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nffactor.c

/* $Id: nffactor.c,v 1.17.2.1 2001/04/09 16:54:18 karim Exp $

Copyright (C) 2000  The PARI group.

This file is part of the PARI/GP package.

PARI/GP is free software; you can redistribute it and/or modify it under the
terms of the GNU General Public License as published by the Free Software
Foundation. It is distributed in the hope that it will be useful, but WITHOUT
ANY WARRANTY WHATSOEVER.

Check the License for details. You should have received a copy of it, along
with the package; see the file 'COPYING'. If not, write to the Free Software
Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA. */

/*******************************************************************/
/*                                                                 */
/*            POLYNOMIAL FACTORIZATION IN A NUMBER FIELD           */
/*                                                                 */
/*******************************************************************/
#include "pari.h"

extern GEN hensel_lift(GEN pol,GEN fk,GEN fkk,GEN p,long e);
extern GEN hensel_lift_fact(GEN pol, GEN fact, GEN p, GEN pev, long e);
extern GEN nf_get_T2(GEN base, GEN polr);
extern GEN nfreducemodpr_i(GEN x, GEN prh);
extern GEN sort_factor(GEN y, int (*cmp)(GEN,GEN));
extern GEN pidealprimeinv(GEN nf, GEN x);

static GEN nf_pol_mul(GEN nf,GEN pol1,GEN pol2);
static GEN nf_pol_sqr(GEN nf,GEN pol1);
static GEN nf_pol_divres(GEN nf,GEN pol1,GEN pol2, GEN *pr);
static GEN nf_pol_subres(GEN nf,GEN pol1,GEN pol2);
static GEN nfmod_pol_reduce(GEN nf,GEN prhall,GEN pol);
static GEN nfmod_pol_mul(GEN nf,GEN prhall,GEN pol1,GEN pol2);
static GEN nfmod_pol_sqr(GEN nf,GEN prhall,GEN pol1);
static GEN nfmod_pol_divres(GEN nf,GEN prhall,GEN pol1,GEN pol2, GEN *pr);
static GEN nfmod_split2(GEN nf,GEN prhall,GEN pmod,GEN pol,GEN exp);
static GEN nfmod_pol_gcd(GEN nf,GEN prhall,GEN pol1,GEN pol2);
static GEN nfmod_pol_pow(GEN nf,GEN prhall,GEN pmod,GEN pol,GEN exp);
static GEN nf_bestlift(GEN id,GEN idinv,GEN den,GEN elt);
static GEN nf_pol_lift(GEN id,GEN idinv,GEN den,GEN pol);
static GEN nfsqff(GEN nf,GEN pol,long fl);
static int nf_combine_factors(GEN nf,long fxn,GEN psf,long dlim,long hint);

typedef struct Nfcmbf /* for use in nfsqff */
{
  GEN pol, h, hinv, fact, res, lt, den;
  long nfact, nfactmod;
} Nfcmbf;
static Nfcmbf nfcmbf;

static GEN
unifpol0(GEN nf,GEN pol,long flag)
{
  static long n = 0;
  static GEN vun = NULL;
  GEN f = (GEN) nf[1];
  long i = lgef(f)-3, av;

  if (i != n)
  {
    n=i; if (vun) free(vun);
    vun = (GEN) gpmalloc((n+1)*sizeof(long));
    vun[0] = evaltyp(t_COL) | evallg(n+1); vun[1]=un;
    for ( ; i>=2; i--) vun[i]=zero;
  }

  av = avma;
  switch(typ(pol))
  {
    case t_INT: case t_FRAC: case t_RFRAC:
      pol = gmul(pol,vun); break;

    case t_POL:
      pol = gmodulcp(pol,f); /* fall through */
    case t_POLMOD:
      pol = algtobasis(nf,pol);
  }
  if (flag) pol = basistoalg(nf, lift(pol));
  return gerepileupto(av,pol);
}

/* Pour pol un polynome a coefficients dans Z, nf ou l'algebre correspondant
 * renvoie le meme polynome avec pour coefficients:
 *  si flag=0: des vecteurs sur la base d'entiers.
 *  si flag=1: des polymods.
 */
GEN
unifpol(GEN nf,GEN pol,long flag)
{
  if (typ(pol)==t_POL && varn(pol) < varn(nf[1]))
  {
    long i, d = lgef(pol);
    GEN p1 = pol;

    pol=cgetg(d,t_POL); pol[1]=p1[1];
    for (i=2; i<d; i++)
      pol[i] = (long) unifpol0(nf,(GEN) p1[i],flag);

    return pol;
  }
  return unifpol0(nf,(GEN) pol, flag);
}

/* cree un polynome unitaire de degre d a entrees au hazard dans Z_nf */
GEN
random_pol(GEN nf,long d)
{
  long i,j, n = lgef(nf[1])-3;
  GEN pl,p;

  pl=cgetg(d+3,t_POL);
  for (i=2; i<d+2; i++)
  {
    p=cgetg(n+1,t_COL); pl[i]=(long)p;
    for (j=1; j<=n; j++)
      p[j] = lstoi(mymyrand()%101 - 50);
  }
  p=cgetg(n+1,t_COL); pl[i]=(long)p;
  p[1]=un; for (i=2; i<=n; i++) p[i]=zero;

  pl[1] = evalsigne(1) | evallgef(d+3) | evalvarn(0);
  return pl;
}

/* multiplication de x par y dans nf (x element accepte) */
static GEN
nf_pol_mul(GEN nf,GEN x,GEN y)
{
  long tetpil,av=avma;
  GEN res = gmul(unifpol(nf,x,1), unifpol(nf,y,1));

  tetpil = avma;
  return gerepile(av,tetpil,unifpol(nf,res,0));
}

/* calcule x^2 dans nf (x element accepte) */
static GEN
nf_pol_sqr(GEN nf,GEN x)
{
  long tetpil,av=avma;
  GEN res = gsqr(unifpol(nf,x,1));

  tetpil = avma;
  return gerepile(av,tetpil,unifpol(nf,res,0));
}

/* Reduction modulo phall des coefficients de pol (pol element accepte) */
static GEN
nfmod_pol_reduce(GEN nf,GEN prhall,GEN pol)
{
  long av=avma,tetpil,i;
  GEN p1;

  if (typ(pol)!=t_POL) return nfreducemodpr(nf,pol,prhall);
  pol=unifpol(nf,pol,0);

  tetpil=avma; i=lgef(pol);
  p1=cgetg(i,t_POL); p1[1]=pol[1];
  for (i--; i>=2; i--)
    p1[i] = (long) nfreducemodpr(nf,(GEN)pol[i],prhall);
  return gerepile(av,tetpil, normalizepol(p1));
}

