/* 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;
}
/* || L ||_p^p in dimension n (L may be a scalar) */
static GEN
normlp(GEN L, long p, long n)
{
long i,l, t = typ(L);
GEN z;
if (!is_vec_t(t)) return gmulsg(n, gpowgs(L, p));
l = lg(L); z = gen_0;
/* assert(n == l-1); */
for (i=1; i<l; i++)
z = gadd(z, gpowgs(gel(L,i), p));
return z;
}
/* 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);
}
/* return a bound for T_2(P), P | polbase
* max |b_i|^2 <= 3^{3/2 + d} / (4 \pi d) [P]_2,
* where [P]_2 is Bombieri's 2-norm
* Sum over conjugates
*/
static GEN
nf_Beauzamy_bound(GEN nf, GEN polbase)
{
GEN lt,C,run,s, G = gmael(nf,5,2), POL, bin;
long i,prec,precnf, d = degpol(polbase), n = degpol(nf[1]);
precnf = gprecision(G);
prec = MEDDEFAULTPREC;
bin = vecbinome(d);
POL = polbase + 2;
/* compute [POL]_2 */
for (;;)
{
run= real_1(prec);
s = real_0(prec);
for (i=0; i<=d; i++)
{
GEN p1 = gnorml2(arch_for_T2(G, gmul(run, gel(POL,i)))); /* T2(POL[i]) */
if (!signe(p1)) continue;
if (lg(p1) == 3) break;
/* s += T2(POL[i]) / binomial(d,i) */
s = addrr(s, gdiv(p1, gel(bin,i+1)));
}
if (i > d) break;
prec = (prec<<1)-2;
if (prec > precnf)
{
nffp_t F; remake_GM(nf, &F, prec); G = F.G;
if (DEBUGLEVEL>1) pari_warn(warnprec, "nf_factor_bound", prec);
}
}
lt = leading_term(polbase);
s = gmul(s, mulis(sqri(lt), n));
C = powrshalf(stor(3,DEFAULTPREC), 3 + 2*d); /* 3^{3/2 + d} */
return gdiv(gmul(C, s), gmulsg(d, mppi(DEFAULTPREC)));
}
