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/*
Copyright (C) 2017 Fredrik Johansson
This file is part of Arb.
Arb is free software: you can redistribute it and/or modify it under
the terms of the GNU Lesser General Public License (LGPL) as published
by the Free Software Foundation; either version 2.1 of the License, or
(at your option) any later version. See <http://www.gnu.org/licenses/>.
*/
#include "arb_fmpz_poly.h"
#include "acb_dirichlet.h"
void
arb_fmpz_poly_gauss_period_minpoly(fmpz_poly_t res, ulong q, ulong n)
{
ulong k, d, e, g, gk, qinv;
ulong * es;
slong prec, initial_prec;
int done, real;
if (n == 0 || !n_is_prime(q) || ((q - 1) % n) != 0 ||
n_gcd(n, (q - 1) / n) != 1)
{
fmpz_poly_zero(res);
return;
}
d = (q - 1) / n;
/* this is much faster */
if (d == 1)
{
fmpz_poly_cyclotomic(res, q);
return;
}
g = n_primitive_root_prime(q);
qinv = n_preinvert_limb(q);
es = flint_malloc(sizeof(ulong) * d);
for (e = 0; e < d; e++)
es[e] = n_powmod2(g, n * e, q);
/* either all roots are real, or all roots are complex */
real = (n % 2) == 1;
/* first estimate precision crudely based on d and n */
initial_prec = n * log(2 * d) * 1.4426950408889 * 1.03 + 20;
initial_prec = FLINT_MAX(initial_prec, 48);
/* if high, start lower to get a good estimate */
if (initial_prec > 200)
initial_prec = 48;
for (prec = initial_prec, done = 0; !done; )
{
acb_dirichlet_roots_t zeta;
arb_poly_t pz;
arb_ptr roots;
acb_ptr croots;
acb_t t, u;
arb_t v;
acb_dirichlet_roots_init(zeta, q, (n * d) / 2, prec);
roots = _arb_vec_init(n);
croots = (acb_ptr) roots;
acb_init(t);
if (!real)
acb_init(u);
else
arb_init(v);
arb_poly_init(pz);
for (k = 0; k < (real ? n : n / 2); k++)
{
gk = n_powmod2(g, k, q);
if (real)
{
arb_zero(v);
for (e = 0; e < d / 2; e++)
{
acb_dirichlet_root(t, zeta, n_mulmod2_preinv(gk, es[e], q, qinv), prec);
arb_add(v, v, acb_realref(t), prec);
}
arb_mul_2exp_si(v, v, 1); /* compute conjugates */
arb_set(roots + k, v);
}
else
{
acb_zero(u);
for (e = 0; e < d; e++)
{
acb_dirichlet_root(t, zeta, n_mulmod2_preinv(gk, es[e], q, qinv), prec);
acb_add(u, u, t, prec);
}
if (arb_contains_zero(acb_imagref(u)))
{
/* todo: could increase precision */
flint_printf("fail! imaginary part should be nonzero\n");
flint_abort();
}
else
{
acb_set(croots + k, u);
}
}
}
if (real)
arb_poly_product_roots(pz, roots, n, prec);
else
arb_poly_product_roots_complex(pz, NULL, 0, croots, n / 2, prec);
done = arb_poly_get_unique_fmpz_poly(res, pz);
if (!done && prec == initial_prec)
{
mag_t m, mmax;
mag_init(m);
mag_init(mmax);
for (k = 0; k < n; k++)
{
arb_get_mag(m, pz->coeffs + k);
mag_max(mmax, mmax, m);
}
prec = mag_get_d_log2_approx(mmax) * 1.03 + 20;
if (prec < 2 * initial_prec)
prec = 2 * initial_prec;
mag_clear(m);
mag_clear(mmax);
}
else if (!done)
{
prec *= 2;
}
acb_dirichlet_roots_clear(zeta);
_arb_vec_clear(roots, n);
acb_clear(t);
if (!real)
acb_clear(u);
else
arb_clear(v);
arb_poly_clear(pz);
}
flint_free(es);
}
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