i,j) = lpileupto(av1,p1);
! 383: }
! 384: t = T;
! 385: }
! 386:
! 387: if (pps)
! 388: {
! 389: long ps = p[2];
! 390: for (i=1; i<=n; i++)
! 391: for (j=1; j<=n; j++)
! 392: {
! 393: coeff(T2,j,i)=(i==j)? ps: 0;
! 394: coeff(T2,j,n+i)=smodis(gcoeff(t,i,j),ps);
! 395: }
! 396: rowred_long(T2,pps);
! 397: }
! 398: else
! 399: {
! 400: for (i=1; i<=n; i++)
! 401: for (j=1; j<=n; j++)
! 402: {
! 403: coeff(T2,j,i)=(i==j)? (long)p: zero;
! 404: coeff(T2,j,n+i)=lmodii(gcoeff(t,i,j),p);
! 405: }
! 406: rowred(T2,pp);
! 407: }
! 408: jp=matinv(T2,p,n);
! 409: if (pps)
! 410: {
! 411: for (k=1; k<=n; k++)
! 412: {
! 413: long av1=avma;
! 414: t = mulmati(mulmati(jp,w[k]), T2);
! 415: for (h=i=1; i<=n; i++)
! 416: for (j=1; j<=n; j++)
! 417: { coeff(Tn,k,h) = itos(divii(gcoeff(t,i,j), p)) % pps; h++; }
! 418: avma=av1;
! 419: }
! 420: avma = av0;
! 421: rowred_long(Tn,pps);
! 422: }
! 423: else
! 424: {
! 425: for (k=1; k<=n; k++)
! 426: {
! 427: t = mulmati(mulmati(jp,w[k]), T2);
! 428: for (h=i=1; i<=n; i++)
! 429: for (j=1; j<=n; j++)
! 430: { coeff(Tn,k,h) = ldivii(gcoeff(t,i,j), p); h++; }
! 431: }
! 432: rowred(Tn,pp);
! 433: }
! 434: for (index=gun,i=1; i<=n; i++)
! 435: index = mulii(index,gcoeff(Tn,i,i));
! 436: if (gcmp1(index)) break;
! 437:
! 438: m = mulmati(matinv(Tn,index,n), m);
! 439: hh = delta = mulii(index,delta);
! 440: for (i=1; i<=n; i++)
! 441: for (j=1; j<=n; j++)
! 442: hh = mppgcd(gcoeff(m,i,j),hh);
! 443: if (!is_pm1(hh))
! 444: {
! 445: m = gdiv(m,hh);
! 446: delta = divii(delta,hh);
! 447: }
! 448: epsilon -= 2 * ggval(index,p);
! 449: if (epsilon < 2) break;
! 450: if (low_stack(limit,stack_lim(av2,1)))
! 451: {
! 452: GEN *gptr[3]; gptr[0]=&m; gptr[1]=δ
! 453: if(DEBUGMEM>1) err(warnmem,"ordmax");
! 454: gerepilemany(av2, gptr,2);
! 455: }
! 456: }
! 457: {
! 458: GEN *gptr[2]; gptr[0]=&m; gptr[1]=δ
! 459: gerepilemany(av,gptr,2);
! 460: }
! 461: *ptdelta=delta; return m;
! 462: }
! 463:
! 464: #if 0
! 465: static void
! 466: to_col(GEN x, GEN col)
! 467: {
! 468: long i,n = lg(col), k = lgef(x)-1;
! 469: x++;
! 470: for (i=1; i<k; i++) col[i] = x[i];
! 471: for ( ; i<n; i++) col[i] = zero;
! 472: }
! 473:
! 474: static GEN
! 475: ordmax2(GEN f, GEN p, long epsilon, GEN *ptdelta)
! 476: {
! 477: long sp,i,n=lgef(f)-3,av=avma, av2,limit;
! 478: GEN col,sym,hard_case_exponent,T2,Tn,m,v,delta,w,a;
! 479: const GEN pp = sqri(p);
! 480:
! 481: if (cmpis(p,n) > 0)
! 482: {
! 483: hard_case_exponent = NULL;
! 484: sym = polsym(f,n-1);
! 485: }
! 486: else
! 487: {
! 488: long k; k = sp = itos(p);
! 489: while (k < n) k *= sp;
! 490: hard_case_exponent = stoi(k);
! 491: }
! 492: col = cgetg(n+1,t_COL);
! 493: T2=cgetg(2*n+1,t_MAT); for (i=1; i<=2*n; i++) T2[i]=lgetg(n+1,t_COL);
! 494: Tn=cgetg(n*n+1,t_MAT); for (i=1; i<=n*n; i++) Tn[i]=lgetg(n+1,t_COL);
! 495: v = new_chunk(n+1);
! 496:
! 497: av2 = avma; limit = stack_lim(av2,1);
! 498: delta=gun; m=idmat(n);
! 499:
! 500: for(;;)
! 501: {
! 502: long j,k,h, av0 = avma;
! 503: GEN hh,index,p1;
! 504:
! 505: if (DEBUGLEVEL > 3)
! 506: fprintferr("ROUND2: epsilon = %ld\tavma = %ld\n",epsilon,avma);
! 507:
! 508: w = mat_to_vecpol(m, 0);
! 509: if (hard_case_exponent)
! 510: {
! 511: for (i=1; i<=n; i++)
! 512: {
! 513: p1 = Fp_pow_mod_pol((GEN)w[i], hard_case_exponent, f,p);
! 514: to_col(p1, (GEN)T2[i]);
! 515: }
! 516: for (i=1; i<=n; i++) /* transpose */
! 517: for (j=1; j<i; j++)
! 518: {
! 519: p1 = gcoeff(T2,i,j);
! 520: coeff(T2,i,j) = coeff(T2,j,i);
! 521: coeff(T2,j,i)= (long)p1;
! 522: }
! 523: }
! 524: else
! 525: {
! 526: for (i=1; i<=n; i++)
! 527: {
! 528: for (j=1; j<i; j++)
! 529: {
! 530: p1 = Fp_res(gmul((GEN)w[i], (GEN)w[j]), f, p);
! 531: coeff(T2,j,i) = coeff(T2,i,j) = lresii(quicktrace(p1,sym), p);
! 532: }
! 533: p1 = Fp_res(gsqr((GEN)w[i]), f, p);
! 534: coeff(T2,i,i) = lresii(quicktrace(p1,sym), p);
! 535: }
! 536: }
! 537: for (i=1; i<=n; i++)
! 538: for (j=1; j<=n; j++)
! 539: coeff(T2,j,n+i)=(i==j)? (long)p : zero;
! 540: rowred(T2,pp);
! 541: a = mat_to_vecpol(matinv(T2,p,n), 0);
! 542: if (2*expi(pp)+2<BITS_IN_LONG)
! 543: {
! 544: for (k=1; k<=n; k++)
! 545: {
! 546: long av1=avma;
! 547: for (h=i=1; i<=n; i++)
! 548: {
! 549: p1 = gres(gmul((GEN)a[i], (GEN)w[k]), f);
! 550: to_col(p1, col);
! 551: for (j=1; j<=n; j++)
! 552: { coeff(Tn,k,h)=itos(divii((GEN)col[j],p)); h++; }
! 553: }
! 554: avma=av1;
! 555: }
! 556: avma = av0;
! 557: rowred_long(Tn,pp[2]);
! 558: }
! 559: else
! 560: {
! 561: for (k=1; k<=n; k++)
! 562: {
! 563: for (h=i=1; i<=n; i++)
! 564: {
! 565: p1 = gres(gmul((GEN)a[i], (GEN)w[k]), f);
! 566: to_col(p1, col);
! 567: for (j=1; j<=n; j++)
! 568: #if 0
! 569: { coeff(Tn,k,h)=ldivii((GEN)col[j],p); h++; }
! 570: #endif
! 571: { coeff(Tn,k,h)=col[j]; h++; }
! 572: }
! 573: }
! 574: rowred(Tn,pp);
! 575: }
! 576: for (index=gun,i=1; i<=n; i++)
! 577: index = mulii(index,gcoeff(Tn,i,i));
! 578: if (gcmp1(index)) break;
! 579:
! 580: m = mulmati(matinv(Tn,index,n), m);
! 581: hh = delta = mulii(index,delta);
! 582: for (i=1; i<=n; i++)
! 583: for (j=1; j<=n; j++)
! 584: hh = mppgcd(gcoeff(m,i,j),hh);
! 585: if (!is_pm1(hh))
! 586: {
! 587: m = gdiv(m,hh);
! 588: delta = divii(delta,hh);
! 589: }
! 590: epsilon -= 2 * ggval(index,p);
! 591: if (epsilon < 2) break;
! 592: if (low_stack(limit,stack_lim(av2,1)))
! 593: {
! 594: GEN *gptr[3]; gptr[0]=&m; gptr[1]=δ
! 595: if(DEBUGMEM>1) err(warnmem,"ordmax");
! 596: gerepilemany(av2, gptr,2);
! 597: }
! 598: }
! 599: {
! 600: GEN *gptr[2]; gptr[0]=&m; gptr[1]=δ
! 601: gerepilemany(av,gptr,2);
! 602: }
! 603: *ptdelta=delta; return m;
! 604: }
! 605: #endif
! 606:
! 607: /* Input:
! 608: * x normalized integral polynomial of degree n, defining K=Q(theta).
! 609: *
! 610: * code 0, 1 or (long)p if we want base, smallbase ou factoredbase (resp.).
! 611: * y is GEN *, which will receive the discriminant of K.
! 612: *
! 613: * Output
! 614: * 1) A t_COL whose n components are rationnal polynomials (with degree
! 615: * 0,1...n-1) : integral basis for K (putting x=theta).
! 616: * Rem: common denominator is in da.
! 617: *
! 618: * 2) discriminant of K (in *y).
! 619: */
! 620: GEN
! 621: allbase(GEN f, long code, GEN *y)
! 622: {
! 623: GEN w1,w2,a,pro,at,bt,b,da,db,q, *cf,*gptr[2];
! 624: long av=avma,tetpil,n,h,j,i,k,r,s,t,v,mf;
! 625:
! 626: allbase_check_args(f,code,y, &w1,&w2);
! 627: v = varn(f); n = lgef(f)-3; h = lg(w1)-1;
! 628: cf = (GEN*)cgetg(n+1,t_VEC);
! 629: cf[2]=companion(f);
! 630: for (i=3; i<=n; i++) cf[i]=mulmati(cf[2],cf[i-1]);
! 631:
! 632: a=idmat(n); da=gun;
! 633: for (i=1; i<=h; i++)
! 634: {
! 635: long av1 = avma;
! 636: mf=itos((GEN)w2[i]); if (mf==1) continue;
! 637: if (DEBUGLEVEL) fprintferr("Treating p^k = %Z^%ld\n",w1[i],mf);
! 638:
! 639: b=ordmax(cf,(GEN)w1[i],mf,&db);
! 640: a=gmul(db,a); b=gmul(da,b);
! 641: da=mulii(db,da);
! 642: at=gtrans(a); bt=gtrans(b);
! 643: for (r=n; r; r--)
! 644: for (s=r; s; s--)
! 645: while (signe(gcoeff(bt,s,r)))
! 646: {
! 647: q=rquot(gcoeff(at,s,s),gcoeff(bt,s,r));
! 648: pro=rtran((GEN)at[s],(GEN)bt[r],q);
! 649: for (t=s-1; t; t--)
! 650: {
! 651: q=rquot(gcoeff(at,t,s),gcoeff(at,t,t));
! 652: pro=rtran(pro,(GEN)at[t],q);
! 653: }
! 654: at[s]=bt[r]; bt[r]=(long)pro;
! 655: }
! 656: for (j=n; j; j--)
! 657: {
! 658: for (k=1; k<j; k++)
! 659: {
! 660: while (signe(gcoeff(at,j,k)))
! 661: {
! 662: q=rquot(gcoeff(at,j,j),gcoeff(at,j,k));
! 663: pro=rtran((GEN)at[j],(GEN)at[k],q);
! 664: at[j]=at[k]; at[k]=(long)pro;
! 665: }
! 666: }
! 667: if (signe(gcoeff(at,j,j))<0)
! 668: for (k=1; k<=j; k++) coeff(at,k,j)=lnegi(gcoeff(at,k,j));
! 669: for (k=j+1; k<=n; k++)
! 670: {
! 671: q=rquot(gcoeff(at,j,k),gcoeff(at,j,j));
! 672: at[k]=(long)rtran((GEN)at[k],(GEN)at[j],q);
! 673: }
! 674: }
! 675: for (j=2; j<=n; j++)
! 676: if (egalii(gcoeff(at,j,j), gcoeff(at,j-1,j-1)))
! 677: {
! 678: coeff(at,1,j)=zero;
! 679: for (k=2; k<=j; k++) coeff(at,k,j)=coeff(at,k-1,j-1);
! 680: }
! 681: tetpil=avma; a=gtrans(at);
! 682: {
! 683: GEN *gptr[2];
! 684: da = icopy(da); gptr[0]=&a; gptr[1]=&da;
! 685: gerepilemanysp(av1,tetpil,gptr,2);
! 686: }
! 687: }
! 688: for (j=1; j<=n; j++)
! 689: *y = divii(mulii(*y,sqri(gcoeff(a,j,j))), sqri(da));
! 690: tetpil=avma; *y=icopy(*y);
! 691: at=cgetg(n+1,t_VEC); v=varn(f);
! 692: for (k=1; k<=n; k++)
! 693: {
! 694: q=cgetg(k+2,t_POL); at[k]=(long)q;
! 695: q[1] = evalsigne(1) | evallgef(2+k) | evalvarn(v);
! 696: for (j=1; j<=k; j++) q[j+1]=ldiv(gcoeff(a,k,j),da);
! 697: }
! 698: gptr[0]=&at; gptr[1]=y;
! 699: gerepilemanysp(av,tetpil,gptr,2);
! 700: return at;
! 701: }
! 702:
! 703: GEN
! 704: base2(GEN x, GEN *y)
! 705: {
! 706: return allbase(x,0,y);
! 707: }
! 708:
! 709: GEN
! 710: discf2(GEN x)
! 711: {
! 712: GEN y;
! 713: long av=avma,tetpil;
! 714:
! 715: allbase(x,0,&y); tetpil=avma;
! 716: return gerepile(av,tetpil,icopy(y));
! 717: }
! 718:
! 719: /*******************************************************************/
! 720: /* */
! 721: /* ROUND 4 */
! 722: /* */
! 723: /*******************************************************************/
! 724:
! 725: static GEN Decomp(GEN p,GEN f,long mf,GEN theta,GEN chi,GEN nu);
! 726: static GEN dbasis(GEN p, GEN f, long mf, GEN alpha, GEN U);
! 727: static GEN eltppm(GEN f,GEN pd,GEN theta,GEN k);
! 728: static GEN maxord(GEN p,GEN f,long mf);
! 729: static GEN nbasis(GEN ibas,GEN pd);
! 730: #if 0
! 731: static GEN nilord(GEN p,GEN fx,long mf,GEN gx);
! 732: #endif
! 733: static GEN nilord2(GEN p,GEN fx,long mf,GEN gx);
! 734: static GEN testd(GEN p,GEN fa,long c,long Da,GEN alph2,long Ma,GEN theta);
! 735: static GEN testb(GEN p,GEN fa,long Da,GEN theta,long Dt);
! 736: static GEN testb2(GEN p,GEN fa,long Fa,GEN theta,long Ft);
! 737: static GEN testc2(GEN p,GEN fa,GEN pmr,GEN alph2,long Ea,GEN thet2,long Et);
! 738:
! 739: static long clcm(long a,long b);
! 740:
! 741: static int
! 742: fnz(GEN x,long j)
! 743: {
! 744: long i=1; while (!signe(x[i])) i++;
! 745: return i==j;
! 746: }
! 747:
! 748: /* retourne la base, dans y le discf et dans ptw la factorisation (peut
! 749: etre partielle) de discf */
! 750: GEN
! 751: allbase4(GEN f,long code, GEN *y, GEN *ptw)
! 752: {
! 753: GEN w,w1,w2,a,da,b,db,bas,q,p1,*gptr[3];
! 754: long v,n,mf,h,lfa,i,j,k,l,first,tetpil,av = avma;
! 755:
! 756: allbase_check_args(f,code,y, &w1,&w2);
! 757: first=1; v = varn(f); n = lgef(f)-3; h = lg(w1)-1;
! 758: for (i=1; i<=h; i++)
! 759: {
! 760: mf=itos((GEN)w2[i]); if (mf == 1) continue;
! 761: if (DEBUGLEVEL) fprintferr("Treating p^k = %Z^%ld\n",w1[i],mf);
! 762:
! 763: b = maxord((GEN)w1[i],f,mf);
! 764: p1=cgetg(n+1,t_VEC); for (j=1; j<=n; j++) p1[j]=coeff(b,j,j);
! 765: db=denom(p1);
! 766: if (! gcmp1(db))
! 767: {
! 768: if (first==1) { da=db; a=gmul(b,db); first=0; }
! 769: else
! 770: {
! 771: da=mulii(da,db); b=gmul(da,b); a=gmul(db,a);
! 772: j=1; while (j<=n && fnz((GEN)a[j],j) && fnz((GEN)b[j],j)) j++;
! 773: k=j-1; p1=cgetg(2*n-k+1,t_MAT);
! 774: for (j=1; j<=k; j++)
! 775: {
! 776: p1[j]=a[j];
! 777: coeff(p1,j,j) = lmppgcd(gcoeff(a,j,j),gcoeff(b,j,j));
! 778: }
! 779: for ( ; j<=n; j++) p1[j]=a[j];
! 780: for ( ; j<=2*n-k; j++) p1[j]=b[j+k-n];
! 781: a=hnfmod(p1,detint(p1));
! 782: }
! 783: }
! 784: if (DEBUGLEVEL>5)
! 785: fprintferr("Result for prime %Z is:\n%Z\n",w1[i],b);
! 786: }
! 787: if (!first)
! 788: {
! 789: for (j=1; j<=n; j++)
! 790: *y = mulii(divii(*y,sqri(da)),sqri(gcoeff(a,j,j)));
! 791: for (j=n-1; j; j--)
! 792: if (cmpis(gcoeff(a,j,j),2) > 0)
! 793: {
! 794: p1=shifti(gcoeff(a,j,j),-1);
! 795: for (k=j+1; k<=n; k++)
! 796: if (cmpii(gcoeff(a,j,k),p1) > 0)
! 797: for (l=1; l<=j; l++)
! 798: coeff(a,l,k)=lsubii(gcoeff(a,l,k),gcoeff(a,l,j));
! 799: }
! 800: }
! 801: if (ptw)
! 802: {
! 803: lfa=0;
! 804: for (j=1; j<=h; j++)
! 805: {
! 806: k=ggval(*y,(GEN)w1[j]);
! 807: if (k) { lfa++; w1[lfa]=w1[j]; w2[lfa]=k; }
! 808: }
! 809: }
! 810: tetpil=avma; *y=icopy(*y);
! 811: bas=cgetg(n+1,t_VEC); v=varn(f);
! 812: for (k=1; k<=n; k++)
! 813: {
! 814: q=cgetg(k+2,t_POL); bas[k]=(long)q;
! 815: q[1] = evalsigne(1) | evallgef(k+2) | evalvarn(v);
! 816: if (!first)
! 817: for (j=1; j<=k; j++) q[j+1]=ldiv(gcoeff(a,j,k),da);
! 818: else
! 819: {
! 820: for (j=2; j<=k; j++) q[j]=zero;
! 821: q[j]=un;
! 822: }
! 823: }
! 824: if (ptw)
! 825: {
! 826: *ptw=w=cgetg(3,t_MAT); w[1]=lgetg(lfa+1,t_COL); w[2]=lgetg(lfa+1,t_COL);
! 827: for (j=1; j<=lfa; j++)
! 828: {
! 829: coeff(w,j,1)=(long)icopy((GEN)w1[j]);
! 830: coeff(w,j,2)=lstoi(w2[j]);
! 831: }
! 832: gptr[2]=ptw;
! 833: }
! 834: gptr[0]=&bas; gptr[1]=y;
! 835: gerepilemanysp(av,tetpil,gptr, ptw?3:2);
! 836: return bas;
! 837: }
! 838:
! 839: /* if y is non-NULL, it receives the discriminant
! 840: * return basis if (ret_basis != 0), discriminant otherwise
! 841: */
! 842: static GEN
! 843: nfbasis00(GEN x, long flag, GEN p, long ret_basis, GEN *y)
