# fmpz_mod_poly_factor.h – factorisation of polynomials over integers mod n¶

Description.

## Types, macros and constants¶

fmpz_mod_poly_factor_struct

A structure representing a polynomial in factorised form as a product of polynomials with associated exponents.

fmpz_mod_poly_factor_t

An array of length 1 of fmpz_mpoly_factor_struct.

## Factorisation¶

void fmpz_mod_poly_factor_init(fmpz_mod_poly_factor_t fac, const fmpz_mod_ctx_t ctx)

Initialises fac for use.

void fmpz_mod_poly_factor_clear(fmpz_mod_poly_factor_t fac, const fmpz_mod_ctx_t ctx)

Frees all memory associated with fac.

void fmpz_mod_poly_factor_realloc(fmpz_mod_poly_factor_t fac, slong alloc, const fmpz_mod_ctx_t ctx)

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

void fmpz_mod_poly_factor_fit_length(fmpz_mod_poly_factor_t fac, slong len, const fmpz_mod_ctx_t ctx)

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

void fmpz_mod_poly_factor_set(fmpz_mod_poly_factor_t res, const fmpz_mod_poly_factor_t fac, const fmpz_mod_ctx_t ctx)

Sets res to the same factorisation as fac.

void fmpz_mod_poly_factor_print(const fmpz_mod_poly_factor_t fac, const fmpz_mod_ctx_t ctx)

Prints the entries of fac to standard output.

void fmpz_mod_poly_factor_insert(fmpz_mod_poly_factor_t fac, const fmpz_mod_poly_t poly, slong exp, const fmpz_mod_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 fmpz_mod_poly_factor_concat(fmpz_mod_poly_factor_t res, const fmpz_mod_poly_factor_t fac, const fmpz_mod_ctx_t ctx)

Concatenates two factorisations.

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

Does not support aliasing between res and fac.

void fmpz_mod_poly_factor_pow(fmpz_mod_poly_factor_t fac, slong exp, const fmpz_mod_ctx_t ctx)

Raises fac to the power exp.

int fmpz_mod_poly_is_irreducible(const fmpz_mod_poly_t f, const fmpz_mod_ctx_t ctx)

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

int fmpz_mod_poly_is_irreducible_ddf(const fmpz_mod_poly_t f, const fmpz_mod_ctx_t ctx)

Returns 1 if the polynomial f is irreducible, otherwise returns 0. Uses fast distinct-degree factorisation.

int fmpz_mod_poly_is_irreducible_rabin(const fmpz_mod_poly_t f, const fmpz_mod_ctx_t ctx)

Returns 1 if the polynomial f is irreducible, otherwise returns 0. Uses Rabin irreducibility test.

int fmpz_mod_poly_is_irreducible_rabin_f(fmpz_t r, const fmpz_mod_poly_t f, const fmpz_mod_ctx_t ctx)

Either sets $$r$$ to $$1$$ and return 1 if the polynomial f is irreducible or $$0$$ otherwise, or set $$r$$ to a nontrivial factor of $$p$$.

This algorithm correctly determines whether $$f$$ to is irreducible over $$\mathbb{Z}/p\mathbb{Z}$$, even for composite $$f$$, or it finds a factor of $$p$$.

int _fmpz_mod_poly_is_squarefree(const fmpz * f, slong len, const fmpz_t p)

Returns 1 if (f, len) is squarefree, and 0 otherwise. As a special case, the zero polynomial is not considered squarefree. There are no restrictions on the length.

int _fmpz_mod_poly_is_squarefree_f(fmpz_t fac, const fmpz * f, slong len, const fmpz_t p)

If $$fac$$ returns with the value $$1$$ then the function operates as per _fmpz_mod_poly_is_squarefree(), otherwise $$f$$ is set to a nontrivial factor of $$p$$.

int fmpz_mod_poly_is_squarefree(const fmpz_mod_poly_t f, const fmpz_mod_ctx_t ctx)

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

int fmpz_mod_poly_is_squarefree_f(fmpz_t fac, const fmpz_mod_poly_t f, const fmpz_mod_ctx_t ctx)

If $$fac$$ returns with the value $$1$$ then the function operates as per fmpz_mod_poly_is_squarefree(), otherwise $$f$$ is set to a nontrivial factor of $$p$$.

int fmpz_mod_poly_factor_equal_deg_prob(fmpz_mod_poly_t factor, flint_rand_t state, const fmpz_mod_poly_t pol, slong d, const fmpz_mod_ctx_t ctx)

Probabilistic equal degree factorisation of pol into irreducible factors of degree d. If it passes, a factor is placed in factor and 1 is returned, otherwise 0 is returned and the value of factor is undetermined.