/* x^2 modulo prhall ds nf (x elt accepte) */
static GEN
nfmod_pol_sqr(GEN nf,GEN prhall,GEN x)
{
  long av=avma,tetpil;
  GEN px;

  px = nfmod_pol_reduce(nf,prhall,x);
  px = unifpol(nf,lift(px),1);
  px = unifpol(nf,nf_pol_sqr(nf,px),0);
  tetpil=avma;
  return gerepile(av,tetpil,nfmod_pol_reduce(nf,prhall,px));
}

/* multiplication des polynomes x et y modulo prhall ds nf (x elt accepte) */
static GEN
nfmod_pol_mul(GEN nf,GEN prhall,GEN x,GEN y)
{
  long av=avma,tetpil;
  GEN px,py;

  px = nfmod_pol_reduce(nf,prhall,x); px = unifpol(nf,lift(px),1);
  py = nfmod_pol_reduce(nf,prhall,y); py = unifpol(nf,lift(py),1);
  px = unifpol(nf,nf_pol_mul(nf,px,py),0);
  tetpil=avma;
  return gerepile(av,tetpil,nfmod_pol_reduce(nf,prhall,px));
}

/* division euclidienne du polynome x par le polynome y */
static GEN
nf_pol_divres(GEN nf,GEN x,GEN y,GEN *pr)
{
  long av = avma,tetpil;
  GEN nq = poldivres(unifpol(nf,x,1),unifpol(nf,y,1),pr);
  GEN *gptr[2];

  tetpil=avma; nq=unifpol(nf,nq,0);
  if (pr) *pr = unifpol(nf,*pr,0);
  gptr[0]=&nq; gptr[1]=pr;
  gerepilemanysp(av,tetpil,gptr,pr ? 2:1);
  return nq;
}

/* division euclidienne du polynome x par le polynome y modulo prhall */
static GEN
nfmod_pol_divres(GEN nf,GEN prhall,GEN x,GEN y, GEN *pr)
{
  long av=avma,dx,dy,dz,i,j,k,l,n,tetpil;
  GEN z,p1,p3,px,py;

  py = nfmod_pol_reduce(nf,prhall,y);
  if (gcmp0(py))
    err(talker, "division by zero in nfmod_pol_divres");

  tetpil=avma;
  px=nfmod_pol_reduce(nf,prhall,x);
  dx=lgef(px)-3; dy=lgef(py)-3; dz=dx-dy;
  if (dx<dy)
  {
    GEN vzero;

    if (pr) *pr = gerepile(av,tetpil,px);
    else avma = av;

    n=lgef(nf[1])-3;
    vzero = cgetg(n+1,t_COL);
    n=lgef(nf[1])-3;
    for (i=1; i<=n; i++) vzero[i]=zero;

    z=cgetg(3,t_POL); z[2]=(long)vzero;
    z[1]=evallgef(2) | evalvarn(varn(px));
    return z;
  }
  p1 = NULL; /* gcc -Wall */

  z=cgetg(dz+3,t_POL); z[1]=evalsigne(1) | evallgef(3+dz);
  setvarn(z,varn(px));
  z[dz+2] = (long) element_divmodpr(nf,(GEN)px[dx+2],(GEN)py[dy+2],prhall);
  for (i=dx-1; i>=dy; --i)
  {
    l=avma; p1=(GEN)px[i+2];
    for (j=i-dy+1; j<=i && j<=dz; j++)
      p1 = gsub(p1, element_mul(nf,(GEN)z[j+2],(GEN)py[i-j+2]));
    p1 = nfreducemodpr(nf,p1,prhall);
    tetpil=avma; p3=element_divmodpr(nf,p1,(GEN)py[dy+2],prhall);
    z[i-dy+2]=lpile(l,tetpil,p3);
    z[i-dy+2]=(long)nfreducemodpr(nf,(GEN)z[i-dy+2],prhall);
  }
  l=avma;
  for (i=dy-1; i>=0; --i)
  {
    l=avma; p1=((GEN)px[i+2]);
    for (j=0; j<=i && j<=dz; j++)
      p1 = gsub(p1, element_mul(nf,(GEN)z[j+2],(GEN)py[i-j+2]));
    p1 = gerepileupto(l, nfreducemodpr(nf,p1,prhall));
    if (!gcmp0(p1)) break;
  }

  if (!pr) { avma = l; return z; }

  if (i<0)
  {
    avma=l;
    p3 = cgetg(3,t_POL); p3[2]=zero;
    p3[1] = evallgef(2) | evalvarn(varn(px));
    *pr=p3; return z;
  }

  p3=cgetg(i+3,t_POL);
  p3[1]=evalsigne(1) | evallgef(i+3) | evalvarn(varn(px));
  p3[i+2]=(long)nfreducemodpr(nf,p1,prhall);
  for (k=i-1; k>=0; --k)
  {
    l=avma; p1=((GEN)px[k+2]);
    for (j=0; j<=k && j<=dz; j++)
      p1 = gsub(p1, element_mul(nf,(GEN)z[j+2],(GEN)py[k-j+2]));
    p3[k+2]=lpileupto(l,nfreducemodpr(nf,p1,prhall));
  }
  *pr=p3; return z;
}

/* PGCD des polynomes x et y, par l'algorithme du sub-resultant */
static GEN
nf_pol_subres(GEN nf,GEN x,GEN y)
{
  long av=avma,tetpil;
  GEN s = srgcd(unifpol(nf,x,1), unifpol(nf,y,1));

  tetpil=avma; return gerepile(av,tetpil,unifpol(nf,s,1));
}

/* PGCD des polynomes x et y modulo prhall */
static GEN
nfmod_pol_gcd(GEN nf,GEN prhall,GEN x,GEN y)
{
  long av=avma;
  GEN p1,p2;

  if (lgef(x)<lgef(y)) { p1=y; y=x; x=p1; }
  p1=nfmod_pol_reduce(nf,prhall,x);
  p2=nfmod_pol_reduce(nf,prhall,y);
  while (!isexactzero(p2))
  {
    GEN p3;

    nfmod_pol_divres(nf,prhall,p1,p2,&p3);
    p1=p2; p2=p3;
  }
  return gerepileupto(av,p1);
}

