.. _fmpz-poly:

**fmpz_poly.h** -- univariate polynomials over the integers
===============================================================================

Introduction
------------

The :type:`fmpz_poly_t` data type represents elements of
`\mathbb{Z}[x]`. The :type:`fmpz_poly` module provides routines for
memory management, basic arithmetic, and conversions from or to other
types.

Each coefficient of an :type:`fmpz_poly_t` is an integer of the FLINT
:type:`fmpz_t` type. There are two advantages of this model. Firstly,
the :type:`fmpz_t` type is memory managed, so the user can manipulate
individual coefficients of a polynomial without having to deal with
tedious memory management. Secondly, a coefficient of an
:type:`fmpz_poly_t` can be changed without changing the size of any of
the other coefficients.

Unless otherwise specified, all functions in this section permit
aliasing between their input arguments and between their input and
output arguments.

.. _simple-example-2:

Simple example
--------------

The following example computes the square of the polynomial `5x^3 -
1`.

.. code:: c

   #include "fmpz_poly.h"

   int main()
   {
       fmpz_poly_t x, y;
       fmpz_poly_init(x);
       fmpz_poly_init(y);
       fmpz_poly_set_coeff_ui(x, 3, 5);
       fmpz_poly_set_coeff_si(x, 0, -1);
       fmpz_poly_mul(y, x, x);
       fmpz_poly_print(x); flint_printf("\n");
       fmpz_poly_print(y); flint_printf("\n");
       fmpz_poly_clear(x);
       fmpz_poly_clear(y);
   }

The output is:

::

   4  -1 0 0 5
   7  1 0 0 -10 0 0 25

Definition of the fmpz_poly_t type
----------------------------------

The :type:`fmpz_poly_t` type is a typedef for an array of length 1 of
:type:`fmpz_poly_struct`'s. This permits passing parameters of type
:type:`fmpz_poly_t` by reference in a manner similar to the way GMP
integers of type ``mpz_t`` can be passed by reference.

In reality one never deals directly with the ``struct`` and simply
deals with objects of type :type:`fmpz_poly_t`. For simplicity we will
think of an :type:`fmpz_poly_t` as a ``struct``, though in practice to
access fields of this ``struct``, one needs to dereference first, e.g.
to access the ``length`` field of an :type:`fmpz_poly_t` called
``poly1`` one writes ``poly1->length``.

An :type:`fmpz_poly_t` is said to be *normalised* if either ``length``
is zero, or if the leading coefficient of the polynomial is non-zero.
All :type:`fmpz_poly` functions expect their inputs to be normalised,
and unless otherwise specified they produce output that is normalised.

It is recommended that users do not access the fields of an
:type:`fmpz_poly_t` or its coefficient data directly, but make use of
the functions designed for this purpose, detailed below.

Functions in ``fmpz_poly`` do all the memory management for the user.
One does not need to specify the maximum length or number of limbs per
coefficient in advance before using a polynomial object. FLINT
reallocates space automatically as the computation proceeds, if more
space is required. Each coefficient is also managed separately, being
resized as needed, independently of the other coefficients.

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

.. type:: fmpz_poly_struct

.. type:: fmpz_poly_t

Memory management
--------------------------------------------------------------------------------


.. function:: void fmpz_poly_init(fmpz_poly_t poly)

    Initialises ``poly`` for use, setting its length to zero.
    A corresponding call to :func:`fmpz_poly_clear` must be made after
    finishing with the ``fmpz_poly_t`` to free the memory used by
    the polynomial.

.. function:: void fmpz_poly_init2(fmpz_poly_t poly, slong alloc)

    Initialises ``poly`` with space for at least ``alloc`` coefficients
    and sets the length to zero.  The allocated coefficients are all set to
    zero.

.. function:: void fmpz_poly_realloc(fmpz_poly_t poly, slong alloc)

    Reallocates the given polynomial to have space for ``alloc``
    coefficients.  If ``alloc`` is zero the polynomial is cleared
    and then reinitialised.  If the current length is greater than
    ``alloc`` the polynomial is first truncated to length ``alloc``.

.. function:: void fmpz_poly_fit_length(fmpz_poly_t poly, slong len)

    If ``len`` is greater than the number of coefficients currently
    allocated, then the polynomial is reallocated to have space for at
    least ``len`` coefficients.  No data is lost when calling this
    function.

    The function efficiently deals with the case where ``fit_length`` is
    called many times in small increments by at least doubling the number
    of allocated coefficients when length is larger than the number of
    coefficients currently allocated.

.. function:: void fmpz_poly_clear(fmpz_poly_t poly)

    Clears the given polynomial, releasing any memory used.  It must
    be reinitialised in order to be used again.

.. function:: void _fmpz_poly_normalise(fmpz_poly_t poly)

    Sets the length of ``poly`` so that the top coefficient is non-zero.
    If all coefficients are zero, the length is set to zero.  This function
    is mainly used internally, as all functions guarantee normalisation.

.. function:: void _fmpz_poly_set_length(fmpz_poly_t poly, slong newlen)

    Demotes the coefficients of ``poly`` beyond ``newlen`` and sets
    the length of ``poly`` to ``newlen``.

.. function:: void fmpz_poly_attach_truncate(fmpz_poly_t trunc, const fmpz_poly_t poly, slong n)

    This function sets the uninitialised polynomial ``trunc`` to the low
    `n` coefficients of ``poly``, or to ``poly`` if the latter doesn't
    have `n` coefficients. The polynomial ``trunc`` not be cleared or used
    as the output of any FLINT functions.

.. function:: void fmpz_poly_attach_shift(fmpz_poly_t trunc, const fmpz_poly_t poly, slong n)

    This function sets the uninitialised polynomial ``trunc`` to the
    high coefficients of ``poly``, i.e. the coefficients not among the low
    `n` coefficients of ``poly``. If the latter doesn't have `n`
    coefficients ``trunc`` is set to the zero polynomial. The polynomial
    ``trunc`` not be cleared or used as the output of any FLINT functions.


Polynomial parameters
--------------------------------------------------------------------------------


.. function:: slong fmpz_poly_length(const fmpz_poly_t poly)

    Returns the length of ``poly``.  The zero polynomial has length zero.

.. function:: slong fmpz_poly_degree(const fmpz_poly_t poly)

    Returns the degree of ``poly``, which is one less than its length.


Assignment and basic manipulation
--------------------------------------------------------------------------------


.. function:: void fmpz_poly_set(fmpz_poly_t poly1, const fmpz_poly_t poly2)

    Sets ``poly1`` to equal ``poly2``.

.. function:: void fmpz_poly_set_si(fmpz_poly_t poly, slong c)

    Sets ``poly`` to the signed integer ``c``.

.. function:: void fmpz_poly_set_ui(fmpz_poly_t poly, ulong c)

    Sets ``poly`` to the unsigned integer ``c``.

.. function:: void fmpz_poly_set_fmpz(fmpz_poly_t poly, const fmpz_t c)

    Sets ``poly`` to the integer ``c``.

.. function:: int _fmpz_poly_set_str(fmpz * poly, const char * str)

    Sets ``poly`` to the polynomial encoded in the null-terminated
    string ``str``.  Assumes that ``poly`` is allocated as a
    sufficiently large array suitable for the number of coefficients
    present in ``str``.

    Returns `0` if no error occurred.  Otherwise, returns a non-zero
    value, in which case the resulting value of ``poly`` is undefined.
    If ``str`` is not null-terminated, calling this method might result
    in a segmentation fault.

.. function:: int fmpz_poly_set_str(fmpz_poly_t poly, const char * str)

    Imports a polynomial from a null-terminated string.  If the string
    ``str`` represents a valid polynomial returns `0`, otherwise
    returns `1`.

    Returns `0` if no error occurred.  Otherwise, returns a non-zero value,
    in which case the resulting value of ``poly`` is undefined.  If
    ``str`` is not null-terminated, calling this method might result in
    a segmentation fault.

.. function:: char * _fmpz_poly_get_str(const fmpz * poly, slong len)

    Returns the plain FLINT string representation of the polynomial
    ``(poly, len)``.

.. function:: char * fmpz_poly_get_str(const fmpz_poly_t poly)

    Returns the plain FLINT string representation of the polynomial
    ``poly``.

.. function:: char * _fmpz_poly_get_str_pretty(const fmpz * poly, slong len, const char * x)

    Returns a pretty representation of the polynomial
    ``(poly, len)`` using the null-terminated string ``x`` as the
    variable name.

.. function:: char * fmpz_poly_get_str_pretty(const fmpz_poly_t poly, const char * x)

    Returns a pretty representation of the polynomial ``poly`` using the
    null-terminated string ``x`` as the variable name.

.. function:: void fmpz_poly_zero(fmpz_poly_t poly)

    Sets ``poly`` to the zero polynomial.

.. function:: void fmpz_poly_one(fmpz_poly_t poly)

    Sets ``poly`` to the constant polynomial one.

.. function:: void fmpz_poly_zero_coeffs(fmpz_poly_t poly, slong i, slong j)

    Sets the coefficients of `x^i, \dotsc, x^{j-1}` to zero.

.. function:: void fmpz_poly_swap(fmpz_poly_t poly1, fmpz_poly_t poly2)

    Swaps ``poly1`` and ``poly2``.  This is done efficiently without
    copying data by swapping pointers, etc.

.. function:: void _fmpz_poly_reverse(fmpz * res, const fmpz * poly, slong len, slong n)

    Sets ``(res, n)`` to the reverse of ``(poly, n)``, where
    ``poly`` is in fact an array of length ``len``.  Assumes that
    ``0 < len <= n``.  Supports aliasing of ``res`` and ``poly``,
    but the behaviour is undefined in case of partial overlap.

.. function:: void fmpz_poly_reverse(fmpz_poly_t res, const fmpz_poly_t poly, slong n)

    This function considers the polynomial ``poly`` to be of length `n`,
    notionally truncating and zero padding if required, and reverses
    the result.  Since the function normalises its result ``res`` may be
    of length less than `n`.

.. function:: void fmpz_poly_truncate(fmpz_poly_t poly, slong newlen)

    If the current length of ``poly`` is greater than ``newlen``, it
    is truncated to have the given length.  Discarded coefficients are not
    necessarily set to zero.

.. function:: void fmpz_poly_set_trunc(fmpz_poly_t res, const fmpz_poly_t poly, slong n)

    Sets ``res`` to a copy of ``poly``, truncated to length ``n``.


Randomisation
--------------------------------------------------------------------------------


.. function:: void fmpz_poly_randtest(fmpz_poly_t f, flint_rand_t state, slong len, flint_bitcnt_t bits)

    Sets `f` to a random polynomial with up to the given length and where
    each coefficient has up to the given number of bits. The coefficients
    are signed randomly.

.. function:: void fmpz_poly_randtest_unsigned(fmpz_poly_t f, flint_rand_t state, slong len, flint_bitcnt_t bits)

    Sets `f` to a random polynomial with up to the given length and where
    each coefficient has up to the given number of bits.

.. function:: void fmpz_poly_randtest_not_zero(fmpz_poly_t f, flint_rand_t state, slong len, flint_bitcnt_t bits)

    As for :func:`fmpz_poly_randtest` except that ``len`` and bits may
    not be zero and the polynomial generated is guaranteed not to be the
    zero polynomial.

.. function:: void fmpz_poly_randtest_no_real_root(fmpz_poly_t p, flint_rand_t state, slong len, flint_bitcnt_t bits)

    Sets ``p`` to a random polynomial without any real root, whose
    length is up to ``len`` and where each coefficient has up to the
    given number of bits.

.. function:: void fmpz_poly_randtest_irreducible1(fmpz_poly_t pol, flint_rand_t state, slong len, flint_bitcnt_t bits)
              void fmpz_poly_randtest_irreducible2(fmpz_poly_t pol, flint_rand_t state, slong len, flint_bitcnt_t bits)
              void fmpz_poly_randtest_irreducible(fmpz_poly_t pol, flint_rand_t state, slong len, flint_bitcnt_t bits)

    Sets ``p`` to a random irreducible polynomial, whose
    length is up to ``len`` and where each coefficient has up to the
    given number of bits. There are two algorithms: *irreducible1*
    generates an irreducible polynomial modulo a random prime number
    and lifts it to the integers; *irreducible2* generates a random
    integer polynomial, factors it, and returns a random factor.
    The default function chooses randomly between these methods.


Getting and setting coefficients
--------------------------------------------------------------------------------


.. function:: void fmpz_poly_get_coeff_fmpz(fmpz_t x, const fmpz_poly_t poly, slong n)

    Sets `x` to the `n`-th coefficient of ``poly``.  Coefficient
    numbering is from zero and if `n` is set to a value beyond the end of
    the polynomial, zero is returned.

.. function:: slong fmpz_poly_get_coeff_si(const fmpz_poly_t poly, slong n)

    Returns coefficient `n` of ``poly`` as a ``slong``. The result is
    undefined if the value does not fit into a ``slong``. Coefficient
    numbering is from zero and if `n` is set to a value beyond the end of
    the polynomial, zero is returned.

.. function:: ulong fmpz_poly_get_coeff_ui(const fmpz_poly_t poly, slong n)

    Returns coefficient `n` of ``poly`` as a ``ulong``.  The result is
    undefined if the value does not fit into a ``ulong``.  Coefficient
    numbering is from zero and if `n` is set to a value beyond the end of the
    polynomial, zero is returned.

.. function:: fmpz * fmpz_poly_get_coeff_ptr(const fmpz_poly_t poly, slong n)

    Returns a reference to the coefficient of `x^n` in the polynomial,
    as an ``fmpz *``.  This function is provided so that individual
    coefficients can be accessed and operated on by functions in the
    ``fmpz`` module.  This function does not make a copy of the
    data, but returns a reference to the actual coefficient.

    Returns ``NULL`` when `n` exceeds the degree of the polynomial.

    This function is implemented as a macro.

.. function:: fmpz * fmpz_poly_lead(const fmpz_poly_t poly)

    Returns a reference to the leading coefficient of the polynomial,
    as an ``fmpz *``.  This function is provided so that the leading
    coefficient can be easily accessed and operated on by functions in
    the ``fmpz`` module.  This function does not make a copy of the
    data, but returns a reference to the actual coefficient.

    Returns ``NULL`` when the polynomial is zero.

    This function is implemented as a macro.

.. function:: void fmpz_poly_set_coeff_fmpz(fmpz_poly_t poly, slong n, const fmpz_t x)

    Sets coefficient `n` of ``poly`` to the ``fmpz`` value ``x``.
    Coefficient numbering starts from zero and if `n` is beyond the current
    length of ``poly`` then the polynomial is extended and zero
    coefficients inserted if necessary.

.. function:: void fmpz_poly_set_coeff_si(fmpz_poly_t poly, slong n, slong x)

    Sets coefficient `n` of ``poly`` to the ``slong`` value ``x``.
    Coefficient numbering starts from zero and if `n` is beyond the current
    length of ``poly`` then the polynomial is extended and zero
    coefficients inserted if necessary.

.. function:: void fmpz_poly_set_coeff_ui(fmpz_poly_t poly, slong n, ulong x)

    Sets coefficient `n` of ``poly`` to the ``ulong`` value
    ``x``.  Coefficient numbering starts from zero and if `n` is beyond
    the current length of ``poly`` then the polynomial is extended and
    zero coefficients inserted if necessary.


Comparison
--------------------------------------------------------------------------------


.. function:: int fmpz_poly_equal(const fmpz_poly_t poly1, const fmpz_poly_t poly2)

    Returns `1` if ``poly1`` is equal to ``poly2``, otherwise
    returns `0`.  The polynomials are assumed to be normalised.

.. function:: int fmpz_poly_equal_trunc(const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)

    Return `1` if ``poly1`` and ``poly2``, notionally truncated to
    length `n` are equal, otherwise return `0`.

.. function:: int fmpz_poly_is_zero(const fmpz_poly_t poly)

    Returns `1` if the polynomial is zero and `0` otherwise.

    This function is implemented as a macro.

.. function:: int fmpz_poly_is_one(const fmpz_poly_t poly)

    Returns `1` if the polynomial is one and `0` otherwise.

.. function:: int fmpz_poly_is_unit(const fmpz_poly_t poly)

    Returns `1` if the polynomial is the constant polynomial `\pm 1`,
    and `0` otherwise.

.. function:: int fmpz_poly_is_gen(const fmpz_poly_t poly)

    Returns `1` if the polynomial is the degree `1` polynomial `x`, and `0`
    otherwise.


Addition and subtraction
--------------------------------------------------------------------------------


.. function:: void _fmpz_poly_add(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)

    Sets ``res`` to the sum of ``(poly1, len1)`` and
    ``(poly2, len2)``.  It is assumed that ``res`` has
    sufficient space for the longer of the two polynomials.

.. function:: void fmpz_poly_add(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

    Sets ``res`` to the sum of ``poly1`` and ``poly2``.

.. function:: void fmpz_poly_add_series(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)

    Notionally truncate ``poly1`` and ``poly2`` to length `n` and then
    set ``res`` to the sum.

.. function:: void _fmpz_poly_sub(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)

    Sets ``res`` to ``(poly1, len1)`` minus ``(poly2, len2)``.  It
    is assumed that ``res`` has sufficient space for the longer of the
    two polynomials.

.. function:: void fmpz_poly_sub(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

    Sets ``res`` to ``poly1`` minus ``poly2``.

.. function:: void fmpz_poly_sub_series(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)

    Notionally truncate ``poly1`` and ``poly2`` to length `n` and then
    set ``res`` to the sum.

.. function:: void fmpz_poly_neg(fmpz_poly_t res, const fmpz_poly_t poly)

    Sets ``res`` to ``-poly``.


Scalar absolute value, multiplication and division
--------------------------------------------------------------------------------


.. function:: void fmpz_poly_scalar_abs(fmpz_poly_t res, const fmpz_poly_t poly)

    Sets ``poly1`` to the polynomial whose coefficients are the absolute
    value of those of ``poly2``.

.. function:: void fmpz_poly_scalar_mul_fmpz(fmpz_poly_t poly1, const fmpz_poly_t poly2, const fmpz_t x)

    Sets ``poly1`` to ``poly2`` times `x`.

.. function:: void fmpz_poly_scalar_mul_si(fmpz_poly_t poly1, const fmpz_poly_t poly2, slong x)

    Sets ``poly1`` to ``poly2`` times the signed ``slong x``.

.. function:: void fmpz_poly_scalar_mul_ui(fmpz_poly_t poly1, const fmpz_poly_t poly2, ulong x)

    Sets ``poly1`` to ``poly2`` times the ``ulong x``.

.. function:: void fmpz_poly_scalar_mul_2exp(fmpz_poly_t poly1, const fmpz_poly_t poly2, ulong exp)

    Sets ``poly1`` to ``poly2`` times ``2^exp``.

