663 lines
24 KiB
Plaintext
663 lines
24 KiB
Plaintext
/*=============================================================================
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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) 2011, 2012 Sebastian Pancratz
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******************************************************************************/
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*******************************************************************************
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Module documentation
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We represent a polynomial in $\mathbf{Q}_p[x]$ as a
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product $p^v f(x)$, where $p$ is a prime number,
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$v \in \mathbf{Z}$ and $f(x) \in \mathbf{Z}[x]$.
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As a data structure, we call this polynomial \emph{normalised}
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if the polynomial $f(x)$ is \emph{normalised}, that is, if the top
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coefficient is non-zero.
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We say this polynomial is in \emph{canonical form} if one of the
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coefficients of $f(x)$ is a $p$-adic unit. If $f(x)$ is the zero
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polynomial, we require that $v = 0$.
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We say this polynomial is \emph{reduced} modulo $p^N$ if it is
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canonical form and if all coefficients lie in the range $[0, p^N)$.
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*******************************************************************************
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*******************************************************************************
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Memory management
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*******************************************************************************
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void padic_poly_init(padic_poly_t poly)
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Initialises \code{poly} for use, setting its length to zero.
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The precision of the polynomial is set to \code{PADIC_DEFAULT_PREC}.
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A corresponding call to \code{padic_poly_clear()} must be made
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after finishing with the \code{padic_poly_t} to free the memory
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used by the polynomial.
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void padic_poly_init2(padic_poly_t poly, slong alloc, slong prec)
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Initialises \code{poly} with space for at least \code{alloc} coefficients
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and sets the length to zero. The allocated coefficients are all set to
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zero. The precision is set to \code{prec}.
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void padic_poly_realloc(padic_poly_t poly, slong alloc, const fmpz_t p)
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Reallocates the given polynomial to have space for \code{alloc}
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coefficients. If \code{alloc} is zero the polynomial is cleared
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and then reinitialised. If the current length is greater than
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\code{alloc} the polynomial is first truncated to length \code{alloc}.
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void padic_poly_fit_length(padic_poly_t poly, slong len)
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If \code{len} is greater than the number of coefficients currently
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allocated, then the polynomial is reallocated to have space for at
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least \code{len} coefficients. No data is lost when calling this
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function.
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The function efficiently deals with the case where \code{fit_length} is
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called many times in small increments by at least doubling the number
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of allocated coefficients when length is larger than the number of
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coefficients currently allocated.
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void _padic_poly_set_length(padic_poly_t poly, slong len)
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Demotes the coefficients of \code{poly} beyond \code{len} and sets
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the length of \code{poly} to \code{len}.
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Note that if the current length is greater than \code{len} the
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polynomial may no slonger be in canonical form.
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void padic_poly_clear(padic_poly_t poly)
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Clears the given polynomial, releasing any memory used. It must
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be reinitialised in order to be used again.
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void _padic_poly_normalise(padic_poly_t poly)
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Sets the length of \code{poly} so that the top coefficient is non-zero.
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If all coefficients are zero, the length is set to zero. This function
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is mainly used internally, as all functions guarantee normalisation.
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void _padic_poly_canonicalise(fmpz *poly, slong *v, slong len, const fmpz_t p)
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void padic_poly_canonicalise(padic_poly_t poly, const fmpz_t p)
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Brings the polynomial \code{poly} into canonical form,
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assuming that it is normalised already. Does \emph{not}
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carry out any reduction.
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void padic_poly_reduce(padic_poly_t poly, const padic_ctx_t ctx)
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Reduces the polynomial \code{poly} modulo $p^N$, assuming
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that it is in canonical form already.
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void padic_poly_truncate(padic_poly_t poly, slong n, const fmpz_t p)
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Truncates the polynomial to length at most~$n$.
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*******************************************************************************
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Polynomial parameters
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*******************************************************************************
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slong padic_poly_degree(padic_poly_t poly)
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Returns the degree of the polynomial \code{poly}.
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slong padic_poly_length(padic_poly_t poly)
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Returns the length of the polynomial \code{poly}.
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slong padic_poly_val(padic_poly_t poly)
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Returns the valuation of the polynomial \code{poly},
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which is defined to be the minimum valuation of all
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its coefficients.
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The valuation of the zero polynomial is~$0$.
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Note that this is implemented as a macro and can be
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used as either a \code{lvalue} or a \code{rvalue}.
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slong padic_poly_prec(padic_poly_t poly)
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Returns the precision of the polynomial \code{poly}.