! 844: {
! 845: GEN disc, basis, lead;
! 846: GEN *gptr[2];
! 847: long k, tetpil, av = avma, n = lgef(x)-3, smll;
! 848:
! 849: if (typ(x)!=t_POL) err(typeer,"nfbasis00");
! 850: if (n<=0) err(zeropoler,"nfbasis00");
! 851: for (k=n+2; k>=2; k--)
! 852: if (typ(x[k])!=t_INT) err(talker,"polynomial not in Z[X] in nfbasis");
! 853:
! 854: x = pol_to_monic(x,&lead);
! 855:
! 856: if (!p || gcmp0(p))
! 857: smll = (flag & 1); /* small basis */
! 858: else
! 859: smll = (long) p; /* factored basis */
! 860:
! 861: if (flag & 2)
! 862: basis = allbase(x,smll,&disc); /* round 2 */
! 863: else
! 864: basis = allbase4(x,smll,&disc,NULL); /* round 4 */
! 865:
! 866: tetpil=avma;
! 867: if (!ret_basis)
! 868: return gerepile(av,tetpil,gcopy(disc));
! 869:
! 870: if (!lead) basis = gcopy(basis);
! 871: else
! 872: {
! 873: long v = varn(x);
! 874: GEN pol = gmul(polx[v],lead);
! 875:
! 876: tetpil = avma; basis = gsubst(basis,v,pol);
! 877: }
! 878: if (!y)
! 879: return gerepile(av,tetpil,basis);
! 880:
! 881: *y = gcopy(disc);
! 882: gptr[0]=&basis; gptr[1]=y;
! 883: gerepilemanysp(av,tetpil,gptr,2);
! 884: return basis;
! 885: }
! 886:
! 887: GEN
! 888: nfbasis(GEN x, GEN *y, long flag, GEN p)
! 889: {
! 890: return nfbasis00(x,flag,p,1,y);
! 891: }
! 892:
! 893: GEN
! 894: nfbasis0(GEN x, long flag, GEN p)
! 895: {
! 896: return nfbasis00(x,flag,p,1,NULL);
! 897: }
! 898:
! 899: GEN
! 900: nfdiscf0(GEN x, long flag, GEN p)
! 901: {
! 902: return nfbasis00(x,flag,p,0,&p);
! 903: }
! 904:
! 905: GEN
! 906: base(GEN x, GEN *y)
! 907: {
! 908: return allbase4(x,0,y,NULL);
! 909: }
! 910:
! 911: GEN
! 912: smallbase(GEN x, GEN *y)
! 913: {
! 914: return allbase4(x,1,y,NULL);
! 915: }
! 916:
! 917: GEN
! 918: factoredbase(GEN x, GEN p, GEN *y)
! 919: {
! 920: return allbase4(x,(long)p,y,NULL);
! 921: }
! 922:
! 923: GEN
! 924: discf(GEN x)
! 925: {
! 926: GEN y;
! 927: long av=avma,tetpil;
! 928:
! 929: allbase4(x,0,&y,NULL); tetpil=avma;
! 930: return gerepile(av,tetpil,icopy(y));
! 931: }
! 932:
! 933: GEN
! 934: smalldiscf(GEN x)
! 935: {
! 936: GEN y;
! 937: long av=avma,tetpil;
! 938:
! 939: allbase4(x,1,&y,NULL); tetpil=avma;
! 940: return gerepile(av,tetpil,icopy(y));
! 941: }
! 942:
! 943: GEN
! 944: factoreddiscf(GEN x, GEN p)
! 945: {
! 946: GEN y;
! 947: long av=avma,tetpil;
! 948:
! 949: allbase4(x,(long)p,&y,NULL); tetpil=avma;
! 950: return gerepile(av,tetpil,icopy(y));
! 951: }
! 952:
! 953: /* return U if Z[alpha] is not maximal or 2*dU < m-1; else return NULL */
! 954: static GEN
! 955: dedek(GEN f, long mf, GEN p,GEN g)
! 956: {
! 957: GEN k,h;
! 958: long dk;
! 959:
! 960: if (DEBUGLEVEL>=3)
! 961: {
! 962: fprintferr(" entering dedek ");
! 963: if (DEBUGLEVEL>5)
! 964: fprintferr("with parameters p=%Z,\n f=%Z",p,f);
! 965: fprintferr("\n");
! 966: }
! 967: h = Fp_deuc(f,g,p);
! 968: k = gdiv(gadd(f, gneg_i(gmul(g,h))), p);
! 969: k = Fp_pol_gcd(k, Fp_pol_gcd(g,h, p), p);
! 970:
! 971: dk = lgef(k)-3;
! 972: if (DEBUGLEVEL>=3) fprintferr(" gcd has degree %ld\n", dk);
! 973: if (2*dk >= mf-1) return Fp_deuc(f,k,p);
! 974: return dk? (GEN)NULL: f;
! 975: }
! 976:
! 977: /* p-maximal order of Af; mf = v_p(Disc(f)) */
! 978: static GEN
! 979: maxord(GEN p,GEN f,long mf)
! 980: {
! 981: long j,r, av = avma, flw = (cmpsi(lgef(f)-3,p) < 0);
! 982: GEN w,g,h,res;
! 983:
! 984: if (flw)
! 985: g = Fp_deuc(f, Fp_pol_gcd(f,derivpol(f), p), p);
! 986: else
! 987: {
! 988: w=(GEN)factmod(f,p)[1]; r=lg(w)-1;
! 989: g = h = lift_intern((GEN)w[r]); /* largest factor */
! 990: for (j=1; j<r; j++) g = Fp_pol_red(gmul(g, lift_intern((GEN)w[j])), p);
! 991: }
! 992: res = dedek(f,mf,p,g);
! 993: if (res)
! 994: res = dbasis(p,f,mf,polx[varn(f)],res);
! 995: else
! 996: {
! 997: if (flw) { w=(GEN)factmod(f,p)[1]; r=lg(w)-1; h=lift_intern((GEN)w[r]); }
! 998: #if 0
! 999: res = (r==1)? nilord(p,f,mf,h): Decomp(p,f,mf,polx[varn(f)],f,h);
! 1000: #else
! 1001: res = (r==1)? nilord2(p,f,mf,h): Decomp(p,f,mf,polx[varn(f)],f,h);
! 1002: #endif
! 1003: }
! 1004: return gerepileupto(av,res);
! 1005: }
! 1006:
! 1007: /* do a centermod on integer or rational number */
! 1008: static GEN
! 1009: polmodiaux(GEN x, GEN y, GEN ys2)
! 1010: {
! 1011: if (typ(x)!=t_INT)
! 1012: x = mulii((GEN)x[1], mpinvmod((GEN)x[2],y));
! 1013: x = modii(x,y);
! 1014: if (cmpii(x,ys2) > 0) x = subii(x,y);
! 1015: return x;
! 1016: }
! 1017:
! 1018: /* x polynomial with integer or rational coeff. Reduce them mod y IN PLACE */
! 1019: GEN
! 1020: polmodi(GEN x, GEN y)
! 1021: {
! 1022: long lx=lgef(x), i;
! 1023: GEN ys2 = shifti(y,-1);
! 1024: for (i=2; i<lx; i++) x[i]=(long)polmodiaux((GEN)x[i],y,ys2);
! 1025: return normalizepol_i(x, lx);
! 1026: }
! 1027:
! 1028: /* same but not in place */
! 1029: GEN
! 1030: polmodi_keep(GEN x, GEN y)
! 1031: {
! 1032: long lx=lgef(x), i;
! 1033: GEN ys2 = shifti(y,-1);
! 1034: GEN z = cgetg(lx,t_POL);
! 1035: for (i=2; i<lx; i++) z[i]=(long)polmodiaux((GEN)x[i],y,ys2);
! 1036: z[1]=x[1]; return normalizepol_i(z, lx);
! 1037: }
! 1038:
! 1039: static GEN
! 1040: dbasis(GEN p, GEN f, long mf, GEN alpha, GEN U)
! 1041: {
! 1042: long n=lgef(f)-3,dU,c,i,dh;
! 1043: GEN b,p1,ha,pd,pdp;
! 1044:
! 1045: if (n == 1) return gscalmat(gun, 1);
! 1046: if (DEBUGLEVEL>=3)
! 1047: {
! 1048: fprintferr(" entering Dedekind Basis ");
! 1049: if (DEBUGLEVEL>5)
! 1050: {
! 1051: fprintferr("with parameters p=%Z\n",p);
! 1052: fprintferr(" f = %Z,\n alpha = %Z",f,alpha);
! 1053: }
! 1054: fprintferr("\n");
! 1055: }
! 1056: ha = pd = gpuigs(p,mf/2); pdp = mulii(pd,p);
! 1057: dU = lgef(U)-3;
! 1058: b = cgetg(n,t_MAT); /* Z[a] + U/p Z[a] is maximal */
! 1059: /* skip first column = gscalcol(pd,n) */
! 1060: for (c=1; c<n; c++)
! 1061: {
! 1062: p1=cgetg(n+1,t_COL); b[c]=(long)p1;
! 1063: if (c == dU)
! 1064: {
! 1065: ha = gdiv(gmul(pd,eleval(f,U,alpha)),p);
! 1066: ha = polmodi(ha,pdp);
! 1067: }
! 1068: else
! 1069: {
! 1070: GEN p2, mod;
! 1071: ha = gmul(ha,alpha);
! 1072: p2 = content(ha); /* to cancel denominator */
! 1073: if (gcmp1(p2)) { p2 = NULL; mod = pdp; }
! 1074: else
! 1075: {
! 1076: ha = gdiv(ha,p2);
! 1077: if (typ(p2)==t_INT)
! 1078: mod = divii(pdp, mppgcd(pdp,p2));
! 1079: else
! 1080: mod = mulii(pdp, (GEN)p2[2]); /* p2 = a / p^e */
! 1081: }
! 1082: ha = Fp_res(ha, f, mod);
! 1083: if (p2) ha = gmul(ha,p2);
! 1084: }
! 1085: dh = lgef(ha)-2;
! 1086: for (i=1; i<=dh; i++) p1[i]=ha[i+1];
! 1087: for ( ; i<=n; i++) p1[i]=zero;
! 1088: }
! 1089: b = hnfmodid(b,pd);
! 1090: if (DEBUGLEVEL>5) fprintferr(" new order: %Z\n",b);
! 1091: return gdiv(b,pd);
! 1092: }
! 1093:
! 1094: static GEN
! 1095: get_partial_order_as_pols(GEN p, GEN f)
! 1096: {
! 1097: long i,j,n=lgef(f)-3, vf = varn(f);
! 1098: GEN b,ib,h,col;
! 1099:
! 1100: b = maxord(p,f, ggval(discsr(f),p));
! 1101: ib = cgetg(n+1,t_VEC);
! 1102: for (i=1; i<=n; i++)
! 1103: {
! 1104: h=cgetg(i+2,t_POL); ib[i]=(long)h; col=(GEN)b[i];
! 1105: h[1]=evalsigne(1)|evallgef(i+2)|evalvarn(vf);
! 1106: for (j=1;j<=i;j++) h[j+1]=col[j];
! 1107: }
! 1108: return ib;
! 1109: }
! 1110:
! 1111: static GEN
! 1112: Decomp(GEN p,GEN f,long mf,GEN theta,GEN chi,GEN nu)
! 1113: {
! 1114: GEN pk,ph,pdr,pmr,unmodp;
! 1115: GEN b1,b2,b3,a1,e,f1,f2,ib1,ib2,ibas;
! 1116: long n1,n2,j;
! 1117:
! 1118: if (DEBUGLEVEL>=3)
! 1119: {
! 1120: fprintferr(" entering Decomp ");
! 1121: if (DEBUGLEVEL>5)
! 1122: {
! 1123: fprintferr("with parameters: p=%Z, expo=%ld\n",p,mf);
! 1124: fprintferr(" f=%Z",f);
! 1125: }
! 1126: fprintferr("\n");
! 1127: }
! 1128: pdr=respm(f,derivpol(f),gpuigs(p,mf));
! 1129:
! 1130: unmodp=gmodulsg(1,p);
! 1131: b1=lift_intern(gmul(chi,unmodp));
! 1132: a1=gun; b2=gun;
! 1133: b3=lift_intern(gmul(nu,unmodp));
! 1134: while (lgef(b3) > 3)
! 1135: {
! 1136: GEN p1;
! 1137: b1 = Fp_deuc(b1,b3, p);
! 1138: b2 = Fp_pol_red(gmul(b2,b3), p);
! 1139: b3 = Fp_pol_extgcd(b2,b1, p, &a1,&p1); /* p1 = junk */
! 1140: p1 = leading_term(b3);
! 1141: if (!gcmp1(p1))
! 1142: { /* Fp_pol_extgcd does not return normalized gcd */
! 1143: p1 = mpinvmod(p1,p);
! 1144: b3 = gmul(b3,p1);
! 1145: a1 = gmul(a1,p1);
! 1146: }
! 1147: }
! 1148: e=eleval(f,Fp_pol_red(gmul(a1,b2), p),theta);
! 1149: e=gdiv(polmodi(gmul(pdr,e), mulii(pdr,p)),pdr);
! 1150:
! 1151: pk=p; pmr=mulii(p,sqri(pdr)); ph=mulii(pdr,pmr);
! 1152: /* E(t)- e(t) belongs to p^k Op, which is contained in p^(k-df)*Zp[xi] */
! 1153: while (cmpii(pk,ph) < 0)
! 1154: {
! 1155: e = gmul(gsqr(e), gsubsg(3,gmul2n(e,1)));
! 1156: e = gres(e,f); pk = sqri(pk);
! 1157: e=gdiv(polmodi(gmul(pdr,e), mulii(pk,pdr)), pdr);
! 1158: }
! 1159: f1 = gcdpm(f,gmul(pdr,gsubsg(1,e)), ph);
! 1160: f1 = Fp_res(f1,f, pmr);
! 1161: f2 = Fp_res(Fp_deuc(f,f1, pmr), f, pmr);
! 1162: f1 = polmodi(f1,pmr);
! 1163: f2 = polmodi(f2,pmr);
! 1164:
! 1165: if (DEBUGLEVEL>=3)
! 1166: {
! 1167: fprintferr(" leaving Decomp");
! 1168: if (DEBUGLEVEL>5)
! 1169: fprintferr(" with parameters: f1 = %Z\nf2 = %Z\ne = %Z\n", f1,f2,e);
! 1170: fprintferr("\n");
! 1171: }
! 1172: ib1 = get_partial_order_as_pols(p,f1); n1=lg(ib1)-1;
! 1173: ib2 = get_partial_order_as_pols(p,f2); n2=lg(ib2)-1;
! 1174: ibas=cgetg(n1+n2+1,t_VEC);
! 1175:
! 1176: for (j=1; j<=n1; j++)
! 1177: ibas[j]=(long)polmodi(gmod(gmul(gmul(pdr,(GEN)ib1[j]),e),f), pdr);
! 1178: e=gsubsg(1,e);
! 1179: for ( ; j<=n1+n2; j++)
! 1180: ibas[j]=(long)polmodi(gmod(gmul(gmul(pdr,(GEN)ib2[j-n1]),e),f), pdr);
! 1181: return nbasis(ibas,pdr);
! 1182: }
! 1183:
! 1184: /* minimum extension valuation: res[0]/res[1] (both are longs) */
! 1185: long *
! 1186: vstar(GEN p,GEN h)
! 1187: {
! 1188: static long res[2];
! 1189: long m,first,j,k,v,w;
! 1190:
! 1191: m=lgef(h)-3; first=1; k=1; v=0;
! 1192: for (j=1; j<=m; j++)
! 1193: if (! gcmp0((GEN)h[m-j+2]))
! 1194: {
! 1195: w = ggval((GEN)h[m-j+2],p);
! 1196: if (first || w*k < v*j) { v=w; k=j; }
! 1197: first=0;
! 1198: }
! 1199: m = cgcd(v,k);
! 1200: res[0]=v/m; res[1]=k/m; return res;
! 1201: }
! 1202:
! 1203: /* Returns [theta,chi,nu] with theta non-primary */
! 1204: static GEN
! 1205: csrch(GEN p,GEN fa,GEN gamma)
! 1206: {
! 1207: GEN b,h,theta,w;
! 1208: long pp,t,v=varn(fa);
! 1209:
! 1210: pp = p[2]; if (lgef(p)>3 || pp<0) pp=0;
! 1211: for (t=1; ; t++)
! 1212: {
! 1213: h = pp? stopoly(t,pp,v): scalarpol(stoi(t),v);
! 1214: theta = gadd(gamma,gmod(h,fa));
! 1215: w=factcp(p,fa,theta); h=(GEN)w[3];
! 1216: if (h[2] > 1)
! 1217: {
! 1218: b=cgetg(5,t_VEC); b[1]=un; b[2]=(long)theta;
! 1219: b[3]=w[1]; b[4]=w[2]; return b;
! 1220: }
! 1221: }
! 1222: }
! 1223:
! 1224: /* Returns
! 1225: * [1,theta,chi,nu] if theta non-primary
! 1226: * [2,phi, * , * ] if D_phi > D_alpha or M_phi > M_alpha
! 1227: */
! 1228: GEN
! 1229: bsrch(GEN p,GEN fa,long ka,GEN eta,long Ma)
! 1230: {
! 1231: long n=lgef(fa)-3,Da=lgef(eta)-3;
! 1232: long c,r,j,MaVb,av=avma;
! 1233: GEN famod,pc,pcc,beta,gamma,delta,pik,w,h;
! 1234:
! 1235: pc=respm(fa,derivpol(fa),gpuigs(p,ka));
! 1236: c=ggval(pc,p); pcc=sqri(pc);
! 1237: famod=polmodi_keep(fa,pcc);
! 1238:
! 1239: r=1+(long)ceil(c/(double)(Da)+gtodouble(gdivsg(c*n-2,mulsi(Da,subis(p,1)))));
! 1240:
! 1241: beta=gdiv(lift_intern(gpuigs(gmodulcp(eta,famod),Ma)),p);
! 1242:
! 1243: for(;;)
! 1244: { /* Compute modulo pc. denom(pik, delta)=1. denom(beta, gamma) | pc */
! 1245: beta=gdiv(polmodi(gmul(pc,beta),pcc), pc);
! 1246: w=testd(p,fa,c,Da,eta,Ma,beta);
! 1247: h=(GEN)w[1]; if (h[2] < 3) return gerepileupto(av,w);
! 1248:
! 1249: w = vstar(p,(GEN)w[3]);
! 1250: MaVb = (w[0]*Ma) / w[1];
! 1251: pik=eltppm(famod,pc,eta,stoi(MaVb));
! 1252:
! 1253: gamma=gmod(gmul(beta,(GEN)(vecbezout(pik,famod))[1]),famod);
! 1254: gamma=gdiv(polmodi(gmul(pc,gamma),pcc),pc);
! 1255: w=testd(p,fa,c,Da,eta,Ma,gamma);
! 1256: h=(GEN)w[1]; if (h[2] < 3) return gerepileupto(av,w);
! 1257:
! 1258: delta=eltppm(famod,pc,gamma,gpuigs(p,r*Da));
! 1259: delta=gdiv(polmodi(gmul(pc,delta),pcc),pc);
! 1260: w=testd(p,fa,c,Da,eta,Ma,delta);
! 1261: h=(GEN)w[1]; if (h[2] < 3) return gerepileupto(av,w);
! 1262:
! 1263: for (j=lgef(delta)-1; j>1; j--)
! 1264: if (typ(delta[j]) != t_INT)
! 1265: {
! 1266: w = csrch(p,fa,gamma);
! 1267: return gerepileupto(av,gcopy(w));
! 1268: }
! 1269: beta=gsub(beta,gmod(gmul(pik,delta),famod));
! 1270: }
! 1271: }
! 1272:
! 1273: static GEN
! 1274: mycaract(GEN f, GEN beta)
! 1275: {
! 1276: GEN chi,p1;
! 1277: long v = varn(f);
! 1278:
! 1279: if (gcmp0(beta)) return zeropol(v);
! 1280: p1 = content(beta);
! 1281: if (gcmp1(p1)) p1 = NULL; else beta = gdiv(beta,p1);
! 1282: chi = caractducos(f,beta,v);
! 1283: if (p1)
! 1284: {
! 1285: chi=poleval(chi,gdiv(polx[v],p1));
! 1286: p1=gpuigs(p1,lgef(f)-3); chi=gmul(chi,p1);
! 1287: }
! 1288: return chi;
! 1289: }
! 1290:
! 1291: /* USED TO Return [theta_1,theta_2,L_theta,M_theta] with theta non-primary */
! 1292: /* Now return theta_2 */
