.. _nmod-poly-factor:

**nmod_poly_factor.h** -- factorisation of univariate polynomials over integers mod n (word-size n)
===================================================================================================

Types, macros and constants
-------------------------------------------------------------------------------

.. type:: nmod_poly_factor_struct

.. type:: nmod_poly_factor_t

Factorisation
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.. function:: void nmod_poly_factor_init(nmod_poly_factor_t fac)

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

.. function:: void nmod_poly_factor_clear(nmod_poly_factor_t fac)

    Frees all memory associated with ``fac``.

.. function:: void nmod_poly_factor_realloc(nmod_poly_factor_t fac, slong alloc)

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

.. function:: void nmod_poly_factor_fit_length(nmod_poly_factor_t fac, slong len)

    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.

.. function:: void nmod_poly_factor_set(nmod_poly_factor_t res, const nmod_poly_factor_t fac)

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

.. function:: void nmod_poly_factor_print(const nmod_poly_factor_t fac)

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

.. function:: void nmod_poly_factor_insert(nmod_poly_factor_t fac, const nmod_poly_t poly, slong exp)

    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 nmod_poly_factor_concat(nmod_poly_factor_t res, const nmod_poly_factor_t fac)

    Concatenates two factorisations.

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

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

.. function:: void nmod_poly_factor_pow(nmod_poly_factor_t fac, slong exp)

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

.. function:: ulong nmod_poly_remove(nmod_poly_t f, const nmod_poly_t p)

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

.. function:: int nmod_poly_is_irreducible(const nmod_poly_t f)

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

.. function:: int nmod_poly_is_irreducible_ddf(const nmod_poly_t f)

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

.. function:: int nmod_poly_is_irreducible_rabin(const nmod_poly_t f)

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

.. function:: int _nmod_poly_is_squarefree(mp_srcptr f, slong len, nmod_t mod)

    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 nmod_poly_is_squarefree(const nmod_poly_t f)

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

.. function:: void nmod_poly_factor_squarefree(nmod_poly_factor_t res, const nmod_poly_t f)

    Sets ``res`` to a square-free factorization of ``f``.

.. function:: int nmod_poly_factor_equal_deg_prob(nmod_poly_t factor, flint_rand_t state, const nmod_poly_t pol, slong d)

    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 nmod_poly_factor_equal_deg(nmod_poly_factor_t factors, const nmod_poly_t pol, slong d)

    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 nmod_poly_factor_distinct_deg(nmod_poly_factor_t res, const nmod_poly_t poly, slong * const *degs)

    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)``.

.. function:: void nmod_poly_factor_distinct_deg_threaded(nmod_poly_factor_t res, const nmod_poly_t poly, slong * const *degs)

    Multithreaded version of :func:`nmod_poly_factor_distinct_deg`.

.. function:: void nmod_poly_factor_cantor_zassenhaus(nmod_poly_factor_t res, const nmod_poly_t f)

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

.. function:: void nmod_poly_factor_berlekamp(nmod_poly_factor_t res, const nmod_poly_t f)

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

.. function:: void nmod_poly_factor_kaltofen_shoup(nmod_poly_factor_t res, const nmod_poly_t poly)

    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. If :func:`flint_get_num_threads` is greater than one
    :func:`nmod_poly_factor_distinct_deg_threaded` is used.

.. function:: mp_limb_t nmod_poly_factor_with_berlekamp(nmod_poly_factor_t res, const nmod_poly_t f)

    Factorises a general polynomial ``f`` into monic irreducible factors
    and returns 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 on all the
    individual square-free factors.

.. function:: mp_limb_t nmod_poly_factor_with_cantor_zassenhaus(nmod_poly_factor_t res, const nmod_poly_t f)

    Factorises a general polynomial ``f`` into monic irreducible factors
    and returns 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:: mp_limb_t nmod_poly_factor_with_kaltofen_shoup(nmod_poly_factor_t res, const nmod_poly_t f)

    Factorises a general polynomial ``f`` into monic irreducible factors
    and returns 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:: mp_limb_t nmod_poly_factor(nmod_poly_factor_t res, const nmod_poly_t f)

    Factorises a general polynomial ``f`` into monic irreducible factors
    and returns 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 either Cantor-Zassenhaus
    or Berlekamp on all the individual square-free factors.
    Currently Cantor-Zassenhaus is used by default unless the modulus is 2, in
    which case Berlekamp is used.

.. function:: void _nmod_poly_interval_poly_worker(void* arg_ptr)

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