/* Calcul pol^e modulo prhall et le polynome pmod */
static GEN
nfmod_pol_pow(GEN nf,GEN prhall,GEN pmod,GEN pol,GEN e)
{
  long i, av = avma, n = lgef(nf[1])-3;
  GEN p1,p2,vun;

  vun=cgetg(n+1,t_COL); vun[1]=un; for (i=2; i<=n; i++) vun[i]=zero;
  p1=gcopy(polun[varn(pol)]); p1[2]=(long)vun;
  if (gcmp0(e)) return p1;

  p2=nfmod_pol_reduce(nf,prhall,pol);
  for(;;)
  {
    if (!vali(e))
    {
      p1=nfmod_pol_mul(nf,prhall,p1,p2);
      nfmod_pol_divres(nf,prhall,p1,pmod,&p1);
    }
    if (gcmp1(e)) break;

    e=shifti(e,-1);
    p2=nfmod_pol_sqr(nf,prhall,p2);
    nfmod_pol_divres(nf,prhall,p2,pmod,&p2);
  }
  return gerepileupto(av,p1);
}

extern GEN trivfact(void);

/* factorisation du polynome x modulo pr */
GEN
nffactormod(GEN nf,GEN pol,GEN pr)
{
  long av = avma, tetpil,lb,nbfact,psim,N,n,i,j,k,d,e,vf,r,kk;
  GEN y,ex,*t,f1,f2,f3,df1,g1,polb,pold,polu,vker;
  GEN Q,f,x,u,v,v2,v3,vz,q,vun,vzero,prhall;

  nf=checknf(nf);
  if (typ(pol)!=t_POL) err(typeer,"nffactormod");
  if (varn(pol) >= varn(nf[1]))
    err(talker,"polynomial variable must have highest priority in nffactormod");

  prhall=nfmodprinit(nf,pr); n=lgef(nf[1])-3;
  vun = gscalcol_i(gun, n);
  vzero = gscalcol_i(gzero, n);
  q = vker = NULL; /* gcc -Wall */

  f=unifpol(nf,pol,0); f=nfmod_pol_reduce(nf,prhall,f);
  if (!signe(f)) err(zeropoler,"nffactormod");
  d=lgef(f)-3; vf=varn(f);
  if (d == 0) return trivfact();
  t  = (GEN*)new_chunk(d+1); ex = cgetg(d+1, t_VECSMALL);
  x=gcopy(polx[vf]); x[3]=(long)vun; x[2]=(long)vzero;
  if (d<=1) { nbfact=2; t[1] = f; ex[1]=1; }
  else
  {
    q = (GEN)pr[1]; psim = VERYBIGINT;
    if (!is_bigint(q)) psim = itos(q);
   /* psim has an effect only when p is small. If too big, set it to a huge
    * number (i.e ignore it) to avoid an error in itos on next line.
    */
    q=gpuigs(q, itos((GEN)pr[4]));
    f1=f; e=1; nbfact=1;
    while (lgef(f1)>3)
    {
      df1=deriv(f1,vf); f2=nfmod_pol_gcd(nf,prhall,f1,df1);
      g1=nfmod_pol_divres(nf,prhall,f1,f2,NULL); k=0;
      while (lgef(g1)>3)
      {
      k++;
      if (k%psim == 0)
      {
        k++; f2=nfmod_pol_divres(nf,prhall,f2,g1,NULL);
      }
      f3=nfmod_pol_gcd(nf,prhall,f2,g1);
      u = nfmod_pol_divres(nf,prhall,g1,f3,NULL);
      f2= nfmod_pol_divres(nf,prhall,f2,f3,NULL);
      g1=f3;
      if (lgef(u)>3)
      {
        N=lgef(u)-3; Q=cgetg(N+1,t_MAT);
        v3=cgetg(N+1,t_COL); Q[1]=(long)v3;
        v3[1]=(long)vun; for (i=2; i<=N; i++) v3[i]=(long)vzero;
      
        v2 = v = nfmod_pol_pow(nf,prhall,u,x,q);
        for (j=2; j<=N; j++)
        {
          v3=cgetg(N+1,t_COL); Q[j]=(long)v3;
          for (i=1; i<=lgef(v2)-2; i++) v3[i]=v2[i+1];
          for (; i<=N; i++) v3[i]=(long)vzero;
          if (j<N)
          {
            v2=nfmod_pol_mul(nf,prhall,v2,v);
            nfmod_pol_divres(nf,prhall,v2,u,&v2);
          }
        }
        for (i=1; i<=N; i++)
          coeff(Q,i,i)=lsub((GEN)coeff(Q,i,i),vun);
        v2=nfkermodpr(nf,Q,prhall); r=lg(v2)-1; t[nbfact]=gcopy(u); kk=1;
        if (r>1)
        {
          vker=cgetg(r+1,t_COL);
          for (i=1; i<=r; i++)
          {
            v3=cgetg(N+2,t_POL);
            v3[1]=evalsigne(1)+evallgef(2+N); setvarn(v3,vf);
            vker[i]=(long)v3; for (j=1; j<=N; j++) v3[j+1]=coeff(v2,j,i);
            normalizepol(v3);
          }
        }
        while (kk<r)
        {
          v=gcopy(polun[vf]); v[2]=(long)vzero;
          for (i=1; i<=r; i++)
          {
            vz=cgetg(n+1,t_COL);
            for (j=1; j<=n; j++)
            vz[j] = lmodsi(mymyrand()>>8, q);
            vz=nfreducemodpr(nf,vz,prhall);
            v=gadd(v,nfmod_pol_mul(nf,prhall,vz,(GEN)vker[i]));
          }
          for (i=1; i<=kk && kk<r; i++)
          {
            polb=t[nbfact+i-1]; lb=lgef(polb);
            if (lb>4)
            {
            if(psim==2)
              polu=nfmod_split2(nf,prhall,polb,v,q);
            else
            {
              polu=nfmod_pol_pow(nf,prhall,polb,v,shifti(q,-1));
                  polu=gsub(polu,vun);
            }
                pold=nfmod_pol_gcd(nf,prhall,polb,polu);
            if (lgef(pold)>3 && lgef(pold)<lb)
            {
              t[nbfact+i-1]=pold; kk++;
              t[nbfact+kk-1]=nfmod_pol_divres(nf,prhall,polb,pold,NULL);
            }
            }
          }
        }
        for (i=nbfact; i<nbfact+r; i++) ex[i]=e*k;
        nbfact+=r;
      }
      }
      e*=psim; j=(lgef(f2)-3)/psim+3; f1=cgetg(j,t_POL);
      f1[1] = evalsigne(1) | evallgef(j) | evalvarn(vf);
      for (i=2; i<j; i++)
      f1[i]=(long)element_powmodpr(nf,(GEN)f2[psim*(i-2)+2],
                             gdiv(q,(GEN)pr[1]),prhall);
    }
  }
  if (nbfact < 2)
    err(talker, "%Z not a prime (nffactormod)", q);
  for (j=1; j<nbfact; j++)
  {
    v = element_divmodpr(nf,vun,gmael(t,j,lgef(t[j])-1),prhall);
    t[j] = unifpol(nf,nfmod_pol_mul(nf,prhall,v,(GEN)t[j]),1);
  }