Requires that pol be monic, non-constant and squarefree.

void fmpz_mod_poly_factor_equal_deg(fmpz_mod_poly_factor_t factors, const fmpz_mod_poly_t pol, slong d, const fmpz_mod_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 fmpz_mod_poly_factor_distinct_deg(fmpz_mod_poly_factor_t res, const fmpz_mod_poly_t poly, slong * const *degs, const fmpz_mod_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 $$f[d]$$ are stored in res, and the degree $$d$$ of the irreducible factors is stored in degs in the same order as the factors.

Requires that degs has enough space for $$(n/2)+1 * sizeof(slong)$$.

void fmpz_mod_poly_factor_distinct_deg_threaded(fmpz_mod_poly_factor_t res, const fmpz_mod_poly_t poly, slong * const *degs, const fmpz_mod_ctx_t ctx)

Multithreaded version of fmpz_mod_poly_factor_distinct_deg().

void fmpz_mod_poly_factor_squarefree(fmpz_mod_poly_factor_t res, const fmpz_mod_poly_t f, const fmpz_mod_ctx_t ctx)

Sets res to a squarefree factorization of f.

void fmpz_mod_poly_factor(fmpz_mod_poly_factor_t res, const fmpz_mod_poly_t f, const fmpz_mod_ctx_t ctx)

Factorises a non-constant polynomial f into monic irreducible factors choosing the best algorithm for given modulo and degree. Choice is based on heuristic measurements.

void fmpz_mod_poly_factor_cantor_zassenhaus(fmpz_mod_poly_factor_t res, const fmpz_mod_poly_t f, const fmpz_mod_ctx_t ctx)

Factorises a non-constant polynomial f into monic irreducible factors using the Cantor-Zassenhaus algorithm.

void fmpz_mod_poly_factor_kaltofen_shoup(fmpz_mod_poly_factor_t res, const fmpz_mod_poly_t poly, const fmpz_mod_ctx_t ctx)

Factorises a non-constant polynomial poly into monic irreducible factors using the fast version of Cantor-Zassenhaus algorithm proposed by Kaltofen and Shoup (1998). More precisely this algorithm uses a baby step/giant step strategy for the distinct-degree factorization step. If flint_get_num_threads() is greater than one fmpz_mod_poly_factor_distinct_deg_threaded() is used.

void fmpz_mod_poly_factor_berlekamp(fmpz_mod_poly_factor_t factors, const fmpz_mod_poly_t f, const fmpz_mod_ctx_t ctx)

Factorises a non-constant polynomial f into monic irreducible factors using the Berlekamp algorithm.

void _fmpz_mod_poly_interval_poly_worker(void* arg_ptr)

Worker function to compute interval polynomials in distinct degree factorisation. Input/output is stored in fmpz_mod_poly_interval_poly_arg_t.

## Root Finding¶

void fmpz_mod_poly_roots(fmpz_mod_poly_factor_t r, const fmpz_mod_poly_t f, int with_multiplicity, const fmpz_mod_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 $$Z/pZ$$. It is expected and not checked that the modulus of $$ctx$$ is prime. 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.

int fmpz_mod_poly_roots_factored(fmpz_mod_poly_factor_t r, const fmpz_mod_poly_t f, int with_multiplicity, const fmpz_factor_t n, const fmpz_mod_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 $$Z/nZ$$. It is expected and not checked that $$n$$ is a prime factorization of the modulus of $$ctx$$. 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 $$f$$. The roots are first found modulo the primes in $$n$$, then lifted to the corresponding prime powers, then combined into roots of the original polynomial $$f$$. A return of $$1$$ indicates the function was successful. A return of $$0$$ indicates the function was not able to find the roots, possibly because there are too many of them. This function throws if $$f$$ is zero.