! 1293: GEN
! 1294: setup(GEN p,GEN f,GEN theta,GEN nut, long *La, long *Ma)
! 1295: {
! 1296: GEN t1,t2,v,dt,pv;
! 1297: long Lt,Mt,r,s,av=avma,tetpil,m,n,k;
! 1298:
! 1299: n=lgef(nut)-1; pv=p;
! 1300: for (m=1; ; m++) /* compute mod p^(2^m) */
! 1301: {
! 1302: t1=gzero; pv = sqri(pv);
! 1303: for (k=n; k>=2; k--)
! 1304: {
! 1305: t1 = gres(gadd(gmul(t1,theta),(GEN)nut[k]), f);
! 1306: dt = denom(content(t1));
! 1307: if (gcmp1(dt))
! 1308: t1 = polmodi(t1,pv);
! 1309: else
! 1310: t1 = gdiv(polmodi(gmul(t1,dt),mulii(dt,pv)),dt);
! 1311: }
! 1312: v = vstar(p, mycaract(f,t1));
! 1313: if (v[0] < (v[1]<<m)) break;
! 1314: }
! 1315: Lt=v[0]; Mt=v[1]; cbezout(Lt,-Mt,&r,&s);
! 1316: if (r<=0) { long q = (-r) / Mt; q++; r += q*Mt; s += q*Lt; }
! 1317: t2 = lift_intern(gpuigs(gmodulcp(t1,f),r));
! 1318: p = gpuigs(p,s); tetpil=avma; *La=Lt; *Ma=Mt;
! 1319: return gerepile(av,tetpil,gdiv(t2,p));
! 1320: }
! 1321:
! 1322: #define RED 1
! 1323:
! 1324: #if 0
! 1325: static GEN
! 1326: nilord(GEN p,GEN fx,long mf,GEN gx)
! 1327: {
! 1328: long La,Ma,first=1,v=varn(fx);
! 1329: GEN h,res,alpha,chi,nu,eta,w,phi,pmf,Dchi,pdr,pmr;
! 1330:
! 1331: if (DEBUGLEVEL>=3)
! 1332: {
! 1333: fprintferr(" entering Nilord");
! 1334: if (DEBUGLEVEL>5)
! 1335: {
! 1336: fprintferr(" with parameters: p=%Z, expo=%ld\n",p,mf);
! 1337: fprintferr(" fx=%Z, gx=%Z",fx,gx);
! 1338: }
! 1339: fprintferr("\n");
! 1340: }
! 1341: pmf=gpuigs(p,mf+1); alpha=polx[v];
! 1342: nu=gx; chi=fx; Dchi=gpuigs(p,mf);
! 1343: #if RED
! 1344: pdr=respm(fx,derivpol(fx), Dchi);
! 1345: pmr=mulii(sqri(pdr),p); chi = dummycopy(chi);
! 1346: #endif
! 1347:
! 1348: for(;;)
! 1349: {
! 1350: #if RED
! 1351: chi = polmodi(chi, pmr);
! 1352: #endif
! 1353: if (first) first=0;
! 1354: else
! 1355: {
! 1356: res=dedek(chi,mf,p,nu);
! 1357: if (res) return dbasis(p,fx,mf,alpha,res);
! 1358: }
! 1359: if (vstar(p,chi)[0] > 0)
! 1360: {
! 1361: alpha = gadd(alpha,gun);
! 1362: chi = poleval(chi, gsub(polx[v],gun));
! 1363: #if RED
! 1364: chi = polmodi(chi, pmr);
! 1365: #endif
! 1366: nu = polmodi(poleval(nu, gsub(polx[v],gun)), p);
! 1367: }
! 1368: eta=setup(p,chi,polx[v],nu, &La,&Ma);
! 1369: if (La>1)
! 1370: alpha=gadd(alpha,eleval(fx,eta,alpha));
! 1371: else
! 1372: {
! 1373: w=bsrch(p,chi,ggval(Dchi,p),eta,Ma);
! 1374: phi=eleval(fx,(GEN)w[2],alpha);
! 1375: if (gcmp1((GEN)w[1]))
! 1376: return Decomp(p,fx,mf,phi,(GEN)w[3],(GEN)w[4]);
! 1377: alpha=gdiv(polmodi(gmul(pmf,phi), mulii(pmf,p)),pmf);
! 1378: }
! 1379:
! 1380: for (;;)
! 1381: {
! 1382: w=factcp(p,fx,alpha); chi=(GEN)w[1]; nu=(GEN)w[2]; h=(GEN)w[3];
! 1383: if (h[2] > 1) return Decomp(p,fx,mf,alpha,chi,nu);
! 1384: #if 0
! 1385: Dchi = respm(chi,derivpol(chi), pmf);
! 1386: #endif
! 1387: Dchi = modii(discsr(polmodi_keep(chi,pmf)), pmf);
! 1388: if (gcmp0(Dchi))
! 1389: {
! 1390: Dchi= discsr(chi);
! 1391: if (gcmp0(Dchi)) { alpha=gadd(alpha,gmul(p,polx[v])); continue; }
! 1392: #if RED
! 1393: pmr = gpowgs(p, 2 * ggval(Dchi,p) + 1);
! 1394: #endif
! 1395: }
! 1396: break;
! 1397: }
! 1398: }
! 1399: }
! 1400: #endif
! 1401:
! 1402: /* reduce the element elt modulo rd, taking first of the denominators */
! 1403: static GEN
! 1404: redelt(GEN elt, GEN rd, GEN pd)
! 1405: {
! 1406: GEN den, relt;
! 1407:
! 1408: den = ggcd(denom(content(elt)), pd);
! 1409: relt = polmodi(gmul(den, elt), gmul(den, rd));
! 1410: return gdiv(relt, den);
! 1411: }
! 1412:
! 1413: /* return the prime element in Zp[phi] */
! 1414: static GEN
! 1415: getprime(GEN p, GEN chi, GEN phi, GEN chip, GEN nup, long *Lp, long *Ep)
! 1416: {
! 1417: long v = varn(chi), L, E, r, s;
! 1418: GEN chin, pip, pp, vn;
! 1419:
! 1420: if (gegal(nup, polx[v]))
! 1421: chin = chip;
! 1422: else
! 1423: chin = mycaract(chip, nup);
! 1424:
! 1425: vn = vstar(p, chin);
! 1426: L = vn[0];
! 1427: E = vn[1];
! 1428:
! 1429: cbezout(L, -E, &r, &s);
! 1430:
! 1431: if (r <= 0)
! 1432: {
! 1433: long q = (-r) / E;
! 1434: q++;
! 1435: r += q*E;
! 1436: s += q*L;
! 1437: }
! 1438:
! 1439: pip = eleval(chi, nup, phi);
! 1440: pip = lift_intern(gpuigs(gmodulcp(pip, chi), r));
! 1441: pp = gpuigs(p, s);
! 1442:
! 1443: *Lp = L;
! 1444: *Ep = E;
! 1445: return gdiv(pip, pp);
! 1446: }
! 1447:
! 1448: static GEN
! 1449: update_alpha(GEN p, GEN fx, GEN alph, GEN chi, GEN pmr, GEN pmf, long mf)
! 1450: {
! 1451: long l, v = varn(fx);
! 1452: GEN nalph, nchi, w, nnu, pdr, npmr, rep;
! 1453:
! 1454: nalph = alph;
! 1455: if (!chi)
! 1456: nchi = mycaract(fx, alph);
! 1457: else
! 1458: nchi = chi;
! 1459:
! 1460: pdr = modii(respm(nchi, derivpol(nchi), pmr), pmr);
! 1461: for (;;)
! 1462: {
! 1463: if (signe(pdr)) break;
! 1464: pdr = modii(respm(nchi, derivpol(nchi), pmf), pmf);
! 1465: if (signe(pdr)) break;
! 1466: if (DEBUGLEVEL >= 6)
! 1467: fprintferr(" non separable polynomial in update_alpha!\n");
! 1468: /* at this point, we assume that chi is not square-free */
! 1469: nalph = gadd(nalph, gmul(p, polx[v]));
! 1470: w = factcp(p, fx, nalph);
! 1471: nchi = (GEN)w[1];
! 1472: nnu = (GEN)w[2];
! 1473: l = itos((GEN)w[3]);
! 1474: if (l > 1) return Decomp(p, fx, mf, nalph, nchi, nnu);
! 1475: pdr = modii(respm(nchi, derivpol(nchi), pmr), pmr);
! 1476: }
! 1477:
! 1478: if (is_pm1(pdr))
! 1479: npmr = gun;
! 1480: else
! 1481: {
! 1482: npmr = mulii(sqri(pdr), p);
! 1483: nchi = polmodi(nchi, npmr);
! 1484: nalph = redelt(nalph, npmr, pmf);
! 1485: }
! 1486:
! 1487: rep = cgetg(5, t_VEC);
! 1488: rep[1] = (long)nalph;
! 1489: rep[2] = (long)nchi;
! 1490: rep[3] = (long)npmr;
! 1491: rep[4] = lmulii(p, pdr);
! 1492:
! 1493: return rep;
! 1494: }
! 1495:
! 1496: static GEN
! 1497: nilord2(GEN p, GEN fx, long mf, GEN gx)
! 1498: {
! 1499: long Fa, La, Ea, oE, Fg, eq, er, v = varn(fx), i, nv, Le, Ee, N, l, vn;
! 1500: GEN p1, alph, chi, nu, w, phi, pmf, pdr, pmr, kapp, pie, chib;
! 1501: GEN gamm, chig, nug, delt, beta, eta, chie, nue, pia, vb, opa;
! 1502:
! 1503: if (DEBUGLEVEL >= 3)
! 1504: {
! 1505: fprintferr(" entering Nilord2");
! 1506: if (DEBUGLEVEL >= 5)
! 1507: {
! 1508: fprintferr(" with parameters: p = %Z, expo = %ld\n", p, mf);
! 1509: fprintferr(" fx = %Z, gx = %Z", fx, gx);
! 1510: }
! 1511: fprintferr("\n");
! 1512: }
! 1513:
! 1514: /* this is quite arbitrary; what is important is that >= mf + 1 */
! 1515: pmf = gpuigs(p, mf + 3);
! 1516: pdr = respm(fx, derivpol(fx), pmf);
! 1517: pmr = mulii(sqri(pdr), p);
! 1518: pdr = mulii(p, pdr);
! 1519: chi = polmodi_keep(fx, pmr);
! 1520:
! 1521: alph = polx[v];
! 1522: nu = gx;
! 1523: N = degree(fx);
! 1524: oE = 0;
! 1525: opa = NULL;
! 1526:
! 1527: for(;;)
! 1528: {
! 1529: /* kappa need to be recomputed */
! 1530: kapp = NULL;
! 1531: Fa = degree(nu);
! 1532: /* the prime element in Zp[alpha] */
! 1533: pia = getprime(p, chi, polx[v], chi, nu, &La, &Ea);
! 1534: pia = redelt(pia, pmr, pmf);
! 1535:
! 1536: if (Ea < oE)
! 1537: {
! 1538: alph = gadd(alph, opa);
! 1539: w = update_alpha(p, fx, alph, NULL, pmr, pmf, mf);
! 1540: alph = (GEN)w[1];
! 1541: chi = (GEN)w[2];
! 1542: pmr = (GEN)w[3];
! 1543: pdr = (GEN)w[4];
! 1544: kapp = NULL;
! 1545: pia = getprime(p, chi, polx[v], chi, nu, &La, &Ea);
! 1546: pia = redelt(pia, pmr, pmf);
! 1547: }
! 1548:
! 1549: oE = Ea; opa = pia;
! 1550:
! 1551: if (DEBUGLEVEL >= 5)
! 1552: fprintferr(" Fa = %ld and Ea = %ld \n", Fa, Ea);
! 1553:
! 1554: /* we change alpha such that nu = pia */
! 1555: if (La > 1)
! 1556: {
! 1557: alph = gadd(alph, eleval(fx, pia, alph));
! 1558:
! 1559: w = update_alpha(p, fx, alph, NULL, pmr, pmf, mf);
! 1560: alph = (GEN)w[1];
! 1561: chi = (GEN)w[2];
! 1562: pmr = (GEN)w[3];
! 1563: pdr = (GEN)w[4];
! 1564: }
! 1565:
! 1566: /* if Ea*Fa == N then O = Zp[alpha] */
! 1567: if (Ea*Fa == N)
! 1568: {
! 1569: alph = redelt(alph, sqri(p), pmf);
! 1570: return dbasis(p, fx, mf, alph, p);
! 1571: }
! 1572:
! 1573: /* during the process beta tends to a factor of chi */
! 1574: beta = lift_intern(gpowgs(gmodulcp(nu, chi), Ea));
! 1575:
! 1576: for (;;)
! 1577: {
! 1578: if (DEBUGLEVEL >= 5)
! 1579: fprintferr(" beta = %Z\n", beta);
! 1580:
! 1581: p1 = gnorm(gmodulcp(beta, chi));
! 1582: if (signe(p1))
! 1583: {
! 1584: chib = NULL;
! 1585: vn = ggval(p1, p);
! 1586: eq = (long)(vn / N);
! 1587: er = (long)(vn*Ea/N - eq*Ea);
! 1588: }
! 1589: else
! 1590: {
! 1591: chib = mycaract(chi, beta);
! 1592: vb = vstar(p, chib);
! 1593: eq = (long)(vb[0] / vb[1]);
! 1594: er = (long)(vb[0]*Ea / vb[1] - eq*Ea);
! 1595: }
! 1596:
! 1597: /* the following code can be used to check if beta approximates
! 1598: a factor of chi well enough to derive a factorization of chi.
! 1599: However, in general, the process will always end before this
! 1600: happens. */
! 1601: #if 0
! 1602: {
! 1603: GEN quo, rem;
! 1604:
! 1605: quo = poldivres(chi, beta, &rem);
! 1606: p1 = content(lift(rem));
! 1607: fprintferr(" val(rem) = %ld\n", ggval(p1, p));
! 1608: p1 = respm(beta, quo, pmr);
! 1609: fprintferr(" val(id) = %ld\n", ggval(p1, p));
! 1610: }
! 1611: #endif
! 1612:
! 1613: /* eq and er are such that gamma = beta.p^-eq.nu^-er is a unit */
! 1614: if (eq) gamm = gdiv(beta, gpowgs(p, eq));
! 1615: else gamm = beta;
! 1616:
! 1617: if (er)
! 1618: {
! 1619: /* kappa = nu^-1 in Zp[alpha] */
! 1620: if (!kapp)
! 1621: {
! 1622: kapp = ginvmod(nu, chi);
! 1623: kapp = redelt(kapp, pmr, pmr);
! 1624: kapp = gmodulcp(kapp, chi);
! 1625: }
! 1626: gamm = lift(gmul(gamm, gpowgs(kapp, er)));
! 1627: gamm = redelt(gamm, p, pmr);
! 1628: }
! 1629:
! 1630: if (DEBUGLEVEL >= 6)
! 1631: fprintferr(" gamma = %Z\n", gamm);
! 1632:
! 1633: if (er || !chib)
! 1634: {
! 1635: p1 = mulii(pdr, ggcd(denom(content(gamm)), pdr));
! 1636: chig = mycaract(redelt(chi, mulii(pdr, p1), pdr), gamm);
! 1637: }
! 1638: else
! 1639: {
! 1640: chig = poleval(chib, gmul(polx[v], gpowgs(p, eq)));
! 1641: chig = gdiv(chig, gpowgs(p, N*eq));
! 1642: }
! 1643:
! 1644: if (!gcmp1(denom(content(chig))))
! 1645: {
! 1646: /* the valuation of beta was wrong... This also means
! 1647: that chi_gamma has more than one factor modulo p */
! 1648: vb = vstar(p, chig);
! 1649: eq = (long)(-vb[0] / vb[1]);
! 1650: er = (long)(-vb[0]*Ea / vb[1] - eq*Ea);
! 1651: if (eq) gamm = gmul(gamm, gpowgs(p, eq));
! 1652: if (er)
! 1653: {
! 1654: gamm = gmul(gamm, gpowgs(nu, er));
! 1655: gamm = gmod(gamm, chi);
! 1656: gamm = redelt(gamm, p, pmr);
! 1657: }
! 1658: p1 = mulii(pdr, ggcd(denom(content(gamm)), pdr));
! 1659: chig = mycaract(redelt(chi, mulii(pdr, p1), pdr), gamm);
! 1660: }
! 1661:
! 1662: chig = polmodi(chig, pmr);
! 1663: nug = (GEN)factmod(chig, p)[1];
! 1664: l = lg(nug) - 1;
! 1665: nug = lift((GEN)nug[l]);
! 1666:
! 1667: if (l > 1)
! 1668: {
! 1669: /* there are at least 2 factors mod. p => chi can be split */
! 1670: phi = eleval(fx, gamm, alph);
! 1671: phi = redelt(phi, p, pmf);
! 1672: return Decomp(p, fx, mf, phi, chig, nug);
! 1673: }
! 1674:
! 1675: Fg = degree(nug);
! 1676: if (Fa%Fg)
! 1677: {
! 1678: if (DEBUGLEVEL >= 5)
! 1679: fprintferr(" Increasing Fa\n");
! 1680: /* we compute a new element such F = lcm(Fa, Fg) */
! 1681: w = testb2(p, chi, Fa, gamm, Fg);
! 1682: if (gcmp1((GEN)w[1]))
! 1683: {
! 1684: /* there are at least 2 factors mod. p => chi can be split */
! 1685: phi = eleval(fx, (GEN)w[2], alph);
! 1686: phi = redelt(phi, p, pmf);
! 1687: return Decomp(p, fx, mf, phi, (GEN)w[3], (GEN)w[4]);
! 1688: }
! 1689: break;
! 1690: }
! 1691:
! 1692: /* we look for a root delta of nug in Fp[alpha] such that
! 1693: vp(gamma - delta) > 0. This root can then be used to
! 1694: improved the approximation given by beta */
! 1695: nv = fetch_var();
! 1696: w = factmod9(nug, p, gsubst(nu, varn(nu), polx[nv]));
! 1697: w = lift(lift((GEN)w[1]));
! 1698:
! 1699: for (i = 1;; i++)
! 1700: if (degree((GEN)w[i]) == 1)
! 1701: {
! 1702: delt = gneg_i(gsubst(gcoeff(w, 2, i), nv, polx[v]));
! 1703: eta = gsub(gamm, delt);
! 1704: if (typ(delt) == t_INT)
! 1705: {
! 1706: chie = poleval(chig, gadd(polx[v], delt));
! 1707: chie = polmodi(chie, pmr);
! 1708: nue = (GEN)factmod(chie, p)[1];
! 1709: l = lg(nue) - 1;
! 1710: nue = lift((GEN)nue[l]);
! 1711: }
! 1712: else
! 1713: {
! 1714: p1 = factcp(p, chi, eta);
! 1715: chie = (GEN)p1[1];
! 1716: chie = polmodi(chie, pmr);
! 1717: nue = (GEN)p1[2];
! 1718: l = itos((GEN)p1[3]);
! 1719: }
! 1720: if (l > 1)
! 1721: {
! 1722: /* there are at least 2 factors mod. p => chi can be split */
! 1723: delete_var();
! 1724: phi = eleval(fx, eta, alph);
! 1725: phi = redelt(phi, p, pmf);
! 1726: return Decomp(p, fx, mf, phi, chie, nue);
! 1727: }
! 1728:
! 1729: /* if vp(eta) = vp(gamma - delta) > 0 */
! 1730: if (gegal(nue, polx[v])) break;
! 1731: }
! 1732: delete_var();
! 1733:
! 1734: pie = getprime(p, chi, eta, chie, nue, &Le, &Ee);
! 1735: if (Ea%Ee)
! 1736: {
! 1737: if (DEBUGLEVEL >= 5)
! 1738: fprintferr(" Increasing Ea\n");
