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/*
Copyright (C) 2011, 2012 Sebastian Pancratz
This file is part of FLINT.
FLINT 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 3 of the License, or
(at your option) any later version. See <https://www.gnu.org/licenses/>.
*/
#include "fmpz_vec.h"
#include "padic.h"
void _padic_teichmuller(fmpz_t rop, const fmpz_t op, const fmpz_t p, slong N)
{
if (fmpz_equal_ui(p, 2))
{
fmpz_one(rop);
}
else if (N == 1)
{
fmpz_mod(rop, op, p);
}
else
{
slong *a, i, n;
fmpz *pow, *u;
fmpz_t s, t;
fmpz_t inv;
fmpz_t pm1;
a = _padic_lifts_exps(&n, N);
pow = _fmpz_vec_init(2 * n);
u = pow + n;
_padic_lifts_pows(pow, a, n, p);
fmpz_init(s);
fmpz_init(t);
fmpz_init(inv);
fmpz_init(pm1);
fmpz_sub_ui(pm1, p, 1);
/* Compute reduced units for (p-1) */
{
fmpz_mod(u + 0, pm1, pow + 0);
}
for (i = 1; i < n; i++)
{
fmpz_mod(u + i, u + (i - 1), pow + i);
}
/* Run Newton iteration */
i = n - 1;
{
fmpz_mod(rop, op, pow + i);
fmpz_set(inv, pm1);
}
for (i--; i >= 0; i--)
{
/* Lift rop */
fmpz_powm(s, rop, p, pow + i);
fmpz_sub(s, s, rop);
fmpz_mul(t, s, inv);
fmpz_sub(rop, rop, t);
fmpz_mod(rop, rop, pow + i);
/* Lift inv */
if (i > 0)
{
fmpz_mul(s, inv, inv);
fmpz_mul(t, u + i, s);
fmpz_mul_2exp(inv, inv, 1);
fmpz_sub(inv, inv, t);
fmpz_mod(inv, inv, pow + i);
}
}
fmpz_clear(s);
fmpz_clear(t);
fmpz_clear(inv);
fmpz_clear(pm1);
_fmpz_vec_clear(pow, 2 * n);
flint_free(a);
}
}
void padic_teichmuller(padic_t rop, const padic_t op, const padic_ctx_t ctx)
{
if (padic_val(op) < 0)
{
flint_throw(FLINT_ERROR, "Exception (padic_teichmuller). op is not a p-adic integer.\n");
}
if (padic_is_zero(op) || padic_val(op) > 0 || padic_prec(rop) <= 0)
{
padic_zero(rop);
}
else
{
_padic_teichmuller(padic_unit(rop), padic_unit(op), ctx->p, padic_prec(rop));
padic_val(rop) = 0;
}
}