  tetpil=avma; y=cgetg(3,t_MAT);
  u=cgetg(nbfact,t_COL); y[1]=(long)u;
  v=cgetg(nbfact,t_COL); y[2]=(long)v;
  for (j=1,k=0; j<nbfact; j++)
    if (ex[j])
    {
      k++;
      u[k]=lcopy((GEN)t[j]);
      v[k]=lstoi(ex[j]);
    }
  return gerepile(av,tetpil,y);
}

/* Calcule pol + pol^2 + ... + pol^(q/2) modulo prhall et le polynome pmod */
static GEN
nfmod_split2(GEN nf,GEN prhall,GEN pmod,GEN pol,GEN exp)
{
  long av = avma;
  GEN p1,p2,q;

  if (cmpis(exp,2)<=0) return pol;
  p2=p1=pol; q=shifti(exp,-1);
  while (!gcmp1(q))
  {
    p2=nfmod_pol_sqr(nf,prhall,p2);
    nfmod_pol_divres(nf,prhall,p2,pmod,&p2);
    q=shifti(q,-1); p1=gadd(p1,p2);
  }
  return gerepileupto(av,p1);
}

/* If p doesn't divide either a or b and has a divisor of degree 1, return it.
 * Return NULL otherwise.
 */
static GEN
p_ok(GEN nf, GEN p, GEN a)
{
  long av,m,i;
  GEN dec;

  if (divise(a,p)) return NULL;
  av = avma; dec = primedec(nf,p); m=lg(dec);
  for (i=1; i<m; i++)
  {
    GEN pr = (GEN)dec[i];
    if (is_pm1(pr[4]))
      return pr;
  }   
  avma = av; return NULL;
}

/* for each new prime ct--, if ct = 0, return NULL */
static GEN
choose_prime(GEN nf, GEN dk, GEN lim, long ct)
{
  GEN p, pr;

  p = nextprime(lim);
  for (;;)
  {
    if ((pr = p_ok(nf,p,dk))) break;
    ct--;
    if (!ct) return NULL;
    p = nextprime(addis(p,2));
  }

  return pr;
}

/* Renvoie les racines de pol contenues dans nf */
GEN
nfroots(GEN nf,GEN pol)
{
  long av=avma,tetpil,i,d=lgef(pol);
  GEN p1,p2,polbase,polmod,den;

  nf=checknf(nf);
  if (typ(pol)!=t_POL) err(talker,"not a polynomial in nfroots");
  if (varn(pol) >= varn(nf[1]))
    err(talker,"polynomial variable must have highest priority in nfroots");

  polbase=unifpol(nf,pol,0);

  if (d==3)
  {
    tetpil=avma; p1=cgetg(1,t_VEC);
    return gerepile(av,tetpil,p1);
  }

  if (d==4)
  {
    tetpil=avma; p1=cgetg(2,t_VEC);
    p1[1] = (long)basistoalg(nf,gneg_i(
      element_div(nf,(GEN)polbase[2],(GEN)polbase[3])));
    return gerepile(av,tetpil,p1);
  }

  p1=element_inv(nf,leading_term(polbase));
  polbase=nf_pol_mul(nf,p1,polbase);

  den=gun;
  for (i=2; i<d; i++)
    if (! gcmp0((GEN)polbase[i]))
      den = glcm(den,denom((GEN)polbase[i]));
  if (! gcmp1(absi(den)))
    for (i=2; i<d; i++)
      polbase[i] = lmul(den,(GEN)polbase[i]);

  polmod=unifpol(nf,polbase,1);

  if (DEBUGLEVEL>=4)
    fprintferr("On teste si le polynome est square-free\n");

  p1=derivpol(polmod);
  p2=nf_pol_subres(nf,polmod,p1);

  if (degree(p2) > 0)
  {
    p1=element_inv(nf,leading_term(p2));
    p2=nf_pol_mul(nf,p1,p2);
    polmod=nf_pol_divres(nf,polmod,p2,NULL);

    p1=element_inv(nf,leading_term(polmod));
    polmod=nf_pol_mul(nf,p1,polmod);

    d=lgef(polmod); den=gun;
    for (i=2; i<d; i++)
      if (!gcmp0((GEN)polmod[i]))
      den = glcm(den,denom((GEN)polmod[i]));
    if (!gcmp1(absi(den)))
      for (i=2; i<d; i++)
      polmod[i] = lmul(den,(GEN)polmod[i]);

    polmod=unifpol(nf,polmod,1);
  }

  p1 = nfsqff(nf,polmod,1);
  tetpil=avma; return gerepile(av, tetpil, gen_sort(p1, 0, cmp_pol));
}

