fmpq_mpoly_factor.h

This module factors polynomials in \mathbb{Q}[x_0, \ldots, x_{n-1}] (see fmpq_mpoly.h). Since an fmpq_mpoly_t is stored as a rational content times an integer polynomial, factoring over \mathbb{Q} amounts to factoring the integer part with fmpz_mpoly_factor and multiplying the rational content into the constant. The factorisation over \mathbb{Q} is unique up to multiplying the bases by nonzero rationals; the two normalising functions below fix this freedom.

Types#

fmpq_mpoly_factor_struct

fmpq_mpoly_factor_t

A factored polynomial c \prod_{i < \mathit{num}} p_i^{e_i} with a rational constant: the fields are constant (an fmpq_t), poly (the bases, of type fmpq_mpoly_struct), exp (the exponents, of type fmpz), num and alloc.

Memory management#

void fmpq_mpoly_factor_init(fmpq_mpoly_factor_t f, const fmpq_mpoly_ctx_t ctx)

Initialises f as the empty product with constant 1.

void fmpq_mpoly_factor_realloc(fmpq_mpoly_factor_t f, slong alloc, const fmpq_mpoly_ctx_t ctx)

void fmpq_mpoly_factor_fit_length(fmpq_mpoly_factor_t f, slong len, const fmpq_mpoly_ctx_t ctx)

Change the allocation to exactly alloc factors (freeing the factors beyond it, and everything if \mathit{alloc} \le 0), respectively ensure room for at least len factors.

void fmpq_mpoly_factor_clear(fmpq_mpoly_factor_t f, const fmpq_mpoly_ctx_t ctx)

Frees the memory used by f.

Basic manipulation#

slong fmpq_mpoly_factor_length(const fmpq_mpoly_factor_t f, const fmpq_mpoly_ctx_t ctx)

Returns the number of bases of f.

void fmpq_mpoly_factor_get_constant_fmpq(fmpq_t c, const fmpq_mpoly_factor_t f, const fmpq_mpoly_ctx_t ctx)

Sets c to the constant of f.

void fmpq_mpoly_factor_get_base(fmpq_mpoly_t p, const fmpq_mpoly_factor_t f, slong i, const fmpq_mpoly_ctx_t ctx)

void fmpq_mpoly_factor_swap_base(fmpq_mpoly_t p, fmpq_mpoly_factor_t f, slong i, const fmpq_mpoly_ctx_t ctx)

Set p to, respectively swap p with, the base of index i.

slong fmpq_mpoly_factor_get_exp_si(fmpq_mpoly_factor_t f, slong i, const fmpq_mpoly_ctx_t ctx)

Returns the exponent of the base of index i, which must fit an slong.

void fmpq_mpoly_factor_sort(fmpq_mpoly_factor_t f, const fmpq_mpoly_ctx_t ctx)

Sorts the factors by increasing exponent, and factors with equal exponents by fmpq_mpoly_cmp.

int fmpq_mpoly_factor_expand(fmpq_mpoly_t A, const fmpq_mpoly_factor_t f, const fmpq_mpoly_ctx_t ctx)

Sets A to the product represented by f. Returns 0 if an exponent is negative or a power cannot be computed, and 1 otherwise.

int fmpq_mpoly_factor_make_monic(fmpq_mpoly_factor_t f, const fmpq_mpoly_ctx_t ctx)

int fmpq_mpoly_factor_make_integral(fmpq_mpoly_factor_t f, const fmpq_mpoly_ctx_t ctx)

Rescale the bases and adjust the constant so that the product is unchanged: make_monic makes every base monic (leading coefficient 1 in the monomial ordering), and make_integral moves the rational content of every base into the constant, leaving the integer part (primitive, with positive leading coefficient). Return 0 if a base is zero or a power of a content cannot be computed, and 1 otherwise.

Factorisation#

The factoring functions return 1 for success and 0 for failure, in which case f is unspecified; failure happens only in the circumstances described for fmpz_mpoly_factor. They overwrite f. The normalisation of the bases is not part of the specification: currently they are integral, primitive and with positive leading coefficients, as after fmpq_mpoly_factor_make_integral, but code that depends on a normalisation should call one of the normalising functions.

int fmpq_mpoly_factor_squarefree(fmpq_mpoly_factor_t f, const fmpq_mpoly_t A, const fmpq_mpoly_ctx_t ctx)

Sets f to a factorisation of A whose bases are squarefree and pairwise coprime, computed with fmpz_mpoly_factor_squarefree on the integer part of A. One multiplicity can occur in several bases; to get one base for each multiplicity, sort with fmpq_mpoly_factor_sort and multiply the bases with equal exponents.

int fmpq_mpoly_factor(fmpq_mpoly_factor_t f, const fmpq_mpoly_t A, const fmpq_mpoly_ctx_t ctx)

Sets f to the factorisation of A into a rational constant and distinct irreducible polynomials, computed with fmpz_mpoly_factor on the integer part of A. See Factoring multivariate polynomials over the integers for the methods.

Internal functions#

void _fmpq_mpoly_factor_swap_fmpz_mpoly_factor(fmpq_mpoly_factor_t f, fmpz_mpoly_factor_t g, const fmpq_t c, const fmpq_mpoly_ctx_t ctx)

Sets f to c times the factorisation g over the integers, moving the bases out of g (which is left with unspecified bases). g must be over the integer context ctx->zctx.