! 1739: pie = redelt(pie, p, pmf);
! 1740: /* we compute a new element such E = lcm(Ea, Ee) */
! 1741: w = testc2(p, chi, pmr, nu, Ea, pie, Ee);
! 1742: if (gcmp1((GEN)w[1]))
! 1743: {
! 1744: /* there are at least 2 factors mod. p => chi can be split */
! 1745: phi = eleval(fx, (GEN)w[2], alph);
! 1746: phi = redelt(phi, p, pmf);
! 1747: return Decomp(p, fx, mf, phi, (GEN)w[3], (GEN)w[4]);
! 1748: }
! 1749: break;
! 1750: }
! 1751:
! 1752: if (eq) delt = gmul(delt, gpowgs(p, eq));
! 1753: if (er) delt = gmul(delt, gpowgs(nu, er));
! 1754: beta = gsub(beta, delt);
! 1755: }
! 1756:
! 1757: /* we replace alpha by a new alpha with a larger F or E */
! 1758: alph = eleval(fx, (GEN)w[2], alph);
! 1759: chi = (GEN)w[3];
! 1760: nu = (GEN)w[4];
! 1761:
! 1762: w = update_alpha(p, fx, alph, chi, pmr, pmf, mf);
! 1763: alph = (GEN)w[1];
! 1764: chi = (GEN)w[2];
! 1765: pmr = (GEN)w[3];
! 1766: pdr = (GEN)w[4];
! 1767:
! 1768: /* that can happen if p does not divide the field discriminant! */
! 1769: if (is_pm1(pmr))
! 1770: return dbasis(p, fx, mf, alph, chi);
! 1771: }
! 1772: }
! 1773:
! 1774: /* Returns [1,phi,chi,nu] if phi non-primary
! 1775: * [2,phi,chi,nu] if D_phi = lcm (D_alpha, D_theta)
! 1776: */
! 1777: static GEN
! 1778: testb(GEN p,GEN fa,long Da,GEN theta,long Dt)
! 1779: {
! 1780: long pp,Dat,t,v=varn(fa);
! 1781: GEN b,w,phi,h;
! 1782:
! 1783: Dat=clcm(Da,Dt)+3; b=cgetg(5,t_VEC);
! 1784: pp = p[2]; if (lgef(p)>3 || pp<0) pp=0;
! 1785: for (t=1; ; t++)
! 1786: {
! 1787: h = pp? stopoly(t,pp,v): scalarpol(stoi(t),v);
! 1788: phi = gadd(theta,gmod(h,fa));
! 1789: w=factcp(p,fa,phi); h=(GEN)w[3];
! 1790: if (h[2] > 1) { b[1]=un; break; }
! 1791: if (lgef(w[2]) == Dat) { b[1]=deux; break; }
! 1792: }
! 1793: b[2]=(long)phi; b[3]=w[1]; b[4]=w[2]; return b;
! 1794: }
! 1795:
! 1796: /* Returns [1,phi,chi,nu] if phi non-primary
! 1797: * [2,phi,chi,nu] with F_phi = lcm (F_alpha, F_theta)
! 1798: * and E_phi = E_alpha
! 1799: */
! 1800: static GEN
! 1801: testb2(GEN p, GEN fa, long Fa, GEN theta, long Ft)
! 1802: {
! 1803: long pp, Dat, t, v = varn(fa);
! 1804: GEN b, w, phi, h;
! 1805:
! 1806: Dat = clcm(Fa, Ft) + 3;
! 1807: b = cgetg(5, t_VEC);
! 1808: pp = p[2];
! 1809: if (lgef(p) > 3 || pp < 0) pp = 0;
! 1810:
! 1811: for (t = 1;; t++)
! 1812: {
! 1813: h = pp? stopoly(t, pp, v): scalarpol(stoi(t), v);
! 1814: phi = gadd(theta, gmod(h, fa));
! 1815: w = factcp(p, fa, phi);
! 1816: h = (GEN)w[3];
! 1817: if (h[2] > 1) { b[1] = un; break; }
! 1818: if (lgef(w[2]) == Dat) { b[1] = deux; break; }
! 1819: }
! 1820:
! 1821: b[2] = (long)phi;
! 1822: b[3] = w[1];
! 1823: b[4] = w[2];
! 1824: return b;
! 1825: }
! 1826:
! 1827: /* Returns [1,phi,chi,nu] if phi non-primary
! 1828: * [2,phi,chi,nu] if M_phi = lcm (M_alpha, M_theta)
! 1829: */
! 1830: static GEN
! 1831: testc(GEN p, GEN fa, long c, GEN alph2, long Ma, GEN thet2, long Mt)
! 1832: {
! 1833: GEN b,pc,ppc,c1,c2,c3,psi,phi,w,h;
! 1834: long r,s,t,v=varn(fa);
! 1835:
! 1836: b=cgetg(5,t_VEC); pc=gpuigs(p,c); ppc=mulii(pc,p);
! 1837:
! 1838: cbezout(Ma,Mt,&r,&s); t=0;
! 1839: while (r<0) { r=r+Mt; t++; }
! 1840: while (s<0) { s=s+Ma; t++; }
! 1841:
! 1842: c1=lift_intern(gpuigs(gmodulcp(alph2,fa),s));
! 1843: c2=lift_intern(gpuigs(gmodulcp(thet2,fa),r));
! 1844: c3=gdiv(gmod(gmul(c1,c2),fa),gpuigs(p,t));
! 1845: psi=gdiv(polmodi(gmul(pc,c3),ppc),pc);
! 1846: phi=gadd(polx[v],psi);
! 1847:
! 1848: w=factcp(p,fa,phi); h=(GEN)w[3];
! 1849: b[1] = (h[2] > 1)? un: deux;
! 1850: b[2]=(long)phi; b[3]=w[1]; b[4]=w[2]; return b;
! 1851: }
! 1852:
! 1853: /* Returns [1, phi, chi, nu] if phi non-primary
! 1854: * [2, phi, chi, nu] if E_phi = lcm (E_alpha, E_theta)
! 1855: */
! 1856: static GEN
! 1857: testc2(GEN p, GEN fa, GEN pmr, GEN alph2, long Ea, GEN thet2, long Et)
! 1858: {
! 1859: GEN b, c1, c2, c3, psi, phi, w, h;
! 1860: long r, s, t, v = varn(fa);
! 1861:
! 1862: b=cgetg(5, t_VEC);
! 1863:
! 1864: cbezout(Ea, Et, &r, &s); t = 0;
! 1865: while (r < 0) { r = r + Et; t++; }
! 1866: while (s < 0) { s = s + Ea; t++; }
! 1867:
! 1868: c1 = lift_intern(gpuigs(gmodulcp(alph2, fa), s));
! 1869: c2 = lift_intern(gpuigs(gmodulcp(thet2, fa), r));
! 1870: c3 = gdiv(gmod(gmul(c1, c2), fa), gpuigs(p, t));
! 1871:
! 1872: psi = redelt(c3, pmr, pmr);
! 1873: phi = gadd(polx[v], psi);
! 1874:
! 1875: w = factcp(p,fa,phi); h = (GEN)w[3];
! 1876: b[1] = (h[2] > 1)? un: deux;
! 1877: b[2] = (long)phi;
! 1878: b[3] = w[1];
! 1879: b[4] = w[2];
! 1880: return b;
! 1881: }
! 1882:
! 1883: /* Returns
! 1884: * [1,phi,chi,nu] if theta non-primary
! 1885: * [2,phi,chi,nu] if D_phi > D_aplha or M_phi > M_alpha
! 1886: * [3,phi,chi,nu] otherwise
! 1887: */
! 1888: static GEN
! 1889: testd(GEN p,GEN fa,long c,long Da,GEN alph2,long Ma,GEN theta)
! 1890: {
! 1891: long Lt,Mt,Dt,av=avma,tetpil;
! 1892: GEN chi,nu,thet2,b,w,h;
! 1893:
! 1894: b=cgetg(5,t_VEC); w=factcp(p,fa,theta);
! 1895: chi=(GEN)w[1]; nu=(GEN)w[2]; h=(GEN)w[3];
! 1896: if (h[2] > 1)
! 1897: {
! 1898: b[1]=un; b[2]=(long)theta; b[3]=(long)chi; b[4]=(long)nu;
! 1899: }
! 1900: else
! 1901: {
! 1902: Dt=lgef(nu)-3;
! 1903: if (Da < clcm(Da,Dt)) b = testb(p,fa,Da,theta,Dt);
! 1904: else
! 1905: {
! 1906: thet2=setup(p,fa,theta,nu, &Lt,&Mt);
! 1907: if (Ma < clcm(Ma,Mt)) b = testc(p,fa,c,alph2,Ma,thet2,Mt);
! 1908: else
! 1909: {
! 1910: b[1]=lstoi(3); b[2]=(long)theta; b[3]=(long)chi; b[4]=(long)nu;
! 1911: }
! 1912: }
! 1913: }
! 1914: tetpil=avma; return gerepile(av,tetpil,gcopy(b));
! 1915: }
! 1916:
! 1917: /* Factor characteristic polynomial of beta mod p */
! 1918: GEN
! 1919: factcp(GEN p,GEN f,GEN beta)
! 1920: {
! 1921: long av,tetpil,l;
! 1922: GEN chi,nu, b = cgetg(4,t_VEC);
! 1923:
! 1924: chi = mycaract(f,beta);
! 1925: av=avma; nu=(GEN)factmod(chi,p)[1]; l=lg(nu)-1;
! 1926: nu=lift_intern((GEN)nu[1]); tetpil=avma;
! 1927: b[1]=(long)chi;
! 1928: b[2]=lpile(av,tetpil,gcopy(nu));
! 1929: b[3]=lstoi(l); return b;
! 1930: }
! 1931:
! 1932: /* evaluate h(a) mod f */
! 1933: GEN
! 1934: eleval(GEN f,GEN h,GEN a)
! 1935: {
! 1936: long n,av,tetpil;
! 1937: GEN y;
! 1938:
! 1939: if (typ(h) != t_POL) return gcopy(h);
! 1940: av = tetpil = avma;
! 1941: n=lgef(h)-1; y=(GEN)h[n];
! 1942: for (n--; n>=2; n--)
! 1943: {
! 1944: y = gadd(gmul(y,a),(GEN)h[n]);
! 1945: tetpil=avma; y = gmod(y,f);
! 1946: }
! 1947: return gerepile(av,tetpil,y);
! 1948: }
! 1949:
! 1950: /* Compute theta^k mod (f,pd) */
! 1951: static GEN
! 1952: eltppm(GEN f,GEN pd,GEN theta,GEN k)
! 1953: {
! 1954: GEN phi,psi,D, q = k;
! 1955: long av = avma, av1, lim = stack_lim(av,2);
! 1956:
! 1957: if (!signe(k)) return polun[varn(f)];
! 1958: D = mulii(pd, sqri(pd)); av1 = avma;
! 1959: phi=pd; psi=gmul(pd,theta);
! 1960:
! 1961: for(;;)
! 1962: {
! 1963: if (mod2(q)) phi = gdivexact(Fp_res(gmul(phi,psi), f, D), pd);
! 1964: q=shifti(q,-1); if (!signe(q)) break;
! 1965: psi = gdivexact(Fp_res(gsqr(psi), f, D), pd);
! 1966: if (low_stack(lim,stack_lim(av,2)))
! 1967: {
! 1968: GEN *gptr[3]; gptr[0]=ψ gptr[1]=φ gptr[2]=&q;
! 1969: if(DEBUGMEM>1) err(warnmem,"eltppm");
! 1970: gerepilemany(av1,gptr,3);
! 1971: }
! 1972: }
! 1973: return gerepileupto(av,gdiv(phi,pd));
! 1974: }
! 1975:
! 1976: /* Sylvester's matrix, mod p^m (assumes f1 monic) */
! 1977: static GEN
! 1978: sylpm(GEN f1,GEN f2,GEN pm)
! 1979: {
! 1980: long n,deg,k,j,v=varn(f1);
! 1981: GEN a,h;
! 1982:
! 1983: n=lgef(f1)-3; a=cgetg(n+1,t_MAT);
! 1984: h = Fp_res(f2,f1,pm);
! 1985: for (j=1; j<=n; j++)
! 1986: {
! 1987: a[j] = lgetg(n+1,t_COL);
! 1988: deg=lgef(h)-3;
! 1989: for (k=1; k<=deg+1; k++) coeff(a,k,j)=h[k+1];
! 1990: for ( ; k<=n; k++) coeff(a,k,j)=zero;
! 1991:
! 1992: if (j<n) h = Fp_res(gmul(polx[v],h),f1,pm);
! 1993: }
! 1994: return hnfmodid(a,pm);
! 1995: }
! 1996:
! 1997: /* polynomial gcd mod p^m (assumes f1 monic) */
! 1998: GEN
! 1999: gcdpm(GEN f1,GEN f2,GEN pm)
! 2000: {
! 2001: long n,c,v=varn(f1),av=avma,tetpil;
! 2002: GEN a,col;
! 2003:
! 2004: n=lgef(f1)-3; a=sylpm(f1,f2,pm);
! 2005: for (c=1; c<=n; c++)
! 2006: if (signe(resii(gcoeff(a,c,c),pm))) break;
! 2007: if (c > n) { avma=av; return zeropol(v); }
! 2008:
! 2009: col = gdiv((GEN)a[c], gcoeff(a,c,c)); tetpil=avma;
! 2010: return gerepile(av,tetpil, gtopolyrev(col,v));
! 2011: }
! 2012:
! 2013: /* reduced resultant mod p^m (assumes x monic) */
! 2014: GEN
! 2015: respm(GEN x,GEN y,GEN pm)
! 2016: {
! 2017: long av=avma,tetpil;
! 2018:
! 2019: x = sylpm(x,y,pm); tetpil=avma;
! 2020: return gerepile(av,tetpil, icopy(gcoeff(x,1,1)));
! 2021: }
! 2022:
! 2023: /* Normalized integral basis */
! 2024: static GEN
! 2025: nbasis(GEN ibas,GEN pd)
! 2026: {
! 2027: long n,j,k,m;
! 2028: GEN a;
! 2029:
! 2030: n=lg(ibas)-1; m=lgef(ibas[1])-2;
! 2031: a=cgetg(n+1,t_MAT);
! 2032: for (k=1; k<=n; k++)
! 2033: {
! 2034: m=lgef(ibas[k])-2; a[k]=lgetg(n+1,t_COL);
! 2035: for (j=1; j<=m; j++) coeff(a,j,k)=coeff(ibas,j+1,k);
! 2036: for ( ; j<=n; j++) coeff(a,j,k)=zero;
! 2037: }
! 2038: return gdiv(hnfmodid(a,pd), pd);
! 2039: }
! 2040:
! 2041: static long
! 2042: clcm(long a,long b)
! 2043: {
! 2044: long d,r,v1;
! 2045:
! 2046: d=a; r=b;
! 2047: for(;;)
! 2048: {
! 2049: if (!r) return (a*b)/d;
! 2050: v1=r; r=d%r; d=labs(v1);
! 2051: }
! 2052: }
! 2053:
! 2054: /*******************************************************************/
! 2055: /* */
! 2056: /* BUCHMANN-LENSTRA ALGORITHM */
! 2057: /* */
! 2058: /*******************************************************************/
! 2059: static GEN lens(GEN nf,GEN p,GEN a);
! 2060: GEN element_powid_mod_p(GEN nf, long I, GEN n, GEN p);
! 2061:
! 2062: /* return a Z basis of Z_K's p-radical, modfrob = x--> x^p-x */
! 2063: static GEN
! 2064: pradical(GEN nf, GEN p, GEN *modfrob)
! 2065: {
! 2066: long i,N=lgef(nf[1])-3;
! 2067: GEN p1,m,frob,rad;
! 2068:
! 2069: frob = cgetg(N+1,t_MAT);
! 2070: for (i=1; i<=N; i++)
! 2071: frob[i] = (long) element_powid_mod_p(nf,i,p, p);
! 2072:
! 2073: /* p1 = smallest power of p st p^k >= N */
! 2074: p1=p; while (cmpis(p1,N)<0) p1=mulii(p1,p);
! 2075: if (p1==p) m = frob;
! 2076: else
! 2077: {
! 2078: m=cgetg(N+1,t_MAT); p1 = divii(p1,p);
! 2079: for (i=1; i<=N; i++)
! 2080: m[i]=(long)element_pow_mod_p(nf,(GEN)frob[i],p1, p);
! 2081: }
! 2082: rad = ker_mod_p(m, p);
! 2083: for (i=1; i<=N; i++)
! 2084: coeff(frob,i,i) = lsubis(gcoeff(frob,i,i), 1);
! 2085: *modfrob = frob; return rad;
! 2086: }
! 2087:
! 2088: static GEN
! 2089: project(GEN algebre, GEN x, long k, long kbar)
! 2090: {
! 2091: x = inverseimage(algebre,x);
! 2092: x += k; x[0] = evaltyp(t_COL) | evallg(kbar+1);
! 2093: return x;
! 2094: }
! 2095:
! 2096: /* Calcule le polynome minimal de alpha dans algebre (coeffs dans Z) */
! 2097: static GEN
! 2098: pol_min(GEN alpha,GEN nf,GEN algebre,long kbar,GEN p)
! 2099: {
! 2100: long av=avma,tetpil,i,N,k;
! 2101: GEN p1,puiss;
! 2102:
! 2103: N = lg(nf[1])-3; puiss=cgetg(N+2,t_MAT);
! 2104: k = N-kbar; p1=alpha;
! 2105: puiss[1] = (long)gscalcol_i(gun,kbar);
! 2106: for (i=2; i<=N+1; i++)
! 2107: {
! 2108: if (i>2) p1 = element_mul(nf,p1,alpha);
! 2109: puiss[i] = (long) project(algebre,p1,k,kbar);
! 2110: }
! 2111: puiss = lift_intern(puiss);
! 2112: p1 = (GEN)ker_mod_p(puiss, p)[1]; tetpil=avma;
! 2113: return gerepile(av,tetpil,gtopolyrev(p1,0));
! 2114: }
! 2115:
! 2116: /* Evalue le polynome pol en alpha,element de nf */
! 2117: static GEN
! 2118: eval_pol(GEN nf,GEN pol,GEN alpha,GEN algebre,GEN algebre1)
! 2119: {
! 2120: long av=avma,tetpil,i,kbar,k, lx = lgef(pol)-1, N = lgef(nf[1])-3;
! 2121: GEN res;
! 2122:
! 2123: kbar = lg(algebre1)-1; k = N-kbar;
! 2124: res = gscalcol_i((GEN)pol[lx], N);
! 2125: for (i=2; i<lx; i++)
! 2126: {
! 2127: res = element_mul(nf,alpha,res);
! 2128: res[1] = ladd((GEN)res[1],(GEN)pol[i]);
! 2129: }
! 2130: res = project(algebre,res,k,kbar); tetpil=avma;
! 2131: return gerepile(av,tetpil,gmul(algebre1,res));
! 2132: }
! 2133:
! 2134: static GEN
! 2135: kerlens2(GEN x, GEN p)
! 2136: {
! 2137: long i,j,k,t,nbc,nbl,av,av1;
! 2138: GEN a,c,l,d,y,q;
! 2139:
! 2140: av=avma; a=gmul(x,gmodulsg(1,p));
! 2141: nbl=nbc=lg(x)-1;
! 2142: c=new_chunk(nbl+1); for (i=1; i<=nbl; i++) c[i]=0;
! 2143: l=new_chunk(nbc+1);
! 2144: d=new_chunk(nbc+1);
! 2145: k = t = 1;
! 2146: while (t<=nbl && k<=nbc)
! 2147: {
! 2148: for (j=1; j<k; j++)
! 2149: for (i=1; i<=nbl; i++)
! 2150: if (i!=l[j])
! 2151: coeff(a,i,k) = lsub(gmul((GEN)d[j],gcoeff(a,i,k)),
! 2152: gmul(gcoeff(a,l[j],k),gcoeff(a,i,j)));
! 2153: t=1; while (t<=nbl && (c[t] || gcmp0(gcoeff(a,t,k)))) t++;
! 2154: if (t<=nbl) { d[k]=coeff(a,t,k); c[t]=k; l[k]=t; k++; }
! 2155: }
! 2156: if (k>nbc) err(bugparier,"kerlens2");
! 2157: y=cgetg(nbc+1,t_COL);
! 2158: y[1]=(k>1)?coeff(a,l[1],k):un;
! 2159: for (q=gun,j=2; j<k; j++)
! 2160: {
! 2161: q=gmul(q,(GEN)d[j-1]);
! 2162: y[j]=lmul(gcoeff(a,l[j],k),q);
! 2163: }