/* Relevement de elt modulo id minimal */
static GEN
nf_bestlift(GEN id,GEN idinv,GEN den,GEN elt)
{
  return gsub(elt,gmul(id,ground(gmul(den,gmul(idinv,elt)))));
}

/* Releve le polynome pol avec des coeff de norme t2 <= C si possible */
static GEN
nf_pol_lift(GEN id,GEN idinv,GEN den,GEN pol)
{
  long i, d = lgef(pol);
  GEN p1 = pol;

  pol=cgetg(d,t_POL); pol[1]=p1[1];
  for (i=2; i<d; i++)
    pol[i] = (long) nf_bestlift(id,idinv,den,(GEN)p1[i]);
  return pol;
}

#if 0
/* Evalue le polynome pol en elt */
static GEN
nf_pol_eval(GEN nf,GEN pol,GEN elt)
{
  long av=avma,tetpil,i;
  GEN p1;

  i=lgef(pol)-1; if (i==2) return gcopy((GEN)pol[2]);

  p1=element_mul(nf,(GEN)pol[i],elt);
  for (i--; i>=3; i--)
    p1=element_mul(nf,elt,gadd((GEN)pol[i],p1));
  tetpil=avma; return gerepile(av,tetpil,gadd(p1,(GEN)pol[2]));
}
#endif

/* Calcule la factorisation du polynome x dans nf */
GEN
nffactor(GEN nf,GEN pol)
{ GEN y,p1,p2,den,p3,p4,quot, rep = cgetg(3,t_MAT);
  long av = avma,tetpil,i,j,d;

  if (DEBUGLEVEL >= 4) timer2();

  nf=checknf(nf);
  if (typ(pol)!=t_POL) err(typeer,"nffactor");
  if (varn(pol) >= varn(nf[1]))
    err(talker,"polynomial variable must have highest priority in nffactor");

  d=lgef(pol);
  if (d==3)
  {
    rep[1]=lgetg(1,t_COL);
    rep[2]=lgetg(1,t_COL);
    return rep;
  }
  if (d==4)
  {
    p1=cgetg(2,t_COL); rep[1]=(long)p1; p1[1]=lcopy(pol);
    p1=cgetg(2,t_COL); rep[2]=(long)p1; p1[1]=un;
    return rep;
  }

  p1=element_inv(nf,leading_term(pol));
  pol=nf_pol_mul(nf,p1,pol);

  p1=unifpol(nf,pol,0); den=gun;
  for (i=2; i<d; i++)
    if (! gcmp0((GEN)p1[i]))
      den = glcm(den,denom((GEN)p1[i]));
  if (! gcmp1(absi(den)))
    for (i=2; i<d; i++)
      p1[i] = lmul(den,(GEN)p1[i]);

  if (DEBUGLEVEL>=4)
    fprintferr("On teste si le polynome est square-free\n");

  p2=derivpol(p1);
  p3=nf_pol_subres(nf,p1,p2);

  if (degree(p3) > 0)
  {
    p4=element_inv(nf,leading_term(p3));
    p3=nf_pol_mul(nf,p4,p3);

    p2=nf_pol_divres(nf,p1,p3,NULL);
    p4=element_inv(nf,leading_term(p2));
    p2=nf_pol_mul(nf,p4,p2);

    d=lgef(p2); den=gun;
    for (i=2; i<d; i++)
      if (!gcmp0((GEN)p2[i]))
      den = glcm(den,denom((GEN)p2[i]));
    if (!gcmp1(absi(den)))
      for (i=2; i<d; i++)
      p2[i] = lmul(den,(GEN)p2[i]);

    p2=unifpol(nf,p2,1); 
    tetpil = avma; y = nfsqff(nf,p2,0);
    i = nfcmbf.nfact;

    quot=nf_pol_divres(nf,p1,p2,NULL);
    p3=(GEN)gpmalloc((i+1) * sizeof(long));
    for (j=i; j>=1; j--)
    {
      GEN fact=(GEN)y[j], quo = quot, rem;
      long e = 0;

      do
      {
      quo = nf_pol_divres(nf,quo,fact,&rem);
      e++;
      }
      while (gcmp0(rem));
      p3[j]=lstoi(e);
    }
    avma = (long)y; y = gerepile(av, tetpil, y);
    p2=cgetg(i+1, t_COL); for (; i>=1; i--) p2[i]=lcopy((GEN)p3[i]);
    free(p3);
  }
  else
  {
    tetpil=avma; y = gerepile(av,tetpil,nfsqff(nf,p1,0));
    i = nfcmbf.nfact;
    p2=cgetg(i+1, t_COL); for (; i>=1; i--) p2[i]=un;
  }
  if (DEBUGLEVEL>=4)
    fprintferr("Nombre de facteur(s) trouve(s) : %ld\n", nfcmbf.nfact);
  rep[1]=(long)y;
  rep[2]=(long)p2; return sort_factor(rep, cmp_pol);
}

/* teste si la matrice M est suffisament orthogonale */
static long
test_mat(GEN M, GEN p, GEN C2, long k)
{
  long av = avma, i, N = lg(M);
  GEN min, prod, L2, R;

  min = prod = gcoeff(M,1,1);
  for (i = 2; i < N; i++)
  {
    L2 = gcoeff(M,i,i); prod = mpmul(prod,L2);
    if (mpcmp(L2,min) < 0) min = L2;
  }
  R = mpmul(min, gpowgs(p, k<<1));
  i = mpcmp(mpmul(C2,prod), R); avma = av;
  return (i < 0);
}

/* calcule la matrice correspondant a pr^e jusqu'a ce que R(pr^e) > C */
static GEN
T2_matrix_pow(GEN nf, GEN pre, GEN p, GEN C, long *kmax, long prec)
{
  long av=avma,av1,lim, k = *kmax, N = lgef((GEN)nf[1])-3;
  int tot_real = !signe(gmael(nf,2,2));
  GEN p1,p2,p3,u,C2,T2, x = (GEN)nf[1];