.. function:: void fmpz_poly_scalar_addmul_si(fmpz_poly_t poly1, const fmpz_poly_t poly2, slong x)

.. function:: void fmpz_poly_scalar_addmul_ui(fmpz_poly_t poly1, const fmpz_poly_t poly2, ulong x)

.. function:: void fmpz_poly_scalar_addmul_fmpz(fmpz_poly_t poly1, const fmpz_poly_t poly2, const fmpz_t x)

    Sets ``poly1`` to ``poly1 + x * poly2``.

.. function:: void fmpz_poly_scalar_submul_fmpz(fmpz_poly_t poly1, const fmpz_poly_t poly2, const fmpz_t x)

    Sets ``poly1`` to ``poly1 - x * poly2``.

.. function:: void fmpz_poly_scalar_fdiv_fmpz(fmpz_poly_t poly1, const fmpz_poly_t poly2, const fmpz_t x)

    Sets ``poly1`` to ``poly2`` divided by the ``fmpz_t x``,
    rounding coefficients down toward `- \infty`.

.. function:: void fmpz_poly_scalar_fdiv_si(fmpz_poly_t poly1, const fmpz_poly_t poly2, slong x)

    Sets ``poly1`` to ``poly2`` divided by the ``slong x``,
    rounding coefficients down toward `- \infty`.

.. function:: void fmpz_poly_scalar_fdiv_ui(fmpz_poly_t poly1, const fmpz_poly_t poly2, ulong x)

    Sets ``poly1`` to ``poly2`` divided by the ``ulong x``,
    rounding coefficients down toward `- \infty`.

.. function:: void fmpz_poly_scalar_fdiv_2exp(fmpz_poly_t poly1, const fmpz_poly_t poly2, ulong x)

    Sets ``poly1`` to ``poly2`` divided by ``2^x``,
    rounding coefficients down toward `- \infty`.

.. function:: void fmpz_poly_scalar_tdiv_fmpz(fmpz_poly_t poly1, const fmpz_poly_t poly2, const fmpz_t x)

    Sets ``poly1`` to ``poly2`` divided by the ``fmpz_t x``,
    rounding coefficients toward `0`.

.. function:: void fmpz_poly_scalar_tdiv_si(fmpz_poly_t poly1, const fmpz_poly_t poly2, slong x)

    Sets ``poly1`` to ``poly2`` divided by the ``slong x``,
    rounding coefficients toward `0`.

.. function:: void fmpz_poly_scalar_tdiv_ui(fmpz_poly_t poly1, const fmpz_poly_t poly2, ulong x)

    Sets ``poly1`` to ``poly2`` divided by the ``ulong x``,
    rounding coefficients toward `0`.

.. function:: void fmpz_poly_scalar_tdiv_2exp(fmpz_poly_t poly1, const fmpz_poly_t poly2, ulong x)

    Sets ``poly1`` to ``poly2`` divided by ``2^x``,
    rounding coefficients toward `0`.

.. function:: void fmpz_poly_scalar_divexact_fmpz(fmpz_poly_t poly1, const fmpz_poly_t poly2, const fmpz_t x)

    Sets ``poly1`` to ``poly2`` divided by the ``fmpz_t x``,
    assuming the division is exact for every coefficient.

.. function:: void fmpz_poly_scalar_divexact_si(fmpz_poly_t poly1, const fmpz_poly_t poly2, slong x)

    Sets ``poly1`` to ``poly2`` divided by the ``slong x``,
    assuming the coefficient is exact for every coefficient.

.. function:: void fmpz_poly_scalar_divexact_ui(fmpz_poly_t poly1, const fmpz_poly_t poly2, ulong x)

    Sets ``poly1`` to ``poly2`` divided by the ``ulong x``,
    assuming the coefficient is exact for every coefficient.

.. function:: void fmpz_poly_scalar_mod_fmpz(fmpz_poly_t poly1, const fmpz_poly_t poly2, const fmpz_t p)

    Sets ``poly1`` to ``poly2``, reducing each coefficient
    modulo `p > 0`.

.. function:: void fmpz_poly_scalar_smod_fmpz(fmpz_poly_t poly1, const fmpz_poly_t poly2, const fmpz_t p)

    Sets ``poly1`` to ``poly2``, symmetrically reducing
    each coefficient modulo `p > 0`, that is, choosing the unique
    representative in the interval `(-p/2, p/2]`.

.. function:: slong _fmpz_poly_remove_content_2exp(fmpz * pol, slong len)

    Remove the 2-content of ``pol`` and return the number `k`
    that is the maximal non-negative integer so that `2^k` divides
    all coefficients of the polynomial. For the zero polynomial,
    `0` is returned.

.. function:: void _fmpz_poly_scale_2exp(fmpz * pol, slong len, slong k)

    Scale ``(pol, len)`` to `p(2^k X)` in-place and divide by the
    2-content (so that the gcd of coefficients is odd). If ``k``
    is negative the polynomial is multiplied by `2^{kd}`.


Bit packing
--------------------------------------------------------------------------------


.. function:: void _fmpz_poly_bit_pack(nn_ptr arr, const fmpz * poly, slong len, flint_bitcnt_t bit_size, int negate)

    Packs the coefficients of ``poly`` into bitfields of the given
    ``bit_size``, negating the coefficients before packing
    if ``negate`` is set to `-1`.

.. function:: int _fmpz_poly_bit_unpack(fmpz * poly, slong len, nn_srcptr arr, flint_bitcnt_t bit_size, int negate)

    Unpacks the polynomial of given length from the array as packed into
    fields of the given ``bit_size``, finally negating the coefficients
    if ``negate`` is set to `-1`. Returns borrow, which is nonzero if a
    leading term with coefficient `\pm1` should be added at
    position ``len`` of ``poly``.

.. function:: void _fmpz_poly_bit_unpack_unsigned(fmpz * poly, slong len, nn_srcptr arr, flint_bitcnt_t bit_size)

    Unpacks the polynomial of given length from the array as packed into
    fields of the given ``bit_size``.  The coefficients are assumed to
    be unsigned.

.. function:: void fmpz_poly_bit_pack(fmpz_t f, const fmpz_poly_t poly, flint_bitcnt_t bit_size)

    Packs ``poly`` into bitfields of size ``bit_size``, writing the
    result to ``f``. The sign of ``f`` will be the same as that of
    the leading coefficient of ``poly``.

.. function:: void fmpz_poly_bit_unpack(fmpz_poly_t poly, const fmpz_t f, flint_bitcnt_t bit_size)

    Unpacks the polynomial with signed coefficients packed into
    fields of size ``bit_size`` as represented by the integer ``f``.

.. function:: void fmpz_poly_bit_unpack_unsigned(fmpz_poly_t poly, const fmpz_t f, flint_bitcnt_t bit_size)

    Unpacks the polynomial with unsigned coefficients packed into
    fields of size ``bit_size`` as represented by the integer ``f``.
    It is required that ``f`` is nonnegative.


Multiplication
--------------------------------------------------------------------------------


.. function:: void _fmpz_poly_mul_classical(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)

    Sets ``(res, len1 + len2 - 1)`` to the product of ``(poly1, len1)``
    and ``(poly2, len2)``.

    Assumes ``len1`` and ``len2`` are positive.  Allows zero-padding
    of the two input polynomials.  No aliasing of inputs with outputs is
    allowed.

.. function:: void fmpz_poly_mul_classical(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

    Sets ``res`` to the product of ``poly1`` and ``poly2``, computed
    using the classical or schoolbook method.

.. function:: void _fmpz_poly_mullow_classical(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2, slong n)

    Sets ``(res, n)`` to the first `n` coefficients of ``(poly1, len1)``
    multiplied by ``(poly2, len2)``.

    Assumes ``0 < n <= len1 + len2 - 1``.  Assumes neither ``len1`` nor
    ``len2`` is zero.

.. function:: void fmpz_poly_mullow_classical(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)

    Sets ``res`` to the first `n` coefficients of ``poly1 * poly2``.

.. function:: void _fmpz_poly_mulhigh_classical(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2, slong start)

    Sets the first ``start`` coefficients of ``res`` to zero and the
    remainder to the corresponding coefficients of
    ``(poly1, len1) * (poly2, len2)``.

    Assumes ``start <= len1 + len2 - 1``.  Assumes neither ``len1`` nor
    ``len2`` is zero.

.. function:: void fmpz_poly_mulhigh_classical(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong start)

    Sets the first ``start`` coefficients of ``res`` to zero and the
    remainder to the corresponding coefficients of the product of ``poly1``
    and ``poly2``.

.. function:: void _fmpz_poly_mulmid_classical(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)

    Sets ``res`` to the middle ``len1 - len2 + 1`` coefficients of
    the product of ``(poly1, len1)`` and ``(poly2, len2)``, i.e. the
    coefficients from degree ``len2 - 1`` to ``len1 - 1`` inclusive.
    Assumes that ``len1 >= len2 > 0``.

.. function:: void fmpz_poly_mulmid_classical(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

    Sets ``res`` to the middle ``len(poly1) - len(poly2) + 1``
    coefficients of ``poly1 * poly2``, i.e. the coefficient from degree
    ``len2 - 1`` to ``len1 - 1`` inclusive.  Assumes that
    ``len1 >= len2``.

.. function:: void _fmpz_poly_mul_karatsuba(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)

    Sets ``(res, len1 + len2 - 1)`` to the product of ``(poly1, len1)``
    and ``(poly2, len2)``.  Assumes ``len1 >= len2 > 0``.  Allows
    zero-padding of the two input polynomials.  No aliasing of inputs with
    outputs is allowed.

.. function:: void fmpz_poly_mul_karatsuba(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

    Sets ``res`` to the product of ``poly1`` and ``poly2``.

.. function:: void _fmpz_poly_mullow_karatsuba_n(fmpz * res, const fmpz * poly1, const fmpz * poly2, slong n)

    Sets ``res`` to the product of ``poly1`` and ``poly2`` and
    truncates to the given length.  It is assumed that ``poly1`` and
    ``poly2`` are precisely the given length, possibly zero padded.
    Assumes `n` is not zero.

.. function:: void fmpz_poly_mullow_karatsuba_n(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)

    Sets ``res`` to the product of ``poly1`` and ``poly2`` and
    truncates to the given length.

.. function:: void _fmpz_poly_mulhigh_karatsuba_n(fmpz * res, const fmpz * poly1, const fmpz * poly2, slong len)

    Sets ``res`` to the product of ``poly1`` and ``poly2`` and
    truncates at the top to the given length.  The first ``len - 1``
    coefficients are set to zero. It is assumed that ``poly1`` and
    ``poly2`` are precisely the given length, possibly zero padded.
    Assumes ``len`` is not zero.

.. function:: void fmpz_poly_mulhigh_karatsuba_n(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong len)

    Sets the first ``len - 1`` coefficients of the result to zero and the
    remaining coefficients to the corresponding coefficients of the product of
    ``poly1`` and ``poly2``.  Assumes ``poly1`` and ``poly2`` are
    at most of the given length.

.. function:: void _fmpz_poly_mul_KS(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)

    Sets ``(res, len1 + len2 - 1)`` to the product of ``(poly1, len1)``
    and ``(poly2, len2)``.

    Places no assumptions on ``len1`` and ``len2``.  Allows zero-padding
    of the two input polynomials.  Supports aliasing of inputs and outputs.

.. function:: void fmpz_poly_mul_KS(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

    Sets ``res`` to the product of ``poly1`` and ``poly2``.

.. function:: void _fmpz_poly_mullow_KS(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2, slong n)

    Sets ``(res, n)`` to the lowest `n` coefficients of the product of
    ``(poly1, len1)`` and ``(poly2, len2)``.

    Assumes that ``len1`` and ``len2`` are positive, but does allow
    for the polynomials to be zero-padded.  The polynomials may be zero,
    too.  Assumes `n` is positive.  Supports aliasing between ``res``,
    ``poly1`` and ``poly2``.

.. function:: void fmpz_poly_mullow_KS(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)

    Sets ``res`` to the lowest `n` coefficients of the product of
    ``poly1`` and ``poly2``.

.. function:: void _fmpz_poly_mul_SS(fmpz * output, const fmpz * input1, slong length1, const fmpz * input2, slong length2)

    Sets ``(output, length1 + length2 - 1)`` to the product of
    ``(input1, length1)`` and ``(input2, length2)``.

    We must have ``len1 > 1`` and ``len2 > 1``.  Allows zero-padding
    of the two input polynomials.  Supports aliasing of inputs and outputs.

.. function:: void fmpz_poly_mul_SS(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

    Sets ``res`` to the product of ``poly1`` and ``poly2``. Uses the
    Schönhage-Strassen algorithm.

.. function:: void _fmpz_poly_mullow_SS(fmpz * output, const fmpz * input1, slong length1, const fmpz * input2, slong length2, slong n)

    Sets ``(res, n)`` to the lowest `n` coefficients of the product of
    ``(poly1, len1)`` and ``(poly2, len2)``.

    Assumes that ``len1`` and ``len2`` are positive, but does allow
    for the polynomials to be zero-padded.  We must have ``len1 > 1``
    and ``len2 > 1``. Assumes `n` is positive. Supports aliasing between
    ``res``, ``poly1`` and ``poly2``.

.. function:: void fmpz_poly_mullow_SS(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)

    Sets ``res`` to the lowest `n` coefficients of the product of
    ``poly1`` and ``poly2``.

.. function:: void _fmpz_poly_mul(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)

    Sets ``(res, len1 + len2 - 1)`` to the product of ``(poly1, len1)``
    and ``(poly2, len2)``.  Assumes ``len1 >= len2 > 0``.  Allows
    zero-padding of the two input polynomials. Does not support aliasing
    between the inputs and the output.


.. function:: void fmpz_poly_mul(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

    Sets ``res`` to the product of ``poly1`` and ``poly2``.  Chooses
    an optimal algorithm from the choices above.

.. function:: void _fmpz_poly_mullow(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2, slong n)

    Sets ``(res, n)`` to the lowest `n` coefficients of the product of
    ``(poly1, len1)`` and ``(poly2, len2)``.

    Assumes ``len1 >= len2 > 0`` and ``0 < n <= len1 + len2 - 1``.
    Allows for zero-padding in the inputs.  Does not support aliasing between
    the inputs and the output.

.. function:: void fmpz_poly_mullow(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)

    Sets ``res`` to the lowest `n` coefficients of the product of
    ``poly1`` and ``poly2``.

.. function:: void fmpz_poly_mulhigh_n(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)

    Sets the high `n` coefficients of ``res`` to the high `n` coefficients
    of the product of ``poly1`` and ``poly2``, assuming the latter are
    precisely `n` coefficients in length, zero padded if necessary.  The
    remaining `n - 1` coefficients may be arbitrary.

.. function:: void _fmpz_poly_mulhigh(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2, slong start)

    Sets all but the low `n` coefficients of `res` to the corresponding
    coefficients of the product of `poly1` of length `len1` and `poly2` of
    length `len2`, the remaining coefficients being arbitrary. It is assumed
    that `len1 >= len2 > 0` and that `0 < n < len1 + len2 - 1`. Aliasing of
    inputs is not permitted.


FFT precached multiplication
--------------------------------------------------------------------------------


.. function:: void fmpz_poly_mul_SS_precache_init(fmpz_poly_mul_precache_t pre, slong len1, slong bits1, const fmpz_poly_t poly2)

    Precompute the FFT of ``poly2`` to enable repeated multiplication of
    ``poly2`` by polynomials whose length does not exceed ``len1`` and
    whose number of bits per coefficient does not exceed ``bits1``.

    The value ``bits1`` may be negative, i.e. it may be the result of
    calling ``fmpz_poly_max_bits``. The function only considers the
    absolute value of ``bits1``.

    Suppose ``len2`` is the length of ``poly2`` and
    ``len = len1 + len2 - 1`` is the maximum output length of a polynomial
    multiplication using ``pre``. Then internally ``len`` is rounded up to
    a power of two, `2^n` say. The truncated FFT algorithm is used to smooth
    performance but note that it can only do this in the range
    `(2^{n-1}, 2^n]`. Therefore, it may be more efficient to recompute `pre`
    for cases where the output length will fall below `2^{n-1} + 1`. Otherwise
    the implementation will zero pad them up to that length.

    Note that the Schoenhage-Strassen algorithm is only efficient for
    polynomials with relatively large coefficients relative to the length of
    the polynomials.

    Also note that there are no restrictions on the polynomials. In particular
    the polynomial whose FFT is being precached does not have to be either
    longer or shorter than the polynomials it is to be multiplied by.

.. function:: void fmpz_poly_mul_precache_clear(fmpz_poly_mul_precache_t pre)

    Clear the space allocated by ``fmpz_poly_mul_SS_precache_init``.

.. function:: void _fmpz_poly_mullow_SS_precache(fmpz * output, const fmpz * input1, slong len1, fmpz_poly_mul_precache_t pre, slong trunc)

    Write into ``output`` the first ``trunc`` coefficients of
    the polynomial ``(input1, len1)`` by the polynomial whose FFT was precached
    by ``fmpz_poly_mul_SS_precache_init`` and stored in ``pre``.

    For performance reasons it is recommended that all polynomials be truncated
    to at most ``trunc`` coefficients if possible.

.. function:: void fmpz_poly_mullow_SS_precache(fmpz_poly_t res, const fmpz_poly_t poly1, fmpz_poly_mul_precache_t pre, slong n)

    Set ``res`` to the product of ``poly1`` by the polynomial whose FFT was
    precached by ``fmpz_poly_mul_SS_precache_init`` (and stored in pre). The
    result is truncated to `n` coefficients (and normalised).

    There are no restrictions on the length of ``poly1`` other than those given
    in the call to ``fmpz_poly_mul_SS_precache_init``.

.. function:: void fmpz_poly_mul_SS_precache(fmpz_poly_t res, const fmpz_poly_t poly1, fmpz_poly_mul_precache_t pre)

    Set ``res`` to the product of ``poly1`` by the polynomial whose FFT was
    precached by ``fmpz_poly_mul_SS_precache_init`` (and stored in pre).

    There are no restrictions on the length of ``poly1`` other than those given
    in the call to ``fmpz_poly_mul_SS_precache_init``.

Squaring
--------------------------------------------------------------------------------


.. function:: void _fmpz_poly_sqr_KS(fmpz * rop, const fmpz * op, slong len)

    Sets ``(rop, 2*len - 1)`` to the square of ``(op, len)``,
    assuming that ``len > 0``.

    Supports zero-padding in ``(op, len)``.  Does not support aliasing.

.. function:: void fmpz_poly_sqr_KS(fmpz_poly_t rop, const fmpz_poly_t op)

    Sets ``rop`` to the square of the polynomial ``op`` using
    Kronecker segmentation.