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Note that this is implemented as a macro and can be
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used as either a \code{lvalue} or a \code{rvalue}.
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Note that increasing the precision might require
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a call to \code{padic_poly_reduce()}.
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*******************************************************************************
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Randomisation
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*******************************************************************************
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void padic_poly_randtest(padic_poly_t f, flint_rand_t state,
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slong len, const padic_ctx_t ctx)
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Sets $f$ to a random polynomial of length at most \code{len}
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with entries reduced modulo $p^N$.
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void padic_poly_randtest_not_zero(padic_poly_t f, flint_rand_t state,
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slong len, const padic_ctx_t ctx)
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Sets $f$ to a non-zero random polynomial of length at most \code{len}
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with entries reduced modulo $p^N$.
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void padic_poly_randtest_val(padic_poly_t f, flint_rand_t state,
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slong val, slong len, const padic_ctx_t ctx)
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Sets $f$ to a random polynomial of length at most \code{len}
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with at most the prescribed valuation \code{val} and entries
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reduced modulo $p^N$.
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Specifically, we aim to set the valuation to be exactly equal
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to \code{val}, but do not check for additional cancellation
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when creating the coefficients.
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*******************************************************************************
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Assignment and basic manipulation
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*******************************************************************************
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void padic_poly_set_padic(padic_poly_t poly, const padic_t x,
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const padic_ctx_t ctx)
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Sets the polynomial \code{poly} to the $p$-adic number $x$,
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reduced to the precision of the polynomial.
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void padic_poly_set(padic_poly_t poly1, const padic_poly_t poly2,
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const padic_ctx_t ctx)
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Sets the polynomial \code{poly1} to the polynomial \code{poly2},
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reduced to the precision of \code{poly1}.
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void padic_poly_set_si(padic_poly_t poly, slong x, const padic_ctx_t ctx)
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Sets the polynomial \code{poly} to the \code{signed slong}
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integer $x$ reduced to the precision of the polynomial.
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void padic_poly_set_ui(padic_poly_t poly, ulong x, const padic_ctx_t ctx)
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Sets the polynomial \code{poly} to the \code{unsigned slong}
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integer $x$ reduced to the precision of the polynomial.
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void padic_poly_set_fmpz(padic_poly_t poly, const fmpz_t x,
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const padic_ctx_t ctx)
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Sets the polynomial \code{poly} to the integer $x$
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reduced to the precision of the polynomial.
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void padic_poly_set_fmpq(padic_poly_t poly, const fmpq_t x,
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const padic_ctx_t ctx)
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Sets the polynomial \code{poly} to the value of the rational $x$,
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reduced to the precision of the polynomial.
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void padic_poly_set_fmpz_poly(padic_poly_t rop, const fmpz_poly_t op,
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const padic_ctx_t ctx)
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Sets the polynomial \code{rop} to the integer polynomial \code{op}
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reduced to the precision of the polynomial.
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void padic_poly_set_fmpq_poly(padic_poly_t rop,
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const fmpq_poly_t op, const padic_ctx_t ctx)
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Sets the polynomial \code{rop} to the value of the rational
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polynomial \code{op}, reduced to the precision of the polynomial.
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int padic_poly_get_fmpz_poly(fmpz_poly_t rop, const padic_poly_t op,
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const padic_ctx_t ctx)
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Sets the integer polynomial \code{rop} to the value of the $p$-adic
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polynomial \code{op} and returns $1$ if the polynomial is $p$-adically
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integral. Otherwise, returns $0$.
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void padic_poly_get_fmpq_poly(fmpq_poly_t rop,
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const padic_poly_t op, const padic_ctx_t ctx)
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Sets \code{rop} to the rational polynomial corresponding to
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the $p$-adic polynomial \code{op}.
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void padic_poly_zero(padic_poly_t poly)
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Sets \code{poly} to the zero polynomial.
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void padic_poly_one(padic_poly_t poly)
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Sets \code{poly} to the constant polynomial $1$,
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reduced to the precision of the polynomial.
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void padic_poly_swap(padic_poly_t poly1, padic_poly_t poly2)
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Swaps the two polynomials \code{poly1} and \code{poly2},
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including their precisions.
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This is done efficiently by swapping pointers.
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*******************************************************************************
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Getting and setting coefficients
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*******************************************************************************
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void padic_poly_get_coeff_padic(padic_t c, const padic_poly_t poly, slong n,
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const padic_ctx_t ctx)
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Sets $c$ to the coefficient of $x^n$ in the polynomial,
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reduced modulo the precision of $c$.
