309 lines
8.0 KiB
C
309 lines
8.0 KiB
C
/*=============================================================================
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This file is part of FLINT.
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FLINT is free software; you can redistribute it and/or modify
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it under the terms of the GNU General Public License as published by
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the Free Software Foundation; either version 2 of the License, or
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(at your option) any later version.
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FLINT is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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GNU General Public License for more details.
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You should have received a copy of the GNU General Public License
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along with FLINT; if not, write to the Free Software
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Foundation, Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA
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=============================================================================*/
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/******************************************************************************
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Copyright (C) 2012 Sebastian Pancratz
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******************************************************************************/
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#include "fmpz_mod_poly.h"
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#include "qadic.h"
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extern slong _padic_log_bound(slong v, slong N, const fmpz_t p);
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/*
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Assumes that P, T are vectors of length 2 d - 1.
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Assumes that 0 < len <= d.
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Assumes that 1 <= lo < hi.
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Does not support aliasing.
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*/
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static void
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_qadic_log_bsplit_series(fmpz *P, fmpz_t B, fmpz *T,
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const fmpz *y, slong len, slong lo, slong hi,
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const fmpz *a, const slong *j, slong lena)
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{
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const slong d = j[lena - 1];
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if (hi - lo == 1)
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{
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_fmpz_vec_set(P, y, len);
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_fmpz_vec_zero(P + len, 2*d - 1 - len);
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fmpz_set_si(B, lo);
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_fmpz_vec_set(T, P, 2*d - 1);
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}
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else if (hi - lo == 2)
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{
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_fmpz_poly_sqr(P, y, len);
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_fmpz_vec_zero(P + (2*len - 1), d - (2*len - 1));
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_fmpz_poly_reduce(P, 2*len - 1, a, j, lena);
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fmpz_set_si(B, lo);
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fmpz_mul_si(B, B, lo + 1);
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_fmpz_vec_scalar_mul_si(T, y, len, lo + 1);
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_fmpz_vec_zero(T + len, d - len);
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_fmpz_vec_scalar_addmul_si(T, P, d, lo);
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}
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else
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{
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const slong m = (lo + hi) / 2;
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fmpz *RP, *RT, *W;
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fmpz_t RB;
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RP = _fmpz_vec_init(2*d - 1);
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RT = _fmpz_vec_init(2*d - 1);
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W = _fmpz_vec_init(2*d - 1);
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fmpz_init(RB);
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_qadic_log_bsplit_series(P, B, T, y, len, lo, m, a, j, lena);
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_qadic_log_bsplit_series(RP, RB, RT, y, len, m, hi, a, j, lena);
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_fmpz_poly_mul(W, RT, d, P, d);
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_fmpz_poly_reduce(W, 2*d - 1, a, j, lena);
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_fmpz_vec_swap(RT, W, d);
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_fmpz_vec_scalar_mul_fmpz(T, T, d, RB);
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_fmpz_vec_scalar_addmul_fmpz(T, RT, d, B);
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_fmpz_poly_mul(W, P, d, RP, d);
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_fmpz_poly_reduce(W, 2*d - 1, a, j, lena);
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_fmpz_vec_swap(P, W, d);
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fmpz_mul(B, B, RB);
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_fmpz_vec_clear(RP, 2*d - 1);
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_fmpz_vec_clear(RT, 2*d - 1);
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_fmpz_vec_clear(W, 2*d - 1);
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fmpz_clear(RB);
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}
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}
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/*
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Sets (z, d) to the sum
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sum_{i=1}^{\infty} y^i / i mod p^N.
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The result may not be reduced modulo p^N, but it is
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reduced modulo f(X) given by the data (a, j, lena).
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Supports aliasing between y and z.
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*/
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static void
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_qadic_log_bsplit(fmpz *z, const fmpz *y, slong v, slong len,
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const fmpz *a, const slong *j, slong lena,
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const fmpz_t p, slong N)
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{
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const slong d = j[lena - 1];
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fmpz *P, *T;
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fmpz_t B, C;
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slong n;
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n = _padic_log_bound(v, N, p);
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n = FLINT_MAX(n, 2);
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P = _fmpz_vec_init(2*d - 1);
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T = _fmpz_vec_init(2*d - 1);
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fmpz_init(B);
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fmpz_init(C);
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_qadic_log_bsplit_series(P, B, T, y, len, 1, n, a, j, lena);
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n = fmpz_remove(B, B, p);
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fmpz_pow_ui(C, p, n);
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_fmpz_vec_scalar_divexact_fmpz(T, T, d, C);
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_padic_inv(B, B, p, N);
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_fmpz_vec_scalar_mul_fmpz(z, T, d, B);
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_fmpz_vec_clear(P, 2*d - 1);
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_fmpz_vec_clear(T, 2*d - 1);
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fmpz_clear(B);
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fmpz_clear(C);
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}
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/*
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Computes
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\begin{equation*}
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z = - \sum_{i = 1}^{\infty} \frac{y^i}{i} \pmod{p^N}.