.. function:: void _fmpz_poly_sqr_karatsuba(fmpz * rop, const fmpz * op, slong len)

    Sets ``(rop, 2*len - 1)`` to the square of ``(op, len)``,
    assuming that ``len > 0``.

    Supports zero-padding in ``(op, len)``.  Does not support aliasing.

.. function:: void fmpz_poly_sqr_karatsuba(fmpz_poly_t rop, const fmpz_poly_t op)

    Sets ``rop`` to the square of the polynomial ``op`` using
    the Karatsuba multiplication algorithm.

.. function:: void _fmpz_poly_sqr_classical(fmpz * rop, const fmpz * op, slong len)

    Sets ``(rop, 2*len - 1)`` to the square of ``(op, len)``,
    assuming that ``len > 0``.

    Supports zero-padding in ``(op, len)``.  Does not support aliasing.

.. function:: void fmpz_poly_sqr_classical(fmpz_poly_t rop, const fmpz_poly_t op)

    Sets ``rop`` to the square of the polynomial ``op`` using
    the classical or schoolbook method.

.. function:: void _fmpz_poly_sqr(fmpz * rop, const fmpz * op, slong len)

    Sets ``(rop, 2*len - 1)`` to the square of ``(op, len)``,
    assuming that ``len > 0``.

    Supports zero-padding in ``(op, len)``.  Does not support aliasing.

.. function:: void fmpz_poly_sqr(fmpz_poly_t rop, const fmpz_poly_t op)

    Sets ``rop`` to the square of the polynomial ``op``.

.. function:: void _fmpz_poly_sqrlow_KS(fmpz * res, const fmpz * poly, slong len, slong n)

    Sets ``(res, n)`` to the lowest `n` coefficients
    of the square of ``(poly, len)``.

    Assumes that ``len`` is positive, but does allow for the polynomial
    to be zero-padded.  The polynomial may be zero, too.  Assumes `n` is
    positive.  Supports aliasing between ``res`` and ``poly``.

.. function:: void fmpz_poly_sqrlow_KS(fmpz_poly_t res, const fmpz_poly_t poly, slong n)

    Sets ``res`` to the lowest `n` coefficients
    of the square of ``poly``.

.. function:: void _fmpz_poly_sqrlow_karatsuba_n(fmpz * res, const fmpz * poly, slong n)

    Sets ``(res, n)`` to the square of ``(poly, n)`` truncated
    to length `n`, which is assumed to be positive.  Allows for ``poly``
    to be zero-padded.

.. function:: void fmpz_poly_sqrlow_karatsuba_n(fmpz_poly_t res, const fmpz_poly_t poly, slong n)

    Sets ``res`` to the square of ``poly`` and
    truncates to the given length.

.. function:: void _fmpz_poly_sqrlow_classical(fmpz * res, const fmpz * poly, slong len, slong n)

    Sets ``(res, n)`` to the first `n` coefficients of the square
    of ``(poly, len)``.

    Assumes that ``0 < n <= 2 * len - 1``.

.. function:: void fmpz_poly_sqrlow_classical(fmpz_poly_t res, const fmpz_poly_t poly, slong n)

    Sets ``res`` to the first `n` coefficients of
    the square of ``poly``.

.. function:: void _fmpz_poly_sqrlow(fmpz * res, const fmpz * poly, slong len, slong n)

    Sets ``(res, n)`` to the lowest `n` coefficients
    of the square of ``(poly, len)``.

    Assumes ``len1 >= len2 > 0`` and ``0 < n <= 2 * len - 1``.
    Allows for zero-padding in the input.  Does not support aliasing
    between the input and the output.

.. function:: void fmpz_poly_sqrlow(fmpz_poly_t res, const fmpz_poly_t poly, slong n)

    Sets ``res`` to the lowest `n` coefficients
    of the square of ``poly``.


Powering
--------------------------------------------------------------------------------


.. function:: void _fmpz_poly_pow_multinomial(fmpz * res, const fmpz * poly, slong len, ulong e)

    Computes ``res = poly^e``.  This uses the J.C.P. Miller pure
    recurrence as follows:

    If `\ell` is the index of the lowest non-zero coefficient in ``poly``,
    as a first step this method zeros out the lowest `e \ell` coefficients of
    ``res``.  The recurrence above is then used to compute the remaining
    coefficients.

    Assumes ``len > 0``, ``e > 0``.  Does not support aliasing.

.. function:: void fmpz_poly_pow_multinomial(fmpz_poly_t res, const fmpz_poly_t poly, ulong e)

    Computes ``res = poly^e`` using a generalisation of binomial expansion
    called the J.C.P. Miller pure recurrence [1], [2].
    If `e` is zero, returns one, so that in particular ``0^0 = 1``.

    The formal statement of the recurrence is as follows.  Write the input
    polynomial as `P(x) = p_0 + p_1 x + \dotsb + p_m x^m` with `p_0 \neq 0`
    and let

    .. math::

        P(x)^n = a(n, 0) + a(n, 1) x + \dotsb + a(n, mn) x^{mn}.

    Then `a(n, 0) = p_0^n` and, for all `1 \leq k \leq mn`,

    .. math::

        a(n, k) =
            (k p_0)^{-1} \sum_{i = 1}^m p_i \bigl( (n + 1) i - k \bigr) a(n, k-i).

    [1] D. Knuth, The Art of Computer Programming Vol. 2, Seminumerical
    Algorithms, Third Edition (Reading, Massachusetts: Addison-Wesley, 1997)

    [2] D. Zeilberger, The J.C.P. Miller Recurrence for Exponentiating a
    Polynomial, and its q-Analog, Journal of Difference Equations and
    Applications, 1995, Vol. 1, pp. 57--60

.. function:: void _fmpz_poly_pow_binomial(fmpz * res, const fmpz * poly, ulong e)

    Computes ``res = poly^e`` when poly is of length 2, using binomial
    expansion.

    Assumes `e > 0`.  Does not support aliasing.

.. function:: void fmpz_poly_pow_binomial(fmpz_poly_t res, const fmpz_poly_t poly, ulong e)

    Computes ``res = poly^e`` when ``poly`` is of length `2`, using
    binomial expansion.

    If the length of ``poly`` is not `2`, raises an exception and aborts.

.. function:: void _fmpz_poly_pow_addchains(fmpz * res, const fmpz * poly, slong len, const int * a, int n)

    Given a star chain `1 = a_0 < a_1 < \dotsb < a_n = e` computes
    ``res = poly^e``.

    A star chain is an addition chain `1 = a_0 < a_1 < \dotsb < a_n` such
    that, for all `i > 0`, `a_i = a_{i-1} + a_j` for some `j < i`.

    Assumes that `e > 2`, or equivalently `n > 1`, and ``len > 0``.  Does
    not support aliasing.

.. function:: void fmpz_poly_pow_addchains(fmpz_poly_t res, const fmpz_poly_t poly, ulong e)

    Computes ``res = poly^e`` using addition chains whenever
    `0 \leq e \leq 148`.

    If `e > 148`, raises an exception and aborts.

.. function:: void _fmpz_poly_pow_binexp(fmpz * res, const fmpz * poly, slong len, ulong e)

    Sets ``res = poly^e`` using left-to-right binary exponentiation as
    described on p. 461 of [Knu1997]_.

    Assumes that ``len > 0``, ``e > 1``.  Assumes that ``res`` is
    an array of length at least ``e*(len - 1) + 1``.  Does not support
    aliasing.

.. function:: void fmpz_poly_pow_binexp(fmpz_poly_t res, const fmpz_poly_t poly, ulong e)

    Computes ``res = poly^e`` using the binary exponentiation algorithm.
    If `e` is zero, returns one, so that in particular ``0^0 = 1``.

.. function:: void _fmpz_poly_pow_small(fmpz * res, const fmpz * poly, slong len, ulong e)

    Sets ``res = poly^e`` whenever `0 \leq e \leq 4`.

    Assumes that ``len > 0`` and that ``res`` is an array of length
    at least ``e*(len - 1) + 1``.  Does not support aliasing.

.. function:: void _fmpz_poly_pow(fmpz * res, const fmpz * poly, slong len, ulong e)

    Sets ``res = poly^e``, assuming that ``e, len > 0`` and that
    ``res`` has space for ``e*(len - 1) + 1`` coefficients.  Does
    not support aliasing.

.. function:: void fmpz_poly_pow(fmpz_poly_t res, const fmpz_poly_t poly, ulong e)

    Computes ``res = poly^e``.  If `e` is zero, returns one,
    so that in particular ``0^0 = 1``.

.. function:: void _fmpz_poly_pow_trunc(fmpz * res, const fmpz * poly, ulong e, slong n)

    Sets ``(res, n)`` to ``(poly, n)`` raised to the power `e` and
    truncated to length `n`.

    Assumes that `e, n > 0`.  Allows zero-padding of ``(poly, n)``.
    Does not support aliasing of any inputs and outputs.

.. function:: void fmpz_poly_pow_trunc(fmpz_poly_t res, const fmpz_poly_t poly, ulong e, slong n)

    Notationally raises ``poly`` to the power `e`, truncates the result
    to length `n` and writes the result in ``res``.  This is computed
    much more efficiently than simply powering the polynomial and truncating.

    Thus, if `n = 0` the result is zero.  Otherwise, whenever `e = 0` the
    result will be the constant polynomial equal to `1`.

    This function can be used to raise power series to a power in an
    efficient way.


Shifting
--------------------------------------------------------------------------------


.. function:: void _fmpz_poly_shift_left(fmpz * res, const fmpz * poly, slong len, slong n)

    Sets ``(res, len + n)`` to ``(poly, len)`` shifted left by
    `n` coefficients.

    Inserts zero coefficients at the lower end.  Assumes that ``len``
    and `n` are positive, and that ``res`` fits ``len + n`` elements.
    Supports aliasing between ``res`` and ``poly``.

.. function:: void fmpz_poly_shift_left(fmpz_poly_t res, const fmpz_poly_t poly, slong n)

    Sets ``res`` to ``poly`` shifted left by `n` coeffs.  Zero
    coefficients are inserted.

.. function:: void _fmpz_poly_shift_right(fmpz * res, const fmpz * poly, slong len, slong n)

    Sets ``(res, len - n)`` to ``(poly, len)`` shifted right by
    `n` coefficients.

    Assumes that ``len`` and `n` are positive, that ``len > n``,
    and that ``res`` fits ``len - n`` elements.  Supports aliasing
    between ``res`` and ``poly``, although in this case the top
    coefficients of ``poly`` are not set to zero.

.. function:: void fmpz_poly_shift_right(fmpz_poly_t res, const fmpz_poly_t poly, slong n)

    Sets ``res`` to ``poly`` shifted right by `n` coefficients.  If `n`
    is equal to or greater than the current length of ``poly``, ``res``
    is set to the zero polynomial.


Bit sizes and norms
--------------------------------------------------------------------------------


.. function:: ulong fmpz_poly_max_limbs(const fmpz_poly_t poly)

    Returns the maximum number of limbs required to store the absolute value
    of coefficients of ``poly``.  If ``poly`` is zero, returns `0`.

.. function:: slong fmpz_poly_max_bits(const fmpz_poly_t poly)

    Computes the maximum number of bits `b` required to store the absolute
    value of coefficients of ``poly``.  If all the coefficients of
    ``poly`` are non-negative, `b` is returned, otherwise `-b` is returned.

.. function:: void fmpz_poly_height(fmpz_t height, const fmpz_poly_t poly)

    Computes the height of ``poly``, defined as the largest of the
    absolute values of the coefficients of ``poly``. Equivalently, this
    gives the infinity norm of the coefficients. If ``poly`` is zero,
    the height is `0`.

.. function:: void _fmpz_poly_2norm(fmpz_t res, const fmpz * poly, slong len)

    Sets ``res`` to the Euclidean norm of ``(poly, len)``, that is,
    the integer square root of the sum of the squares of the coefficients
    of ``poly``.

.. function:: void fmpz_poly_2norm(fmpz_t res, const fmpz_poly_t poly)

    Sets ``res`` to the Euclidean norm of ``poly``, that is, the
    integer square root of the sum of the squares of the coefficients of
    ``poly``.

.. function:: ulong _fmpz_poly_2norm_normalised_bits(const fmpz * poly, slong len)

    Returns an upper bound on the number of bits of the normalised
    Euclidean norm of ``(poly, len)``, i.e. the number of bits of
    the Euclidean norm divided by the absolute value of the leading
    coefficient. The returned value will be no more than 1 bit too
    large.

    This is used in the computation of the Landau-Mignotte bound.

    It is assumed that ``len > 0``. The result only makes sense
    if the leading coefficient is nonzero.


Greatest common divisor
--------------------------------------------------------------------------------


.. function:: void _fmpz_poly_gcd_subresultant(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)

    Computes the greatest common divisor ``(res, len2)`` of
    ``(poly1, len1)`` and ``(poly2, len2)``, assuming
    ``len1 >= len2 > 0``.  The result is normalised to have
    positive leading coefficient.  Aliasing between ``res``,
    ``poly1`` and ``poly2`` is supported.

.. function:: void fmpz_poly_gcd_subresultant(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

    Computes the greatest common divisor ``res`` of ``poly1`` and
    ``poly2``, normalised to have non-negative leading coefficient.

    This function uses the subresultant algorithm as described
    in Algorithm 3.3.1 of [Coh1996]_.

.. function:: int _fmpz_poly_gcd_heuristic(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)

    Computes the greatest common divisor ``(res, len2)`` of
    ``(poly1, len1)`` and ``(poly2, len2)``, assuming
    ``len1 >= len2 > 0``.  The result is normalised to have
    positive leading coefficient.  Aliasing between ``res``,
    ``poly1`` and ``poly2`` is not supported. The function
    may not always succeed in finding the GCD. If it fails, the
    function returns 0, otherwise it returns 1.

.. function:: int fmpz_poly_gcd_heuristic(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

    Computes the greatest common divisor ``res`` of ``poly1`` and
    ``poly2``, normalised to have non-negative leading coefficient.

    The function may not always succeed in finding the GCD. If it fails,
    the function returns 0, otherwise it returns 1.

    This function uses the heuristic GCD algorithm (GCDHEU). The basic
    strategy is to remove the content of the polynomials, pack them
    using Kronecker segmentation (given a bound on the size of the
    coefficients of the GCD) and take the integer GCD. Unpack the
    result and test divisibility.

.. function:: void _fmpz_poly_gcd_modular(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)

    Computes the greatest common divisor ``(res, len2)`` of
    ``(poly1, len1)`` and ``(poly2, len2)``, assuming
    ``len1 >= len2 > 0``.  The result is normalised to have
    positive leading coefficient.  Aliasing between ``res``,
    ``poly1`` and ``poly2`` is not supported.

.. function:: void fmpz_poly_gcd_modular(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

    Computes the greatest common divisor ``res`` of ``poly1`` and
    ``poly2``, normalised to have non-negative leading coefficient.

    This function uses the modular GCD algorithm. The basic
    strategy is to remove the content of the polynomials, reduce them
    modulo sufficiently many primes and do CRT reconstruction until
    some bound is reached (or we can prove with trial division that
    we have the GCD).

.. function:: void _fmpz_poly_gcd(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)

    Computes the greatest common divisor ``res`` of ``(poly1, len1)``
    and ``(poly2, len2)``, assuming ``len1 >= len2 > 0``.  The result
    is normalised to have positive leading coefficient.

    Assumes that ``res`` has space for ``len2`` coefficients.
    Aliasing between ``res``, ``poly1`` and ``poly2`` is not
    supported.

.. function:: void fmpz_poly_gcd(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

    Computes the greatest common divisor ``res`` of ``poly1`` and
    ``poly2``, normalised to have non-negative leading coefficient.

.. function:: void _fmpz_poly_xgcd_modular(fmpz_t r, fmpz * s, fmpz * t, const fmpz * f, slong len1, const fmpz * g, slong len2)

    Set `r` to the resultant of ``(f, len1)`` and ``(g, len2)``.
    If the resultant is zero, the function returns immediately. Otherwise it
    finds polynomials `s` and `t` such that ``s*f + t*g = r``. The length
    of `s` will be no greater than ``len2`` and the length of `t` will be
    no greater than ``len1`` (both are zero padded if necessary).

    It is assumed that ``len1 >= len2 > 0``. No aliasing of inputs and
    outputs is permitted.

    The function assumes that `f` and `g` are primitive (have Gaussian content
    equal to 1). The result is undefined otherwise.

    Uses a multimodular algorithm. The resultant is first computed and
    extended GCDs modulo various primes `p` are computed and combined using
    CRT. When the CRT stabilises the resulting polynomials are simply reduced
    modulo further primes until a proven bound is reached.

.. function:: void fmpz_poly_xgcd_modular(fmpz_t r, fmpz_poly_t s, fmpz_poly_t t, const fmpz_poly_t f, const fmpz_poly_t g)

    Set `r` to the resultant of `f` and `g`. If the resultant is zero, the
    function then returns immediately, otherwise `s` and `t` are found such
    that ``s*f + t*g = r``.

    The function assumes that `f` and `g` are primitive (have Gaussian content
    equal to 1). The result is undefined otherwise.

    Uses the multimodular algorithm.

.. function:: void _fmpz_poly_xgcd(fmpz_t r, fmpz * s, fmpz * t, const fmpz * f, slong len1, const fmpz * g, slong len2)

    Set `r` to the resultant of ``(f, len1)`` and ``(g, len2)``.
    If the resultant is zero, the function returns immediately. Otherwise it
    finds polynomials `s` and `t` such that ``s*f + t*g = r``. The length
    of `s` will be no greater than ``len2`` and the length of `t` will be
    no greater than ``len1`` (both are zero padded if necessary).

    The function assumes that `f` and `g` are primitive (have Gaussian content
    equal to 1). The result is undefined otherwise.

    It is assumed that ``len1 >= len2 > 0``. No aliasing of inputs and
    outputs is permitted.

.. function:: void fmpz_poly_xgcd(fmpz_t r, fmpz_poly_t s, fmpz_poly_t t, const fmpz_poly_t f, const fmpz_poly_t g)

    Set `r` to the resultant of `f` and `g`. If the resultant is zero, the
    function then returns immediately, otherwise `s` and `t` are found such
    that ``s*f + t*g = r``.

    The function assumes that `f` and `g` are primitive (have Gaussian content
    equal to 1). The result is undefined otherwise.

.. function:: void _fmpz_poly_lcm(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)

    Sets ``(res, len1 + len2 - 1)`` to the least common multiple
    of the two polynomials ``(poly1, len1)`` and ``(poly2, len2)``,
    normalised to have non-negative leading coefficient.

    Assumes that ``len1 >= len2 > 0``.

    Does not support aliasing.

