fmpz_poly_factor.h – factorisation of polynomials over the integers

Description.

Types, macros and constants

fmpz_poly_factor_struct
fmpz_poly_factor_t

Description.

Memory management

void fmpz_poly_factor_init(fmpz_poly_factor_t fac)

Initialises a new factor structure.

void fmpz_poly_factor_init2(fmpz_poly_factor_t fac, slong alloc)

Initialises a new factor structure, providing space for at least alloc factors.

void fmpz_poly_factor_realloc(fmpz_poly_factor_t fac, slong alloc)

Reallocates the factor structure to provide space for precisely alloc factors.

void fmpz_poly_factor_fit_length(fmpz_poly_factor_t fac, slong len)

Ensures that the factor structure has space for at least len factors. This functions takes care of the case of repeated calls by always at least doubling the number of factors the structure can hold.

void fmpz_poly_factor_clear(fmpz_poly_factor_t fac)

Releases all memory occupied by the factor structure.

Manipulating factors

void fmpz_poly_factor_set(fmpz_poly_factor_t res, const fmpz_poly_factor_t fac)

Sets res to the same factorisation as fac.

void fmpz_poly_factor_insert(fmpz_poly_factor_t fac, const fmpz_poly_t p, slong e)

Adds the primitive polynomial \(p^e\) to the factorisation fac.

Assumes that \(\deg(p) \geq 2\) and \(e \neq 0\).

void fmpz_poly_factor_concat(fmpz_poly_factor_t res, const fmpz_poly_factor_t fac)

Concatenates two factorisations.

This is equivalent to calling fmpz_poly_factor_insert() repeatedly with the individual factors of fac.

Does not support aliasing between res and fac.

Input and output

void fmpz_poly_factor_print(const fmpz_poly_factor_t fac)

Prints the entries of fac to standard output.

Factoring algorithms

void fmpz_poly_factor_squarefree(fmpz_poly_factor_t fac, fmpz_poly_t F)

Takes as input a polynomial \(F\) and a freshly initialized factor structure fac. Updates fac to contain a factorization of \(F\) into (not necessarily irreducible) factors that themselves have no repeated factors. None of the returned factors will have the same exponent. That is we return \(g_i\) and unique \(e_i\) such that

\[F = c \prod_{i} g_i^{e_i}\]

where \(c\) is the signed content of \(F\) and \(\gcd(g_i, g_i') = 1\).

void fmpz_poly_factor_zassenhaus_recombination(fmpz_poly_factor_t final_fac, const fmpz_poly_factor_t lifted_fac, const fmpz_poly_t F, const fmpz_t P, slong exp)

Takes as input a factor structure lifted_fac containing a squarefree factorization of the polynomial \(F \bmod p\). The algorithm does a brute force search for irreducible factors of \(F\) over the integers, and each factor is raised to the power exp.

The impact of the algorithm is to augment a factorization of F^exp to the factor structure final_fac.

void _fmpz_poly_factor_zassenhaus(fmpz_poly_factor_t final_fac, slong exp, fmpz_poly_t f, slong cutoff, int use_van_hoeij)

This is the internal wrapper of Zassenhaus.

It will attempt to find a small prime such that \(f\) modulo \(p\) has a minimal number of factors. If it cannot find a prime giving less than cutoff factors it aborts. Then it decides a \(p\)-adic precision to lift the factors to, hensel lifts, and finally calls Zassenhaus recombination.

Assumes that \(\operatorname{len}(f) \geq 2\).

Assumes that \(f\) is primitive.

Assumes that the constant coefficient of \(f\) is non-zero. Note that this can be easily achieved by taking out factors of the form \(x^k\) before calling this routine.

If the final flag is set, the function will use the van Hoeij factorisation algorithm with gradual feeding and mod \(2^k\) data truncation to find factors when the number of local factors is large.

void fmpz_poly_factor_zassenhaus(fmpz_poly_factor_t final_fac, fmpz_poly_t F)

A wrapper of the Zassenhaus factoring algorithm, which takes as input any polynomial \(F\), and stores a factorization in final_fac.

The complexity will be exponential in the number of local factors we find for the components of a squarefree factorization of \(F\).

void _fmpz_poly_factor_quadratic(fmpz_poly_factor_t fac, const fmpz_poly_t f, slong exp)
void _fmpz_poly_factor_cubic(fmpz_poly_factor_t fac, const fmpz_poly_t f, slong exp)

Inserts the factorisation of the quadratic (resp. cubic) polynomial f into fac with multiplicity exp. This function requires that the content of f has been removed, and does not update the content of fac. The factorzation is calculated over \(\mathbb{R}\) or \(\mathbb{Q}_2\) and then tested over \(\mathbb{Z}\).

void fmpz_poly_factor(fmpz_poly_factor_t final_fac, fmpz_poly_t F)

A wrapper of the Zassenhaus and van Hoeij factoring algorithms, which takes as input any polynomial \(F\), and stores a factorization in final_fac.