  C2 = gdiv(gmul2n(C,2), absi((GEN)nf[3]));
  p1 = gmul(pre,lllintpartial(pre)); av1 = avma;
  T2 = tot_real? gmael(nf,5,4)
               : nf_get_T2((GEN) nf[7], roots(x,prec));
  p3 = qf_base_change(T2,p1,1);

  if (N <= 6 && test_mat(p3,p,C2,k))
  {
    avma = av1; return gerepileupto(av,p1);
  }

  av1=avma; lim = stack_lim(av1,2);
  for (;;)
  {
    if (DEBUGLEVEL>2) fprintferr("exponent: %ld\n",k);

    for(;;)
    {
      u = tot_real? lllgramint(p3): lllgramintern(p3,100,1,prec);
      if (u) break;

      prec=(prec<<1)-2;
      if (DEBUGLEVEL > 1) err(warnprec,"nffactor[1]",prec);
      T2 = nf_get_T2((GEN) nf[7], roots(x,prec));
      p3 = qf_base_change(T2,p1,1);
    }
    if (DEBUGLEVEL>2) msgtimer("lllgram + base change");
    p3 = qf_base_change(p3,u,1);
    if (test_mat(p3,p,C2,k))
    {
      *kmax = k;
      return gerepileupto(av,gmul(p1,u));
    }

    /* il faut augmenter la precision en meme temps */
    p2=shifti(gceil(mulsr(k, glog(p,DEFAULTPREC))),-1);
    prec += (long)(itos(p2)*pariK1);
    if (DEBUGLEVEL > 1) err(warnprec,"nffactor[2]",prec);
    k = k<<1; p1 = idealmullll(nf,p1,p1);
    if (low_stack(lim, stack_lim(av1,2)))
    {
      if (DEBUGMEM>1) err(warnmem,"T2_matrix_pow");
      p1 = gerepileupto(av1,p1);
    }
    if (!tot_real) T2 = nf_get_T2((GEN) nf[7], roots(x,prec));
    p3 = qf_base_change(T2,p1,1);
  }
}

/* Calcule la factorisation du polynome x qui est sans facteurs carres dans nf,
   Le polynome est a coefficients dans Z_k et son terme dominant est un entier
   rationnel, si fl=1 renvoie seulement les racines du polynome contenues
   dans le corps */
static GEN
nfsqff(GEN nf,GEN pol, long fl)
{
  long d=lgef(pol),i,k,m,n,av=avma,tetpil,newprec,prec,nbf=BIGINT,anbf,ct=5;
  GEN p1,pr,p2,rep,k2,C,h,dk,dki,p,prh,p3,T2,polbase,fact,pk,ap,apr;
  GEN polmod,polred,hinv,lt,minp,den,maxp=shifti(gun,32),run,aprh;

  dk=absi((GEN)nf[3]);
  dki=mulii(dk,(GEN)nf[4]);
  n=lgef(nf[1])-3;

  polbase = unifpol(nf,pol,0);
  polmod  = unifpol(nf,pol,1);
  dki=mulii(dki,gnorm((GEN)polmod[d-1]));

  prec = DEFAULTPREC;
  for (;;)
  {
    if (prec <= gprecision(nf))
      T2 = gprec_w(gmael(nf,5,3), prec);
    else
      T2 = nf_get_T2((GEN)nf[7], roots((GEN)nf[1], prec));

    run=realun(prec);
    p1=realzero(prec);
    for (i=2; i<d; i++)
    { 
      p2 = gmul(run, (GEN)polbase[i]);
      p1 = addrr(p1, qfeval(T2, p2));
      if (signe(p1) < 0) break;
    }

    if (signe(p1) > 0) break;

    prec = (prec<<1)-2;
    if (DEBUGLEVEL > 1) err(warnprec, "nfsqff", prec);
  }

  if (DEBUGLEVEL>=4)
    fprintferr("La norme de ce polynome est : %Z\n", p1);

  lt=(GEN)leading_term(polbase)[1];
  p2=mulis(sqri(lt),n);
  C=mpadd(p1,p2);
  C=mpadd(C,gsqrt(gmul(p1,p2),prec));
          
  if (!fl)
    C=mpmul(C,sqri(binome(shifti(stoi(n-1),-1),(n-1)>>2)));
    
  C=gmul(C,sqri(lt));
  
  if (DEBUGLEVEL>=4)
    fprintferr("La borne de la norme des coeff du diviseur est : %Z\n", C);
  
  /* this is the theoretical bound for the exponent */
  /* k2=gadd(glog(gdivgs(C,n),DEFAULTPREC), mulsr(n*(n-1), dbltor(0.347)));
     k2=gmul2n(gmulgs(k2,n),-1); */

  /* we use instead the following heuristic bound */
  k2=gmul2n(gmulgs(glog(gdivgs(gmul2n(C,2),n),DEFAULTPREC),n),-1); 
  k2=mulrr(k2,dbltor(1.2));

  minp=gmin(gexp(gmul2n(k2,-6),BIGDEFAULTPREC), maxp);
  minp=gceil(minp);
  
  if (DEBUGLEVEL>=4)
  {
    fprintferr("borne inf. sur les nombres premiers : %Z\n", minp);
    msgtimer("Calcul des bornes");
  }

  p = rep = polred = NULL; /* gcc -Wall */
  pr=NULL;
  for (;;)
  {
    apr = choose_prime(nf,dki,minp, pr?30:0); 
    if (!apr) break;

    ap=(GEN)apr[1];
    aprh=prime_to_ideal(nf,apr);

    polred=gcopy(polbase);
    lt=(GEN)leading_term(polbase)[1];
    lt=mpinvmod(lt,ap);

    for (i=2; i<d; i++)
      polred[i] = nfreducemodpr_i(gmul(lt,(GEN)polbase[i]), aprh)[1];

    if (!divise(discsr(polred),ap))
    {
      rep=(GEN)simplefactmod(polred,ap)[1];
      anbf=lg(rep)-1;
      ct--;
      if (anbf < nbf)
      {
      nbf=anbf;
      pr=gcopy(apr);
      p=gcopy(ap);
      if (DEBUGLEVEL>=4)
      {
        fprintferr("Ideal premier pr considere pour decomposition: %Z\n", pr);
        fprintferr("Nombre de facteurs irreductibles modulo pr = %ld\n", nbf);
      }
      if (nbf == 1) break;
      }
    }
    if (pr && !ct) break;

    minp = addis(ap,1);
  }

  k = itos(gceil(gdiv(k2,glog(p,BIGDEFAULTPREC))));   