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void padic_poly_set_coeff_padic(padic_poly_t f, slong n, const padic_t c,
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const padic_ctx_t ctx)
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Sets the coefficient of $x^n$ in the polynomial $f$ to $c$,
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reduced to the precision of the polynomial $f$.
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Note that this operation can take linear time in the length
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of the polynomial.
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*******************************************************************************
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Comparison
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*******************************************************************************
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int padic_poly_equal(const padic_poly_t poly1, const padic_poly_t poly2)
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Returns whether the two polynomials \code{poly1} and \code{poly2}
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are equal.
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int padic_poly_is_zero(const padic_poly_t poly)
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Returns whether the polynomial \code{poly} is the zero polynomial.
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int padic_poly_is_one(const padic_poly_t poly, const padic_ctx_t ctx)
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Returns whether the polynomial \code{poly} is equal
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to the constant polynomial~$1$, taking the precision
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of the polynomial into account.
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*******************************************************************************
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Addition and subtraction
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*******************************************************************************
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void _padic_poly_add(fmpz *rop, slong *rval, slong N,
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const fmpz *op1, slong val1, slong len1, slong N1,
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const fmpz *op2, slong val2, slong len2, slong N2,
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const padic_ctx_t ctx)
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Sets \code{(rop, *val, FLINT_MAX(len1, len2)} to the sum of
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\code{(op1, val1, len1)} and \code{(op2, val2, len2)}.
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Assumes that the input is reduced and guarantees that this is
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also the case for the output.
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Assumes that $\min\{v_1, v_2\} < N$.
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Supports aliasing between the output and input arguments.
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void padic_poly_add(padic_poly_t f,
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const padic_poly_t g, const padic_poly_t h,
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const padic_ctx_t ctx);
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Sets $f$ to the sum $g + h$.
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void _padic_poly_sub(fmpz *rop, slong *rval,
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const fmpz *op1, slong val1, slong len1,
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const fmpz *op2, slong val2, slong len2,
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const padic_ctx_t ctx);
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Sets \code{(rop, *val, FLINT_MAX(len1, len2)} to the difference of
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\code{(op1, val1, len1)} and \code{(op2, val2, len2)}.
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Assumes that the input is reduced and guarantees that this is
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also the case for the output.
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Assumes that $\min\{v_1, v_2\} < N$.
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Support aliasing between the output and input arguments.
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void padic_poly_sub(padic_poly_t f,
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const padic_poly_t g, const padic_poly_t h,
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const padic_ctx_t ctx);
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Sets $f$ to the difference $g - h$.
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void padic_poly_neg(padic_poly_t f, const padic_poly_t g,
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const padic_ctx_t ctx);
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Sets $f$ to $-g$.
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*******************************************************************************
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Scalar multiplication
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*******************************************************************************
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void _padic_poly_scalar_mul_padic(fmpz *rop, slong *rval,
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const fmpz *op, slong val, slong len,
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const padic_t c, const padic_ctx_t ctx)
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Sets \code{(rop, *rval, len)} to \code{(op, val, len)} multiplied
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by the scalar $c$.
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The result will only be correctly reduced if the polynomial
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is non-zero. Otherwise, the array \code{(rop, len)} will be
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set to zero but the valuation \code{*rval} might be wrong.
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void padic_poly_scalar_mul_padic(padic_poly_t rop, const padic_poly_t op,
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const padic_t c, const padic_ctx_t ctx)
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Sets the polynomial \code{rop} to the product of the
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polynomial \code{op} and the $p$-adic number $c$,
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reducing the result modulo $p^N$.
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*******************************************************************************
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Multiplication
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*******************************************************************************
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void _padic_poly_mul(fmpz *rop, slong *rval, slong N,
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const fmpz *op1, slong val1, slong len1,
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const fmpz *op2, slong val2, slong len2,
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const padic_ctx_t ctx)
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Sets \code{(rop, *rval, len1 + len2 - 1)} to the product of
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\code{(op1, val1, len1)} and \code{(op2, val2, len2)}.
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Assumes that the resulting valuation \code{*rval}, which is
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the sum of the valuations \code{val1} and \code{val2}, is less
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than the precision~$N$ of the context.
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Assumes that \code{len1 >= len2 > 0}.
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void padic_poly_mul(padic_poly_t res,
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const padic_poly_t poly1, const padic_poly_t poly2,
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const padic_ctx_t ctx)
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Sets the polynomial \code{res} to the product of the two polynomials
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\code{poly1} and \code{poly2}, reduced modulo $p^N$.