.. function:: void fmpz_poly_lcm(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

    Sets ``res`` to the least common multiple of the two
    polynomials ``poly1`` and ``poly2``, normalised to
    have non-negative leading coefficient.

    If either of the two polynomials is zero, sets ``res``
    to zero.

    This ensures that the equality

    .. math::

        f g = \gcd(f, g) \operatorname{lcm}(f, g)

    holds up to sign.

.. function:: void _fmpz_poly_resultant_modular(fmpz_t res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)

    Sets ``res`` to the resultant of ``(poly1, len1)`` and
    ``(poly2, len2)``, assuming that ``len1 >= len2 > 0``.

.. function:: void fmpz_poly_resultant_modular(fmpz_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

    Computes the resultant of ``poly1`` and ``poly2``.

    For two non-zero polynomials `f(x) = a_m x^m + \dotsb + a_0` and
    `g(x) = b_n x^n + \dotsb + b_0` of degrees `m` and `n`, the resultant
    is defined to be

    .. math::

        a_m^n b_n^m \prod_{(x, y) : f(x) = g(y) = 0} (x - y).

    For convenience, we define the resultant to be equal to zero if either
    of the two polynomials is zero.

    This function uses the modular algorithm described
    in [Col1971]_.

.. function:: void fmpz_poly_resultant_modular_div(fmpz_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, const fmpz_t div, slong nbits)

    Computes the resultant of ``poly1`` and ``poly2`` divided by
    ``div`` using a slight modification of the above function. It is assumed that
    the resultant is exactly divisible by ``div`` and the result ``res``
    has at most ``nbits`` bits.
    This bypasses the computation of general bounds.


.. function:: void _fmpz_poly_resultant_euclidean(fmpz_t res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)

    Sets ``res`` to the resultant of ``(poly1, len1)`` and
    ``(poly2, len2)``, assuming that ``len1 >= len2 > 0``.

.. function:: void fmpz_poly_resultant_euclidean(fmpz_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

    Computes the resultant of ``poly1`` and ``poly2``.

    For two non-zero polynomials `f(x) = a_m x^m + \dotsb + a_0` and
    `g(x) = b_n x^n + \dotsb + b_0` of degrees `m` and `n`, the resultant
    is defined to be

    .. math::

        a_m^n b_n^m \prod_{(x, y) : f(x) = g(y) = 0} (x - y).

    For convenience, we define the resultant to be equal to zero if either
    of the two polynomials is zero.

    This function uses the algorithm described
    in Algorithm 3.3.7 of [Coh1996]_.

.. function:: void _fmpz_poly_resultant(fmpz_t res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)

    Sets ``res`` to the resultant of ``(poly1, len1)`` and
    ``(poly2, len2)``, assuming that ``len1 >= len2 > 0``.

.. function:: void fmpz_poly_resultant(fmpz_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

    Computes the resultant of ``poly1`` and ``poly2``.

    For two non-zero polynomials `f(x) = a_m x^m + \dotsb + a_0` and
    `g(x) = b_n x^n + \dotsb + b_0` of degrees `m` and `n`, the resultant
    is defined to be

    .. math::

        a_m^n b_n^m \prod_{(x, y) : f(x) = g(y) = 0} (x - y).

    For convenience, we define the resultant to be equal to zero if either
    of the two polynomials is zero.


Discriminant
--------------------------------------------------------------------------------


.. function:: void _fmpz_poly_discriminant(fmpz_t res, const fmpz * poly, slong len)

    Set ``res`` to the discriminant of ``(poly, len)``. Assumes
    ``len > 1``.

.. function:: void fmpz_poly_discriminant(fmpz_t res, const fmpz_poly_t poly)

    Set ``res`` to the discriminant of ``poly``. We normalise the
    discriminant so that `\operatorname{disc}(f) = (-1)^{(n(n-1)/2)}
    \operatorname{res}(f, f')/\operatorname{lc}(f)`, thus
    `\operatorname{disc}(f) = \operatorname{lc}(f)^{(2n - 2)} \prod_{i < j} (r_i
    - r_j)^2`, where `\operatorname{lc}(f)` is the leading coefficient of `f`,
    `n` is the degree of `f` and `r_i` are the roots of `f`.


Gaussian content
--------------------------------------------------------------------------------


.. function:: void _fmpz_poly_content(fmpz_t res, const fmpz * poly, slong len)

    Sets ``res`` to the non-negative content of ``(poly, len)``.
    Aliasing between ``res`` and the coefficients of ``poly`` is
    not supported.

.. function:: void fmpz_poly_content(fmpz_t res, const fmpz_poly_t poly)

    Sets ``res`` to the non-negative content of ``poly``.  The content
    of the zero polynomial is defined to be zero.  Supports aliasing, that is,
    ``res`` is allowed to be one of the coefficients of ``poly``.

.. function:: void _fmpz_poly_primitive_part(fmpz * res, const fmpz * poly, slong len)

    Sets ``(res, len)`` to ``(poly, len)`` divided by the content
    of ``(poly, len)``, and normalises the result to have non-negative
    leading coefficient.

    Assumes that ``(poly, len)`` is non-zero.  Supports aliasing of
    ``res`` and ``poly``.

.. function:: void fmpz_poly_primitive_part(fmpz_poly_t res, const fmpz_poly_t poly)

    Sets ``res`` to ``poly`` divided by the content of ``poly``,
    and normalises the result to have non-negative leading coefficient.
    If ``poly`` is zero, sets ``res`` to zero.


Square-free
--------------------------------------------------------------------------------


.. function:: int _fmpz_poly_is_squarefree(const fmpz * poly, slong len)

    Returns whether the polynomial ``(poly, len)`` is square-free.

.. function:: int fmpz_poly_is_squarefree(const fmpz_poly_t poly)

    Returns whether the polynomial ``poly`` is square-free.  A non-zero
    polynomial is defined to be square-free if it has no non-unit square
    factors.  We also define the zero polynomial to be square-free.

    Returns `1` if the length of ``poly`` is at most `2`.  Returns whether
    the discriminant is zero for quadratic polynomials.  Otherwise, returns
    whether the greatest common divisor of ``poly`` and its derivative has
    length `1`.


Euclidean division
--------------------------------------------------------------------------------


.. function:: int _fmpz_poly_divrem_basecase(fmpz * Q, fmpz * R, const fmpz * A, slong lenA, const fmpz * B, slong lenB, int exact)

    Computes ``(Q, lenA - lenB + 1)``, ``(R, lenA)`` such that
    `A = B Q + R` and each coefficient of `R` beyond ``lenB`` is reduced
    modulo the leading coefficient of `B`.
    If the leading coefficient of `B` is `\pm 1` or the division is exact,
    this is the same thing as division over `\mathbb{Q}`.

    Assumes that `\operatorname{len}(A), \operatorname{len}(B) > 0`.  Allows zero-padding in
    ``(A, lenA)``.  `R` and `A` may be aliased, but apart from this no
    aliasing of input and output operands is allowed.

    If the flag ``exact`` is `1`, the function stops if an inexact division
    is encountered, upon which the function will return `0`. If no inexact
    division is encountered, the function returns `1`. Note that this does not
    guarantee the remainder of the polynomial division is zero, merely that
    its length is less than that of B. This feature is useful for series
    division and for divisibility testing (upon testing the remainder).

    For ordinary use set the flag ``exact`` to `0`. In this case, no checks
    or early aborts occur and the function always returns `1`.

.. function:: void fmpz_poly_divrem_basecase(fmpz_poly_t Q, fmpz_poly_t R, const fmpz_poly_t A, const fmpz_poly_t B)

    Computes `Q`, `R` such that `A = B Q + R` and each coefficient of `R`
    beyond `\operatorname{len}(B) - 1` is reduced modulo the leading coefficient of `B`.
    If the leading coefficient of `B` is `\pm 1` or the division is exact,
    this is the same thing as division over `\mathbb{Q}`.  An exception is raised
    if `B` is zero.

.. function:: int _fmpz_poly_divrem_divconquer_recursive(fmpz * Q, fmpz * BQ, fmpz * W, const fmpz * A, const fmpz * B, slong lenB, int exact)

    Computes ``(Q, lenB)``, ``(BQ, 2 lenB - 1)`` such that
    `BQ = B \times Q` and `A = B Q + R` where each coefficient of `R` beyond
    `\operatorname{len}(B) - 1` is reduced modulo the leading coefficient of `B`.  We
    assume that `\operatorname{len}(A) = 2 \operatorname{len}(B) - 1`.  If the leading coefficient
    of `B` is `\pm 1` or the division is exact, this is the same as division
    over `\mathbb{Q}`.

    Assumes `\operatorname{len}(B) > 0`.  Allows zero-padding in ``(A, lenA)``.  Requires
    a temporary array ``(W, 2 lenB - 1)``.  No aliasing of input and output
    operands is allowed.

    This function does not read the bottom `\operatorname{len}(B) - 1` coefficients from
    `A`, which means that they might not even need to exist in allocated
    memory.

    If the flag ``exact`` is `1`, the function stops if an inexact division
    is encountered, upon which the function will return `0`. If no inexact
    division is encountered, the function returns `1`. Note that this does not
    guarantee the remainder of the polynomial division is zero, merely that
    its length is less than that of B. This feature is useful for series
    division and for divisibility testing (upon testing the remainder).

    For ordinary use set the flag ``exact`` to `0`. In this case, no checks
    or early aborts occur and the function always returns `1`.

.. function:: int _fmpz_poly_divrem_divconquer(fmpz * Q, fmpz * R, const fmpz * A, slong lenA, const fmpz * B, slong lenB, int exact)

    Computes ``(Q, lenA - lenB + 1)``, ``(R, lenA)`` such that
    `A = B Q + R` and each coefficient of `R` beyond `\operatorname{len}(B) - 1` is
    reduced modulo the leading coefficient of `B`.  If the leading
    coefficient of `B` is `\pm 1` or the division is exact, this is
    the same as division over `\mathbb{Q}`.

    Assumes `\operatorname{len}(A) \geq \operatorname{len}(B) > 0`.  Allows zero-padding in
    ``(A, lenA)``.  No aliasing of input and output operands is
    allowed.

    If the flag ``exact`` is `1`, the function stops if an inexact division
    is encountered, upon which the function will return `0`. If no inexact
    division is encountered, the function returns `1`. Note that this does not
    guarantee the remainder of the polynomial division is zero, merely that
    its length is less than that of B. This feature is useful for series
    division and for divisibility testing (upon testing the remainder).

    For ordinary use set the flag ``exact`` to `0`. In this case, no checks
    or early aborts occur and the function always returns `1`.

.. function:: void fmpz_poly_divrem_divconquer(fmpz_poly_t Q, fmpz_poly_t R, const fmpz_poly_t A, const fmpz_poly_t B)

    Computes `Q`, `R` such that `A = B Q + R` and each coefficient of `R`
    beyond `\operatorname{len}(B) - 1` is reduced modulo the leading coefficient of `B`.
    If the leading coefficient of `B` is `\pm 1` or the division is exact,
    this is the same as division over `\mathbb{Q}`.  An exception is raised if `B`
    is zero.

.. function:: int _fmpz_poly_divrem(fmpz * Q, fmpz * R, const fmpz * A, slong lenA, const fmpz * B, slong lenB, int exact)

    Computes ``(Q, lenA - lenB + 1)``, ``(R, lenA)`` such that
    `A = B Q + R` and each coefficient of `R` beyond `\operatorname{len}(B) - 1` is
    reduced modulo the leading coefficient of `B`.  If the leading
    coefficient of `B` is `\pm 1` or the division is exact, this is
    the same thing as division over `\mathbb{Q}`.

    Assumes `\operatorname{len}(A) \geq \operatorname{len}(B) > 0`.  Allows zero-padding in
    ``(A, lenA)``.  No aliasing of input and output operands is
    allowed.

    If the flag ``exact`` is `1`, the function stops if an inexact division
    is encountered, upon which the function will return `0`. If no inexact
    division is encountered, the function returns `1`. Note that this does not
    guarantee the remainder of the polynomial division is zero, merely that
    its length is less than that of B. This feature is useful for series
    division and for divisibility testing (upon testing the remainder).

    For ordinary use set the flag ``exact`` to `0`. In this case, no checks
    or early aborts occur and the function always returns `1`.

.. function:: void fmpz_poly_divrem(fmpz_poly_t Q, fmpz_poly_t R, const fmpz_poly_t A, const fmpz_poly_t B)

    Computes `Q`, `R` such that `A = B Q + R` and each coefficient of `R`
    beyond `\operatorname{len}(B) - 1` is reduced modulo the leading coefficient of `B`.
    If the leading coefficient of `B` is `\pm 1` or the division is exact,
    this is the same as division over `\mathbb{Q}`.  An exception is raised if `B`
    is zero.

.. function:: int _fmpz_poly_div_basecase(fmpz * Q, fmpz * R, const fmpz * A, slong lenA, const fmpz * B, slong lenB, int exact)

    Computes the quotient ``(Q, lenA - lenB + 1)`` of ``(A, lenA)``
    divided by ``(B, lenB)``.

    Notationally, computes `Q`, `R` such that `A = B Q + R` and each
    coefficient of `R` beyond `\operatorname{len}(B) - 1` is reduced modulo the leading
    coefficient of `B`.

    If the leading coefficient of `B` is `\pm 1` or the division is exact,
    this is the same as division over `\mathbb{Q}`.

    Assumes `\operatorname{len}(A), \operatorname{len}(B) > 0`.  Allows zero-padding in ``(A, lenA)``.
    Requires a temporary array `R` of size at least the (actual) length
    of `A`. For convenience, `R` may be ``NULL``.  `R` and `A` may be
    aliased, but apart from this no aliasing of input and output operands
    is allowed.

    If the flag ``exact`` is `1`, the function stops if an inexact division
    is encountered, upon which the function will return `0`. If no inexact
    division is encountered, the function returns `1`. Note that this does not
    guarantee the remainder of the polynomial division is zero, merely that
    its length is less than that of B. This feature is useful for series
    division and for divisibility testing (upon testing the remainder).

    For ordinary use set the flag ``exact`` to `0`. In this case, no checks
    or early aborts occur and the function always returns `1`.

.. function:: void fmpz_poly_div_basecase(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_poly_t B)

    Computes the quotient `Q` of `A` divided by `Q`.

    Notationally, computes `Q`, `R` such that `A = B Q + R` and each
    coefficient of `R` beyond `\operatorname{len}(B) - 1` is reduced modulo the leading
    coefficient of `B`.

    If the leading coefficient of `B` is `\pm 1` or the division is exact,
    this is the same as division over `\mathbb{Q}`.  An exception is raised if `B`
    is zero.

.. function:: int _fmpz_poly_divremlow_divconquer_recursive(fmpz * Q, fmpz * BQ, const fmpz * A, const fmpz * B, slong lenB, int exact)

    Divide and conquer division of ``(A, 2 lenB - 1)`` by ``(B, lenB)``,
    computing only the bottom `\operatorname{len}(B) - 1` coefficients of `B Q`.

    Assumes `\operatorname{len}(B) > 0`.  Requires `B Q` to have length at least
    `2 \operatorname{len}(B) - 1`, although only the bottom `\operatorname{len}(B) - 1` coefficients will
    carry meaningful output.  Does not support any aliasing.  Allows
    zero-padding in `A`, but not in `B`.

    If the flag ``exact`` is `1`, the function stops if an inexact division
    is encountered, upon which the function will return `0`. If no inexact
    division is encountered, the function returns `1`. Note that this does not
    guarantee the remainder of the polynomial division is zero, merely that
    its length is less than that of B. This feature is useful for series
    division and for divisibility testing (upon testing the remainder).

    For ordinary use set the flag ``exact`` to `0`. In this case, no checks
    or early aborts occur and the function always returns `1`.

.. function:: int _fmpz_poly_div_divconquer_recursive(fmpz * Q, fmpz * temp, const fmpz * A, const fmpz * B, slong lenB, int exact)

    Recursive short division in the balanced case.

    Computes the quotient ``(Q, lenB)`` of ``(A, 2 lenB - 1)`` upon
    division by ``(B, lenB)``.  Requires `\operatorname{len}(B) > 0`.  Needs a
    temporary array ``temp`` of length `2 \operatorname{len}(B) - 1`.  Does not support
    any aliasing.

    For further details, see [Mul2000]_.

    If the flag ``exact`` is `1`, the function stops if an inexact division
    is encountered, upon which the function will return `0`. If no inexact
    division is encountered, the function returns `1`. Note that this does not
    guarantee the remainder of the polynomial division is zero, merely that
    its length is less than that of B. This feature is useful for series
    division and for divisibility testing (upon testing the remainder).

    For ordinary use set the flag ``exact`` to `0`. In this case, no checks
    or early aborts occur and the function always returns `1`.

.. function:: int _fmpz_poly_div_divconquer(fmpz * Q, const fmpz * A, slong lenA, const fmpz * B, slong lenB, int exact)

    Computes the quotient ``(Q, lenA - lenB + 1)`` of ``(A, lenA)``
    upon division by ``(B, lenB)``.  Assumes that
    `\operatorname{len}(A) \geq \operatorname{len}(B) > 0`.  Does not support aliasing.

    If the flag ``exact`` is `1`, the function stops if an inexact division
    is encountered, upon which the function will return `0`. If no inexact
    division is encountered, the function returns `1`. Note that this does not
    guarantee the remainder of the polynomial division is zero, merely that
    its length is less than that of B. This feature is useful for series
    division and for divisibility testing (upon testing the remainder).

    For ordinary use set the flag ``exact`` to `0`. In this case, no checks
    or early aborts occur and the function always returns `1`.

.. function:: void fmpz_poly_div_divconquer(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_poly_t B)

    Computes the quotient `Q` of `A` divided by `B`.

    Notationally, computes `Q`, `R` such that `A = B Q + R` and each
    coefficient of `R` beyond `\operatorname{len}(B) - 1` is reduced modulo the leading
    coefficient of `B`.

    If the leading coefficient of `B` is `\pm 1` or the division is exact,
    this is the same as division over `\mathbb{Q}`.  An exception is raised if `B`
    is zero.

.. function:: int _fmpz_poly_div(fmpz * Q, const fmpz * A, slong lenA, const fmpz * B, slong lenB, int exact)

    Computes the quotient ``(Q, lenA - lenB + 1)`` of ``(A, lenA)``
    divided by ``(B, lenB)``.

    Notationally, computes `Q`, `R` such that `A = B Q + R` and each
    coefficient of `R` beyond `\operatorname{len}(B) - 1` is reduced modulo the leading
    coefficient of `B`.  If the leading coefficient of `B` is `\pm 1` or
    the division is exact, this is the same as division over `\mathbb{Q}`.

