/* return C in Z[X]/(p,T), C[ D[1] ] = 1, C[ D[i] ] = 0 otherwise. H is the
* list of degree 1 polynomials X - D[i] (come directly from factorization) */
static GEN
interpol(GEN H, GEN T, GEN p)
{
long i, m = lg(H);
GEN X = pol_x[0],d,p1,p2,a;
p1=pol_1[0]; p2=gen_1; a = gneg(constant_term(gel(H,1))); /* = D[1] */
for (i=2; i<m; i++)
{
d = constant_term(gel(H,i)); /* -D[i] */
p1 = FpXQX_mul(p1,gadd(X,d), T,p);
p2 = Fq_mul(p2, gadd(a,d), T,p);
}
return FqX_Fq_mul(p1,Fq_inv(p2, T,p), T,p);
}
static GEN
polsimplify(GEN x)
{
long i,lx = lg(x);
for (i=2; i<lx; i++)
if (typ(x[i]) == t_POL) gel(x,i) = constant_term(gel(x,i));
return x;
}
/* Return a subfield, gen_0 [ change p ] or NULL [ not a subfield ] */
static GEN
subfield(GEN A, blockdata *B)
{
long N, i, j, d, lf, m = lg(A)-1;
GEN M, pe, pol, fhk, g, e, d_1_term, delta, listdelta, whichdelta;
GEN T = B->S->T, p = B->S->p, firstroot = B->S->firstroot;
pol= (GEN)B->DATA[1]; N = degpol(pol); d = N/m; /* m | N */
pe = (GEN)B->DATA[2];
fhk= (GEN)B->DATA[3];
M = (GEN)B->DATA[8];
delta = cgetg(m+1,t_VEC);
whichdelta = cgetg(N+1, t_VECSMALL);
d_1_term = gen_0;
for (i=1; i<=m; i++)
{
GEN Ai = gel(A,i), p1 = (GEN)fhk[Ai[1]];
for (j=2; j<=d; j++)
p1 = Fq_mul(p1, (GEN)fhk[Ai[j]], T, pe);
gel(delta,i) = p1;
if (DEBUGLEVEL>2) fprintferr("delta[%ld] = %Z\n",i,p1);
/* g = prod (X - delta[i])
* if g o h = 0 (pol), we'll have h(Ai[j]) = delta[i] for all j */
/* fk[k] belongs to block number whichdelta[k] */
for (j=1; j<=d; j++) whichdelta[Ai[j]] = i;
if (typ(p1) == t_POL) p1 = constant_term(p1);
d_1_term = addii(d_1_term, p1);
}
d_1_term = centermod(d_1_term, pe); /* Tr(g) */
if (absi_cmp(d_1_term, gel(M,3)) > 0) {
if (DEBUGLEVEL>1) fprintferr("d-1 test failed\n");
return NULL;
}
g = FqV_roots_to_pol(delta, T, pe, 0);
g = centermod(polsimplify(g), pe); /* assume g in Z[X] */
if (DEBUGLEVEL>2) fprintferr("pol. found = %Z\n",g);
if (!ok_coeffs(g,M)) {
if (DEBUGLEVEL>1) fprintferr("coeff too big for pol g(x)\n");
return NULL;
}
if (!FpX_is_squarefree(g, p)) {
if (DEBUGLEVEL>1) fprintferr("changing f(x): p divides disc(g)\n");
compute_data(B);
return subfield(A, B);
}
lf = lg(firstroot); listdelta = cgetg(lf, t_VEC);
for (i=1; i<lf; i++) listdelta[i] = delta[whichdelta[firstroot[i]]];
if (DEBUGLEVEL) fprintferr("candidate = %Z\n", g);
e = embedding(g, B->DATA, B->S, B->PD->den, listdelta);
if (!e) return NULL;
if (DEBUGLEVEL) fprintferr("embedding = %Z\n", e);
return _subfield(g, e);
}
/* return U list of polynomials s.t U[i] = 1 mod fk[i] and 0 mod fk[j] for all
* other j */
static GEN
get_bezout(GEN pol, GEN fk, GEN p)
{
long i, l = lg(fk);
GEN A, B, d, u, v, U = cgetg(l, t_VEC);
for (i=1; i<l; i++)
{
A = gel(fk,i);
B = FpX_div(pol, A, p);
d = FpX_extgcd(A,B,p, &u, &v);
if (degpol(d) > 0) pari_err(talker, "relatively prime polynomials expected");
d = constant_term(d);
if (!gcmp1(d)) v = FpX_Fp_mul(v, Fp_inv(d, p), p);
gel(U,i) = FpX_mul(B,v, p);
}
return U;
}
static GEN
unifpol0(GEN nf,GEN x,long flag)
{
switch(typ(x))
{
case t_INT: case t_FRAC:
return gcopy(x);
case t_POLMOD:
x = gel(x,2); /* fall through */
if (typ(x) != t_POL) return gcopy(x); /* scalar */
case t_POL:
if (!degpol(x)) return gcopy(constant_term(x));
return (flag == t_COL)? algtobasis(nf, x): gmodulo(x, gel(nf,1));
default: /* t_COL */
return (flag == t_COL)? gcopy(x): basistoalg(nf, x);
}
}
static term_t convert_unint(term_table_t *terms, value_table_t *vtbl, value_t v) {
value_unint_t *u;
u = vtbl_unint(vtbl, v);
return constant_term(terms, u->type, u->index);
}