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\end{equation*}
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Note that this can be used to compute the $p$-adic logarithm
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via the equation
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\begin{align*}
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\log(x) & = \sum_{i=1}^{\infty} (-1)^{i-1} \frac{(x-1)^i}{i} \\
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& = - \sum_{i=1}^{\infty} \frac{(1-x)^i}{i}.
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\end{align*}
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Assumes that $y = 1 - x$ is non-zero and that $v = \ord_p(y)$
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is at least $1$ when $p$ is odd and at least $2$ when $p = 2$
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so that the series converges.
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Assumes that $v < N$.
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Sets $(z, d)$.
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Ensures that the result is reduced modulo $p^N$.
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Does not support aliasing between $y$ and $z$.
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*/
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void _qadic_log_balanced(fmpz *z, const fmpz *y, slong len,
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const fmpz *a, const slong *j, slong lena,
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const fmpz_t p, slong N, const fmpz_t pN)
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{
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const slong d = j[lena - 1];
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fmpz_t pv;
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fmpz *r, *s, *t, *u;
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slong i, w;
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r = _fmpz_vec_init(d);
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s = _fmpz_vec_init(2*d - 1);
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t = _fmpz_vec_init(d);
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u = _fmpz_vec_init(d);
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fmpz_init(pv);
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fmpz_set(pv, p);
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_fmpz_vec_scalar_mod_fmpz(t, y, len, pN);
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_fmpz_vec_zero(z, d);
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w = 1;
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while (!_fmpz_vec_is_zero(t, d))
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{
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fmpz_mul(pv, pv, pv);
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for (i = 0; i < d; i++)
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{
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fmpz_fdiv_qr(t + i, r + i, t + i, pv);
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}
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if (!_fmpz_vec_is_zero(t, d))
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{
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_fmpz_vec_scalar_mul_fmpz(t, t, d, pv);
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/* Temporarily set r = 1 - r to compute u = (1-r)^{-1} */
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fmpz_sub_ui(r + 0, r + 0, 1);
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_fmpz_vec_neg(r, r, d);
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_qadic_inv(u, r, d, a, j, lena, p, N);
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_fmpz_vec_neg(r, r, d);
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fmpz_add_ui(r + 0, r + 0, 1);
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_fmpz_poly_mul(s, t, d, u, d);
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_fmpz_poly_reduce(s, 2 * d - 1, a, j, lena);
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_fmpz_vec_scalar_mod_fmpz(t, s, d, pN);
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}
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if (!_fmpz_vec_is_zero(r, d))
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{
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_qadic_log_bsplit(r, r, w, d, a, j, lena, p, N);
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_fmpz_vec_sub(z, z, r, d);
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_fmpz_vec_scalar_mod_fmpz(z, z, d, pN);
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}
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w *= 2;
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}
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_fmpz_vec_clear(r, d);
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_fmpz_vec_clear(s, 2*d - 1);
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_fmpz_vec_clear(t, d);
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_fmpz_vec_clear(u, d);
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fmpz_clear(pv);
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}
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int qadic_log_balanced(qadic_t rop, const qadic_t op, const qadic_ctx_t ctx)
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{
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const fmpz *p = (&ctx->pctx)->p;
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const slong d = qadic_ctx_degree(ctx);
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const slong N = qadic_prec(rop);
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const slong len = op->length;
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if (op->val < 0)
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{
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return 0;
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}
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else
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{
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fmpz *x;
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fmpz_t pN;
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int alloc, ans;
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x = _fmpz_vec_init(len + 1);
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alloc = _padic_ctx_pow_ui(pN, N, &ctx->pctx);
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/* Set x := (1 - op) mod p^N */
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fmpz_pow_ui(x + len, p, op->val);
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_fmpz_vec_scalar_mul_fmpz(x, op->coeffs, len, x + len);
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fmpz_sub_ui(x + 0, x + 0, 1);
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_fmpz_vec_neg(x, x, len);
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_fmpz_vec_scalar_mod_fmpz(x, x, len, pN);
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if (_fmpz_vec_is_zero(x, len))
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{
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padic_poly_zero(rop);
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ans = 1;
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}
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else
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{
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const slong v = _fmpz_vec_ord_p(x, len, p);
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if (v >= 2 || (*p != WORD(2) && v >= 1))
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{
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if (v >= N)
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{
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padic_poly_zero(rop);
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}
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else
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{
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padic_poly_fit_length(rop, d);
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_qadic_log_balanced(rop->coeffs, x, len,
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ctx->a, ctx->j, ctx->len, p, N, pN);
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rop->val = 0;
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_padic_poly_set_length(rop, d);
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_padic_poly_normalise(rop);
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padic_poly_canonicalise(rop, p);
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}
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ans = 1;
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}
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else
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{
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ans = 0;
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}
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}
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_fmpz_vec_clear(x, len + 1);
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if (alloc)
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fmpz_clear(pN);
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return ans;
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}
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}
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