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*******************************************************************************
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Powering
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*******************************************************************************
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void _padic_poly_pow(fmpz *rop, slong *rval, slong N,
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const fmpz *op, slong val, slong len, ulong e,
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const padic_ctx_t ctx)
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Sets the polynomial \code{(rop, *rval, e (len - 1) + 1)} to the
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polynomial \code{(op, val, len)} raised to the power~$e$.
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Assumes that $e > 1$ and \code{len > 0}.
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Does not support aliasing between the input and output arguments.
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void padic_poly_pow(padic_poly_t rop, const padic_poly_t op, ulong e,
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const padic_ctx_t ctx)
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Sets the polynomial \code{rop} to the polynomial \code{op} raised
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to the power~$e$, reduced to the precision in \code{rop}.
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In the special case $e = 0$, sets \code{rop} to the constant
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polynomial one reduced to the precision of \code{rop}.
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Also note that when $e = 1$, this operation sets \code{rop} to
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\code{op} and then reduces \code{rop}.
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When the valuation of the input polynomial is negative,
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this results in a loss of $p$-adic precision. Suppose
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that the input polynomial is given to precision~$N$ and
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has valuation~$v < 0$. The result then has valuation
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$e v < 0$ but is only correct to precision $N + (e - 1) v$.
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*******************************************************************************
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Series inversion
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*******************************************************************************
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void padic_poly_inv_series(padic_poly_t g, const padic_poly_t f, slong n,
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const padic_ctx_t ctx)
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Computes the power series inverse $g$ of $f$ modulo $X^n$,
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where $n \geq 1$.
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Given the polynomial $f \in \mathbf{Q}[X] \subset \mathbf{Q}_p[X]$,
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there exists a unique polynomial $f^{-1} \in \mathbf{Q}[X]$ such that
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$f f^{-1} = 1$ modulo $X^n$. This function sets $g$ to $f^{-1}$
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reduced modulo $p^N$.
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Assumes that the constant coefficient of $f$ is non-zero.
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Moreover, assumes that the valuation of the constant coefficient
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of $f$ is minimal among the coefficients of $f$.
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Note that the result $g$ is zero if and only if $- \ord_p(f) \geq N$.
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*******************************************************************************
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Derivative
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*******************************************************************************
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void _padic_poly_derivative(fmpz *rop, slong *rval, slong N,
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const fmpz *op, slong val, slong len,
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const padic_ctx_t ctx)
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Sets \code{(rop, rval)} to the derivative of \code{(op, val)} reduced
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modulo $p^N$.
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Supports aliasing of the input and the output parameters.
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void padic_poly_derivative(padic_poly_t rop,
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const padic_poly_t op, const padic_ctx_t ctx)
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Sets \code{rop} to the derivative of \code{op}, reducing the
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result modulo the precision of \code{rop}.
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*******************************************************************************
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Shifting
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*******************************************************************************
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void padic_poly_shift_left(padic_poly_t rop, const padic_poly_t op, slong n,
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const padic_ctx_t ctx)
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Notationally, sets the polynomial \code{rop} to the polynomial \code{op}
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multiplied by $x^n$, where $n \geq 0$, and reduces the result.
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void padic_poly_shift_right(padic_poly_t rop, const padic_poly_t op, slong n)
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Notationally, sets the polynomial \code{rop} to the polynomial
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\code{op} after floor division by $x^n$, where $n \geq 0$, ensuring
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the result is reduced.
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*******************************************************************************
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Evaluation
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*******************************************************************************
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void _padic_poly_evaluate_padic(fmpz_t u, slong *v, slong N,
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const fmpz *poly, slong val, slong len,
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const fmpz_t a, slong b, const padic_ctx_t ctx)
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void padic_poly_evaluate_padic(padic_t y, const padic_poly_t poly,
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const padic_t a, const padic_ctx_t ctx)
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Sets the $p$-adic number \code{y} to \code{poly} evaluated at $a$,
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reduced in the given context.
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Suppose that the polynomial can be written as $F(X) = p^w f(X)$
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with $\ord_p(f) = 1$, that $\ord_p(a) = b$ and that both are
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defined to precision~$N$. Then $f$ is defined to precision
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$N-w$ and so $f(a)$ is defined to precision $N-w$ when $a$ is
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integral and $N-w+(n-1)b$ when $b < 0$, where $n = \deg(f)$. Thus,
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$y = F(a)$ is defined to precision $N$ when $a$ is integral and
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$N+(n-1)b$ when $b < 0$.