  if (DEBUGLEVEL>=4) msgtimer("Choix de l'ideal premier");

  if (lg(rep)==2)
  {
    if (fl) { avma=av; return cgetg(1,t_VEC); }
    rep=cgetg(2,t_VEC); rep[1]=lcopy(polmod);
    nfcmbf.nfact=1; return gerepileupto(av, rep);
  }

  p2=cgetr(DEFAULTPREC);
  affir(mulii(absi(dk),gpowgs(p,k)),p2);
  p2=shifti(gceil(mplog(p2)),-1);

  newprec = MEDDEFAULTPREC + (long)(itos(p2)*pariK1);
  if (DEBUGLEVEL>=4)
    fprintferr("nouvelle precision : %ld\n",newprec);

  prh = idealpows(nf,pr,k); m = k;
  h = T2_matrix_pow(nf,prh,p,C, &k, newprec);
  if (m != k) prh=idealpows(nf,pr,k); 

  if (DEBUGLEVEL>=4)
  {
    fprintferr("un exposant convenable est : %ld\n",(long)k);
    msgtimer("Calcul de H");
  }

  pk = gcoeff(prh,1,1);
  lt=(GEN)leading_term(polbase)[1];
  lt=mpinvmod(lt,pk);

  polred[1] = polbase[1];
  for (i=2; i<d; i++)
    polred[i] = nfreducemodpr_i(gmul(lt,(GEN)polbase[i]), prh)[1];

  fact = lift_intern((GEN)factmod(polred,p)[1]);
  rep = hensel_lift_fact(polred,fact,p,pk,k);

  if (DEBUGLEVEL >= 4) msgtimer("Calcul de la factorisation pr-adique");

  den=ginv(det(h));
  hinv=adj(h);
  lt=(GEN)leading_term(polbase)[1];

  if (fl)
  {
    long x_a[] = { evaltyp(t_POL)|m_evallg(4), 0,0,0 };
    GEN mlt = negi(lt), rem;
    x_a[1] = polbase[1]; setlgef(x_a, 4);
    x_a[3] = un;
    p1=cgetg(lg(rep)+1,t_VEC);
    polbase = unifpol(nf,polbase,1);
    for (m=1,i=1; i<lg(rep); i++)
    {
      p2=(GEN)rep[i]; if (lgef(p2)!=4) break;

      p3 = algtobasis(nf, gmul(mlt,(GEN)p2[2]));
      p3 = nf_bestlift(h,hinv,den,p3);
      p3 = gdiv(p3,lt);
      x_a[2] = lneg(p3); /* check P(p3) == 0 */
      p2 = poldivres(polbase, unifpol(nf,x_a,1), &rem);
      if (!signe(rem)) { polbase = p2; p1[m++] = (long)p3; }
    }
    tetpil=avma; rep=cgetg(m,t_VEC);
    for (i=1; i<m; i++) rep[i]=(long)basistoalg(nf,(GEN)p1[i]);
    return gerepile(av,tetpil,rep);
  }

  for (i=1; i<lg(rep); i++)
    rep[i] = (long)unifpol(nf,(GEN)rep[i],0);

  nfcmbf.pol      = polmod;
  nfcmbf.lt       = lt;
  nfcmbf.h        = h;
  nfcmbf.hinv     = hinv;
  nfcmbf.den      = den;
  nfcmbf.fact     = rep;
  nfcmbf.res      = cgetg(lg(rep)+1,t_VEC);
  nfcmbf.nfact    = 0;
  nfcmbf.nfactmod = lg(rep)-1;
  nf_combine_factors(nf,1,NULL,d-3,1);

  if (DEBUGLEVEL >= 4) msgtimer("Reconnaissance des facteurs");

  i = nfcmbf.nfact;
  if (lgef(nfcmbf.pol)>3)
  {
    nfcmbf.res[++i] = (long) nf_pol_divres(nf,nfcmbf.pol,nfcmbf.lt,NULL);
    nfcmbf.nfact = i;
  }

  tetpil=avma; rep=cgetg(i+1,t_VEC);
  for (  ; i>=1; i--)
    rep[i]=(long)unifpol(nf,(GEN)nfcmbf.res[i],1);
  return gerepile(av,tetpil,rep);
}

static int
nf_combine_factors(GEN nf,long fxn,GEN psf,long dlim,long hint)
{
  int val=0; /* assume failure */
  GEN newf, newpsf = NULL;
  long newd,ltop,i;

  if (dlim<=0) return 0;
  if (fxn > nfcmbf.nfactmod) return 0;
  /* first, try deeper factors without considering the current one */
  if (fxn != nfcmbf.nfactmod)
  {
    val=nf_combine_factors(nf,fxn+1,psf,dlim,hint);
    if (val && psf) return 1;
  }

  /* second, try including the current modular factor in the product */
  newf=(GEN)nfcmbf.fact[fxn];
  if (!newf) return val; /* modular factor already used */
  newd=lgef(newf)-3;
  if (newd>dlim) return val; /* degree of new factor is too large */

  if (newd%hint == 0)
  {
    GEN p, quot,rem;

    newpsf = nf_pol_mul(nf, (psf)? psf: nfcmbf.lt, newf);
    newpsf = nf_pol_lift(nfcmbf.h,nfcmbf.hinv,nfcmbf.den,newpsf);
    /* try out the new combination */
    ltop=avma;
    quot=nf_pol_divres(nf,nfcmbf.pol,newpsf,&rem);
    if (gcmp0(rem))  /* found a factor */
    {
      p = nf_pol_mul(nf,element_inv(nf,leading_term(newpsf)),newpsf);
      nfcmbf.res[++nfcmbf.nfact] = (long) p; /* store factor */
      nfcmbf.fact[fxn]=0;                    /* remove used modular factor */

      /* fix up target */
      p=gun; quot=unifpol(nf,quot,0);
      for (i=2; i<lgef(quot); i++)
      if (!gcmp0((GEN)quot[i]))
        p = glcm(p, denom((GEN)quot[i]));

      nfcmbf.pol = nf_pol_mul(nf,p,quot);
      nfcmbf.lt  = leading_term(nfcmbf.pol);
      return 1;
    }
    avma=ltop;
  }