! 2164: if (k>1) y[k]=lneg(gmul(q,(GEN)d[k-1]));
! 2165: for (j=k+1; j<=nbc; j++) y[j]=zero;
! 2166: av1=avma; return gerepile(av,av1,lift(y));
! 2167: }
! 2168:
! 2169: static GEN
! 2170: kerlens(GEN x, GEN pgen)
! 2171: {
! 2172: long av = avma, i,j,k,t,nbc,nbl,p,q,*c,*l,*d,**a;
! 2173: GEN y;
! 2174:
! 2175: if (cmpis(pgen, MAXHALFULONG>>1) > 0)
! 2176: return kerlens2(x,pgen);
! 2177: /* ici p <= (MAXHALFULONG>>1) ==> long du C */
! 2178: p=itos(pgen); nbl=nbc=lg(x)-1;
! 2179: a=(long**)new_chunk(nbc+1);
! 2180: for (j=1; j<=nbc; j++)
! 2181: {
! 2182: c=a[j]=new_chunk(nbl+1);
! 2183: for (i=1; i<=nbl; i++) c[i]=smodis(gcoeff(x,i,j),p);
! 2184: }
! 2185: c=new_chunk(nbl+1); for (i=1; i<=nbl; i++) c[i]=0;
! 2186: l=new_chunk(nbc+1);
! 2187: d=new_chunk(nbc+1);
! 2188: k = t = 1;
! 2189: while (t<=nbl && k<=nbc)
! 2190: {
! 2191: for (j=1; j<k; j++)
! 2192: for (i=1; i<=nbl; i++)
! 2193: if (i!=l[j])
! 2194: a[k][i] = (d[j]*a[k][i] - a[j][i]*a[k][l[j]]) % p;
! 2195: t=1; while (t<=nbl && (c[t] || !a[k][t])) t++;
! 2196: if (t<=nbl) { d[k]=a[k][t]; c[t]=k; l[k++]=t; }
! 2197: }
! 2198: if (k>nbc) err(bugparier,"kerlens");
! 2199: avma=av; y=cgetg(nbc+1,t_COL);
! 2200: t=(k>1) ? a[k][l[1]]:1;
! 2201: y[1]=(t>0)? lstoi(t):lstoi(t+p);
! 2202: for (q=1,j=2; j<k; j++)
! 2203: {
! 2204: q = (q*d[j-1]) % p;
! 2205: t = (a[k][l[j]]*q) % p;
! 2206: y[j] = (t>0) ? lstoi(t) : lstoi(t+p);
! 2207: }
! 2208: if (k>1)
! 2209: {
! 2210: t = (q*d[k-1]) % p;
! 2211: y[k] = (t>0) ? lstoi(p-t) : lstoi(-t);
! 2212: }
! 2213: for (j=k+1; j<=nbc; j++) y[j]=zero;
! 2214: return y;
! 2215: }
! 2216:
! 2217: /* Calcule la constante de lenstra de l'ideal p.Z_K+a.Z_K ou a est un
! 2218: vecteur sur la base d'entiers */
! 2219: static GEN
! 2220: lens(GEN nf, GEN p, GEN a)
! 2221: {
! 2222: long av=avma,tetpil,N=lgef(nf[1])-3,j;
! 2223: GEN mat=cgetg(N+1,t_MAT);
! 2224: for (j=1; j<=N; j++) mat[j]=(long)element_mulid(nf,a,j);
! 2225: tetpil=avma; return gerepile(av,tetpil,kerlens(mat,p));
! 2226: }
! 2227:
! 2228: GEN det_mod_P_n(GEN a, GEN N, GEN P);
! 2229: GEN sylvestermatrix_i(GEN x, GEN y);
! 2230:
! 2231: /* check if p^va doesnt divide norm x (or norm(x+p)) */
! 2232: #if 0
! 2233: /* compute norm mod p^whatneeded using Sylvester's matrix */
! 2234: /* looks slower than the new subresultant. Have to re-check this */
! 2235: static GEN
! 2236: prime_check_elt(GEN a, GEN pol, GEN p, GEN pf)
! 2237: {
! 2238: GEN M,mod,x, c = denom(content(a));
! 2239: long v = pvaluation(c, p, &x); /* x is junk */
! 2240:
! 2241: mod = mulii(pf, gpowgs(p, (lgef(pol)-3)*v + 1));
! 2242:
! 2243: x = Fp_pol_red(gmul(a,c), mod);
! 2244: M = sylvestermatrix_i(pol,x);
! 2245: if (det_mod_P_n(M,mod,p) == gzero)
! 2246: {
! 2247: x[2] = ladd((GEN)x[2], mulii(p,c));
! 2248: M = sylvestermatrix_i(pol,x);
! 2249: if (det_mod_P_n(M,mod,p) == gzero) return NULL;
! 2250: a[2] = ladd((GEN)a[2], p);
! 2251: }
! 2252: return a;
! 2253: }
! 2254: #else
! 2255: /* use subres to compute norm */
! 2256: static GEN
! 2257: prime_check_elt(GEN a, GEN pol, GEN p, GEN pf)
! 2258: {
! 2259: GEN norme=subres(pol,a);
! 2260: if (resii(divii(norme,pf),p) != gzero) return a;
! 2261: a=gadd(a,p); norme=subres(pol,a);
! 2262: if (resii(divii(norme,pf),p) != gzero) return a;
! 2263: return NULL;
! 2264: }
! 2265: #endif
! 2266:
! 2267: #if 0
! 2268: GEN
! 2269: prime_two_elt_loop(GEN beta, GEN pol, GEN p, GEN pf)
! 2270: {
! 2271: long av, m = lg(beta)-1;
! 2272: int i,j,K, *x = (int*)new_chunk(m+1);
! 2273: GEN a;
! 2274:
! 2275: K = 1; av = avma;
! 2276: for(;;)
! 2277: { /* x runs through strictly increasing sequences of length K,
! 2278: * 1 <= x[i] <= m */
! 2279: nextK:
! 2280: if (DEBUGLEVEL) fprintferr("K = %d\n", K);
! 2281: for (i=1; i<=K; i++) x[i] = i;
! 2282: for(;;)
! 2283: {
! 2284: if (DEBUGLEVEL > 1)
! 2285: {
! 2286: for (i=1; i<=K; i++) fprintferr("%d ",x[i]);
! 2287: fprintferr("\n"); flusherr();
! 2288: }
! 2289: a = (GEN)beta[x[1]];
! 2290: for (i=2; i<=K; i++) a = gadd(a, (GEN)beta[x[i]]);
! 2291: if ((a = prime_check_elt(a,pol,p,pf))) return a;
! 2292: avma = av;
! 2293:
! 2294: /* start: i = K+1; */
! 2295: do
! 2296: {
! 2297: if (--i == 0)
! 2298: {
! 2299: if (++K > m) return NULL; /* fail */
! 2300: goto nextK;
! 2301: }
! 2302: x[i]++;
! 2303: } while (x[i] > m - K + i);
! 2304: for (j=i; j<K; j++) x[j+1] = x[j]+1;
! 2305: }
! 2306: }
! 2307: }
! 2308: #endif
! 2309:
! 2310: GEN
! 2311: random_prime_two_elt_loop(GEN beta, GEN pol, GEN p, GEN pf)
! 2312: {
! 2313: long av = avma, z,i, m = lg(beta)-1;
! 2314: long keep = getrand();
! 2315: int c = 0;
! 2316: GEN a;
! 2317:
! 2318: for(i=1; i<=m; i++)
! 2319: if ((a = prime_check_elt((GEN)beta[i],pol,p,pf))) return a;
! 2320: (void)setrand(1);
! 2321: if (DEBUGLEVEL) fprintferr("prime_two_elt_loop, hard case: ");
! 2322: for(;;avma=av)
! 2323: {
! 2324: if (DEBUGLEVEL) fprintferr("%d ", ++c);
! 2325: a = gzero;
! 2326: for (i=1; i<=m; i++)
! 2327: {
! 2328: z = mymyrand() >> (BITS_IN_RANDOM-5); /* in [0,15] */
! 2329: if (z >= 9) z -= 7;
! 2330: a = gadd(a,gmulsg(z,(GEN)beta[i]));
! 2331: }
! 2332: if ((a = prime_check_elt(a,pol,p,pf)))
! 2333: {
! 2334: if (DEBUGLEVEL) fprintferr("\n");
! 2335: (void)setrand(keep); return a;
! 2336: }
! 2337: }
! 2338: }
! 2339:
! 2340: /* Input: an ideal mod p (!= Z_K)
! 2341: * Output: a 2-elt representation [p, x] */
! 2342: static GEN
! 2343: prime_two_elt(GEN nf, GEN p, GEN ideal)
! 2344: {
! 2345: GEN beta,a,pf, pol = (GEN)nf[1];
! 2346: long av,tetpil,f, N=lgef(pol)-3, m=lg(ideal)-1;
! 2347:
! 2348: if (!m) return gscalcol_i(p,N);
! 2349:
! 2350: /* we want v_p(Norm(beta)) = p^f, f = N-m */
! 2351: av = avma; f = N-m; pf = gpuigs(p,f);
! 2352: ideal = centerlift(ideal);
! 2353: ideal = concatsp(gscalcol(p,N), ideal);
! 2354: ideal = ideal_better_basis(nf, ideal, p);
! 2355: beta = gmul((GEN)nf[7], ideal);
! 2356:
! 2357: #if 0
! 2358: a = prime_two_elt_loop(beta,pol,p,pf);
! 2359: if (!a) err(bugparier, "prime_two_elt (failed)");
! 2360: #else
! 2361: a = random_prime_two_elt_loop(beta,pol,p,pf);
! 2362: #endif
! 2363:
! 2364: a = centermod(algtobasis_intern(nf,a), p);
! 2365: if (resii(divii(subres(gmul((GEN)nf[7],a),pol),pf),p) == gzero)
! 2366: a[1] = laddii((GEN)a[1],p);
! 2367: tetpil = avma; return gerepile(av,tetpil,gcopy(a));
! 2368: }
! 2369:
! 2370: static GEN
! 2371: apply_kummer(GEN nf,GEN pol,GEN e,GEN p,long N,GEN *beta)
! 2372: {
! 2373: GEN T,p1, res = cgetg(6,t_VEC);
! 2374: long f = lgef(pol)-3;
! 2375:
! 2376: res[1]=(long)p;
! 2377: res[3]=(long)e;
! 2378: res[4]=lstoi(f);
! 2379: if (f == N) /* inert */
! 2380: {
! 2381: res[2]=(long)gscalcol_i(p,N);
! 2382: res[5]=(long)gscalcol_i(gun,N);
! 2383: }
! 2384: else
! 2385: {
! 2386: T = (GEN) nf[1];
! 2387: if (ggval(subres(pol,T),p) > f)
! 2388: pol[2] = laddii((GEN)pol[2],p);
! 2389: res[2] = (long) algtobasis_intern(nf,pol);
! 2390:
! 2391: p1 = Fp_deuc(T,pol,p);
! 2392: res[5] = (long) centermod(algtobasis_intern(nf,p1), p);
! 2393:
! 2394: if (beta)
! 2395: *beta = *beta? Fp_deuc(*beta,pol,p): p1;
! 2396: }
! 2397: return res;
! 2398: }
! 2399:
! 2400: /* prime ideal decomposition of p sorted by increasing residual degree */
! 2401: GEN
! 2402: primedec(GEN nf, GEN p)
! 2403: {
! 2404: long av=avma,tetpil,i,j,k,kbar,np,c,indice,N,lp;
! 2405: GEN ex,f,list,ip,elth,h;
! 2406: GEN modfrob,algebre,algebre1,b,mat1,T,nfp;
! 2407: GEN alpha,beta,p1,p2,unmodp,zmodp,idmodp;
! 2408:
! 2409: if (DEBUGLEVEL>=3) timer2();
! 2410: nf=checknf(nf); T=(GEN)nf[1]; N=lgef(T)-3;
! 2411: f=factmod(T,p); ex=(GEN)f[2];
! 2412: f=centerlift((GEN)f[1]); np=lg(f);
! 2413: if (DEBUGLEVEL>=6) msgtimer("factmod");
! 2414:
! 2415: if (signe(modii((GEN)nf[4],p))) /* p doesn't divide index */
! 2416: {
! 2417: list=cgetg(np,t_VEC);
! 2418: for (i=1; i<np; i++)
! 2419: list[i]=(long)apply_kummer(nf,(GEN)f[i],(GEN)ex[i],p,N, NULL);
! 2420: if (DEBUGLEVEL>=6) msgtimer("simple primedec");
! 2421: p1=stoi(4); tetpil=avma;
! 2422: return gerepile(av,tetpil,vecsort(list,p1));
! 2423: }
! 2424:
! 2425: p1 = (GEN)f[1];
! 2426: for (i=2; i<np; i++)
! 2427: p1 = Fp_pol_red(gmul(p1, (GEN)f[i]), p);
! 2428: p1 = Fp_pol_red(gdiv(gadd(gmul(p1, Fp_deuc(T,p1,p)), gneg(T)), p), p);
! 2429: list = cgetg(N+1,t_VEC);
! 2430: indice=1; beta=NULL;
! 2431: for (i=1; i<np; i++) /* e = 1 or f[i] does not divide p1 (mod p) */
! 2432: if (is_pm1(ex[i]) || signe(Fp_res(p1,(GEN)f[i],p)))
! 2433: list[indice++] = (long)apply_kummer(nf,(GEN)f[i],(GEN)ex[i],p,N,&beta);
! 2434: if (DEBUGLEVEL>=3) msgtimer("unramified factors");
! 2435:
! 2436: /* modfrob = modified Frobenius: x -> x^p - x mod p */
! 2437: ip = pradical(nf,p,&modfrob);
! 2438: if (DEBUGLEVEL>=3) msgtimer("pradical");
! 2439:
! 2440: if (beta)
! 2441: {
! 2442: beta = algtobasis_intern(nf,beta);
! 2443: lp=lg(ip)-1; p1=cgetg(2*lp+N+1,t_MAT);
! 2444: for (i=1; i<=N; i++) p1[i]=(long)element_mulid(nf,beta,i);
! 2445: for ( ; i<=N+lp; i++)
! 2446: {
! 2447: p2 = (GEN) ip[i-N];
! 2448: p1[i+lp] = (long) p2;
! 2449: p1[i] = ldiv(element_mul(nf,lift(p2),beta),p);
! 2450: }
! 2451: ip = image_mod_p(p1, p);
! 2452: if (lg(ip)>N) err(bugparier,"primedec (bad pradical)");
! 2453: }
! 2454: unmodp=gmodulsg(1,p); zmodp=gmodulsg(0,p);
! 2455: idmodp = idmat_intern(N,unmodp,zmodp);
! 2456: ip = gmul(ip, unmodp);
! 2457: nfp = gscalcol_i(p,N);
! 2458:
! 2459: h=cgetg(N+1,t_VEC); h[1]=(long)ip;
! 2460: for (c=1; c; c--)
! 2461: {
! 2462: elth=(GEN)h[c]; k=lg(elth)-1; kbar=N-k;
! 2463: p1 = concatsp(elth,(GEN)idmodp[1]);
! 2464: algebre = suppl_intern(p1,idmodp);
! 2465: algebre1 = cgetg(kbar+1,t_MAT);
! 2466: for (i=1; i<=kbar; i++) algebre1[i]=algebre[i+k];
! 2467: b = gmul(modfrob,algebre1);
! 2468: for (i=1;i<=kbar;i++)
! 2469: b[i] = (long) project(algebre,(GEN) b[i],k,kbar);
! 2470: mat1 = ker_mod_p(lift_intern(b), p);
! 2471: if (lg(mat1)>2)
! 2472: {
! 2473: GEN mat2 = cgetg(k+N+1,t_MAT);
! 2474: for (i=1; i<=k; i++) mat2[i]=elth[i];
! 2475: alpha=gmul(algebre1,(GEN)mat1[2]);
! 2476: p1 = pol_min(alpha,nf,algebre,kbar,p);
! 2477: p1 = (GEN)factmod(p1,p)[1];
! 2478: for (i=1; i<lg(p1); i++)
! 2479: {
! 2480: beta = eval_pol(nf,(GEN)p1[i],alpha,algebre,algebre1);
! 2481: beta = lift_intern(beta);
! 2482: for (j=1; j<=N; j++)
! 2483: mat2[k+j] = (long)Fp_vec(element_mulid(nf,beta,j), p);
! 2484: h[c] = (long) image(mat2); c++;
! 2485: }
! 2486: }
! 2487: else
! 2488: {
! 2489: long av1; p1 = cgetg(6,t_VEC);
! 2490: list[indice++] = (long)p1;
! 2491: p1[1]=(long)p; p1[4]=lstoi(kbar);
! 2492: p1[2]=(long)prime_two_elt(nf,p,elth);
! 2493: p1[5]=(long)lens(nf,p,(GEN)p1[2]);
! 2494: av1=avma;
! 2495: i = int_elt_val(nf,nfp,p,(GEN)p1[5],N);
! 2496: avma=av1;
! 2497: p1[3]=lstoi(i);
! 2498: }
! 2499: if (DEBUGLEVEL>=3) msgtimer("h[%ld]",c);
! 2500: }
! 2501: setlg(list, indice); tetpil=avma;
! 2502: return gerepile(av,tetpil,gen_sort(list,0,cmp_prime_over_p));
! 2503: }
! 2504:
! 2505: /* REDUCTION Modulo a prime ideal */
! 2506:
! 2507: /* x integral, reduce mod prh in HNF */
! 2508: GEN
! 2509: nfreducemodpr_i(GEN x, GEN prh)
! 2510: {
! 2511: GEN p = gcoeff(prh,1,1);
! 2512: long i,j;
! 2513:
! 2514: x = dummycopy(x);
! 2515: for (i=lg(x)-1; i>=2; i--)
! 2516: {
! 2517: GEN t = (GEN)prh[i], p1 = resii((GEN)x[i], p);
! 2518: x[i] = (long)p1;
! 2519: if (signe(p1) && is_pm1(t[i]))
! 2520: {
! 2521: for (j=1; j<i; j++)
! 2522: x[j] = lsubii((GEN)x[j], mulii(p1, (GEN)t[j]));
! 2523: x[i] = zero;
! 2524: }
! 2525: }
! 2526: x[1] = lresii((GEN)x[1], p); return x;
! 2527: }
! 2528:
! 2529: /* for internal use */
! 2530: GEN
! 2531: nfreducemodpr(GEN nf, GEN x, GEN prhall)
! 2532: {
! 2533: long i,v;
! 2534: GEN p,prh,den;
! 2535:
! 2536: for (i=lg(x)-1; i>0; i--)
! 2537: if (typ(x[i]) == t_INTMOD) { x=lift_intern(x); break; }
! 2538: prh=(GEN)prhall[1]; p=gcoeff(prh,1,1);
! 2539: den=denom(x);
! 2540: if (!gcmp1(den))
! 2541: {
! 2542: v=ggval(den,p);
! 2543: if (v) x=element_mul(nf,x,element_pow(nf,(GEN)prhall[2],stoi(v)));
! 2544: x = gmod(x,p);
! 2545: }
! 2546: return Fp_vec(nfreducemodpr_i(x, prh), p);
! 2547: }
! 2548:
! 2549: /* public function */
! 2550: GEN
! 2551: nfreducemodpr2(GEN nf, GEN x, GEN prhall)
! 2552: {
! 2553: long av = avma; checkprhall(prhall);
! 2554: if (typ(x) != t_COL) x = algtobasis(nf,x);
! 2555: return gerepileupto(av, nfreducemodpr(nf,x,prhall));
! 2556: }
! 2557:
! 2558: /* relative ROUND 2
! 2559: *
! 2560: * input: nf = base field K
! 2561: * x monic polynomial, coefficients in Z_K, degree n defining a relative
! 2562: * extension L=K(theta).
! 2563: * One MUST have varn(x) < varn(nf[1]).
! 2564: * output: a pseudo-basis [A,I] of Z_L, where A is in M_n(K) in HNF form and
! 2565: * I a vector of n ideals.
! 2566: */
! 2567:
! 2568: /* given MODULES x and y by their pseudo-bases in HNF, gives a
! 2569: * pseudo-basis of the module generated by x and y. For internal use.