    Assumes `\operatorname{len}(A) \geq \operatorname{len}(B) > 0`.  Allows zero-padding in
    ``(A, lenA)``.  Aliasing of input and output operands is not
    allowed.

    If the flag ``exact`` is `1`, the function stops if an inexact division
    is encountered, upon which the function will return `0`. If no inexact
    division is encountered, the function returns `1`. Note that this does not
    guarantee the remainder of the polynomial division is zero, merely that
    its length is less than that of B. This feature is useful for series
    division and for divisibility testing (upon testing the remainder).

    For ordinary use set the flag ``exact`` to `0`. In this case, no checks
    or early aborts occur and the function always returns `1`.

.. function:: void fmpz_poly_div(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_poly_t B)

    Computes the quotient `Q` of `A` divided by `B`.

    Notationally, computes `Q`, `R` such that `A = B Q + R` and each
    coefficient of `R` beyond `\operatorname{len}(B) - 1` is reduced modulo the leading
    coefficient of `B`.  If the leading coefficient of `B` is `\pm 1` or
    the division is exact, this is the same as division over `Q`.  An
    exception is raised if `B` is zero.

.. function:: void _fmpz_poly_rem_basecase(fmpz * R, const fmpz * A, slong lenA, const fmpz * B, slong lenB)

    Computes the remainder ``(R, lenA)`` of ``(A, lenA)`` upon
    division by ``(B, lenB)``.

    Notationally, computes `Q`, `R` such that `A = B Q + R` and each
    coefficient of `R` beyond `\operatorname{len}(B) - 1` is reduced modulo the leading
    coefficient of `B`.  If the leading coefficient of `B` is `\pm 1` or
    the division is exact, this is the same thing as division over `\mathbb{Q}`.

    Assumes that `\operatorname{len}(A), \operatorname{len}(B) > 0`.  Allows zero-padding in
    ``(A, lenA)``.  `R` and `A` may be aliased, but apart from this no
    aliasing of input and output operands is allowed.

.. function:: void fmpz_poly_rem_basecase(fmpz_poly_t R, const fmpz_poly_t A, const fmpz_poly_t B)

    Computes the remainder `R` of `A` upon division by `B`.

    Notationally, computes `Q`, `R` such that `A = B Q + R` and each
    coefficient of `R` beyond `\operatorname{len}(B) - 1` is reduced modulo the leading
    coefficient of `B`.  If the leading coefficient of `B` is `\pm 1` or
    the division is exact, this is the same as division over `\mathbb{Q}`.  An
    exception is raised if `B` is zero.

.. function:: void _fmpz_poly_rem(fmpz * R, const fmpz * A, slong lenA, const fmpz * B, slong lenB)

    Computes the remainder ``(R, lenA)`` of ``(A, lenA)`` upon division
    by ``(B, lenB)``.

    Notationally, computes `Q`, `R` such that `A = B Q + R` and each
    coefficient of `R` beyond `\operatorname{len}(B) - 1` is reduced modulo the leading
    coefficient of `B`.  If the leading coefficient of `B` is `\pm 1` or
    the division is exact, this is the same thing as division over `\mathbb{Q}`.

    Assumes that `\operatorname{len}(A) \geq \operatorname{len}(B) > 0`.  Allows zero-padding in
    ``(A, lenA)``.  Aliasing of input and output operands is not allowed.

.. function:: void fmpz_poly_rem(fmpz_poly_t R, const fmpz_poly_t A, const fmpz_poly_t B)

    Computes the remainder `R` of `A` upon division by `B`.

    Notationally, computes `Q`, `R` such that `A = B Q + R` and each
    coefficient of `R` beyond `\operatorname{len}(B) - 1` is reduced modulo the leading
    coefficient of `B`.  If the leading coefficient of `B` is `\pm 1` or
    the division is exact, this is the same as division over `\mathbb{Q}`.  An
    exception is raised if `B` is zero.

.. function:: void _fmpz_poly_div_root(fmpz * Q, const fmpz * A, slong len, const fmpz_t c)

    Computes the quotient ``(Q, len-1)`` of ``(A, len)`` upon
    division by `x - c`.

    Supports aliasing of ``Q`` and ``A``, but the result is
    undefined in case of partial overlap.

.. function:: void fmpz_poly_div_root(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_t c)

    Computes the quotient ``(Q, len-1)`` of ``(A, len)`` upon
    division by `x - c`.

.. function:: void _fmpz_poly_divexact(fmpz * Q, const fmpz * A, slong lenA, const fmpz * B, slong lenB)
              void fmpz_poly_divexact(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_poly_t B)

    Like :func:`fmpz_poly_div`, but assumes that the division is exact.

Division with precomputed inverse
--------------------------------------------------------------------------------


.. function:: void _fmpz_poly_preinvert(fmpz * B_inv, const fmpz * B, slong n)

    Given a monic polynomial ``B`` of length ``n``, compute a precomputed
    inverse ``B_inv`` of length ``n`` for use in the functions below. No
    aliasing of ``B`` and ``B_inv`` is permitted. We assume ``n`` is not zero.

.. function:: void fmpz_poly_preinvert(fmpz_poly_t B_inv, const fmpz_poly_t B)

    Given a monic polynomial ``B``, compute a precomputed inverse
    ``B_inv`` for use in the functions below. An exception is raised if
    ``B`` is zero.

.. function:: void _fmpz_poly_div_preinv(fmpz * Q, const fmpz * A, slong len1, const fmpz * B, const fmpz * B_inv, slong len2)

    Given a precomputed inverse ``B_inv`` of the polynomial ``B`` of
    length ``len2``, compute the quotient ``Q`` of ``A`` by ``B``.
    We assume the length ``len1`` of ``A`` is at least ``len2``. The
    polynomial ``Q`` must have space for ``len1 - len2 + 1``
    coefficients. No aliasing of operands is permitted.

.. function:: void fmpz_poly_div_preinv(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_poly_t B, const fmpz_poly_t B_inv)

    Given a precomputed inverse ``B_inv`` of the polynomial ``B``,
    compute the quotient ``Q`` of ``A`` by ``B``. Aliasing of ``B``
    and ``B_inv`` is not permitted.

.. function:: void _fmpz_poly_divrem_preinv(fmpz * Q, fmpz * A, slong len1, const fmpz * B, const fmpz * B_inv, slong len2)

    Given a precomputed inverse ``B_inv`` of the polynomial ``B`` of
    length ``len2``, compute the quotient ``Q`` of ``A`` by ``B``.
    The remainder is then placed in ``A``. We assume the length ``len1``
    of ``A`` is at least ``len2``. The polynomial ``Q`` must have
    space for ``len1 - len2 + 1`` coefficients. No aliasing of operands is
    permitted.

.. function:: void fmpz_poly_divrem_preinv(fmpz_poly_t Q, fmpz_poly_t R, const fmpz_poly_t A, const fmpz_poly_t B, const fmpz_poly_t B_inv)

    Given a precomputed inverse ``B_inv`` of the polynomial ``B``,
    compute the quotient ``Q`` of ``A`` by ``B`` and the remainder
    ``R``. Aliasing of ``B`` and ``B_inv`` is not permitted.

.. function:: fmpz ** _fmpz_poly_powers_precompute(const fmpz * B, slong len)

    Computes ``2*len - 1`` powers of `x` modulo the polynomial `B` of
    the given length. This is used as a kind of precomputed inverse in
    the remainder routine below.

.. function:: void fmpz_poly_powers_precompute(fmpz_poly_powers_precomp_t pinv, fmpz_poly_t poly)

    Computes ``2*len - 1`` powers of `x` modulo the polynomial `B` of
    the given length. This is used as a kind of precomputed inverse in
    the remainder routine below.

.. function:: void _fmpz_poly_powers_clear(fmpz ** powers, slong len)

    Clean up resources used by precomputed powers which have been computed
    by ``_fmpz_poly_powers_precompute``.

.. function:: void fmpz_poly_powers_clear(fmpz_poly_powers_precomp_t pinv)

    Clean up resources used by precomputed powers which have been computed
    by ``fmpz_poly_powers_precompute``.

.. function:: void _fmpz_poly_rem_powers_precomp(fmpz * A, slong m, const fmpz * B, slong n, fmpz ** const powers)

    Set `A` to the remainder of `A` divide `B` given precomputed powers mod `B`
    provided by ``_fmpz_poly_powers_precompute``. No aliasing is allowed.

.. function:: void fmpz_poly_rem_powers_precomp(fmpz_poly_t R, const fmpz_poly_t A, const fmpz_poly_t B, const fmpz_poly_powers_precomp_t B_inv)

    Set `R` to the remainder of `A` divide `B` given precomputed powers mod `B`
    provided by ``fmpz_poly_powers_precompute``.


Divisibility testing
--------------------------------------------------------------------------------


.. function:: int _fmpz_poly_divides(fmpz * Q, const fmpz * A, slong lenA, const fmpz * B, slong lenB)

    Returns 1 if ``(B, lenB)`` divides ``(A, lenA)`` exactly and
    sets `Q` to the quotient, otherwise returns 0.

    It is assumed that `\operatorname{len}(A) \geq \operatorname{len}(B) > 0` and that `Q` has space
    for `\operatorname{len}(A) - \operatorname{len}(B) + 1` coefficients.

    Aliasing of `Q` with either of the inputs is not permitted.

    This function is currently unoptimised and provided for convenience
    only.

.. function:: int fmpz_poly_divides(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_poly_t B)

    Returns 1 if `B` divides `A` exactly and sets `Q` to the quotient,
    otherwise returns 0.

    This function is currently unoptimised and provided for convenience
    only.

.. function:: slong fmpz_poly_remove(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

    Set ``res`` to ``poly1`` divided by the highest power of ``poly2`` that
    divides it and return the power. The divisor ``poly2`` must not be zero or
    `\pm 1`, otherwise an exception is raised.

Division mod p
--------------------------------------------------------------------------------


.. function:: void fmpz_poly_divlow_smodp(fmpz * res, const fmpz_poly_t f, const fmpz_poly_t g, const fmpz_t p, slong n)

    Compute the `n` lowest coefficients of `f` divided by `g`, assuming the
    division is exact modulo `p`. The computed coefficients are reduced modulo
    `p` using the symmetric remainder system. We require `f` to be at least `n`
    in length. The function can handle trailing zeroes, but the low nonzero
    coefficient of `g` must be coprime to `p`. This is a bespoke function used
    by factoring.

.. function:: void fmpz_poly_divhigh_smodp(fmpz * res, const fmpz_poly_t f, const fmpz_poly_t g, const fmpz_t p, slong n)

    Compute the `n` highest coefficients of `f` divided by `g`, assuming the
    division is exact modulo `p`. The computed coefficients are reduced modulo
    `p` using the symmetric remainder system. We require `f` to be as output
    by ``fmpz_poly_mulhigh_n`` given polynomials `g` and a polynomial of
    length `n` as inputs. The leading coefficient of `g` must be coprime to
    `p`. This is a bespoke function used by factoring.


Power series division
--------------------------------------------------------------------------------


.. function:: void _fmpz_poly_inv_series_basecase(fmpz * Qinv, const fmpz * Q, slong Qlen, slong n)

    Computes the first `n` terms of the inverse power series of
    ``(Q, lenQ)`` using a recurrence.

    Assumes that `n \geq 1` and that `Q` has constant term `\pm 1`.
    Does not support aliasing.

.. function:: void fmpz_poly_inv_series_basecase(fmpz_poly_t Qinv, const fmpz_poly_t Q, slong n)

    Computes the first `n` terms of the inverse power series of `Q`
    using a recurrence, assuming that `Q` has constant term `\pm 1`
    and `n \geq 1`.

.. function:: void _fmpz_poly_inv_series_newton(fmpz * Qinv, const fmpz * Q, slong Qlen, slong n)

    Computes the first `n` terms of the inverse power series of
    ``(Q, lenQ)`` using Newton iteration.

    Assumes that `n \geq 1` and that `Q` has constant term `\pm 1`.
    Does not support aliasing.

.. function:: void fmpz_poly_inv_series_newton(fmpz_poly_t Qinv, const fmpz_poly_t Q, slong n)

    Computes the first `n` terms of the inverse power series of `Q` using
    Newton iteration, assuming `Q` has constant term `\pm 1` and `n \geq 1`.

.. function:: void _fmpz_poly_inv_series(fmpz * Qinv, const fmpz * Q, slong Qlen, slong n)

    Computes the first `n` terms of the inverse power series of
    ``(Q, lenQ)``.

    Assumes that `n \geq 1` and that `Q` has constant term `\pm 1`.
    Does not support aliasing.

.. function:: void fmpz_poly_inv_series(fmpz_poly_t Qinv, const fmpz_poly_t Q, slong n)

    Computes the first `n` terms of the inverse power series of `Q`,
    assuming `Q` has constant term `\pm 1` and `n \geq 1`.

.. function:: void _fmpz_poly_div_series_basecase(fmpz * Q, const fmpz * A, slong Alen, const fmpz * B, slong Blen, slong n)

.. function:: void _fmpz_poly_div_series_divconquer(fmpz * Q, const fmpz * A, slong Alen, const fmpz * B, slong Blen, slong n)

.. function:: void _fmpz_poly_div_series(fmpz * Q, const fmpz * A, slong Alen, const fmpz * B, slong Blen, slong n)

    Divides ``(A, Alen)`` by ``(B, Blen)`` as power series over `\mathbb{Z}`,
    assuming `B` has constant term `\pm 1` and `n \geq 1`.
    Aliasing is not supported.

.. function:: void fmpz_poly_div_series_basecase(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_poly_t B, slong n)

.. function:: void fmpz_poly_div_series_divconquer(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_poly_t B, slong n)

.. function:: void fmpz_poly_div_series(fmpz_poly_t Q, const fmpz_poly_t A, const fmpz_poly_t B, slong n)

    Performs power series division in `\mathbb{Z}[[x]] / (x^n)`.  The function
    considers the polynomials `A` and `B` as power series of length `n`
    starting with the constant terms.  The function assumes that `B` has
    constant term `\pm 1` and `n \geq 1`.


Pseudo division
--------------------------------------------------------------------------------


.. function:: void _fmpz_poly_pseudo_divrem_basecase(fmpz * Q, fmpz * R, ulong * d, const fmpz * A, slong lenA, const fmpz * B, slong lenB, const fmpz_preinvn_t inv)

    If `\ell` is the leading coefficient of `B`, then computes `Q`, `R` such
    that `\ell^d A = Q B + R`.  This function is used for simulating division
    over `\mathbb{Q}`.

    Assumes that `\operatorname{len}(A) \geq \operatorname{len}(B) > 0`.  Assumes that `Q` can fit
    `\operatorname{len}(A) - \operatorname{len}(B) + 1` coefficients, and that `R` can fit `\operatorname{len}(A)`
    coefficients.  Supports aliasing of ``(R, lenA)`` and ``(A, lenA)``.
    But other than this,  no aliasing of the inputs and outputs is supported.

    An optional precomputed inverse of the leading coefficient of `B` from
    ``fmpz_preinvn_init`` can be supplied. Otherwise ``inv`` should be
    ``NULL``.

    Note: ``fmpz.h`` has to be included before ``fmpz_poly.h`` in order for
    ``fmpz_poly.h`` to declare this function.

.. function:: void fmpz_poly_pseudo_divrem_basecase(fmpz_poly_t Q, fmpz_poly_t R, ulong * d, const fmpz_poly_t A, const fmpz_poly_t B)

    If `\ell` is the leading coefficient of `B`, then computes `Q`, `R` such
    that `\ell^d A = Q B + R`.  This function is used for simulating division
    over `\mathbb{Q}`.

.. function:: void _fmpz_poly_pseudo_divrem_divconquer(fmpz * Q, fmpz * R, ulong * d, const fmpz * A, slong lenA, const fmpz * B, slong lenB, const fmpz_preinvn_t inv)

    Computes ``(Q, lenA - lenB + 1)``, ``(R, lenA)`` such that
    `\ell^d A = B Q + R`, only setting the bottom `\operatorname{len}(B) - 1` coefficients
    of `R` to their correct values.  The remaining top coefficients of
    ``(R, lenA)`` may be arbitrary.

    Assumes `\operatorname{len}(A) \geq \operatorname{len}(B) > 0`.  Allows zero-padding in
    ``(A, lenA)``.  No aliasing of input and output operands is allowed.

    An optional precomputed inverse of the leading coefficient of `B` from
    ``fmpz_preinvn_init`` can be supplied. Otherwise ``inv`` should be
    ``NULL``.

    Note: ``fmpz.h`` has to be included before ``fmpz_poly.h`` in order for
    ``fmpz_poly.h`` to declare this function.

.. function:: void fmpz_poly_pseudo_divrem_divconquer(fmpz_poly_t Q, fmpz_poly_t R, ulong * d, const fmpz_poly_t A, const fmpz_poly_t B)

    Computes `Q`, `R`, and `d` such that `\ell^d A = B Q + R`, where `R` has
    length less than the length of `B` and `\ell` is the leading coefficient
    of `B`.  An exception is raised if `B` is zero.

.. function:: void _fmpz_poly_pseudo_divrem_cohen(fmpz * Q, fmpz * R, const fmpz * A, slong lenA, const fmpz * B, slong lenB)

    Assumes that `\operatorname{len}(A) \geq \operatorname{len}(B) > 0`.  Assumes that `Q` can fit
    `\operatorname{len}(A) - \operatorname{len}(B) + 1` coefficients, and that `R` can fit `\operatorname{len}(A)`
    coefficients.  Supports aliasing of ``(R, lenA)`` and ``(A, lenA)``.
    But other than this, no aliasing of the inputs and outputs is supported.

.. function:: void fmpz_poly_pseudo_divrem_cohen(fmpz_poly_t Q, fmpz_poly_t R, const fmpz_poly_t A, const fmpz_poly_t B)

    This is a variant of ``fmpz_poly_pseudo_divrem`` which computes
    polynomials `Q` and `R` such that `\ell^d A = B Q + R`.  However, the
    value of `d` is fixed at `\max{\{0, \operatorname{len}(A) - \operatorname{len}(B) + 1\}}`.

    This function is faster when the remainder is not well behaved, i.e.
    where it is not expected to be close to zero.  Note that this function
    is not asymptotically fast.  It is efficient only for short polynomials,
    e.g. when `\operatorname{len}(B) < 32`.

.. function:: void _fmpz_poly_pseudo_rem_cohen(fmpz * R, const fmpz * A, slong lenA, const fmpz * B, slong lenB)

    Assumes that `\operatorname{len}(A) \geq \operatorname{len}(B) > 0`.  Assumes that `R` can fit
    `\operatorname{len}(A)` coefficients.  Supports aliasing of ``(R, lenA)`` and
    ``(A, lenA)``.  But other than this, no aliasing of the inputs and
    outputs is supported.