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*******************************************************************************
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Composition
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*******************************************************************************
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void _padic_poly_compose(fmpz *rop, slong *rval, slong N,
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const fmpz *op1, slong val1, slong len1,
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const fmpz *op2, slong val2, slong len2,
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const padic_ctx_t ctx)
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Sets \code{(rop, *rval, (len1-1)*(len2-1)+1)} to the composition
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of the two input polynomials, reducing the result modulo $p^N$.
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Assumes that \code{len1} is non-zero.
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Does not support aliasing.
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void padic_poly_compose(padic_poly_t rop,
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const padic_poly_t op1, const padic_poly_t op2,
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const padic_ctx_t ctx)
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Sets \code{rop} to the composition of \code{op1} and \code{op2},
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reducing the result in the given context.
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To be clear about the order of composition, let $f(X)$ and $g(X)$
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denote the polynomials \code{op1} and \code{op2}, respectively.
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Then \code{rop} is set to $f(g(X))$.
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void _padic_poly_compose_pow(fmpz *rop, slong *rval, slong N,
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const fmpz *op, slong val, slong len, slong k,
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const padic_ctx_t ctx)
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Sets \code{(rop, *rval, (len - 1)*k + 1)} to the composition of
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\code{(op, val, len)} and the monomial $x^k$, where $k \geq 1$.
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Assumes that \code{len} is positive.
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Supports aliasing between the input and output polynomials.
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void padic_poly_compose_pow(padic_poly_t rop, const padic_poly_t op, slong k,
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const padic_ctx_t ctx)
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Sets \code{rop} to the composition of \code{op} and the monomial $x^k$,
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where $k \geq 1$.
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Note that no reduction takes place.
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*******************************************************************************
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Input and output
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*******************************************************************************
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int padic_poly_debug(const padic_poly_t poly)
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Prints the data defining the $p$-adic polynomial \code{poly}
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in a simple format useful for debugging purposes.
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In the current implementation, always returns $1$.
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int _padic_poly_fprint(FILE *file, const fmpz *poly, slong val, slong len,
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const padic_ctx_t ctx)
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int padic_poly_fprint(FILE *file, const padic_poly_t poly,
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const padic_ctx_t ctx)
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Prints a simple representation of the polynomial \code{poly}
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to the stream \code{file}.
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A non-zero polynomial is represented by the number of coeffients,
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two spaces, followed by a list of the coefficients, which are printed
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in a way depending on the print mode,
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\begin{itemize}
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\item In the \code{PADIC_TERSE} mode, the coefficients are printed as
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rational numbers.
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\item The \code{PADIC_SERIES} mode is currently not supported and will
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raise an abort signal.
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\item In the \code{PADIC_VAL_UNIT} mode, the coefficients are printed
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in the form $p^v u$.
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\end{itemize}
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The zero polynomial is represented by \code{"0"}.
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In the current implementation, always returns $1$.
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int _padic_poly_print(const fmpz *poly, slong val, slong len,
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const padic_ctx_t ctx)
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int padic_poly_print(const padic_poly_t poly, const padic_ctx_t ctx)
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Prints a simple representation of the polynomial \code{poly}
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to \code{stdout}.
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In the current implementation, always returns $1$.
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int _padic_poly_fprint_pretty(FILE *file,
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|
const fmpz *poly, slong val, slong len,
|
|
const char *var,
|
|
const padic_ctx_t ctx)
|
|
|
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int padic_poly_fprint_pretty(FILE *file,
|
|
const padic_poly_t poly, const char *var,
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|
const padic_ctx_t ctx)
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|
|
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int _padic_poly_print_pretty(FILE *file,
|
|
const fmpz *poly, slong val, slong len,
|
|
const char *var,
|
|
const padic_ctx_t ctx)
|
|
|
|
int padic_poly_print_pretty(const padic_poly_t poly, const char *var,
|
|
const padic_ctx_t ctx)
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|
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|
*******************************************************************************
|
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|
|
Testing
|
|
|
|
*******************************************************************************
|
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|
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int _padic_poly_is_canonical(const fmpz *op, slong val, slong len,
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|
const padic_ctx_t ctx);
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|
|
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int padic_poly_is_canonical(const padic_poly_t op, const padic_ctx_t ctx);
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int _padic_poly_is_reduced(const fmpz *op, slong val, slong len, slong N,
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|
const padic_ctx_t ctx);
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|
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int padic_poly_is_reduced(const padic_poly_t op, const padic_ctx_t ctx);
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|