! 2570: */
! 2571: static GEN
! 2572: rnfjoinmodules(GEN nf, GEN x, GEN y)
! 2573: {
! 2574: long i,lx,ly;
! 2575: GEN p1,p2,z,Hx,Hy,Ix,Iy;
! 2576:
! 2577: if (x == NULL) return y;
! 2578: Hx=(GEN)x[1]; lx=lg(Hx); Ix=(GEN)x[2];
! 2579: Hy=(GEN)y[1]; ly=lg(Hy); Iy=(GEN)y[2];
! 2580: i = lx+ly-1;
! 2581: z = (GEN)gpmalloc(sizeof(long*)*(3+2*i));
! 2582: *z = evaltyp(t_VEC)|evallg(3);
! 2583: p1 = z+3; z[1]=(long)p1; *p1 = evaltyp(t_MAT)|evallg(i);
! 2584: p2 = p1+i; z[2]=(long)p2; *p2 = evaltyp(t_VEC)|evallg(i);
! 2585:
! 2586: for (i=1; i<lx; i++) { p1[i]=Hx[i]; p2[i]=Ix[i]; }
! 2587: for ( ; i<lx+ly-1; i++) { p1[i]=Hy[i-lx+1]; p2[i]=Iy[i-lx+1]; }
! 2588: x = nfhermite(nf,z); free(z); return x;
! 2589: }
! 2590:
! 2591: /* a usage interne, pas de gestion de pile : x et y sont des vecteurs dont
! 2592: * les coefficients sont les composantes sur nf[7]; avec reduction mod pr sauf
! 2593: * si prhall=NULL
! 2594: */
! 2595: static GEN
! 2596: rnfelement_mulidmod(GEN nf, GEN multab, GEN unnf, GEN x, long h, GEN prhall)
! 2597: {
! 2598: long j,k,N;
! 2599: GEN p1,c,v,s,znf;
! 2600:
! 2601: if (h==1) return gcopy(x);
! 2602: N = lg(x)-1; multab += (h-1)*N;
! 2603: x = lift(x); v = cgetg(N+1,t_COL);
! 2604: znf = gscalcol_i(gzero,lg(unnf)-1);
! 2605: for (k=1; k<=N; k++)
! 2606: {
! 2607: s = gzero;
! 2608: for (j=1; j<=N; j++)
! 2609: if (!gcmp0(p1 = (GEN)x[j]) && !gcmp0(c = gcoeff(multab,k,j)))
! 2610: {
! 2611: if (!gegal(c,unnf)) p1 = element_mul(nf,p1,c);
! 2612: s = gadd(s,p1);
! 2613: }
! 2614: if (s == gzero) s = znf;
! 2615: else
! 2616: if (prhall) s = nfreducemodpr(nf,s,prhall);
! 2617: v[k] = (long)s;
! 2618: }
! 2619: return v;
! 2620: }
! 2621:
! 2622: /* a usage interne, pas de gestion de pile : x est un vecteur dont
! 2623: * les coefficients sont les composantes sur nf[7]
! 2624: */
! 2625: static GEN
! 2626: rnfelement_sqrmod(GEN nf, GEN multab, GEN unnf, GEN x, GEN prhall)
! 2627: {
! 2628: long i,j,k,n;
! 2629: GEN p1,c,z,s;
! 2630:
! 2631: n=lg(x)-1; x=lift(x); z=cgetg(n+1,t_COL);
! 2632: for (k=1; k<=n; k++)
! 2633: {
! 2634: if (k == 1)
! 2635: s = element_sqr(nf,(GEN)x[1]);
! 2636: else
! 2637: s = gmul2n(element_mul(nf,(GEN)x[1],(GEN)x[k]), 1);
! 2638: for (i=2; i<=n; i++)
! 2639: {
! 2640: c = gcoeff(multab,k,(i-1)*n+i);
! 2641: if (!gcmp0(c))
! 2642: {
! 2643: p1=element_sqr(nf,(GEN)x[i]);
! 2644: if (!gegal(c,unnf)) p1 = element_mul(nf,p1,c);
! 2645: s = gadd(s,p1);
! 2646: }
! 2647: for (j=i+1; j<=n; j++)
! 2648: {
! 2649: c = gcoeff(multab,k,(i-1)*n+j);
! 2650: if (!gcmp0(c))
! 2651: {
! 2652: p1=gmul2n(element_mul(nf,(GEN)x[i],(GEN)x[j]),1);
! 2653: if (!gegal(c,unnf)) p1 = element_mul(nf,p1,c);
! 2654: s = gadd(s,p1);
! 2655: }
! 2656: }
! 2657: }
! 2658: if (prhall) s = nfreducemodpr(nf,s,prhall);
! 2659: z[k]=(long)s;
! 2660: }
! 2661: return z;
! 2662: }
! 2663:
! 2664: /* Compute x^n mod pr in the extension, assume n >= 0 */
! 2665: static GEN
! 2666: rnfelementid_powmod(GEN nf, GEN multab, GEN matId, long h, GEN n, GEN prhall)
! 2667: {
! 2668: long i,m,av=avma,tetpil;
! 2669: GEN y, unrnf=(GEN)matId[1], unnf=(GEN)unrnf[1];
! 2670: ulong j;
! 2671:
! 2672: if (!signe(n)) return unrnf;
! 2673: y=(GEN)matId[h]; i = lgefint(n)-1; m=n[i]; j=HIGHBIT;
! 2674: while ((m&j)==0) j>>=1;
! 2675: for (j>>=1; j; j>>=1)
! 2676: {
! 2677: y = rnfelement_sqrmod(nf,multab,unnf,y,prhall);
! 2678: if (m&j) y = rnfelement_mulidmod(nf,multab,unnf,y,h,prhall);
! 2679: }
! 2680: for (i--; i>=2; i--)
! 2681: for (m=n[i],j=HIGHBIT; j; j>>=1)
! 2682: {
! 2683: y = rnfelement_sqrmod(nf,multab,unnf,y,prhall);
! 2684: if (m&j) y = rnfelement_mulidmod(nf,multab,unnf,y,h,prhall);
! 2685: }
! 2686: tetpil=avma; return gerepile(av,tetpil,gcopy(y));
! 2687: }
! 2688:
! 2689: GEN
! 2690: mymod(GEN x, GEN p)
! 2691: {
! 2692: long i,lx, tx = typ(x);
! 2693: GEN y,p1;
! 2694:
! 2695: if (tx == t_INT) return resii(x,p);
! 2696: if (tx == t_FRAC)
! 2697: {
! 2698: p1 = resii((GEN)x[2], p);
! 2699: if (p1 != gzero) { cgiv(p1); return gmod(x,p); }
! 2700: return x;
! 2701: }
! 2702: if (!is_matvec_t(tx))
! 2703: err(bugparier, "mymod (missing type)");
! 2704: lx = lg(x); y = cgetg(lx,tx);
! 2705: for (i=1; i<lx; i++) y[i] = (long)mymod((GEN)x[i],p);
! 2706: return y;
! 2707: }
! 2708:
! 2709: static GEN
! 2710: rnfordmax(GEN nf, GEN pol, GEN pr, GEN unnf, GEN id, GEN matId)
! 2711: {
! 2712: long av=avma,tetpil,av1,lim,i,j,k,n,v1,v2,vpol,m,cmpt,sep;
! 2713: GEN p,q,q1,prhall,A,Aa,Aaa,A1,I,R,p1,p2,p3,multab,multabmod,Aainv;
! 2714: GEN pip,baseIp,baseOp,alpha,matprod,alphainv,matC,matG,vecpro,matH;
! 2715: GEN neworder,H,Hid,alphalistinv,alphalist,prhinv;
! 2716:
! 2717: if (DEBUGLEVEL>1) fprintferr(" treating %Z\n",pr);
! 2718: prhall=nfmodprinit(nf,pr);
! 2719: q=cgetg(3,t_VEC); q[1]=(long)pr;q[2]=(long)prhall;
! 2720: p1=rnfdedekind(nf,pol,q);
! 2721: if (gcmp1((GEN)p1[1]))
! 2722: {tetpil=avma; return gerepile(av,tetpil,gcopy((GEN)p1[2]));}
! 2723:
! 2724: sep=itos((GEN)p1[3]);
! 2725: A=gmael(p1,2,1);
! 2726: I=gmael(p1,2,2);
! 2727:
! 2728: n=lgef(pol)-3; vpol=varn(pol);
! 2729: p=(GEN)pr[1]; q=powgi(p,(GEN)pr[4]); pip=(GEN)pr[2];
! 2730: q1=q; while (cmpis(q1,n)<0) q1=mulii(q1,q);
! 2731:
! 2732: multab=cgetg(n*n+1,t_MAT);
! 2733: for (j=1; j<=n*n; j++) multab[j]=lgetg(n+1,t_COL);
! 2734: prhinv = idealinv(nf,(GEN)prhall[1]);
! 2735: alphalistinv=cgetg(n+1,t_VEC);
! 2736: alphalist=cgetg(n+1,t_VEC);
! 2737: A1=cgetg(n+1,t_MAT);
! 2738: av1=avma; lim=stack_lim(av1,1);
! 2739: for(cmpt=1; ; cmpt++)
! 2740: {
! 2741: if (DEBUGLEVEL>1)
! 2742: {
! 2743: fprintferr(" %ld%s pass\n",cmpt,eng_ord(cmpt));
! 2744: flusherr();
! 2745: }
! 2746: for (i=1; i<=n; i++)
! 2747: {
! 2748: if (gegal((GEN)I[i],id)) alphalist[i]=alphalistinv[i]=(long)unnf;
! 2749: else
! 2750: {
! 2751: p1=ideal_two_elt(nf,(GEN)I[i]);
! 2752: v1=gcmp0((GEN)p1[1])? EXP220
! 2753: : element_val(nf,(GEN)p1[1],pr);
! 2754: v2=element_val(nf,(GEN)p1[2],pr);
! 2755: if (v1>v2) p2=(GEN)p1[2]; else p2=(GEN)p1[1];
! 2756: alphalist[i]=(long)p2;
! 2757: alphalistinv[i]=(long)element_inv(nf,p2);
! 2758: }
! 2759: }
! 2760: for (j=1; j<=n; j++)
! 2761: {
! 2762: p1=cgetg(n+1,t_COL); A1[j]=(long)p1;
! 2763: for (i=1; i<=n; i++)
! 2764: p1[i] = (long)element_mul(nf,gcoeff(A,i,j),(GEN)alphalist[j]);
! 2765: }
! 2766: Aa=basistoalg(nf,A1); Aainv=lift_intern(ginv(Aa));
! 2767: Aaa=mat_to_vecpol(Aa,vpol);
! 2768: for (i=1; i<=n; i++) for (j=i; j<=n; j++)
! 2769: {
! 2770: long tp;
! 2771: p1 = lift_intern(gres(gmul((GEN)Aaa[i],(GEN)Aaa[j]), pol));
! 2772: tp = typ(p1);
! 2773: if (is_scalar_t(tp) || (tp==t_POL && varn(p1)>vpol))
! 2774: p2 = gmul(p1, (GEN)Aainv[1]);
! 2775: else
! 2776: p2 = gmul(Aainv, pol_to_vec(p1, n));
! 2777:
! 2778: p3 = algtobasis(nf,p2);
! 2779: for (k=1; k<=n; k++)
! 2780: {
! 2781: coeff(multab,k,(i-1)*n+j) = p3[k];
! 2782: coeff(multab,k,(j-1)*n+i) = p3[k];
! 2783: }
! 2784: }
! 2785: R=cgetg(n+1,t_MAT); multabmod = mymod(multab,p);
! 2786: R[1] = matId[1];
! 2787: for (j=2; j<=n; j++)
! 2788: R[j] = (long) rnfelementid_powmod(nf,multabmod,matId, j,q1,prhall);
! 2789: baseIp = nfkermodpr(nf,R,prhall);
! 2790: baseOp = lift_intern(nfsuppl(nf,baseIp,n,prhall));
! 2791: alpha=cgetg(n+1,t_MAT);
! 2792: for (j=1; j<lg(baseIp); j++) alpha[j]=baseOp[j];
! 2793: for ( ; j<=n; j++)
! 2794: {
! 2795: p1=cgetg(n+1,t_COL); alpha[j]=(long)p1;
! 2796: for (i=1; i<=n; i++)
! 2797: p1[i]=(long)element_mul(nf,pip,gcoeff(baseOp,i,j));
! 2798: }
! 2799: matprod=cgetg(n+1,t_MAT);
! 2800: for (j=1; j<=n; j++)
! 2801: {
! 2802: p1=cgetg(n+1,t_COL); matprod[j]=(long)p1;
! 2803: for (i=1; i<=n; i++)
! 2804: {
! 2805: p2 = rnfelement_mulidmod(nf,multab,unnf, (GEN)alpha[i],j, NULL);
! 2806: for (k=1; k<=n; k++)
! 2807: p2[k] = lmul((GEN)nf[7], (GEN)p2[k]);
! 2808: p1[i] = (long)p2;
! 2809: }
! 2810: }
! 2811: alphainv = lift_intern(ginv(basistoalg(nf,alpha)));
! 2812: matC = cgetg(n+1,t_MAT);
! 2813: for (j=1; j<=n; j++)
! 2814: {
! 2815: p1=cgetg(n*n+1,t_COL); matC[j]=(long)p1;
! 2816: for (i=1; i<=n; i++)
! 2817: {
! 2818: p2 = gmul(alphainv, gcoeff(matprod,i,j));
! 2819: for (k=1; k<=n; k++)
! 2820: p1[(i-1)*n+k]=(long)nfreducemodpr(nf,algtobasis(nf,(GEN)p2[k]),prhall);
! 2821: }
! 2822: }
! 2823: matG=nfkermodpr(nf,matC,prhall); m=lg(matG)-1;
! 2824: vecpro=cgetg(3,t_VEC);
! 2825: p1=cgetg(n+m+1,t_MAT); vecpro[1]=(long)p1;
! 2826: p2=cgetg(n+m+1,t_VEC); vecpro[2]=(long)p2;
! 2827: for (j=1; j<=m; j++)
! 2828: {
! 2829: p1[j] = llift((GEN)matG[j]);
! 2830: p2[j] = (long)prhinv;
! 2831: }
! 2832: p1 += m;
! 2833: p2 += m;
! 2834: for (j=1; j<=n; j++)
! 2835: {
! 2836: p1[j] = matId[j];
! 2837: p2[j] = (long)idealmul(nf,(GEN)I[j],(GEN)alphalistinv[j]);
! 2838: }
! 2839: matH=nfhermite(nf,vecpro);
! 2840: p1=algtobasis(nf,gmul(Aa,basistoalg(nf,(GEN)matH[1])));
! 2841: p2=(GEN)matH[2];
! 2842:
! 2843: tetpil=avma; neworder=cgetg(3,t_VEC);
! 2844: H=cgetg(n+1,t_MAT); Hid=cgetg(n+1,t_VEC);
! 2845: for (j=1; j<=n; j++)
! 2846: {
! 2847: p3=cgetg(n+1,t_COL); H[j]=(long)p3;
! 2848: for (i=1; i<=n; i++)
! 2849: p3[i]=(long)element_mul(nf,gcoeff(p1,i,j),(GEN)alphalistinv[j]);
! 2850: Hid[j]=(long)idealmul(nf,(GEN)p2[j],(GEN)alphalist[j]);
! 2851: }
! 2852: if (DEBUGLEVEL>3)
! 2853: { fprintferr(" new order:\n"); outerr(H); outerr(Hid); }
! 2854: if (sep == 2 || gegal(I,Hid))
! 2855: {
! 2856: neworder[1]=(long)H; neworder[2]=(long)Hid;
! 2857: return gerepile(av,tetpil,neworder);
! 2858: }
! 2859:
! 2860: A=H; I=Hid;
! 2861: if (low_stack(lim, stack_lim(av1,1)))
! 2862: {
! 2863: GEN *gptr[2]; gptr[0]=&A; gptr[1]=&I;
! 2864: if(DEBUGMEM>1) err(warnmem,"rnfordmax");
! 2865: gerepilemany(av1,gptr,2);
! 2866: }
! 2867: }
! 2868: }
! 2869:
! 2870: static void
! 2871: check_pol(GEN x, long v)
! 2872: {
! 2873: long i,tx, lx = lg(x);
! 2874: if (varn(x) != v)
! 2875: err(talker,"incorrect variable in rnf function");
! 2876: for (i=2; i<lx; i++)
! 2877: {
! 2878: tx = typ(x[i]);
! 2879: if (!is_scalar_t(tx) || tx == t_POLMOD)
! 2880: err(talker,"incorrect polcoeff in rnf function");
! 2881: }
! 2882: }
! 2883:
! 2884: GEN
! 2885: fix_relative_pol(GEN nf, GEN x)
! 2886: {
! 2887: GEN xnf = (typ(nf) == t_POL)? nf: (GEN)nf[1];
! 2888: long i, vnf = varn(xnf), lx = lg(x);
! 2889: if (typ(x) != t_POL || varn(x) >= vnf)
! 2890: err(talker,"incorrect polynomial in rnf function");
! 2891: x = dummycopy(x);
! 2892: for (i=2; i<lx; i++)
! 2893: if (typ(x[i]) == t_POL)
! 2894: {
! 2895: check_pol((GEN)x[i], vnf);
! 2896: x[i] = lmodulcp((GEN)x[i], xnf);
! 2897: }
! 2898: return x;
! 2899: }
! 2900:
! 2901: static GEN
! 2902: rnfround2all(GEN nf, GEN pol, long all)
! 2903: {
! 2904: long av=avma,tetpil,i,j,n,N,nbidp,vpol,*ep;
! 2905: GEN p1,p2,p3,p4,polnf,list,unnf,id,matId,I,W,pseudo,y,discpol,d,D,sym;
! 2906:
! 2907: nf=checknf(nf); polnf=(GEN)nf[1]; vpol=varn(pol);
! 2908: pol = fix_relative_pol(nf,pol);
! 2909: N=lgef(polnf)-3; n=lgef(pol)-3; discpol=discsr(pol);
! 2910: list=idealfactor(nf,discpol); ep=(long*)list[2]; list=(GEN)list[1];
! 2911: nbidp=lg(list)-1; for(i=1;i<=nbidp;i++) ep[i]=itos((GEN)ep[i]);
! 2912: if (DEBUGLEVEL>1)
! 2913: {
! 2914: fprintferr("Ideals to consider:\n");
! 2915: for (i=1; i<=nbidp; i++)
! 2916: if (ep[i]>1) fprintferr("%Z^%ld\n",list[i],ep[i]);
! 2917: flusherr();
! 2918: }
! 2919: id=idmat(N); unnf=gscalcol_i(gun,N);
! 2920: matId=idmat_intern(n,unnf, gscalcol_i(gzero,N));
! 2921: pseudo = NULL;
! 2922: for (i=1; i<=nbidp; i++)
! 2923: if (ep[i] > 1)
! 2924: {
! 2925: y=rnfordmax(nf,pol,(GEN)list[i],unnf,id,matId);
! 2926: pseudo = rnfjoinmodules(nf,pseudo,y);
! 2927: }
! 2928: if (!pseudo)
! 2929: {
! 2930: I=cgetg(n+1,t_VEC); for (i=1; i<=n; i++) I[i]=(long)id;
! 2931: pseudo=cgetg(3,t_VEC); pseudo[1]=(long)matId; pseudo[2]=(long)I;
! 2932: }
! 2933: W=gmodulcp(mat_to_vecpol(basistoalg(nf,(GEN)pseudo[1]),vpol),pol);
! 2934: p2=cgetg(n+1,t_MAT); for (j=1; j<=n; j++) p2[j]=lgetg(n+1,t_COL);
! 2935: sym=polsym(pol,n-1);
! 2936: for (j=1; j<=n; j++)
! 2937: for (i=j; i<=n; i++)
! 2938: {
! 2939: p1 = lift_intern(gmul((GEN)W[i],(GEN)W[j]));
! 2940: coeff(p2,j,i)=coeff(p2,i,j)=(long)quicktrace(p1,sym);
! 2941: }
! 2942: d = algtobasis_intern(nf,det(p2));
! 2943:
! 2944: I=(GEN)pseudo[2];
! 2945: i=1; while (i<=n && gegal((GEN)I[i],id)) i++;
! 2946: if (i>n) D=id;
! 2947: else
! 2948: {
! 2949: D = (GEN)I[i];
! 2950: for (i++; i<=n; i++)
! 2951: if (!gegal((GEN)I[i],id)) D = idealmul(nf,D,(GEN)I[i]);
! 2952: D = idealpow(nf,D,gdeux);
! 2953: }
! 2954: p4=gun; p3=auxdecomp(content(d),0);
! 2955: for (i=1; i<lg(p3[1]); i++)
! 2956: p4 = gmul(p4, gpuigs(gcoeff(p3,i,1), itos(gcoeff(p3,i,2))>>1));
! 2957: p4 = gsqr(p4); tetpil=avma;
! 2958: i = all? 2: 0;
! 2959: p1=cgetg(3 + i,t_VEC);
! 2960: if (i) { p1[1]=lcopy((GEN)pseudo[1]); p1[2]=lcopy(I); }
! 2961: p1[1+i] = (long)idealmul(nf,D,d);
! 2962: p1[2+i] = ldiv(d,p4);
! 2963: return gerepile(av,tetpil,p1);
! 2964: }
! 2965:
! 2966: GEN
! 2967: rnfpseudobasis(GEN nf, GEN pol)
! 2968: {
! 2969: return rnfround2all(nf,pol,1);
! 2970: }
! 2971:
! 2972: GEN
! 2973: rnfdiscf(GEN nf, GEN pol)
! 2974: {
! 2975: return rnfround2all(nf,pol,0);
! 2976: }
! 2977:
! 2978: /* given bnf as output by buchinit and a pseudo-basis of an order
! 2979: * in HNF [A,I] (or [A,I,D,d] it does not matter), tries to simplify the
! 2980: * HNF as much as possible. The resulting matrix will be upper triangular
! 2981: * but the diagonal coefficients will not be equal to 1. The ideals
! 2982: * are guaranteed to be integral and primitive.