.. function:: void fmpz_poly_pseudo_rem_cohen(fmpz_poly_t R, const fmpz_poly_t A, const fmpz_poly_t B)

    This is a variant of :func:`fmpz_poly_pseudo_rem` which computes
    polynomials `Q` and `R` such that `\ell^d A = B Q + R`, but only
    returns `R`.  However, the value of `d` is fixed at
    `\max{\{0, \operatorname{len}(A) - \operatorname{len}(B) + 1\}}`.

    This function is faster when the remainder is not well behaved, i.e.
    where it is not expected to be close to zero.  Note that this function
    is not asymptotically fast.  It is efficient only for short polynomials,
    e.g. when `\operatorname{len}(B) < 32`.

    This function uses the algorithm described
    in Algorithm 3.1.2 of [Coh1996]_.

.. function:: void _fmpz_poly_pseudo_divrem(fmpz * Q, fmpz * R, ulong * d, const fmpz * A, slong lenA, const fmpz * B, slong lenB, const fmpz_preinvn_t inv)

    If `\ell` is the leading coefficient of `B`, then computes
    ``(Q, lenA - lenB + 1)``, ``(R, lenB - 1)`` and `d` such that
    `\ell^d A = B Q + R`.  This function is used for simulating division
    over `\mathbb{Q}`.

    Assumes that `\operatorname{len}(A) \geq \operatorname{len}(B) > 0`.  Assumes that `Q` can fit
    `\operatorname{len}(A) - \operatorname{len}(B) + 1` coefficients, and that `R` can fit `\operatorname{len}(A)`
    coefficients, although on exit only the bottom `\operatorname{len}(B)` coefficients
    will carry meaningful data.

    Supports aliasing of ``(R, lenA)`` and ``(A, lenA)``.  But other
    than this, no aliasing of the inputs and outputs is supported.

    An optional precomputed inverse of the leading coefficient of `B` from
    ``fmpz_preinvn_init`` can be supplied. Otherwise ``inv`` should be
    ``NULL``.

    Note: ``fmpz.h`` has to be included before ``fmpz_poly.h`` in order for
    ``fmpz_poly.h`` to declare this function.

.. function:: void fmpz_poly_pseudo_divrem(fmpz_poly_t Q, fmpz_poly_t R, ulong * d, const fmpz_poly_t A, const fmpz_poly_t B)

    Computes `Q`, `R`, and `d` such that `\ell^d A = B Q + R`.

.. function:: void _fmpz_poly_pseudo_div(fmpz * Q, ulong * d, const fmpz * A, slong lenA, const fmpz * B, slong lenB, const fmpz_preinvn_t inv)

    Pseudo-division, only returning the quotient.

    Note: ``fmpz.h`` has to be included before ``fmpz_poly.h`` in order for
    ``fmpz_poly.h`` to declare this function.

.. function:: void fmpz_poly_pseudo_div(fmpz_poly_t Q, ulong * d, const fmpz_poly_t A, const fmpz_poly_t B)

    Pseudo-division, only returning the quotient.

.. function:: void _fmpz_poly_pseudo_rem(fmpz * R, ulong * d, const fmpz * A, slong lenA, const fmpz * B, slong lenB, const fmpz_preinvn_t inv)

    Pseudo-division, only returning the remainder.

    Note: ``fmpz.h`` has to be included before ``fmpz_poly.h`` in order for
    ``fmpz_poly.h`` to declare this function.

.. function:: void fmpz_poly_pseudo_rem(fmpz_poly_t R, ulong * d, const fmpz_poly_t A, const fmpz_poly_t B)

    Pseudo-division, only returning the remainder.


Derivative
--------------------------------------------------------------------------------


.. function:: void _fmpz_poly_derivative(fmpz * rpoly, const fmpz * poly, slong len)

    Sets ``(rpoly, len - 1)`` to the derivative of ``(poly, len)``.
    Also handles the cases where ``len`` is `0` or `1` correctly.
    Supports aliasing of ``rpoly`` and ``poly``.

.. function:: void fmpz_poly_derivative(fmpz_poly_t res, const fmpz_poly_t poly)

    Sets ``res`` to the derivative of ``poly``.

.. function:: void _fmpz_poly_nth_derivative(fmpz * rpoly, const fmpz * poly, ulong n, slong len)

    Sets ``(rpoly, len - n)`` to the nth derivative of ``(poly, len)``.
    Also handles the cases where ``len <= n`` correctly.
    Supports aliasing of ``rpoly`` and ``poly``.

.. function:: void fmpz_poly_nth_derivative(fmpz_poly_t res, const fmpz_poly_t poly, ulong n)

    Sets ``res`` to the nth derivative of ``poly``.


Evaluation
--------------------------------------------------------------------------------


.. function:: void _fmpz_poly_evaluate_divconquer_fmpz(fmpz_t res, const fmpz * poly, slong len, const fmpz_t a)

    Evaluates the polynomial ``(poly, len)`` at the integer `a` using
    a divide and conquer approach.  Assumes that the length of the polynomial
    is at least one.  Allows zero padding.  Does not allow aliasing between
    ``res`` and ``x``.

.. function:: void fmpz_poly_evaluate_divconquer_fmpz(fmpz_t res, const fmpz_poly_t poly, const fmpz_t a)

    Evaluates the polynomial ``poly`` at the integer `a` using a divide
    and conquer approach.

    Aliasing between ``res`` and ``a`` is supported, however,
    ``res`` may not be part of ``poly``.

.. function:: void _fmpz_poly_evaluate_horner_fmpz(fmpz_t res, const fmpz * f, slong len, const fmpz_t a)

    Evaluates the polynomial ``(f, len)`` at the integer `a` using
    Horner's rule, and sets ``res`` to the result.  Aliasing between
    ``res`` and `a` or any of the coefficients of `f` is not supported.

.. function:: void fmpz_poly_evaluate_horner_fmpz(fmpz_t res, const fmpz_poly_t f, const fmpz_t a)

    Evaluates the polynomial `f` at the integer `a` using Horner's rule, and
    sets ``res`` to the result.

    As expected, aliasing between ``res`` and ``a`` is supported.
    However, ``res`` may not be aliased with a coefficient of `f`.

.. function:: void _fmpz_poly_evaluate_fmpz(fmpz_t res, const fmpz * f, slong len, const fmpz_t a)

    Evaluates the polynomial ``(f, len)`` at the integer `a` and sets
    ``res`` to the result.  Aliasing between ``res`` and `a` or any
    of the coefficients of `f` is not supported.

.. function:: void fmpz_poly_evaluate_fmpz(fmpz_t res, const fmpz_poly_t f, const fmpz_t a)

    Evaluates the polynomial `f` at the integer `a` and sets ``res``
    to the result.

    As expected, aliasing between ``res`` and `a` is supported.  However,
    ``res`` may not be aliased with a coefficient of `f`.


.. function:: void _fmpz_poly_evaluate_divconquer_fmpq(fmpz_t rnum, fmpz_t rden, const fmpz * f, slong len, const fmpz_t anum, const fmpz_t aden)

    Evaluates the polynomial ``(f, len)`` at the rational
    ``(anum, aden)`` using a divide and conquer approach, and sets
    ``(rnum, rden)`` to the result in lowest terms. Assumes that
    the length of the polynomial is at least one.

    Aliasing between ``(rnum, rden)`` and ``(anum, aden)`` or any of
    the coefficients of `f` is not supported.

.. function:: void fmpz_poly_evaluate_divconquer_fmpq(fmpq_t res, const fmpz_poly_t f, const fmpq_t a)

    Evaluates the polynomial `f` at the rational `a` using a divide
    and conquer approach, and sets ``res`` to the result.

.. function:: void _fmpz_poly_evaluate_horner_fmpq(fmpz_t rnum, fmpz_t rden, const fmpz * f, slong len, const fmpz_t anum, const fmpz_t aden)

    Evaluates the polynomial ``(f, len)`` at the rational
    ``(anum, aden)`` using Horner's rule, and sets ``(rnum, rden)`` to
    the result in lowest terms.

    Aliasing between ``(rnum, rden)`` and ``(anum, aden)`` or any of
    the coefficients of `f` is not supported.

.. function:: void fmpz_poly_evaluate_horner_fmpq(fmpq_t res, const fmpz_poly_t f, const fmpq_t a)

    Evaluates the polynomial `f` at the rational `a` using Horner's rule, and
    sets ``res`` to the result.

.. function:: void _fmpz_poly_evaluate_fmpq(fmpz_t rnum, fmpz_t rden, const fmpz * f, slong len, const fmpz_t anum, const fmpz_t aden)

    Evaluates the polynomial ``(f, len)`` at the rational
    ``(anum, aden)`` and sets ``(rnum, rden)`` to the result in lowest
    terms.

    Aliasing between ``(rnum, rden)`` and ``(anum, aden)`` or any of
    the coefficients of `f` is not supported.

.. function:: void fmpz_poly_evaluate_fmpq(fmpq_t res, const fmpz_poly_t f, const fmpq_t a)

    Evaluates the polynomial `f` at the rational `a`, and
    sets ``res`` to the result.

.. function:: ulong _fmpz_poly_evaluate_mod(const fmpz * poly, slong len, ulong a, ulong n, ulong ninv)

    Evaluates ``(poly, len)`` at the value `a` modulo `n` and
    returns the result.  The last argument ``ninv`` must be set
    to the precomputed inverse of `n`, which can be obtained using
    the function :func:`n_preinvert_limb`.

.. function:: ulong fmpz_poly_evaluate_mod(const fmpz_poly_t poly, ulong a, ulong n)

    Evaluates ``poly`` at the value `a` modulo `n` and returns the result.

.. function:: void fmpz_poly_evaluate_fmpz_vec(fmpz * res, const fmpz_poly_t f, const fmpz * a, slong n)

    Evaluates ``f`` at the `n` values given in the vector ``f``,
    writing the results to ``res``.

.. function:: double _fmpz_poly_evaluate_horner_d(const fmpz * poly, slong n, double d)

    Evaluate ``(poly, n)`` at the double `d`. No attempt is made to do this
    efficiently or in a numerically stable way. It is currently only used in
    FLINT for quick and dirty evaluations of polynomials with all coefficients
    positive.

.. function:: double fmpz_poly_evaluate_horner_d(const fmpz_poly_t poly, double d)

    Evaluate ``poly`` at the double `d`. No attempt is made to do this
    efficiently or in a numerically stable way. It is currently only used in
    FLINT for quick and dirty evaluations of polynomials with all coefficients
    positive.

.. function:: double _fmpz_poly_evaluate_horner_d_2exp(slong * exp, const fmpz * poly, slong n, double d)

    Evaluate ``(poly, n)`` at the double `d`. Return the result as a double
    and an exponent ``exp`` combination. No attempt is made to do this
    efficiently or in a numerically stable way. It is currently only used in
    FLINT for quick and dirty evaluations of polynomials with all coefficients
    positive.

.. function:: double fmpz_poly_evaluate_horner_d_2exp(slong * exp, const fmpz_poly_t poly, double d)

    Evaluate ``poly`` at the double `d`. Return the result as a double
    and an exponent ``exp`` combination. No attempt is made to do this
    efficiently or in a numerically stable way. It is currently only used in
    FLINT for quick and dirty evaluations of polynomials with all coefficients
    positive.

.. function:: double _fmpz_poly_evaluate_horner_d_2exp2(slong * exp, const fmpz * poly, slong n, double d, slong dexp)

    Evaluate ``poly`` at ``d*2^dexp``. Return the result as a double
    and an exponent ``exp`` combination. No attempt is made to do this
    efficiently or in a numerically stable way. It is currently only used in
    FLINT for quick and dirty evaluations of polynomials with all coefficients
    positive.


Newton basis
--------------------------------------------------------------------------------


.. function:: void _fmpz_poly_monomial_to_newton(fmpz * poly, const fmpz * roots, slong n)

    Converts ``(poly, n)`` in-place from its coefficients given
    in the standard monomial basis to the Newton basis
    for the roots `r_0, r_1, \ldots, r_{n-2}`.
    In other words, this determines output coefficients `c_i` such that
    `c_0 + c_1(x-r_0) + c_2(x-r_0)(x-r_1) + \ldots + c_{n-1}(x-r_0)(x-r_1)\cdots(x-r_{n-2})`
    is equal to the input polynomial.
    Uses repeated polynomial division.

.. function:: void _fmpz_poly_newton_to_monomial(fmpz * poly, const fmpz * roots, slong n)

    Converts ``(poly, n)`` in-place from its coefficients given
    in the Newton basis for the roots `r_0, r_1, \ldots, r_{n-2}`
    to the standard monomial basis. In other words, this evaluates
    `c_0 + c_1(x-r_0) + c_2(x-r_0)(x-r_1) + \ldots + c_{n-1}(x-r_0)(x-r_1)\cdots(x-r_{n-2})`
    where `c_i` are the input coefficients for ``poly``.
    Uses Horner's rule.


Interpolation
--------------------------------------------------------------------------------


.. function:: void fmpz_poly_interpolate_fmpz_vec(fmpz_poly_t poly, const fmpz * xs, const fmpz * ys, slong n)

    Sets ``poly`` to the unique interpolating polynomial of degree at
    most `n - 1` satisfying `f(x_i) = y_i` for every pair `x_i, y_u` in
    ``xs`` and ``ys``, assuming that this polynomial has integer
    coefficients.

    If an interpolating polynomial with integer coefficients does not
    exist, a ``FLINT_INEXACT`` exception is thrown.

    It is assumed that the `x` values are distinct.


Composition
--------------------------------------------------------------------------------


.. function:: void _fmpz_poly_compose_horner(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)

    Sets ``res`` to the composition of ``(poly1, len1)`` and
    ``(poly2, len2)``.

    Assumes that ``res`` has space for ``(len1-1)*(len2-1) + 1``
    coefficients.  Assumes that ``poly1`` and ``poly2`` are non-zero
    polynomials.  Does not support aliasing between any of the inputs and
    the output.

.. function:: void fmpz_poly_compose_horner(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

    Sets ``res`` to the composition of ``poly1`` and ``poly2``.
    To be more precise, denoting ``res``, ``poly1``, and ``poly2``
    by `f`, `g`, and `h`, sets `f(t) = g(h(t))`.

    This implementation uses Horner's method.

.. function:: void _fmpz_poly_compose_divconquer(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)

    Computes the composition of ``(poly1, len1)`` and ``(poly2, len2)``
    using a divide and conquer approach and places the result into ``res``,
    assuming ``res`` can hold the output of length
    ``(len1 - 1) * (len2 - 1) + 1``.

    Assumes ``len1, len2 > 0``.  Does not support aliasing between
    ``res`` and any of ``(poly1, len1)`` and ``(poly2, len2)``.

.. function:: void fmpz_poly_compose_divconquer(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

    Sets ``res`` to the composition of ``poly1`` and ``poly2``.
    To be precise about the order of composition, denoting ``res``,
    ``poly1``, and ``poly2`` by `f`, `g`, and `h`, respectively,
    sets `f(t) = g(h(t))`.

.. function:: void _fmpz_poly_compose(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2)

    Sets ``res`` to the composition of ``(poly1, len1)`` and
    ``(poly2, len2)``.

    Assumes that ``res`` has space for ``(len1-1)*(len2-1) + 1``
    coefficients.  Assumes that ``poly1`` and ``poly2`` are non-zero
    polynomials.  Does not support aliasing between any of the inputs and
    the output.

.. function:: void fmpz_poly_compose(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2)

    Sets ``res`` to the composition of ``poly1`` and ``poly2``.
    To be precise about the order of composition, denoting ``res``,
    ``poly1``, and ``poly2`` by `f`, `g`, and `h`, respectively,
    sets `f(t) = g(h(t))`.


Inflation and deflation
--------------------------------------------------------------------------------


.. function:: void fmpz_poly_inflate(fmpz_poly_t result, const fmpz_poly_t input, ulong inflation)

    Sets ``result`` to the inflated polynomial `p(x^n)` where
    `p` is given by ``input`` and `n` is given by ``inflation``.

.. function:: void fmpz_poly_deflate(fmpz_poly_t result, const fmpz_poly_t input, ulong deflation)

    Sets ``result`` to the deflated polynomial `p(x^{1/n})` where
    `p` is given by ``input`` and `n` is given by ``deflation``.
    Requires `n > 0`.

.. function:: ulong fmpz_poly_deflation(const fmpz_poly_t input)

    Returns the largest integer by which ``input`` can be deflated.
    As special cases, returns 0 if ``input`` is the zero polynomial
    and 1 if ``input`` is a constant polynomial.


Taylor shift
--------------------------------------------------------------------------------


.. function:: void _fmpz_poly_taylor_shift_horner(fmpz * poly, const fmpz_t c, slong n)

    Performs the Taylor shift composing ``poly`` by `x+c` in-place.
    Uses an efficient version Horner's rule.

.. function:: void fmpz_poly_taylor_shift_horner(fmpz_poly_t g, const fmpz_poly_t f, const fmpz_t c)

    Performs the Taylor shift composing ``f`` by `x+c`.

.. function:: void _fmpz_poly_taylor_shift_divconquer(fmpz * poly, const fmpz_t c, slong n)

    Performs the Taylor shift composing ``poly`` by `x+c` in-place.
    Uses the divide-and-conquer polynomial composition algorithm.

.. function:: void fmpz_poly_taylor_shift_divconquer(fmpz_poly_t g, const fmpz_poly_t f, const fmpz_t c)

    Performs the Taylor shift composing ``f`` by `x+c`.
    Uses the divide-and-conquer polynomial composition algorithm.

.. function:: void _fmpz_poly_taylor_shift_multi_mod(fmpz * poly, const fmpz_t c, slong n)

    Performs the Taylor shift composing ``poly`` by `x+c` in-place.
    Uses a multimodular algorithm, distributing the computation
    across :func:`flint_get_num_threads` threads.

.. function:: void fmpz_poly_taylor_shift_multi_mod(fmpz_poly_t g, const fmpz_poly_t f, const fmpz_t c)

    Performs the Taylor shift composing ``f`` by `x+c`.
    Uses a multimodular algorithm, distributing the computation
    across :func:`flint_get_num_threads` threads.

.. function:: void _fmpz_poly_taylor_shift(fmpz * poly, const fmpz_t c, slong n)

    Performs the Taylor shift composing ``poly`` by `x+c` in-place.

.. function:: void fmpz_poly_taylor_shift(fmpz_poly_t g, const fmpz_poly_t f, const fmpz_t c)

    Performs the Taylor shift composing ``f`` by `x+c`.


Power series composition
--------------------------------------------------------------------------------


.. function:: void _fmpz_poly_compose_series_horner(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2, slong n)

    Sets ``res`` to the composition of ``poly1`` and ``poly2``
    modulo `x^n`, where the constant term of ``poly2`` is required
    to be zero.

