GEN
nffactormod(GEN nf, GEN x, GEN pr)
{
long j, l, vx = varn(x), vn;
pari_sp av = avma;
GEN F, E, rep, xrd, modpr, T, p;
nf = checknf(nf);
vn = varn(nf[1]);
if (typ(x)!=t_POL) pari_err(typeer,"nffactormod");
if (varncmp(vx,vn) >= 0)
pari_err(talker,"polynomial variable must have highest priority in nffactormod");
modpr = nf_to_ff_init(nf, &pr, &T, &p);
xrd = modprX(x, nf, modpr);
rep = FqX_factor(xrd,T,p);
settyp(rep, t_MAT);
F = gel(rep,1); l = lg(F);
E = gel(rep,2); settyp(E, t_COL);
for (j = 1; j < l; j++) {
gel(F,j) = modprX_lift(gel(F,j), modpr);
gel(E,j) = stoi(E[j]);
}
return gerepilecopy(av, rep);
}
/* Let x be a polynomial with coefficients in Z or nf (vectors or polymods)
* return the same polynomial with coefficients expressed:
* if flag=t_COL: as vectors (on the integral basis).
* if flag=t_POLMOD: as polmods.
*/
GEN
unifpol(GEN nf, GEN x, long flag)
{
if (typ(x)==t_POL && varncmp(varn(x), varn(nf[1])) < 0)
{
long i, d = lg(x);
GEN y = cgetg(d,t_POL); y[1] = x[1];
for (i=2; i<d; i++) gel(y,i) = unifpol0(nf, gel(x,i), flag);
return y;
}
return unifpol0(nf, x, flag);
}
/* return the characteristic polynomial of alpha over nf, where alpha
is an element of the algebra nf[X]/(T) given as a polynomial in X */
GEN
rnfcharpoly(GEN nf, GEN T, GEN alpha, long v)
{
long vnf, vT, lT;
pari_sp av = avma;
GEN p1;
nf=checknf(nf); vnf = varn(nf[1]);
if (v<0) v = 0;
T = fix_relative_pol(nf,T,1);
if (typ(alpha) == t_POLMOD) alpha = lift_to_pol(alpha);
lT = lg(T);
if (typ(alpha) != t_POL || varn(alpha) == vnf)
return gerepileupto(av, gpowgs(gsub(pol_x[v], alpha), lT - 3));
vT = varn(T);
if (varn(alpha) != vT || varncmp(v, vnf)>=0)
pari_err(talker,"incorrect variables in rnfcharpoly");
if (lg(alpha) >= lT) alpha = RgX_rem(alpha, T);
if (lT <= 4)
return gerepileupto(av, gsub(pol_x[v], alpha));
p1 = caract2(T, unifpol(nf,alpha, t_POLMOD), v);
return gerepileupto(av, unifpol(nf, p1, t_POLMOD));
}
/* return the roots of pol in nf */
GEN
nfroots(GEN nf,GEN pol)
{
pari_sp av = avma;
GEN A,g, T;
long d;
if (!nf) return nfrootsQ(pol);
nf = checknf(nf); T = gel(nf,1);
if (typ(pol) != t_POL) pari_err(notpoler,"nfroots");
if (varncmp(varn(pol), varn(T)) >= 0)
pari_err(talker,"polynomial variable must have highest priority in nfroots");
d = degpol(pol);
if (d == 0) return cgetg(1,t_VEC);
if (d == 1)
{
A = gneg_i(gdiv(gel(pol,2),gel(pol,3)));
return gerepilecopy(av, mkvec( basistoalg(nf,A) ));
}
A = fix_relative_pol(nf,pol,0);
A = Q_primpart( lift_intern(A) );
if (DEBUGLEVEL>3) fprintferr("test if polynomial is square-free\n");
g = nfgcd(A, derivpol(A), T, gel(nf,4));
if (degpol(g))
{ /* not squarefree */
g = QXQX_normalize(g, T);
A = RgXQX_div(A,g,T);
}
A = QXQX_normalize(A, T);
A = Q_primpart(A);
A = nfsqff(nf,A,1);
A = RgXQV_to_mod(A, T);
return gerepileupto(av, gen_sort(A, 0, cmp_pol));
}
/* return the factorization of x in nf */
GEN
nffactor(GEN nf,GEN pol)
{
GEN A,g,y,p1,T, rep = cgetg(3, t_MAT);
long l, j, dA;
pari_sp av = avma;
pari_timer ti;
if (DEBUGLEVEL>2) { TIMERstart(&ti); fprintferr("\nEntering nffactor:\n"); }
nf = checknf(nf); T = gel(nf,1);
if (typ(pol) != t_POL) pari_err(notpoler,"nffactor");
if (varncmp(varn(pol), varn(T)) >= 0)
pari_err(talker,"polynomial variable must have highest priority in nffactor");
A = fix_relative_pol(nf,pol,0);
dA = degpol(A);
if (dA <= 0) {
avma = (pari_sp)(rep + 3);
return dA == 0? trivfact(): zerofact(varn(pol));
}
A = Q_primpart( QXQX_normalize(A, T) );
if (dA == 1) {
GEN c;
A = gerepilecopy(av, A); c = gel(A,2);
if (typ(c) == t_POL && degpol(c) > 0) gel(A,2) = mkpolmod(c, gcopy(T));
gel(rep,1) = mkcol(A);
gel(rep,2) = mkcol(gen_1); return rep;
}
if (degpol(T) == 1)
return gerepileupto(av, factpol(Q_primpart(simplify(pol)), 0));
A = Q_primpart( lift_intern(A) );
g = nfgcd(A, derivpol(A), T, gel(nf,4));
A = QXQX_normalize(A, T);
A = Q_primpart(A);
if (DEBUGLEVEL>2) msgTIMER(&ti, "squarefree test");
if (degpol(g))
{ /* not squarefree */
pari_sp av1;
GEN ex;
g = QXQX_normalize(g, T);
A = RgXQX_div(A,g, T);
y = nfsqff(nf,A,0); av1 = avma;
l = lg(y);
ex=(GEN)gpmalloc(l * sizeof(long));
for (j=l-1; j>=1; j--)
{
GEN fact = lift(gel(y,j)), quo = g, q;
long e = 0;
for(e = 1;; e++)
{
q = RgXQX_divrem(quo,fact,T, ONLY_DIVIDES);
if (!q) break;
quo = q;
}
ex[j] = e;
}
avma = av1; y = gerepileupto(av, RgXQXV_to_mod(y,T));
p1 = cgetg(l, t_COL); for (j=l-1; j>=1; j--) gel(p1,j) = utoipos(ex[j]);
free(ex);
}
else
{
y = gerepileupto(av, RgXQXV_to_mod(nfsqff(nf,A,0), T));
l = lg(y);
p1 = cgetg(l, t_COL); for (j=l-1; j>=1; j--) gel(p1,j) = gen_1;
}
if (DEBUGLEVEL>3)
fprintferr("number of factor(s) found: %ld\n", lg(y)-1);
gel(rep,1) = y;
gel(rep,2) = p1; return sort_factor(rep, cmp_pol);
}
static int
cmp_by_var(void *E,GEN x, GEN y)
{ (void)E; return varncmp((long)x,(long)y); }
static int
_lower(void *E, hashentry *e)
{ long v = (long)e->val; return (varncmp(v, (long)E) > 0); }
