fq_default_poly_factor.h – factorisation of univariate polynomials over finite fields

Description.

Types, macros and constants

fq_default_poly_factor_t

Description.

Memory Management

void fq_default_poly_factor_init(fq_default_poly_factor_t fac, const fq_default_ctx_t ctx)

Initialises fac for use. An fq_default_poly_factor_t represents a polynomial in factorised form as a product of polynomials with associated exponents.

void fq_default_poly_factor_clear(fq_default_poly_factor_t fac, const fq_default_ctx_t ctx)

Frees all memory associated with fac.

void fq_default_poly_factor_realloc(fq_default_poly_factor_t fac, slong alloc, const fq_default_ctx_t ctx)

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

void fq_default_poly_factor_fit_length(fq_default_poly_factor_t fac, slong len, const fq_default_ctx_t ctx)

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.

Basic Operations

void fq_default_poly_factor_set(fq_default_poly_factor_t res, const fq_default_poly_factor_t fac, const fq_default_ctx_t ctx)

Sets res to the same factorisation as fac.

void fq_default_poly_factor_print_pretty(const fq_default_poly_factor_t fac, const fq_default_ctx_t ctx)

Pretty-prints the entries of fac to standard output.

void fq_default_poly_factor_print(const fq_default_poly_factor_t fac, const fq_default_ctx_t ctx)

Prints the entries of fac to standard output.

void fq_default_poly_factor_insert(fq_default_poly_factor_t fac, const fq_default_poly_t poly, slong exp, const fq_default_ctx_t ctx)

Inserts the factor poly with multiplicity exp into the factorisation fac.

If fac already contains poly, then exp simply gets added to the exponent of the existing entry.

void fq_default_poly_factor_concat(fq_default_poly_factor_t res, const fq_default_poly_factor_t fac, const fq_default_ctx_t ctx)

Concatenates two factorisations.

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

Does not support aliasing between res and fac.

void fq_default_poly_factor_pow(fq_default_poly_factor_t fac, slong exp, const fq_default_ctx_t ctx)

Raises fac to the power exp.

ulong fq_default_poly_remove(fq_default_poly_t f, const fq_default_poly_t p, const fq_default_ctx_t ctx)

Removes the highest possible power of p from f and returns the exponent.

slong fq_default_poly_factor_length(fq_default_poly_factor_t fac, const fq_default_ctx_t ctx)

Return the number of factors, not including the unit.

void fq_default_poly_factor_get_poly(fq_default_poly_t poly, const fq_default_poly_factor_t fac, slong i, const fq_default_ctx_t ctx)

Set poly to factor i of fac (numbering starts at zero).

slong fq_default_poly_factor_exp(fq_default_poly_factor_t fac, slong i, const fq_default_ctx_t ctx)

Return the exponent of factor i of fac.

Irreducibility Testing

int fq_default_poly_is_irreducible(const fq_default_poly_t f, const fq_default_ctx_t ctx)

Returns 1 if the polynomial f is irreducible, otherwise returns 0.

int fq_default_poly_is_squarefree(const fq_default_poly_t f, const fq_default_ctx_t ctx)

Returns 1 if f is squarefree, and 0 otherwise. As a special case, the zero polynomial is not considered squarefree.

Factorisation

void fq_default_poly_factor_equal_deg(fq_default_poly_factor_t factors, const fq_default_poly_t pol, slong d, const fq_default_ctx_t ctx)

Assuming pol is a product of irreducible factors all of degree d, finds all those factors and places them in factors. Requires that pol be monic, non-constant and squarefree.

void fq_default_poly_factor_split_single(fq_default_poly_t linfactor, const fq_default_poly_t input, const fq_default_ctx_t ctx)

Assuming input is a product of factors all of degree 1, finds a single linear factor of input and places it in linfactor. Requires that input be monic and non-constant.

void fq_default_poly_factor_distinct_deg(fq_default_poly_factor_t res, const fq_default_poly_t poly, slong * const * degs, const fq_default_ctx_t ctx)

Factorises a monic non-constant squarefree polynomial poly of degree n into factors \(f[d]\) such that for \(1 \leq d \leq n\) \(f[d]\) is the product of the monic irreducible factors of poly of degree \(d\). Factors are stored in res, associated powers of irreducible polynomials are stored in degs in the same order as factors.

Requires that degs have enough space for irreducible polynomials’ powers (maximum space required is n * sizeof(slong)).

void fq_default_poly_factor_squarefree(fq_default_poly_factor_t res, const fq_default_poly_t f, const fq_default_ctx_t ctx)

Sets res to a squarefree factorization of f.

void fq_default_poly_factor(fq_default_poly_factor_t res, fq_default_t lead, const fq_default_poly_t f, const fq_default_ctx_t ctx)

Factorises a non-constant polynomial f into monic irreducible factors choosing the best algorithm for given modulo and degree. The output lead is set to the leading coefficient of \(f\) upon return. Choice of algorithm is based on heuristic measurements.

Root Finding

void fq_default_poly_roots(fq_default_poly_factor_t r, const fq_default_poly_t f, int with_multiplicity, const fq_default_ctx_t ctx)

Fill \(r\) with factors of the form \(x - r_i\) where the \(r_i\) are the distinct roots of a nonzero \(f\) in \(F_q\). If \(with_multiplicity\) is zero, the exponent \(e_i\) of the factor \(x - r_i\) is \(1\). Otherwise, it is the largest \(e_i\) such that \((x-r_i)^e_i\) divides \(f\). This function throws if \(f\) is zero, but is otherwise always successful.