/* log N_{F_P/Q_p}(x) / deg_F P */
static GEN
vtilde_i(GEN K, GEN x, GEN T, GEN deg, GEN ell, long prec)
{
GEN L, N, cx;
if (typ(x) != t_POL) x = nf_to_scalar_or_alg(K, x);
x = Q_primitive_part(x,&cx);
N = RgXQ_norm(x, T);
L = Qp_log(cvtop(N,ell,prec));
if (cx)
{
Q_pvalrem(cx, ell, &cx);
if (!isint1(cx))
L = gadd(L, gmulsg(degpol(T), Qp_log(cvtop(cx,ell,prec))));
}
return gdiv(L, deg);
}
/* min(v, v(Log_p Norm_{F_\p/Q_p}(x))) */
static long
vlognorm(GEN nf, GEN T, GEN x, GEN p, long v)
{
GEN a = nf_to_scalar_or_alg(nf, x);
GEN N = RgXQ_norm(a, T);
if (typ(N) != t_PADIC) N = cvtop(N, p, v);
return minss(v, valp( Qp_log(N) ));
}
/* v[i] = deg S[i] mod p^prec */
static GEN
get_vdegS(GEN Ftilde, GEN ell, long prec)
{
long i, l = lg(Ftilde);
GEN v = cgetg(l, t_VEC), degell = Qp_log( cvtop(ell1(ell), ell, prec) );
for (i = 1; i < l; i++) gel(v,i) = gmulsg(Ftilde[i], degell);
return v;
}
/* Assume a = 1 [4] */
static GEN
Z2XQ_invnorm(GEN a, GEN T, long e)
{
pari_timer ti;
GEN pe = int2n(e), s;
if (degpol(a)==0)
return Fp_inv(Fp_powu(gel(a,2), get_FpX_degree(T), pe), pe);
if (DEBUGLEVEL>=3) timer_start(&ti);
s = ZpXQ_log(a, T, gen_2, e);
if (DEBUGLEVEL>=3) timer_printf(&ti,"Z2XQ_log");
s = Fp_neg(FpXQ_trace(s, T, pe), pe);
if (DEBUGLEVEL>=3) timer_printf(&ti,"FpXQ_trace");
s = modii(gel(Qp_exp(cvtop(s, gen_2, e-2)),4),pe);
if (DEBUGLEVEL>=3) timer_printf(&ti,"Qp_exp");
return s;
}