! 2983: */
! 2984: GEN
! 2985: rnfsimplifybasis(GEN bnf, GEN order)
! 2986: {
! 2987: long av=avma,tetpil,j,N,n;
! 2988: GEN p1,id,Az,Iz,nf,A,I;
! 2989:
! 2990: bnf = checkbnf(bnf);
! 2991: if (typ(order)!=t_VEC || lg(order)<3)
! 2992: err(talker,"not a pseudo-basis in nfsimplifybasis");
! 2993: A=(GEN)order[1]; I=(GEN)order[2]; n=lg(A)-1; nf=(GEN)bnf[7];
! 2994: N=lgef(nf[1])-3; id=idmat(N); Iz=cgetg(n+1,t_VEC); Az=cgetg(n+1,t_MAT);
! 2995: for (j=1; j<=n; j++)
! 2996: {
! 2997: if (gegal((GEN)I[j],id)) { Iz[j]=(long)id; Az[j]=A[j]; }
! 2998: else
! 2999: {
! 3000: p1=content((GEN)I[j]);
! 3001: if (!gcmp1(p1))
! 3002: {
! 3003: Iz[j]=(long)gdiv((GEN)I[j],p1); Az[j]=lmul((GEN)A[j],p1);
! 3004: }
! 3005: else Az[j]=A[j];
! 3006: if (!gegal((GEN)Iz[j],id))
! 3007: {
! 3008: p1=isprincipalgen(bnf,(GEN)Iz[j]);
! 3009: if (gcmp0((GEN)p1[1]))
! 3010: {
! 3011: p1=(GEN)p1[2]; Iz[j]=(long)id;
! 3012: Az[j]=(long)element_mulvec(nf,p1,(GEN)Az[j]);
! 3013: }
! 3014: }
! 3015: }
! 3016: }
! 3017: tetpil=avma; p1=cgetg(lg(order),t_VEC); p1[1]=lcopy(Az); p1[2]=lcopy(Iz);
! 3018: for (j=3; j<lg(order); j++) p1[j]=lcopy((GEN)order[j]);
! 3019: return gerepile(av,tetpil,p1);
! 3020: }
! 3021:
! 3022: GEN
! 3023: rnfdet2(GEN nf, GEN A, GEN I)
! 3024: {
! 3025: long av,tetpil,i;
! 3026: GEN p1;
! 3027:
! 3028: nf=checknf(nf); av = tetpil = avma;
! 3029: p1=idealhermite(nf,det(matbasistoalg(nf,A)));
! 3030: for(i=1;i<lg(I);i++) { tetpil=avma; p1=idealmul(nf,p1,(GEN)I[i]); }
! 3031: tetpil=avma; return gerepile(av,tetpil,p1);
! 3032: }
! 3033:
! 3034: GEN
! 3035: rnfdet(GEN nf, GEN order)
! 3036: {
! 3037: if (typ(order)!=t_VEC || lg(order)<3)
! 3038: err(talker,"not a pseudo-matrix in rnfdet");
! 3039: return rnfdet2(nf,(GEN)order[1],(GEN)order[2]);
! 3040: }
! 3041:
! 3042: GEN
! 3043: rnfdet0(GEN nf, GEN x, GEN y)
! 3044: {
! 3045: return y? rnfdet2(nf,x,y): rnfdet(nf,x);
! 3046: }
! 3047:
! 3048: /* given a pseudo-basis of an order in HNF [A,I] (or [A,I,D,d] it does
! 3049: * not matter), gives an nxn matrix (not in HNF) of a pseudo-basis and
! 3050: * an ideal vector [id,id,...,id,I] such that order=nf[7]^(n-1)xI.
! 3051: * Since it uses the approximation theorem, can be long.
! 3052: */
! 3053: GEN
! 3054: rnfsteinitz(GEN nf, GEN order)
! 3055: {
! 3056: long av=avma,tetpil,N,j,n;
! 3057: GEN id,A,I,p1,p2,a,b;
! 3058:
! 3059: nf=checknf(nf);
! 3060: N=lgef(nf[1])-3; id=idmat(N);
! 3061: if (typ(order)==t_POL) order=rnfpseudobasis(nf,order);
! 3062: if (typ(order)!=t_VEC || lg(order)<3)
! 3063: err(talker,"not a pseudo-matrix in rnfsteinitz");
! 3064: A=gcopy((GEN)order[1]); I=gcopy((GEN)order[2]); n=lg(A)-1;
! 3065: for (j=1; j<=n-1; j++)
! 3066: {
! 3067: a=(GEN)I[j];
! 3068: if (!gegal(a,id))
! 3069: {
! 3070: b=(GEN)I[j+1];
! 3071: if (gegal(b,id))
! 3072: {
! 3073: p1=(GEN)A[j]; A[j]=A[j+1]; A[j+1]=lneg(p1);
! 3074: I[j]=(long)b; I[j+1]=(long)a;
! 3075: }
! 3076: else
! 3077: {
! 3078: p2=nfidealdet1(nf,a,b);
! 3079: p1=gadd(element_mulvec(nf,(GEN)p2[1],(GEN)A[j]),
! 3080: element_mulvec(nf,(GEN)p2[2],(GEN)A[j+1]));
! 3081: A[j+1]= (long) gadd(element_mulvec(nf,(GEN)p2[3],(GEN)A[j]),
! 3082: element_mulvec(nf,(GEN)p2[4],(GEN)A[j+1]));
! 3083: A[j]=(long)p1;
! 3084: I[j]=(long)id; I[j+1]=(long)idealmul(nf,a,b);
! 3085: p1=content((GEN)I[j+1]);
! 3086: if (!gcmp1(p1))
! 3087: {
! 3088: I[j+1] = (long) gdiv((GEN)I[j+1],p1);
! 3089: A[j+1]=lmul(p1,(GEN)A[j+1]);
! 3090: }
! 3091: }
! 3092: }
! 3093: }
! 3094: tetpil=avma; p1=cgetg(lg(order),t_VEC);
! 3095: p1[1]=lcopy(A); p1[2]=lcopy(I);
! 3096: for (j=3; j<lg(order); j++) p1[j]=lcopy((GEN)order[j]);
! 3097: return gerepile(av,tetpil,p1);
! 3098: }
! 3099:
! 3100: /* Given bnf as output by buchinit and either an order as output by
! 3101: * rnfpseudobasis or a polynomial, and outputs a basis if it is free,
! 3102: * an n+1-generating set if it is not
! 3103: */
! 3104: GEN
! 3105: rnfbasis(GEN bnf, GEN order)
! 3106: {
! 3107: long av=avma,tetpil,j,N,n;
! 3108: GEN nf,A,I,classe,p1,p2,id;
! 3109:
! 3110: bnf = checkbnf(bnf);
! 3111: nf=(GEN)bnf[7]; N=lgef(nf[1])-3; id=idmat(N);
! 3112: if (typ(order)==t_POL) order=rnfpseudobasis(nf,order);
! 3113: if (typ(order)!=t_VEC || lg(order)<3)
! 3114: err(talker,"not a pseudo-matrix in rnfbasis");
! 3115: A=(GEN)order[1]; I=(GEN)order[2]; n=lg(A)-1;
! 3116: j=1; while (j<n && gegal((GEN)I[j],id)) j++;
! 3117: if (j<n) order=rnfsteinitz(nf,order);
! 3118: A=(GEN)order[1]; I=(GEN)order[2]; classe=(GEN)I[n];
! 3119: p1=isprincipalgen(bnf,classe);
! 3120: if (gcmp0((GEN)p1[1]))
! 3121: {
! 3122: p2=cgetg(n+1,t_MAT);
! 3123: p2[n]=(long)element_mulvec(nf,(GEN)p1[2],(GEN)A[n]);
! 3124: }
! 3125: else
! 3126: {
! 3127: p1=ideal_two_elt(nf,classe);
! 3128: p2=cgetg(n+2,t_MAT);
! 3129: p2[n]=lmul((GEN)p1[1],(GEN)A[n]);
! 3130: p2[n+1]=(long)element_mulvec(nf,(GEN)p1[2],(GEN)A[n]);
! 3131: }
! 3132: for (j=1; j<n; j++) p2[j]=A[j];
! 3133: tetpil = avma; return gerepile(av,tetpil,gcopy(p2));
! 3134: }
! 3135:
! 3136: /* Given bnf as output by buchinit and either an order as output by
! 3137: * rnfpseudobasis or a polynomial, and outputs a basis (not pseudo)
! 3138: * in Hermite Normal Form if it exists, zero if not
! 3139: */
! 3140: GEN
! 3141: rnfhermitebasis(GEN bnf, GEN order)
! 3142: {
! 3143: long av=avma,tetpil,j,N,n;
! 3144: GEN nf,A,I,p1,id;
! 3145:
! 3146: bnf = checkbnf(bnf); nf=(GEN)bnf[7];
! 3147: N=lgef(nf[1])-3; id=idmat(N);
! 3148: if (typ(order)==t_POL)
! 3149: {
! 3150: order=rnfpseudobasis(nf,order);
! 3151: A=(GEN)order[1];
! 3152: }
! 3153: else
! 3154: {
! 3155: if (typ(order)!=t_VEC || lg(order)<3)
! 3156: err(talker,"not a pseudo-matrix in rnfbasis");
! 3157: A=gcopy((GEN)order[1]);
! 3158: }
! 3159: I=(GEN)order[2]; n=lg(A)-1;
! 3160: for (j=1; j<=n; j++)
! 3161: {
! 3162: if (!gegal((GEN)I[j],id))
! 3163: {
! 3164: p1=isprincipalgen(bnf,(GEN)I[j]);
! 3165: if (gcmp0((GEN)p1[1]))
! 3166: A[j]=(long)element_mulvec(nf,(GEN)p1[2],(GEN)A[j]);
! 3167: else { avma=av; return gzero; }
! 3168: }
! 3169: }
! 3170: tetpil=avma; return gerepile(av,tetpil,gcopy(A));
! 3171: }
! 3172:
! 3173: long
! 3174: rnfisfree(GEN bnf, GEN order)
! 3175: {
! 3176: long av=avma,n,N,j;
! 3177: GEN nf,p1,id,I;
! 3178:
! 3179: bnf = checkbnf(bnf);
! 3180: if (gcmp1(gmael3(bnf,8,1,1))) return 1;
! 3181:
! 3182: nf=(GEN)bnf[7]; N=lgef(nf[1])-3; id=idmat(N);
! 3183: if (typ(order)==t_POL) order=rnfpseudobasis(nf,order);
! 3184: if (typ(order)!=t_VEC || lg(order)<3)
! 3185: err(talker,"not a pseudo-matrix in rnfisfree");
! 3186:
! 3187: I=(GEN)order[2]; n=lg(I)-1;
! 3188: j=1; while (j<=n && gegal((GEN)I[j],id)) j++;
! 3189: if (j>n) { avma=av; return 1; }
! 3190:
! 3191: p1=(GEN)I[j];
! 3192: for (j++; j<=n; j++)
! 3193: if (!gegal((GEN)I[j],id)) p1=idealmul(nf,p1,(GEN)I[j]);
! 3194: j = gcmp0(isprincipal(bnf,p1));
! 3195: avma=av; return j;
! 3196: }
! 3197:
! 3198: /**********************************************************************/
! 3199: /** **/
! 3200: /** COMPOSITUM OF TWO NUMBER FIELDS **/
! 3201: /** **/
! 3202: /**********************************************************************/
! 3203:
! 3204: #define nexta(a) (a>0 ? -a : 1-a)
! 3205:
! 3206: GEN
! 3207: polcompositum0(GEN pol1, GEN pol2, long flall)
! 3208: {
! 3209: long av=avma,tetpil,i,v,a,l;
! 3210: GEN pro1,p1,p2,p3,p4,p5,fa,rk,y;
! 3211:
! 3212: if (typ(pol1)!=t_POL || typ(pol2)!=t_POL) err(typeer,"polcompositum0");
! 3213: v=varn(pol1);
! 3214: if (varn(pol2)!=v) err(talker,"not the same variable in compositum");
! 3215: if (lgef(pol1)<=3 || lgef(pol2)<=3)
! 3216: err(constpoler,"compositum");
! 3217: if (lgef(ggcd(pol1,derivpol(pol1)))>3 || lgef(ggcd(pol2,derivpol(pol2)))>3)
! 3218: err(talker,"not a separable polynomial in compositum");
! 3219:
! 3220: for (a=1; ; a=nexta(a))
! 3221: {
! 3222: avma=av;
! 3223: if (DEBUGLEVEL>=2)
! 3224: {
! 3225: fprintferr("trying beta ");
! 3226: if (a>0) fprintferr("- "); else fprintferr("+ ");
! 3227: if (labs(a)>1) fprintferr("%ld ",labs(a));
! 3228: fprintferr("alpha\n"); flusherr();
! 3229: }
! 3230: pro1 = gadd(polx[MAXVARN],gmulsg(a,polx[v]));
! 3231: p1 = gsubst(pol2,v,pro1);
! 3232: p2 = subresall(pol1,p1,&rk);
! 3233: if (lgef(ggcd(p2,deriv(p2,MAXVARN)))==3)
! 3234: {
! 3235: p2 = gsubst(p2,MAXVARN,polx[v]);
! 3236: fa = factor(p2); fa = (GEN)fa[1];
! 3237: if (typ(rk)==t_POL && lgef(rk)==4)
! 3238: {
! 3239: if (flall)
! 3240: {
! 3241: l=lg(fa); y=cgetg(l,t_VEC);
! 3242: for (i=1; i<l; i++)
! 3243: {
! 3244: p3=cgetg(5,t_VEC); p3[1]=fa[i]; y[i]=(long)p3;
! 3245: p4=gmodulcp(polx[v],(GEN)fa[i]);
! 3246: p5=gneg_i(gdiv(gsubst((GEN)rk[2],MAXVARN,p4),
! 3247: gsubst((GEN)rk[3],MAXVARN,p4)));
! 3248: p3[2]=(long)p5;
! 3249: p3[3]=ladd(p4,gmulsg(a,p5));
! 3250: p3[4]=lstoi(-a);
! 3251: }
! 3252: }
! 3253: else y=fa;
! 3254: tetpil=avma; return gerepile(av,tetpil,gcopy(y));
! 3255: }
! 3256: }
! 3257: }
! 3258: }
! 3259:
! 3260: GEN
! 3261: compositum(GEN pol1,GEN pol2)
! 3262: {
! 3263: return polcompositum0(pol1,pol2,0);
! 3264: }
! 3265:
! 3266: GEN
! 3267: compositum2(GEN pol1,GEN pol2)
! 3268: {
! 3269: return polcompositum0(pol1,pol2,1);
! 3270: }
! 3271:
! 3272: GEN
! 3273: rnfequation0(GEN nf, GEN pol2, long flall)
! 3274: {
! 3275: long av=avma,av1,tetpil,v,vpol,a,l1,l2;
! 3276: GEN pol1,pro1,p1,p2,p4,p5,rk,y;
! 3277:
! 3278: if (typ(nf)==t_POL) pol1=nf; else { nf=checknf(nf); pol1=(GEN)nf[1]; }
! 3279: pol2 = fix_relative_pol(nf,pol2);
! 3280: v=varn(pol1); vpol=varn(pol2);
! 3281:
! 3282: l1=lgef(pol1); l2=lgef(pol2);
! 3283: if (l1<=3 || l2<=3) err(constpoler,"rnfequation");
! 3284:
! 3285: p2=cgetg(l2,t_POL); p2[1]=pol2[1];
! 3286: for (a=2; a<l2; a++)
! 3287: p2[a] = (lgef(pol2[a]) < l1)? pol2[a]: lres((GEN)pol2[a],pol1);
! 3288: pol2=p2;
! 3289: if (lgef(ggcd(pol2,derivpol(pol2)))>3)
! 3290: err(talker,"not a separable relative equation in rnfequation");
! 3291: pol2=lift_intern(pol2);
! 3292:
! 3293: a=0; av1=avma;
! 3294: for(;;)
! 3295: {
! 3296: avma=av1;
! 3297: if (DEBUGLEVEL>=2)
! 3298: {
! 3299: fprintferr("trying beta ");
! 3300: if (a)
! 3301: {
! 3302: if (a>0) fprintferr("- "); else fprintferr("+ ");
! 3303: if (labs(a)>1) fprintferr("%ld alpha\n",labs(a));
! 3304: else fprintferr("alpha\n");
! 3305: }
! 3306: flusherr();
! 3307: }
! 3308: pro1=gadd(polx[MAXVARN],gmulsg(a,polx[v]));
! 3309: p1=poleval(pol2,pro1);
! 3310: p2=subresall(pol1,p1,&rk);
! 3311: if (rk != gzero && lgef(rk)==4 && lgef(ggcd(p2,deriv(p2,MAXVARN)))==3)
! 3312: {
! 3313: p2=gsubst(p2,MAXVARN,polx[vpol]);
! 3314: if (gsigne(leadingcoeff(p2))<0) p2=gneg_i(p2);
! 3315: if (flall)
! 3316: {
! 3317: y=cgetg(4,t_VEC); y[1]=(long)p2;
! 3318: p4=gmodulcp(polx[vpol],p2);
! 3319: p5=gneg_i(gdiv(gsubst((GEN)rk[2],MAXVARN,p4),
! 3320: gsubst((GEN)rk[3],MAXVARN,p4)));
! 3321: y[3]=(long)stoi(-a);
! 3322: y[2]=lmul(gmodulcp(polun[vpol],p2),p5);
! 3323: }
! 3324: else y=p2;
! 3325: if (DEBUGLEVEL>=2) fprintferr("ok! leaving rnfequation\n");
! 3326: tetpil=avma; return gerepile(av,tetpil,gcopy(y));
! 3327: }
! 3328: a=nexta(a);
! 3329: }
! 3330: }
! 3331:
! 3332: GEN
! 3333: rnfequation(GEN nf,GEN pol2)
! 3334: {
! 3335: return rnfequation0(nf,pol2,0);
! 3336: }
! 3337:
! 3338: GEN
! 3339: rnfequation2(GEN nf,GEN pol2)
! 3340: {
! 3341: return rnfequation0(nf,pol2,1);
! 3342: }
! 3343:
! 3344: static GEN
! 3345: nftau(long r1, GEN x)
! 3346: {
! 3347: long i, ru = lg(x);
! 3348: GEN s;
! 3349:
! 3350: s = r1 ? (GEN)x[1] : gmul2n(greal((GEN)x[1]),1);
! 3351: for (i=2; i<=r1; i++) s=gadd(s,(GEN)x[i]);
! 3352: for ( ; i<ru; i++) s=gadd(s,gmul2n(greal((GEN)x[i]),1));
! 3353: return s;
! 3354: }
! 3355:
! 3356: static GEN
! 3357: nftocomplex(GEN nf, GEN x)
! 3358: {
! 3359: long ru,vnf,k;
! 3360: GEN p2,p3,ronf;
! 3361:
! 3362: p2 = (typ(x)==t_POLMOD)? (GEN)x[2]: gmul((GEN)nf[7],x);
! 3363: vnf=varn(nf[1]);
! 3364: ronf=(GEN)nf[6]; ru=lg(ronf); p3=cgetg(ru,t_COL);
! 3365: for (k=1; k<ru; k++) p3[k]=lsubst(p2,vnf,(GEN)ronf[k]);
! 3366: return p3;
! 3367: }
! 3368:
! 3369: static GEN
! 3370: rnfscal(GEN mth, GEN xth, GEN yth)
! 3371: {
! 3372: long n,ru,i,j,kk;
! 3373: GEN x,y,m,res,p1,p2;
! 3374:
! 3375: n=lg(mth)-1; ru=lg(gcoeff(mth,1,1));
! 3376: res=cgetg(ru,t_COL);
! 3377: for (kk=1; kk<ru; kk++)
! 3378: {
! 3379: m=cgetg(n+1,t_MAT);
! 3380: for (j=1; j<=n; j++)
! 3381: {
! 3382: p1=cgetg(n+1,t_COL); m[j]=(long)p1;
! 3383: for (i=1; i<=n; i++) { p2=gcoeff(mth,i,j); p1[i]=p2[kk]; }
! 3384: }
! 3385: x=cgetg(n+1,t_VEC);
! 3386: for (j=1; j<=n; j++) x[j]=(long)gconj((GEN)((GEN)xth[j])[kk]);
! 3387: y=cgetg(n+1,t_COL);
! 3388: for (j=1; j<=n; j++) y[j]=((GEN)yth[j])[kk];
! 3389: res[kk]=(long)gmul(x,gmul(m,y));
! 3390: }
! 3391: return res;
! 3392: }
! 3393:
! 3394: static GEN
! 3395: rnfdiv(GEN x, GEN y)
! 3396: {
! 3397: long i, ru = lg(x);
! 3398: GEN z;
! 3399:
! 3400: z=cgetg(ru,t_COL);
! 3401: for (i=1; i<ru; i++) z[i]=(long)gdiv((GEN)x[i],(GEN)y[i]);
! 3402: return z;
! 3403: }
! 3404:
! 3405: static GEN
! 3406: rnfmul(GEN x, GEN y)
! 3407: {
! 3408: long i, ru = lg(x);
! 3409: GEN z;
! 3410:
! 3411: z=cgetg(ru,t_COL);
! 3412: for (i=1; i<ru; i++) z[i]=(long)gmul((GEN)x[i],(GEN)y[i]);
! 3413: return z;
! 3414: }
! 3415:
! 3416: static GEN
! 3417: rnfvecmul(GEN x, GEN v)
! 3418: {
! 3419: long i, lx = lg(v);
! 3420: GEN y;
! 3421:
! 3422: y=cgetg(lx,typ(v));
! 3423: for (i=1; i<lx; i++) y[i]=(long)rnfmul(x,(GEN)v[i]);
! 3424: return y;
! 3425: }
! 3426:
! 3427: static GEN
! 3428: allonge(GEN v, long N)
! 3429: {
! 3430: long r,r2,i;
! 3431: GEN y;
! 3432:
! 3433: r=lg(v)-1; r2=N-r;
! 3434: y=cgetg(N+1,t_COL);
! 3435: for (i=1; i<=r; i++) y[i]=v[i];
! 3436: for ( ; i<=N; i++) y[i]=(long)gconj((GEN)v[i-r2]);
! 3437: return y;
! 3438: }
! 3439:
! 3440: static GEN
! 3441: findmin(GEN nf, GEN ideal, GEN muf,long prec)
! 3442: {
! 3443: long av=avma,N,tetpil,i;
! 3444: GEN m,y;
! 3445:
! 3446: m = qf_base_change(gmael(nf,5,3), ideal, 0); /* nf[5][3] = T2 */
! 3447: m = lllgramintern(m,4,1,prec);
! 3448: if (!m)
! 3449: {
! 3450: m = lllint(ideal);
! 3451: m = qf_base_change(gmael(nf,5,3), gmul(ideal,m), 0);
! 3452: m = lllgramintern(m,4,1,prec);
! 3453: if (!m) err(talker,"precision too low in rnflllgram");
! 3454: }
! 3455: ideal=gmul(ideal,m);
! 3456: N=lg(ideal)-1; y=cgetg(N+1,t_MAT);
! 3457: for (i=1; i<=N; i++)
! 3458: y[i] = (long) allonge(nftocomplex(nf,(GEN)ideal[i]),N);
! 3459: m=ground(greal(gauss(y,allonge(muf,N))));
! 3460: tetpil=avma; return gerepile(av,tetpil,gmul(ideal,m));
! 3461: }
! 3462:
! 3463: #define swap(x,y) { long _t=x; x=y; y=_t; }
! 3464:
! 3465: /* given a base field nf (e.g main variable y), a polynomial pol with
! 3466: * coefficients in nf (e.g main variable x), and an order as output
! 3467: * by rnfpseudobasis, outputs a reduced order.