    Assumes that ``len1, len2, n > 0``, that ``len1, len2 <= n``,
    and that ``(len1-1) * (len2-1) + 1 <= n``, and that ``res`` has
    space for ``n`` coefficients. Does not support aliasing between any
    of the inputs and the output.

    This implementation uses the Horner scheme.

.. function:: void fmpz_poly_compose_series_horner(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)

    Sets ``res`` to the composition of ``poly1`` and ``poly2``
    modulo `x^n`, where the constant term of ``poly2`` is required
    to be zero.

    This implementation uses the Horner scheme.

.. function:: void _fmpz_poly_compose_series_brent_kung(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2, slong n)

    Sets ``res`` to the composition of ``poly1`` and ``poly2``
    modulo `x^n`, where the constant term of ``poly2`` is required
    to be zero.

    Assumes that ``len1, len2, n > 0``, that ``len1, len2 <= n``,
    and that ``(len1-1) * (len2-1) + 1 <= n``, and that ``res`` has
    space for ``n`` coefficients. Does not support aliasing between any
    of the inputs and the output.

    This implementation uses Brent-Kung algorithm 2.1 [BrentKung1978]_.

.. function:: void fmpz_poly_compose_series_brent_kung(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)

    Sets ``res`` to the composition of ``poly1`` and ``poly2``
    modulo `x^n`, where the constant term of ``poly2`` is required
    to be zero.

    This implementation uses Brent-Kung algorithm 2.1 [BrentKung1978]_.

.. function:: void _fmpz_poly_compose_series(fmpz * res, const fmpz * poly1, slong len1, const fmpz * poly2, slong len2, slong n)

    Sets ``res`` to the composition of ``poly1`` and ``poly2``
    modulo `x^n`, where the constant term of ``poly2`` is required
    to be zero.

    Assumes that ``len1, len2, n > 0``, that ``len1, len2 <= n``,
    and that ``(len1-1) * (len2-1) + 1 <= n``, and that ``res`` has
    space for ``n`` coefficients. Does not support aliasing between any
    of the inputs and the output.

    This implementation automatically switches between the Horner scheme
    and Brent-Kung algorithm 2.1 depending on the size of the inputs.

.. function:: void fmpz_poly_compose_series(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_poly_t poly2, slong n)

    Sets ``res`` to the composition of ``poly1`` and ``poly2``
    modulo `x^n`, where the constant term of ``poly2`` is required
    to be zero.

    This implementation automatically switches between the Horner scheme
    and Brent-Kung algorithm 2.1 depending on the size of the inputs.


Power series reversion
--------------------------------------------------------------------------------

.. function:: void _fmpz_poly_revert_series(fmpz * Qinv, const fmpz * Q, slong Qlen, slong n)
              void fmpz_poly_revert_series(fmpz_poly_t Qinv, const fmpz_poly_t Q, slong n)

    Sets ``Qinv`` to the compositional inverse or reversion of ``Q``
    as a power series, i.e. computes `Q^{-1}` such that
    `Q(Q^{-1}(x)) = Q^{-1}(Q(x)) = x \bmod x^n`.
    It is required that `Q_0 = 0` and `Q_1 = \pm 1`.

    Wraps :func:`_gr_poly_revert_series` which chooses automatically
    between various algorithms.


Square root
--------------------------------------------------------------------------------


.. function:: int _fmpz_poly_sqrtrem_classical(fmpz * res, fmpz * r, const fmpz * poly, slong len)

    Returns 1 if ``(poly, len)`` can be written in the form `A^2 + R` where
    deg(`R`) < deg(``poly``), otherwise returns `0`. If it can be so
    written, ``(res, m - 1)`` is set to `A` and ``(res, m)`` is set to
    `R`, where `m = \deg(\mathtt{poly})/2 + 1`.

    For efficiency reasons, ``r`` must have room for ``len``
    coefficients, and may alias ``poly``.

.. function:: int fmpz_poly_sqrtrem_classical(fmpz_poly_t b, fmpz_poly_t r, const fmpz_poly_t a)

    If `a` can be written as `b^2 + r` with `\deg(r) < \deg(a)/2`, return
    `1` and set `b` and `r` appropriately. Otherwise return `0`.

.. function:: int _fmpz_poly_sqrtrem_divconquer(fmpz * res, fmpz * r, const fmpz * poly, slong len, fmpz * temp)

    Returns 1 if ``(poly, len)`` can be written in the form `A^2 + R` where
    deg(`R`) < deg(``poly``), otherwise returns `0`. If it can be so
    written, ``(res, m - 1)`` is set to `A` and ``(res, m)`` is set to
    `R`, where `m = \deg(\mathtt{poly})/2 + 1`.

    For efficiency reasons, ``r`` must have room for ``len``
    coefficients, and may alias ``poly``. Temporary space of ``len``
    coefficients is required.

.. function:: int fmpz_poly_sqrtrem_divconquer(fmpz_poly_t b, fmpz_poly_t r, const fmpz_poly_t a)

    If `a` can be written as `b^2 + r` with `\deg(r) < \deg(a)/2`, return
    `1` and set `b` and `r` appropriately. Otherwise return `0`.

.. function:: int _fmpz_poly_sqrt_classical(fmpz * res, const fmpz * poly, slong len, int exact)

    If ``exact`` is `1` and ``(poly, len)`` is a perfect square, sets
    ``(res, len / 2 + 1)`` to the square root of ``poly`` with positive
    leading coefficient and returns 1. Otherwise returns 0.

    If ``exact`` is `0`, allows a remainder after the square root, which is
    not computed.

    This function first uses various tests to detect nonsquares quickly.
    Then, it computes the square root iteratively from top to bottom,
    requiring `O(n^2)` coefficient operations.

.. function:: int fmpz_poly_sqrt_classical(fmpz_poly_t b, const fmpz_poly_t a)

    If ``a`` is a perfect square, sets ``b`` to the square root of
    ``a`` with positive leading coefficient and returns 1.
    Otherwise returns 0.

.. function:: int _fmpz_poly_sqrt_KS(fmpz * res, const fmpz * poly, slong len)

    Heuristic square root. If the return value is `-1`, the function failed,
    otherwise it succeeded and the following applies.

    If ``(poly, len)`` is a perfect square, sets
    ``(res, len / 2 + 1)`` to the square root of ``poly`` with positive
    leading coefficient and returns 1. Otherwise returns 0.

    This function first uses various tests to detect nonsquares quickly.
    Then, it computes the square root iteratively from top to bottom.

.. function:: int fmpz_poly_sqrt_KS(fmpz_poly_t b, const fmpz_poly_t a)

    Heuristic square root. If the return value is `-1`, the function failed,
    otherwise it succeeded and the following applies.

    If ``a`` is a perfect square, sets ``b`` to the square root of
    ``a`` with positive leading coefficient and returns 1.
    Otherwise returns 0.

.. function:: int _fmpz_poly_sqrt_divconquer(fmpz * res, const fmpz * poly, slong len, int exact)

    If ``exact`` is `1` and ``(poly, len)`` is a perfect square, sets
    ``(res, len / 2 + 1)`` to the square root of ``poly`` with positive
    leading coefficient and returns 1. Otherwise returns 0.

    If ``exact`` is `0`, allows a remainder after the square root, which is
    not computed.

    This function first uses various tests to detect nonsquares quickly.
    Then, it computes the square root iteratively from top to bottom.

.. function:: int fmpz_poly_sqrt_divconquer(fmpz_poly_t b, const fmpz_poly_t a)

    If ``a`` is a perfect square, sets ``b`` to the square root of
    ``a`` with positive leading coefficient and returns 1.
    Otherwise returns 0.

.. function:: int _fmpz_poly_sqrt(fmpz * res, const fmpz * poly, slong len)

    If ``(poly, len)`` is a perfect square, sets ``(res, len / 2 + 1)``
    to the square root of ``poly`` with positive leading coefficient
    and returns 1. Otherwise returns 0.

.. function:: int fmpz_poly_sqrt(fmpz_poly_t b, const fmpz_poly_t a)

    If ``a`` is a perfect square, sets ``b`` to the square root of
    ``a`` with positive leading coefficient and returns 1.
    Otherwise returns 0.

.. function:: int _fmpz_poly_sqrt_series(fmpz * res, const fmpz * poly, slong len, slong n)

    Set ``(res, n)`` to the square root of the series ``(poly, n)``, if it
    exists, and return `1`, otherwise, return `0`.

    If the valuation of ``poly`` is not zero, ``res`` is zero padded
    to make up for the fact that the square root may not be known to precision
    `n`.

.. function:: int fmpz_poly_sqrt_series(fmpz_poly_t b, const fmpz_poly_t a, slong n)

    Set ``b`` to the square root of the series ``a``, where the latter
    is taken to be a series of precision `n`. If such a square root exists,
    return `1`, otherwise, return `0`.

    Note that if the valuation of ``a`` is not zero, ``b`` will
    not have precision ``n``. It is given only to the precision to which
    the square root can be computed.


Power sums
--------------------------------------------------------------------------------


.. function:: void _fmpz_poly_power_sums_naive(fmpz * res, const fmpz * poly, slong len, slong n)

    Compute the (truncated) power sums series of the monic polynomial
    ``(poly,len)`` up to length `n` using Newton identities.

.. function:: void fmpz_poly_power_sums_naive(fmpz_poly_t res, const fmpz_poly_t poly, slong n)

    Compute the (truncated) power sum series of the monic polynomial
    ``poly`` up to length `n` using Newton identities.

.. function:: void fmpz_poly_power_sums(fmpz_poly_t res, const fmpz_poly_t poly, slong n)

    Compute the (truncated) power sums series of the monic polynomial ``poly``
    up to length `n`. That is the power series whose coefficient of degree `i` is
    the sum of the `i`-th power of all (complex) roots of the polynomial
    ``poly``.

.. function:: void _fmpz_poly_power_sums_to_poly(fmpz * res, const fmpz * poly, slong len)

    Compute the (monic) polynomial given by its power sums series ``(poly,len)``.

.. function:: void fmpz_poly_power_sums_to_poly(fmpz_poly_t res, const fmpz_poly_t Q)

    Compute the (monic) polynomial given its power sums series ``(Q)``.


Signature
--------------------------------------------------------------------------------


.. function:: void _fmpz_poly_signature(slong * r1, slong * r2, const fmpz * poly, slong len)

    Computes the signature `(r_1, r_2)` of the polynomial
    ``(poly, len)``.  Assumes that the polynomial is squarefree over `\mathbb{Q}`.

.. function:: void fmpz_poly_signature(slong * r1, slong * r2, const fmpz_poly_t poly)

    Computes the signature `(r_1, r_2)` of the polynomial ``poly``,
    which is assumed to be square-free over `\mathbb{Q}`.  The values of `r_1` and
    `2 r_2` are the number of real and complex roots of the polynomial,
    respectively.  For convenience, the zero polynomial is allowed, in which
    case the output is `(0, 0)`.

    If the polynomial is not square-free, the behaviour is undefined and an
    exception may be raised.

    This function uses the algorithm described
    in Algorithm 4.1.11 of [Coh1996]_.


Hensel lifting
--------------------------------------------------------------------------------


.. function:: void fmpz_poly_hensel_build_tree(slong * link, fmpz_poly_t * v, fmpz_poly_t * w, const nmod_poly_factor_t fac)

    Initialises and builds a Hensel tree consisting of two arrays `v`, `w`
    of polynomials and an array of links, called ``link``.

    The caller supplies a set of `r` local factors (in the factor structure
    ``fac``) of some polynomial `F` over `\mathbf{Z}`. They also supply
    two arrays of initialised polynomials `v` and `w`, each of length
    `2r - 2` and an array ``link``, also of length `2r - 2`.

    We will have five arrays: a `v` of ``fmpz_poly_t``'s and a `V` of
    ``nmod_poly_t``'s and also a `w` and a `W` and ``link``.  Here's
    the idea: we sort each leaf and node of a factor tree by degree, in
    fact choosing to multiply the two smallest factors, then the next two
    smallest (factors or products) etc. until a tree is made.  The tree
    will be stored in the `v`'s. The first two elements of `v` will be the
    smallest modular factors, the last two elements of `v` will multiply to
    form `F` itself.  Since `v` will be rearranging the original factors we
    will need to be able to recover the original order. For this we use the
    array ``link`` which has nonnegative even numbers and negative numbers.
    It is an array of ``slong``\s which aligns with `V` and `v` if
    ``link`` has a negative number in spot `j` that means `V_j` is an
    original modular factor which has been lifted, if ``link[j]`` is a
    nonnegative even number then `V_j` stores a product of the two entries
    at ``V[link[j]]`` and ``V[link[j]+1]``.
    `W` and `w` play the role of the extended GCD, at `V_0`, `V_2`, `V_4`,
    etc. we have a new product, `W_0`, `W_2`, `W_4`, etc. are the XGCD
    cofactors of the `V`'s. For example,
    `V_0 W_0 + V_1 W_1 \equiv 1 \pmod{p^{\ell}}` for some `\ell`.  These
    will be lifted along with the entries in `V`.  It is not enough to just
    lift each factor, we have to lift the entire tree and the tree of
    XGCD cofactors.

.. function:: void fmpz_poly_hensel_lift(fmpz_poly_t G, fmpz_poly_t H, fmpz_poly_t A, fmpz_poly_t B, const fmpz_poly_t f, const fmpz_poly_t g, const fmpz_poly_t h, const fmpz_poly_t a, const fmpz_poly_t b, const fmpz_t p, const fmpz_t p1)

    This is the main Hensel lifting routine, which performs a Hensel step
    from polynomials mod `p` to polynomials mod `P = p p_1`. One starts with
    polynomials `f`, `g`, `h` such that `f = gh \pmod p`. The polynomials
    `a`, `b` satisfy `ag + bh = 1 \pmod p`.

    The lifting formulae are

    .. math::

        G = \biggl( \bigl( \frac{f-gh}{p} \bigr) b \bmod g \biggr) p + g

        H = \biggl( \bigl( \frac{f-gh}{p} \bigr) a \bmod h \biggr) p + h

        B = \biggl( \bigl( \frac{1-aG-bH}{p} \bigr) b \bmod g \biggr) p + b

        A = \biggl( \bigl( \frac{1-aG-bH}{p} \bigr) a \bmod h \biggr) p + a

    Upon return we have `A G + B H = 1 \pmod P` and `f = G H \pmod P`,
    where `G = g \pmod p` etc.

    We require that `1 < p_1 \leq p` and that the input polynomials `f, g, h`
    have degree at least `1` and that the input polynomials `a` and `b` are
    non-zero.

    The output arguments `G, H, A, B` may only be aliased with
    the input arguments `g, h, a, b`, respectively.

.. function:: void fmpz_poly_hensel_lift_without_inverse(fmpz_poly_t Gout, fmpz_poly_t Hout, const fmpz_poly_t f, const fmpz_poly_t g, const fmpz_poly_t h, const fmpz_poly_t a, const fmpz_poly_t b, const fmpz_t p, const fmpz_t p1)

    Given polynomials such that `f = gh \pmod p` and `ag + bh = 1 \pmod p`,
    lifts only the factors `g` and `h` modulo `P = p p_1`.

    See :func:`fmpz_poly_hensel_lift`.

.. function:: void fmpz_poly_hensel_lift_only_inverse(fmpz_poly_t Aout, fmpz_poly_t Bout, const fmpz_poly_t G, const fmpz_poly_t H, const fmpz_poly_t a, const fmpz_poly_t b, const fmpz_t p, const fmpz_t p1)

    Given polynomials such that `f = gh \pmod p` and `ag + bh = 1 \pmod p`,
    lifts only the cofactors `a` and `b` modulo `P = p p_1`.

    See :func:`fmpz_poly_hensel_lift`.

.. function:: void fmpz_poly_hensel_lift_tree_recursive(slong * link, fmpz_poly_t * v, fmpz_poly_t * w, fmpz_poly_t f, slong j, slong inv, const fmpz_t p0, const fmpz_t p1)

    Takes a current Hensel tree ``(link, v, w)`` and a pair `(j,j+1)`
    of entries in the tree and lifts the tree from mod `p_0` to
    mod `P = p_0 p_1`, where `1 < p_1 \leq p_0`.

    Set ``inv`` to `-1` if restarting Hensel lifting, `0` if stopping
    and `1` otherwise.

    Here `f = g h` is the polynomial whose factors we are trying to lift.
    We will have that ``v[j]`` is the product of ``v[link[j]]`` and
    ``v[link[j] + 1]`` as described above.

    Does support aliasing of `f` with one of the polynomials in
    the lists `v` and `w`.  But the polynomials in these two lists
    are not allowed to be aliases of each other.

.. function:: void fmpz_poly_hensel_lift_tree(slong * link, fmpz_poly_t * v, fmpz_poly_t * w, fmpz_poly_t f, slong r, const fmpz_t p, slong e0, slong e1, slong inv)

    Computes `p_0 = p^{e_0}` and `p_1 = p^{e_1 - e_0}` for a small prime `p`
    and `P = p^{e_1}`.

    If we aim to lift to `p^b` then `f` is the polynomial whose factors we
    wish to lift, made monic mod `p^b`. As usual, ``(link, v, w)`` is an
    initialised tree.

    This starts the recursion on lifting the *product tree* for lifting
    from `p^{e_0}` to `p^{e_1}`. The value of ``inv`` corresponds to that
    given for the function :func:`fmpz_poly_hensel_lift_tree_recursive`. We
    set `r` to the number of local factors of `f`.

    In terms of the notation, above `P = p^{e_1}`, `p_0 = p^{e_0}` and
    `p_1 = p^{e_1-e_0}`.

    Assumes that `f` is monic.

    Assumes that `1 < p_1 \leq p_0`, that is, `0 < e_1 \leq e_0`.

.. function:: slong _fmpz_poly_hensel_start_lift(fmpz_poly_factor_t lifted_fac, slong * link, fmpz_poly_t * v, fmpz_poly_t * w, const fmpz_poly_t f, const nmod_poly_factor_t local_fac, slong N)

    This function takes the local factors in ``local_fac``
    and Hensel lifts them until they are known mod `p^N`, where
    `N \geq 1`.

    These lifted factors will be stored (in the same ordering) in
    ``lifted_fac``. It is assumed that ``link``, ``v``, and
    ``w`` are initialized arrays of ``fmpz_poly_t``'s with at least
    `2*r - 2` entries and that `r \geq 2`.  This is done outside of
    this function so that you can keep them for restarting Hensel lifting
    later. The product of local factors must be squarefree.

    The return value is an exponent which must be passed to the function
    :func:`_fmpz_poly_hensel_continue_lift` as ``prev_exp`` if the
    Hensel lifting is to be resumed.