  /* newpsf needs more; try for it */
  if (newd==dlim) return val; /* no more room in degree limit */
  if (fxn==nfcmbf.nfactmod) return val; /* no more modular factors to try */

  if (nf_combine_factors(nf,fxn+1,newpsf,dlim-newd,hint))
  {
    nfcmbf.fact[fxn]=0; /* remove used modular factor */
    return 1;
  }
  return val;
}

/* Calcule le polynome caracteristique de alpha sur nf ou alpha est un
   element de l'algebre nf[X]/(T) exprime comme polynome en x */
GEN
rnfcharpoly(GEN nf,GEN T,GEN alpha,int v)
{
  long av = avma, vnf, vT, lT;
  GEN p1;

  nf=checknf(nf); vnf = varn(nf[1]);
  if (v<0) v = 0;
  T = fix_relative_pol(nf,T);
  if (typ(alpha) == t_POLMOD) alpha = lift_to_pol(alpha);
  lT = lgef(T);
  if (typ(alpha) != t_POL || varn(alpha) == vnf)
    return gerepileupto(av, gpowgs(gsub(polx[v], alpha), lT - 3));
  vT = varn(T);
  if (varn(alpha) != vT || v >= vnf)
    err(talker,"incorrect variables in rnfcharpoly");
  if (lgef(alpha) >= lT) alpha = gmod(alpha,T);
  if (lT <= 4)
    return gerepileupto(av, gsub(polx[v], alpha));
  p1 = caract2(unifpol(nf,T,1), unifpol(nf,alpha,1), v);
  return gerepileupto(av, unifpol(nf,p1,1));
}

#if 0
/* Calcule le polynome minimal de alpha sur nf ou alpha est un
   element de l'algebre nf[X]/(T) exprime comme polynome en x */
GEN
rnfminpoly(GEN nf,GEN T,GEN alpha,int n)
{
  long av=avma,tetpil;
  GEN p1,p2;

  nf=checknf(nf); p1=rnfcharpoly(nf,T,alpha,n);
  tetpil=avma; p2=nf_pol_subres(nf,p1,deriv(p1,varn(T)));
  if (lgef(p2)==3) { avma=tetpil; return p1; }

  p1 = nf_pol_divres(nf,p1,p2,NULL);
  p2 = element_inv(nf,leading_term(p1));
  tetpil=avma; return gerepile(av,tetpil,unifpol(nf,nf_pol_mul(nf,p2,p1),1));
}
#endif

/* relative Dedekind criterion over nf, applied to the order defined by a
 * root of irreducible polynomial T, modulo the prime ideal pr. Returns
 * [flag,basis,val], where basis is a pseudo-basis of the enlarged order,
 * flag is 1 iff this order is pr-maximal, and val is the valuation in pr of
 * the order discriminant
 */
GEN
rnfdedekind(GEN nf,GEN T,GEN pr)
{
  long av=avma,vt,r,d,da,n,m,i,j;
  GEN p1,p2,p,tau,g,vecun,veczero,matid;
  GEN prhall,res,h,k,base,Ca;

  nf=checknf(nf); Ca=unifpol(nf,T,0);
  res=cgetg(4,t_VEC);
  if (typ(pr)==t_VEC && lg(pr)==3)
  { prhall = (GEN)pr[2]; pr = (GEN)pr[1]; }
  else
    prhall=nfmodprinit(nf,pr);
  p=(GEN)pr[1]; tau=gdiv((GEN)pr[5],p);
  n=lgef(nf[1])-3; m=lgef(T)-3;

  vecun = gscalcol_i(gun,n);
  veczero = zerocol(n);

  p1=(GEN)nffactormod(nf,Ca,pr)[1];
  r=lg(p1); if (r < 2) err(constpoler,"rnfdedekind");
  g=lift((GEN)p1[1]);
  for (i=2; i<r; i++)
    g = nf_pol_mul(nf,g, lift((GEN)p1[i]));
  h=nfmod_pol_divres(nf,prhall,Ca,g,NULL);
  k=nf_pol_mul(nf,tau,gsub(Ca, nf_pol_mul(nf,lift(g),lift(h))));
  p2=nfmod_pol_gcd(nf,prhall,g,h);
  k= nfmod_pol_gcd(nf,prhall,p2,k);

  d=lgef(k)-3;  /* <= m */
  vt = idealval(nf,discsr(T),pr) - 2*d;
  res[3]=lstoi(vt);
  if (!d || vt<=1) res[1]=un; else res[1]=zero;

  base=cgetg(3,t_VEC);
  p1=cgetg(m+d+1,t_MAT); base[1]=(long)p1;
  p2=cgetg(m+d+1,t_VEC); base[2]=(long)p2;
 /* if d > 0, base[2] temporarily multiplied by p, for the final nfhermitemod
  * [ which requires integral ideals ] */
  matid = gscalmat(d? p: gun, n);
  for (j=1; j<=m; j++)
  {
    p2[j]=(long)matid;
    p1[j]=lgetg(m+1,t_COL);
    for (i=1; i<=m; i++)
      coeff(p1,i,j) = (i==j)?(long)vecun:(long)veczero;
  }
  if (d)
  {
    GEN pal = lift(nfmod_pol_divres(nf,prhall,Ca,k,NULL));
    GEN prinvp = pidealprimeinv(nf,pr); /* again multiplied by p */
    GEN nfx = unifpol(nf,polx[varn(T)],0);

    for (   ; j<=m+d; j++)
    {
      p1[j]=lgetg(m+1,t_COL);
      da=lgef(pal)-3;
      for (i=1; i<=da+1; i++) coeff(p1,i,j)=pal[i+1];
      for (   ; i<=m; i++) coeff(p1,i,j)=(long)veczero;
      p2[j]=(long)prinvp;
      nf_pol_divres(nf,nf_pol_mul(nf,pal,nfx),T,&pal);
    }
    /* the modulus is integral */
    base = nfhermitemod(nf,base, gmul(gpowgs(p, m-d), 
                              idealpows(nf, prinvp, d))); 
    base[2] = ldiv((GEN)base[2], p); /* cancel the factor p */
  }
  res[2]=(long)base; return gerepileupto(av, gcopy(res));
}

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