.. _fq-poly-factor:

**fq_poly_factor.h** -- factorisation of univariate polynomials over finite fields
==================================================================================

Types, macros and constants
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.. type:: fq_poly_factor_struct

.. type:: fq_poly_factor_t

Memory Management
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.. function:: void fq_poly_factor_init(fq_poly_factor_t fac, const fq_ctx_t ctx)

    Initialises ``fac`` for use. An :type:`fq_poly_factor_t`
    represents a polynomial in factorised form as a product of
    polynomials with associated exponents.

.. function:: void fq_poly_factor_clear(fq_poly_factor_t fac, const fq_ctx_t ctx)

    Frees all memory associated with ``fac``.

.. function:: void fq_poly_factor_realloc(fq_poly_factor_t fac, slong alloc, const fq_ctx_t ctx)

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

.. function:: void fq_poly_factor_fit_length(fq_poly_factor_t fac, slong len, const fq_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.


Basic Operations
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.. function:: void fq_poly_factor_set(fq_poly_factor_t res, const fq_poly_factor_t fac, const fq_ctx_t ctx)

    Sets ``res`` to the same factorisation as ``fac``.

.. function:: void fq_poly_factor_print_pretty(const fq_poly_factor_t fac, const char * var, const fq_ctx_t ctx)

    Pretty-prints the entries of ``fac`` to standard output.

.. function:: void fq_poly_factor_print(const fq_poly_factor_t fac, const fq_ctx_t ctx)

    Prints the entries of ``fac`` to standard output.

.. function:: void fq_poly_factor_insert(fq_poly_factor_t fac, const fq_poly_t poly, slong exp, const fq_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.

.. function:: void fq_poly_factor_concat(fq_poly_factor_t res, const fq_poly_factor_t fac, const fq_ctx_t ctx)

    Concatenates two factorisations.

    This is equivalent to calling :func:`fq_poly_factor_insert`
    repeatedly with the individual factors of ``fac``.

    Does not support aliasing between ``res`` and ``fac``.

.. function:: void fq_poly_factor_pow(fq_poly_factor_t fac, slong exp, const fq_ctx_t ctx)

    Raises ``fac`` to the power ``exp``.

.. function:: ulong fq_poly_remove(fq_poly_t f, const fq_poly_t p, const fq_ctx_t ctx)

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


Irreducibility Testing
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.. function:: int fq_poly_is_irreducible(const fq_poly_t f, const fq_ctx_t ctx)

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

.. function:: int fq_poly_is_irreducible_ddf(const fq_poly_t f, const fq_ctx_t ctx)

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

.. function:: int fq_poly_is_irreducible_ben_or(const fq_poly_t f, const fq_ctx_t ctx)

    Returns 1 if the polynomial ``f`` is irreducible, otherwise returns 0.
    Uses Ben-Or's irreducibility test.

.. function:: int _fq_poly_is_squarefree(const fq_struct * f, slong len, const fq_ctx_t ctx)

    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.

.. function:: int fq_poly_is_squarefree(const fq_poly_t f, const fq_ctx_t ctx)

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



Factorisation
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.. function:: int fq_poly_factor_equal_deg_prob(fq_poly_t factor, flint_rand_t state, const fq_poly_t pol, slong d, const fq_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.

.. function:: void fq_poly_factor_equal_deg(fq_poly_factor_t factors, const fq_poly_t pol, slong d, const fq_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.

.. function:: void fq_poly_factor_split_single(fq_poly_t linfactor, const fq_poly_t input, const fq_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.

.. function:: void fq_poly_factor_distinct_deg(fq_poly_factor_t res, const fq_poly_t poly, slong * const * degs, const fq_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)``).

.. function:: void fq_poly_factor_squarefree(fq_poly_factor_t res, const fq_poly_t f, const fq_ctx_t ctx)

    Sets ``res`` to a squarefree factorization of ``f``.

.. function:: void fq_poly_factor(fq_poly_factor_t res, fq_t lead, const fq_poly_t f, const fq_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.

.. function:: void fq_poly_factor_cantor_zassenhaus(fq_poly_factor_t res, const fq_poly_t f, const fq_ctx_t ctx)

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

.. function:: void fq_poly_factor_kaltofen_shoup(fq_poly_factor_t res, const fq_poly_t poly, const fq_ctx_t ctx)

    Factorises a non-constant polynomial ``f`` 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.

.. function:: void fq_poly_factor_berlekamp(fq_poly_factor_t factors, const fq_poly_t f, const fq_ctx_t ctx)

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

.. function:: void fq_poly_factor_with_berlekamp(fq_poly_factor_t res, fq_t leading_coeff, const fq_poly_t f, const fq_ctx_t ctx)

    Factorises a general polynomial ``f`` into monic irreducible
    factors and sets ``leading_coeff`` to the leading coefficient
    of ``f``, or 0 if ``f`` is the zero polynomial.

    This function first checks for small special cases, deflates
    ``f`` if it is of the form `p(x^m)` for some `m > 1`, then
    performs a square-free factorisation, and finally runs Berlekamp
    factorisation on all the individual square-free factors.

.. function:: void fq_poly_factor_with_cantor_zassenhaus(fq_poly_factor_t res, fq_t leading_coeff, const fq_poly_t f, const fq_ctx_t ctx)

    Factorises a general polynomial ``f`` into monic irreducible
    factors and sets ``leading_coeff`` to the leading coefficient
    of ``f``, or 0 if ``f`` is the zero polynomial.

    This function first checks for small special cases, deflates
    ``f`` if it is of the form `p(x^m)` for some `m > 1`, then
    performs a square-free factorisation, and finally runs
    Cantor-Zassenhaus on all the individual square-free factors.

.. function:: void fq_poly_factor_with_kaltofen_shoup(fq_poly_factor_t res, fq_t leading_coeff, const fq_poly_t f, const fq_ctx_t ctx)

    Factorises a general polynomial ``f`` into monic irreducible
    factors and sets ``leading_coeff`` to the leading coefficient
    of ``f``, or 0 if ``f`` is the zero polynomial.

    This function first checks for small special cases, deflates
    ``f`` if it is of the form `p(x^m)` for some `m > 1`, then
    performs a square-free factorisation, and finally runs
    Kaltofen-Shoup on all the individual square-free factors.

.. function:: void fq_poly_iterated_frobenius_preinv(fq_poly_t * rop, slong n, const fq_poly_t v, const fq_poly_t vinv, const fq_ctx_t ctx)

    Sets ``rop[i]`` to be `x^{q^i}\bmod v` for `0 \le i < n`.

    It is required that ``vinv`` is the inverse of the reverse of
    ``v`` mod ``x^lenv``.

Root Finding
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.. function:: void fq_poly_roots(fq_poly_factor_t r, const fq_poly_t f, int with_multiplicity, const fq_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.