! 3468: */
! 3469: GEN
! 3470: rnflllgram(GEN nf, GEN pol, GEN order,long prec)
! 3471: {
! 3472: long av=avma,tetpil,i,j,k,l,kk,kmax,r1,ru,lx,n,vnf;
! 3473: GEN p1,p2,M,I,U,ronf,poll,unro,roorder,powreorder,mth,s,MC,MPOL,MCS;
! 3474: GEN B,mu,Bf,temp,ideal,x,xc,xpol,muf,mufc,muno,y,z,Ikk_inv;
! 3475:
! 3476: /* Initializations and verifications */
! 3477:
! 3478: nf=checknf(nf);
! 3479: if (typ(order)!=t_VEC || lg(order)<3)
! 3480: err(talker,"not a pseudo-matrix in rnflllgram");
! 3481: M=(GEN)order[1]; I=gcopy((GEN)order[2]); lx=lg(I); n=lg(I)-1;
! 3482:
! 3483: /* Initialize U to the n x n identity matrix with coefficients in nf in
! 3484: the form of polymods */
! 3485:
! 3486: U=cgetg(n+1,t_MAT);
! 3487: for (j=1; j<=n; j++)
! 3488: {
! 3489: p1=cgetg(n+1,t_COL); U[j]=(long)p1;
! 3490: for (i=1; i<=n; i++) p1[i]=(i==j)?un:zero;
! 3491: }
! 3492:
! 3493: /* Compute the relative T2 matrix of powers of theta */
! 3494:
! 3495: vnf=varn(nf[1]); ronf=(GEN)nf[6]; ru=lg(ronf); poll=lift(pol);
! 3496: r1=itos(gmael(nf,2,1));
! 3497: unro=cgetg(n+1,t_COL); for (i=1; i<=n; i++) unro[i]=un;
! 3498: roorder=cgetg(ru,t_VEC);
! 3499: for (i=1; i<ru; i++)
! 3500: roorder[i]=lroots(gsubst(poll,vnf,(GEN)ronf[i]),prec);
! 3501: powreorder=cgetg(n+1,t_MAT);
! 3502: p1=cgetg(ru,t_COL); powreorder[1]=(long)p1;
! 3503: for (i=1; i<ru; i++) p1[i]=(long)unro;
! 3504: for (k=2; k<=n; k++)
! 3505: {
! 3506: p1=cgetg(ru,t_COL); powreorder[k]=(long)p1;
! 3507: for (i=1; i<ru; i++)
! 3508: {
! 3509: p2=cgetg(n+1,t_COL); p1[i]=(long)p2;
! 3510: for (j=1; j<=n; j++)
! 3511: p2[j] = lmul(gmael(roorder,i,j),gmael3(powreorder,k-1,i,j));
! 3512: }
! 3513: }
! 3514: mth=cgetg(n+1,t_MAT);
! 3515: for (l=1; l<=n; l++)
! 3516: {
! 3517: p1=cgetg(n+1,t_COL); mth[l]=(long)p1;
! 3518: for (k=1; k<=n; k++)
! 3519: {
! 3520: p2=cgetg(ru,t_COL); p1[k]=(long)p2;
! 3521: for (i=1; i<ru; i++)
! 3522: {
! 3523: s=gzero;
! 3524: for (j=1; j<=n; j++)
! 3525: s = gadd(s,gmul(gconj(gmael3(powreorder,k,i,j)),
! 3526: gmael3(powreorder,l,i,j)));
! 3527: p2[i]=(long)s;
! 3528: }
! 3529: }
! 3530: }
! 3531:
! 3532: /* Transform the matrix M into a matrix with coefficients in K and also
! 3533: with coefficients polymod */
! 3534:
! 3535: MC=cgetg(lx,t_MAT); MPOL=cgetg(lx,t_MAT);
! 3536: for (j=1; j<=n; j++)
! 3537: {
! 3538: p1=cgetg(lx,t_COL); MC[j]=(long)p1;
! 3539: p2=cgetg(lx,t_COL); MPOL[j]=(long)p2;
! 3540: for (i=1; i<=n; i++)
! 3541: {
! 3542: p2[i]=(long)basistoalg(nf,gcoeff(M,i,j));
! 3543: p1[i]=(long)nftocomplex(nf,(GEN)p2[i]);
! 3544: }
! 3545: }
! 3546: MCS=cgetg(lx,t_MAT);
! 3547:
! 3548: /* Start LLL algorithm */
! 3549:
! 3550: mu=cgetg(lx,t_MAT); B=cgetg(lx,t_COL);
! 3551: for (j=1; j<lx; j++)
! 3552: {
! 3553: p1=cgetg(lx,t_COL); mu[j]=(long)p1; for (i=1; i<lx; i++) p1[i]=zero;
! 3554: B[j]=zero;
! 3555: }
! 3556: kk=2; if (DEBUGLEVEL) fprintferr("kk = %ld ",kk);
! 3557: kmax=1; B[1]=lreal(rnfscal(mth,(GEN)MC[1],(GEN)MC[1]));
! 3558: MCS[1]=lcopy((GEN)MC[1]);
! 3559: do
! 3560: {
! 3561: if (kk>kmax)
! 3562: {
! 3563: /* Incremental Gram-Schmidt */
! 3564: kmax=kk; MCS[kk]=lcopy((GEN)MC[kk]);
! 3565: for (j=1; j<kk; j++)
! 3566: {
! 3567: coeff(mu,kk,j) = (long) rnfdiv(rnfscal(mth,(GEN)MCS[j],(GEN)MC[kk]),
! 3568: (GEN) B[j]);
! 3569: MCS[kk] = lsub((GEN) MCS[kk], rnfvecmul(gcoeff(mu,kk,j),(GEN)MCS[j]));
! 3570: }
! 3571: B[kk] = lreal(rnfscal(mth,(GEN)MCS[kk],(GEN)MCS[kk]));
! 3572: if (gcmp0((GEN)B[kk])) err(lllger3);
! 3573: }
! 3574:
! 3575: /* RED(k,k-1) */
! 3576: l=kk-1; Ikk_inv=idealinv(nf, (GEN)I[kk]);
! 3577: ideal=idealmul(nf,(GEN)I[l],Ikk_inv);
! 3578: x=findmin(nf,ideal,gcoeff(mu,kk,l),2*prec-2);
! 3579: if (!gcmp0(x))
! 3580: {
! 3581: xpol=basistoalg(nf,x); xc=nftocomplex(nf,xpol);
! 3582: MC[kk]=lsub((GEN)MC[kk],rnfvecmul(xc,(GEN)MC[l]));
! 3583: U[kk]=lsub((GEN)U[kk],gmul(xpol,(GEN)U[l]));
! 3584: coeff(mu,kk,l)=lsub(gcoeff(mu,kk,l),xc);
! 3585: for (i=1; i<l; i++)
! 3586: coeff(mu,kk,i)=lsub(gcoeff(mu,kk,i),rnfmul(xc,gcoeff(mu,l,i)));
! 3587: }
! 3588: /* Test LLL condition */
! 3589: p1=nftau(r1,gadd((GEN) B[kk],
! 3590: gmul(gnorml2(gcoeff(mu,kk,kk-1)),(GEN)B[kk-1])));
! 3591: p2=gdivgs(gmulsg(9,nftau(r1,(GEN)B[kk-1])),10);
! 3592: if (gcmp(p1,p2)<=0)
! 3593: {
! 3594: /* Execute SWAP(k) */
! 3595: k=kk;
! 3596: swap(MC[k-1],MC[k]);
! 3597: swap(U[k-1],U[k]);
! 3598: swap(I[k-1],I[k]);
! 3599: for (j=1; j<=k-2; j++) swap(coeff(mu,k-1,j),coeff(mu,k,j));
! 3600: muf=gcoeff(mu,k,k-1);
! 3601: mufc=gconj(muf); muno=greal(rnfmul(muf,mufc));
! 3602: Bf=gadd((GEN)B[k],rnfmul(muno,(GEN)B[k-1]));
! 3603: p1=rnfdiv((GEN)B[k-1],Bf);
! 3604: coeff(mu,k,k-1)=(long)rnfmul(mufc,p1);
! 3605: temp=(GEN)MCS[k-1];
! 3606: MCS[k-1]=ladd((GEN)MCS[k],rnfvecmul(muf,(GEN)MCS[k-1]));
! 3607: MCS[k]=lsub(rnfvecmul(rnfdiv((GEN)B[k],Bf),temp),
! 3608: rnfvecmul(gcoeff(mu,k,k-1),(GEN)MCS[k]));
! 3609: B[k]=(long)rnfmul((GEN)B[k],p1); B[k-1]=(long)Bf;
! 3610: for (i=k+1; i<=kmax; i++)
! 3611: {
! 3612: temp=gcoeff(mu,i,k);
! 3613: coeff(mu,i,k)=lsub(gcoeff(mu,i,k-1),rnfmul(muf,gcoeff(mu,i,k)));
! 3614: coeff(mu,i,k-1) = ladd(temp, rnfmul(gcoeff(mu,k,k-1),gcoeff(mu,i,k)));
! 3615: }
! 3616: if (kk>2) { kk--; if (DEBUGLEVEL) fprintferr("%ld ",kk); }
! 3617: }
! 3618: else
! 3619: {
! 3620: for (l=kk-2; l; l--)
! 3621: {
! 3622: /* RED(k,l) */
! 3623: ideal=idealmul(nf,(GEN)I[l],Ikk_inv);
! 3624: x=findmin(nf,ideal,gcoeff(mu,kk,l),2*prec-2);
! 3625: if (!gcmp0(x))
! 3626: {
! 3627: xpol=basistoalg(nf,x); xc=nftocomplex(nf,xpol);
! 3628: MC[kk]=(long)gsub((GEN)MC[kk],rnfvecmul(xc,(GEN)MC[l]));
! 3629: U[kk]=(long)gsub((GEN)U[kk],gmul(xpol,(GEN)U[l]));
! 3630: coeff(mu,kk,l)=lsub(gcoeff(mu,kk,l),xc);
! 3631: for (i=1; i<l; i++)
! 3632: coeff(mu,kk,i) = lsub(gcoeff(mu,kk,i), rnfmul(xc,gcoeff(mu,l,i)));
! 3633: }
! 3634: }
! 3635: kk++; if (DEBUGLEVEL) fprintferr("%ld ",kk);
! 3636: }
! 3637: }
! 3638: while (kk<=n);
! 3639: if (DEBUGLEVEL) fprintferr("\n");
! 3640: p1=gmul(MPOL,U); tetpil=avma;
! 3641: y=cgetg(3,t_VEC); z=cgetg(3,t_VEC); y[1]=(long)z;
! 3642: z[2]=lcopy(I); z[1]=(long)algtobasis(nf,p1);
! 3643: y[2]=(long)algtobasis(nf,U);
! 3644: return gerepile(av,tetpil,y);
! 3645: }
! 3646:
! 3647: GEN
! 3648: rnfpolred(GEN nf, GEN pol, long prec)
! 3649: {
! 3650: long av=avma,tetpil,i,j,k,n,N,vpol,flbnf;
! 3651: GEN id,id2,newid,newor,p1,p2,al,newpol,w,z;
! 3652: GEN bnf,zk,newideals,ideals,order,neworder;
! 3653:
! 3654: if (typ(nf)!=t_VEC) err(idealer1);
! 3655: switch(lg(nf))
! 3656: {
! 3657: case 10: flbnf=0; break;
! 3658: case 11: flbnf=1; bnf=nf; nf=checknf((GEN)nf[7]); break;
! 3659: default: err(idealer1);
! 3660: }
! 3661: id=rnfpseudobasis(nf,pol); N=lgef(nf[1])-3;
! 3662: if (flbnf && gcmp1(gmael3(bnf,8,1,1))) /* if bnf is principal */
! 3663: {
! 3664: ideals=(GEN)id[2]; n=lg(ideals)-1; order=(GEN)id[1];
! 3665: newideals=cgetg(n+1,t_VEC); neworder=cgetg(n+1,t_MAT);
! 3666: zk=idmat(N);
! 3667: for (j=1; j<=n; j++)
! 3668: {
! 3669: newideals[j]=(long)zk; p1=cgetg(n+1,t_COL); neworder[j]=(long)p1;
! 3670: p2=(GEN)order[j];
! 3671: al=(GEN)isprincipalgen(bnf,(GEN)ideals[j])[2];
! 3672: for (k=1; k<=n; k++)
! 3673: p1[k]=(long)element_mul(nf,(GEN)p2[k],al);
! 3674: }
! 3675: id=cgetg(3,t_VEC); id[1]=(long)neworder; id[2]=(long)newideals;
! 3676: }
! 3677: id2=rnflllgram(nf,pol,id,prec);
! 3678: z=(GEN)id2[1]; newid=(GEN)z[2]; newor=(GEN)z[1];
! 3679: n=lg(newor)-1; w=cgetg(n+1,t_VEC); vpol=varn(pol);
! 3680: for (j=1; j<=n; j++)
! 3681: {
! 3682: p1=(GEN)newid[j]; al=gmul(gcoeff(p1,1,1),(GEN)newor[j]);
! 3683: p1=basistoalg(nf,(GEN)al[n]);
! 3684: for (i=n-1; i; i--)
! 3685: p1=gadd(basistoalg(nf,(GEN)al[i]),gmul(polx[vpol],p1));
! 3686: newpol=gtopoly(gmodulcp(gtovec(caract2(lift(pol),lift(p1),vpol)),
! 3687: (GEN) nf[1]), vpol);
! 3688: p1 = ggcd(newpol, derivpol(newpol));
! 3689: if (degree(p1)>0)
! 3690: {
! 3691: newpol=gdiv(newpol,p1);
! 3692: newpol=gdiv(newpol,leading_term(newpol));
! 3693: }
! 3694: w[j]=(long)newpol;
! 3695: if (DEBUGLEVEL>=4) outerr(newpol);
! 3696: }
! 3697: tetpil=avma; return gerepile(av,tetpil,gcopy(w));
! 3698: }
! 3699:
! 3700: GEN
! 3701: makebasis(GEN nf,GEN pol)
! 3702: /* Etant donne un corps de nombres nf et un polynome relatif relpol,
! 3703: construit une pseudo-base de l'extension puis calcule une base absolue
! 3704: de cette extension pour une racine \theta de relpol. Renvoie le
! 3705: polynome irreductible de theta sur Q et la matrice de la base */
! 3706: {
! 3707: GEN elts,ids,polabs,plg,B,bs,p1,colonne,p2,rep,a;
! 3708: GEN den,vbs,vbspro,mpro,vpro,rnf;
! 3709: long av=avma,tetpil,n,N,m,i,j,k,v1,v2;
! 3710:
! 3711: v1=varn((GEN)nf[1]); v2=varn(pol);
! 3712: p1=rnfequation2(nf,pol);
! 3713: polabs=(GEN)p1[1]; plg=(GEN)p1[2];
! 3714: a=(GEN)p1[3];
! 3715: rnf=cgetg(12,t_VEC); rnf[1]=(long)pol;
! 3716: for (i=2;i<=9;i++) rnf[i]=zero;
! 3717: rnf[10]=(long)nf;
! 3718: p2=cgetg(4,t_VEC); p2[1] = p2[2] = zero;
! 3719: p2[3]=(long)a; rnf[11]=(long)p2;
! 3720: if (signe(a))
! 3721: pol=gsubst(pol,v2,gsub(polx[v2],
! 3722: gmul(a,gmodulcp(polx[v1],(GEN)nf[1]))));
! 3723: p1=rnfpseudobasis(nf,pol);
! 3724: if (DEBUGLEVEL>=2) { fprintferr("relative basis computed\n"); flusherr(); }
! 3725: elts=(GEN)p1[1];ids=(GEN)p1[2];
! 3726: N=lgef(pol)-3;n=lgef((GEN)nf[1])-3;m=n*N;
! 3727: den=denom(content(lift(plg)));
! 3728: vbs=cgetg(n+1,t_VEC);
! 3729: vbs[1]=un;vbs[2]=(long)plg;
! 3730: vbspro=gmul(den,plg);
! 3731: for(i=3;i<=n;i++)
! 3732: vbs[i]=ldiv(gmul((GEN)vbs[i-1],vbspro),den);
! 3733: mpro=cgetg(n+1,t_MAT);
! 3734: for(j=1;j<=n;j++)
! 3735: {
! 3736: p2=cgetg(n+1,t_COL);mpro[j]=(long)p2;
! 3737: for(i=1;i<=n;i++)
! 3738: p2[i]=(long)truecoeff(gmael(nf,7,j),i-1);
! 3739: }
! 3740: bs=gmul(vbs,mpro); B=idmat(m);
! 3741: vpro=cgetg(N+1,t_VEC);
! 3742: for (i=1;i<=N;i++)
! 3743: {
! 3744: p1=cgetg(3,t_POLMOD);
! 3745: p1[1]=(long)polabs;
! 3746: p1[2]=lpuigs(polx[v2],i-1); vpro[i]=(long)p1;
! 3747: }
! 3748: vpro=gmul(vpro,elts);
! 3749: for(i=1;i<=N;i++)
! 3750: for(j=1;j<=n;j++)
! 3751: {
! 3752: colonne=gmul(bs,element_mul(nf,(GEN)vpro[i],gmael(ids,i,j)));
! 3753: p1=gtovec(lift_intern(colonne));
! 3754: p2=cgetg(m+1,t_COL);
! 3755: for(k=1;k<lg(p1);k++) p2[lg(p1)-k]=p1[k];
! 3756: for( ;k<=m;k++) p2[k]=zero;
! 3757: B[(i-1)*n+j]=(long)p2;
! 3758: }
! 3759: rep=cgetg(4,t_VEC);
! 3760: rep[1]=(long)polabs;
! 3761: rep[2]=(long)B;
! 3762: rep[3]=(long)rnf;
! 3763: tetpil=avma;
! 3764: return gerepile(av,tetpil,gcopy(rep));
! 3765: }
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