    Currently, supports the case when `N = 1` for convenience,
    although it is preferable in this case to simply iterate
    over the local factors and convert them to polynomials over
    `\mathbf{Z}`.

.. function:: slong _fmpz_poly_hensel_continue_lift(fmpz_poly_factor_t lifted_fac, slong * link, fmpz_poly_t * v, fmpz_poly_t * w, const fmpz_poly_t f, slong prev, slong curr, slong N, const fmpz_t p)

    This function restarts a stopped Hensel lift.

    It lifts from ``curr`` to `N`. It also requires ``prev``
    (to lift the cofactors) given as the return value of the function
    :func:`_fmpz_poly_hensel_start_lift` or the function
    :func:`_fmpz_poly_hensel_continue_lift`. The current lifted factors
    are supplied in ``lifted_fac`` and upon return are updated
    there. As usual ``link``, ``v``, and ``w`` describe the
    current Hensel tree, `r` is the number of local factors and `p` is
    the small prime modulo whose power we are lifting to. It is required
    that ``curr`` be at least `1` and that ``N > curr``.

    Currently, supports the case when ``prev`` and ``curr``
    are equal.

.. function:: void fmpz_poly_hensel_lift_once(fmpz_poly_factor_t lifted_fac, const fmpz_poly_t f, const nmod_poly_factor_t local_fac, slong N)

    This function does a Hensel lift.

    It lifts local factors stored in ``local_fac`` of `f` to `p^N`,
    where `N \geq 2`. The lifted factors will be stored in ``lifted_fac``.
    This lift cannot be restarted. This function is a convenience function
    intended for end users. The product of local factors must be squarefree.


Input and output
--------------------------------------------------------------------------------

The functions in this section are not intended to be particularly fast.
They are intended mainly as a debugging aid.

For the string output functions there are two variants.  The first uses a
simple string representation of polynomials which prints only the length
of the polynomial and the integer coefficients, whilst the latter variant,
appended with ``_pretty``, uses a more traditional string
representation of polynomials which prints a variable name as part of the
representation.

The first string representation is given by a sequence of integers, in
decimal notation, separated by white space.  The first integer gives the
length of the polynomial; the remaining integers are the coefficients.
For example `5x^3 - x + 1` is represented by the string
``"4  1 -1 0 5"``, and the zero polynomial is represented by ``"0"``.
The coefficients may be signed and arbitrary precision.

The string representation of the functions appended by ``_pretty``
includes only the non-zero terms of the polynomial, starting with the
one of highest degree.  Each term starts with a coefficient, prepended
with a sign, followed by the character ``*``, followed by a variable
name, which must be passed as a string parameter to the function,
followed by a caret ``^`` followed by a non-negative exponent.

If the sign of the leading coefficient is positive, it is omitted. Also
the exponents of the degree 1 and 0 terms are omitted, as is the
variable and the ``*`` character in the case of the degree 0
coefficient.  If the coefficient is plus or minus one, the coefficient
is omitted, except for the sign.

Some examples of the ``_pretty`` representation are::

    5*x^3+7*x-4
    x^2+3
    -x^4+2*x-1
    x+1
    5

.. function:: int _fmpz_poly_print(const fmpz * poly, slong len)

    Prints the polynomial ``(poly, len)`` to ``stdout``.

    In case of success, returns a positive value.  In case of failure,
    returns a non-positive value.

.. function:: int fmpz_poly_print(const fmpz_poly_t poly)

    Prints the polynomial to ``stdout``.

    In case of success, returns a positive value.  In case of failure,
    returns a non-positive value.

.. function:: int _fmpz_poly_print_pretty(const fmpz * poly, slong len, const char * x)

    Prints the pretty representation of ``(poly, len)`` to ``stdout``,
    using the string ``x`` to represent the indeterminate.

    In case of success, returns a positive value.  In case of failure,
    returns a non-positive value.

.. function:: int fmpz_poly_print_pretty(const fmpz_poly_t poly, const char * x)

    Prints the pretty representation of ``poly`` to ``stdout``,
    using the string ``x`` to represent the indeterminate.

    In case of success, returns a positive value.  In case of failure,
    returns a non-positive value.

.. function:: int _fmpz_poly_fprint(FILE * file, const fmpz * poly, slong len)

    Prints the polynomial ``(poly, len)`` to the stream ``file``.

    In case of success, returns a positive value.  In case of failure,
    returns a non-positive value.

.. function:: int fmpz_poly_fprint(FILE * file, const fmpz_poly_t poly)

    Prints the polynomial to the stream ``file``.

    In case of success, returns a positive value.  In case of failure,
    returns a non-positive value.

.. function:: int _fmpz_poly_fprint_pretty(FILE * file, const fmpz * poly, slong len, const char * x)

    Prints the pretty representation of ``(poly, len)`` to the stream
    ``file``, using the string ``x`` to represent the indeterminate.

    In case of success, returns a positive value.  In case of failure,
    returns a non-positive value.

.. function:: int fmpz_poly_fprint_pretty(FILE * file, const fmpz_poly_t poly, const char * x)

    Prints the pretty representation of ``poly`` to the stream ``file``,
    using the string ``x`` to represent the indeterminate.

    In case of success, returns a positive value.  In case of failure,
    returns a non-positive value.

.. function:: int fmpz_poly_read(fmpz_poly_t poly)

    Reads a polynomial from ``stdin``, storing the result in ``poly``.

    In case of success, returns a positive number.  In case of failure,
    returns a non-positive value.

.. function:: int fmpz_poly_read_pretty(fmpz_poly_t poly, char **x)

    Reads a polynomial in pretty format from ``stdin``.

    For further details, see the documentation for the function
    :func:`fmpz_poly_fread_pretty`.

.. function:: int fmpz_poly_fread(FILE * file, fmpz_poly_t poly)

    Reads a polynomial from the stream ``file``, storing the result
    in ``poly``.

    In case of success, returns a positive number.  In case of failure,
    returns a non-positive value.

.. function:: int fmpz_poly_fread_pretty(FILE * file, fmpz_poly_t poly, char **x)

    Reads a polynomial from the file ``file`` and sets ``poly``
    to this polynomial.  The string ``*x`` is set to the variable
    name that is used in the input.

    Returns a positive value, equal to the number of characters read from
    the file, in case of success.  Returns a non-positive value in case of
    failure, which could either be a read error or the indicator of a
    malformed input.


Modular reduction and reconstruction
--------------------------------------------------------------------------------


.. function:: void fmpz_poly_get_nmod_poly(nmod_poly_t Amod, const fmpz_poly_t A)

    Sets the coefficients of ``Amod`` to the coefficients in ``A``,
    reduced by the modulus of ``Amod``.

.. function:: void fmpz_poly_set_nmod_poly(fmpz_poly_t A, const nmod_poly_t Amod)

    Sets the coefficients of ``A`` to the residues in ``Amod``,
    normalised to the interval `-m/2 \le r < m/2` where `m` is the modulus.

.. function:: void fmpz_poly_set_nmod_poly_unsigned(fmpz_poly_t A, const nmod_poly_t Amod)

    Sets the coefficients of ``A`` to the residues in ``Amod``,
    normalised to the interval `0 \le r < m` where `m` is the modulus.

.. function:: void _fmpz_poly_CRT_ui_precomp(fmpz * res, const fmpz * poly1, slong len1, const fmpz_t m1, nn_srcptr poly2, slong len2, ulong m2, ulong m2inv, fmpz_t m1m2, ulong c, int sign)

    Sets the coefficients in ``res`` to the CRT reconstruction modulo
    `m_1m_2` of the residues ``(poly1, len1)`` and ``(poly2, len2)``
    which are images modulo `m_1` and `m_2` respectively.
    The caller must supply the precomputed product of the input moduli as
    `m_1m_2`, the inverse of `m_1` modulo `m_2` as `c`, and
    the precomputed inverse of `m_2` (in the form computed by
    ``n_preinvert_limb``) as ``m2inv``.

    If ``sign`` = 0, residues `0 \le r < m_1 m_2` are computed, while
    if ``sign`` = 1, residues `-m_1 m_2/2 \le r < m_1 m_2/2` are computed.

    Coefficients of ``res`` are written up to the maximum of
    ``len1`` and ``len2``.

.. function:: void _fmpz_poly_CRT_ui(fmpz * res, const fmpz * poly1, slong len1, const fmpz_t m1, nn_srcptr poly2, slong len2, ulong m2, ulong m2inv, int sign)

    This function is identical to ``_fmpz_poly_CRT_ui_precomp``,
    apart from automatically computing `m_1m_2` and `c`. It also
    aborts if `c` cannot be computed.

.. function:: void fmpz_poly_CRT_ui(fmpz_poly_t res, const fmpz_poly_t poly1, const fmpz_t m, const nmod_poly_t poly2, int sign)

    Given ``poly1`` with coefficients modulo ``m`` and ``poly2``
    with modulus `n`, sets ``res`` to the CRT reconstruction modulo `mn`
    with coefficients satisfying `-mn/2 \le c < mn/2` (if sign = 1)
    or `0 \le c < mn` (if sign = 0).


Products
--------------------------------------------------------------------------------


.. function:: void _fmpz_poly_product_roots_fmpz_vec(fmpz * poly, const fmpz * xs, slong n)

    Sets ``(poly, n + 1)`` to the monic polynomial which is the product
    of `(x - x_0)(x - x_1) \cdots (x - x_{n-1})`, the roots `x_i` being
    given by ``xs``.

    Aliasing of the input and output is not allowed.


.. function:: void fmpz_poly_product_roots_fmpz_vec(fmpz_poly_t poly, const fmpz * xs, slong n)

    Sets ``poly`` to the monic polynomial which is the product
    of `(x - x_0)(x - x_1) \cdots (x - x_{n-1})`, the roots `x_i` being
    given by ``xs``.

.. function:: void _fmpz_poly_product_roots_fmpq_vec(fmpz * poly, const fmpq * xs, slong n)

    Sets ``(poly, n + 1)`` to the product of
    `(q_0 x - p_0)(q_1 x - p_1) \cdots (q_{n-1} x - p_{n-1})`, the roots
    `p_i/q_i` being given by ``xs``.

.. function:: void fmpz_poly_product_roots_fmpq_vec(fmpz_poly_t poly, const fmpq * xs, slong n)

    Sets ``poly`` to the polynomial which is the product
    of `(q_0 x - p_0)(q_1 x - p_1) \cdots (q_{n-1} x - p_{n-1})`, the roots
    `p_i/q_i` being given by ``xs``.


Roots
--------------------------------------------------------------------------------


.. function:: void _fmpz_poly_bound_roots(fmpz_t bound, const fmpz * poly, slong len)
              void fmpz_poly_bound_roots(fmpz_t bound, const fmpz_poly_t poly)

    Computes a nonnegative integer ``bound`` that bounds the absolute
    value of all complex roots of ``poly``. Uses Fujiwara's bound

    .. math::

        2 \max \left(
            \left|\frac{a_{n-1}}{a_n}\right|,
            \left|\frac{a_{n-2}}{a_n}\right|^{\frac{1}{2}}, \dotsc,
            \left|\frac{a_1}{a_n}\right|^{\frac{1}{n-1}},
            \left|\frac{a_0}{2a_n}\right|^{\frac{1}{n}}
        \right)

    where the coefficients of the polynomial are `a_0, \ldots, a_n`.

.. function:: void _fmpz_poly_num_real_roots_sturm(slong * n_neg, slong * n_pos, const fmpz * pol, slong len)

    Sets ``n_neg`` and ``n_pos`` to the number of negative and
    positive roots of the polynomial ``(pol, len)`` using Sturm
    sequence. The Sturm sequence is computed via subresultant
    remainders obtained by repeated call to the function
    ``_fmpz_poly_pseudo_rem_cohen``.

    The polynomial is assumed to be squarefree, of degree larger than 1
    and with non-zero constant coefficient.

.. function:: slong fmpz_poly_num_real_roots_sturm(const fmpz_poly_t pol)

    Returns the number of real roots of the squarefree polynomial ``pol``
    using Sturm sequence.

    The polynomial is assumed to be squarefree.

.. function:: slong _fmpz_poly_num_real_roots(const fmpz * pol, slong len)

    Returns the number of real roots of the squarefree polynomial
    ``(pol, len)``.

    The polynomial is assumed to be squarefree.

.. function:: slong fmpz_poly_num_real_roots(const fmpz_poly_t pol)

    Returns the number of real roots of the squarefree polynomial ``pol``.

    The polynomial is assumed to be squarefree.


Minimal polynomials
--------------------------------------------------------------------------------


.. function:: void _fmpz_poly_cyclotomic(fmpz * a, ulong n, nn_ptr factors, slong num_factors, ulong phi)

    Sets ``a`` to the lower half of the cyclotomic polynomial `\Phi_n(x)`,
    given `n \ge 3` which must be squarefree.

    A precomputed array containing the prime factors of `n` must be provided,
    as well as the value of the Euler totient function `\phi(n)` as ``phi``.
    If `n` is even, 2 must be the first factor in the list.

    The degree of `\Phi_n(x)` is exactly `\phi(n)`. Only the low
    `(\phi(n) + 1) / 2` coefficients are written; the high coefficients
    can be obtained afterwards by copying the low coefficients
    in reverse order, since  `\Phi_n(x)` is a palindrome for `n \ne 1`.

    We use the sparse power series algorithm described as Algorithm 4
    [ArnoldMonagan2011]_. The algorithm is based on the identity

    .. math::

        \Phi_n(x) = \prod_{d|n} (x^d - 1)^{\mu(n/d)}.

    Treating the polynomial as a power series, the multiplications and
    divisions can be done very cheaply using repeated additions and
    subtractions. The complexity is `O(2^k \phi(n))` where `k` is the
    number of prime factors in `n`.

    To improve efficiency for small `n`, we treat the ``fmpz``
    coefficients as machine integers when there is no risk of overflow.
    The following bounds are given in Table 6 of [ArnoldMonagan2011]_:

    For `n < 10163195`, the largest coefficient in any `\Phi_n(x)`
    has 27 bits, so machine arithmetic is safe on 32 bits.

    For `n < 169828113`, the largest coefficient in any `\Phi_n(x)`
    has 60 bits, so machine arithmetic is safe on 64 bits.

    Further, the coefficients are always `\pm 1` or 0 if there are
    exactly two prime factors, so in this case machine arithmetic can be
    used as well.

    Finally, we handle two special cases: if there is exactly one prime
    factor `n = p`, then `\Phi_n(x) = 1 + x + x^2 + \ldots + x^{n-1}`,
    and if `n = 2m`, we use `\Phi_n(x) = \Phi_m(-x)` to fall back
    to the case when `n` is odd.

.. function:: void fmpz_poly_cyclotomic(fmpz_poly_t poly, ulong n)

    Sets ``poly`` to the `n`-th cyclotomic polynomial, defined as
    `\Phi_n(x) = \prod_{\omega} (x-\omega)`
    where `\omega` runs over all the `n`-th primitive roots of unity.

    We factor `n` into `n = qs` where `q` is squarefree,
    and compute `\Phi_q(x)`. Then `\Phi_n(x) = \Phi_q(x^s)`.

.. function:: ulong _fmpz_poly_is_cyclotomic(const fmpz * poly, slong len)
              ulong fmpz_poly_is_cyclotomic(const fmpz_poly_t poly)

    If ``poly`` is a cyclotomic polynomial, returns the index `n` of this
    cyclotomic polynomial. If ``poly`` is not a cyclotomic polynomial,
    returns 0.

.. function:: void _fmpz_poly_cos_minpoly(fmpz * coeffs, ulong n)
              void fmpz_poly_cos_minpoly(fmpz_poly_t poly, ulong n)

    Sets ``poly`` to the minimal polynomial of `2 \cos(2 \pi / n)`.
    For suitable choice of `n`, this gives the minimal polynomial
    of `2 \cos(a \pi)` or `2 \sin(a \pi)` for any rational `a`.

    The cosine is multiplied by a factor two since this gives
    a monic polynomial with integer coefficients. One can obtain
    the minimal polynomial for `\cos(2 \pi / n)` by making
    the substitution `x \to x / 2`.

    For `n > 2`, the degree of the polynomial is `\varphi(n) / 2`.
    For `n = 1, 2`, the degree is 1. For `n = 0`, we define
    the output to be the constant polynomial 1.

    See [WaktinsZeitlin1993]_.

.. function:: void _fmpz_poly_swinnerton_dyer(fmpz * coeffs, ulong n)
              void fmpz_poly_swinnerton_dyer(fmpz_poly_t poly, ulong n)

    Sets ``poly`` to the Swinnerton-Dyer polynomial `S_n`, defined as
    the integer polynomial
    `S_n = \prod (x \pm \sqrt{2} \pm \sqrt{3} \pm \sqrt{5} \pm \ldots \pm \sqrt{p_n})`
    where `p_n` denotes the `n`-th prime number and all combinations
    of signs are taken. This polynomial has degree `2^n` and is
    irreducible over the integers (it is the minimal polynomial
    of `\sqrt{2} + \ldots + \sqrt{p_n}`).


Orthogonal polynomials
--------------------------------------------------------------------------------

.. function:: void _fmpz_poly_chebyshev_t(fmpz * coeffs, ulong n)
              void fmpz_poly_chebyshev_t(fmpz_poly_t poly, ulong n)

    Sets ``poly`` to the Chebyshev polynomial of the first kind `T_n(x)`,
    defined by `T_n(x) = \cos(n \cos^{-1}(x))`, for `n\ge0`. The coefficients are
    calculated using a hypergeometric recurrence.

.. function:: void _fmpz_poly_chebyshev_u(fmpz * coeffs, ulong n)
              void fmpz_poly_chebyshev_u(fmpz_poly_t poly, ulong n)

    Sets ``poly`` to the Chebyshev polynomial of the first kind `U_n(x)`,
    defined by `(n+1) U_n(x) = T'_{n+1}(x)`, for `n\ge0`.
    The coefficients are calculated using a hypergeometric recurrence.

.. function:: void _fmpz_poly_legendre_pt(fmpz * coeffs, ulong n)

    Sets ``coeffs`` to the coefficient array of the shifted Legendre
    polynomial `\tilde{P_n}(x)`, defined by `\tilde{P_n}(x) = P_n(2x-1)`, for `n\ge0`.
    The coefficients are calculated using a hypergeometric recurrence.
    The length of the array will be ``n+1``.
    See ``fmpq_poly`` for the Legendre polynomials.

.. function:: void fmpz_poly_legendre_pt(fmpz_poly_t poly, ulong n)

    Sets ``poly`` to the shifted Legendre polynomial `\tilde{P_n}(x)`,
    defined by `\tilde{P_n}(x) = P_n(2x-1)`, for `n\ge0`. The coefficients are
    calculated using a hypergeometric recurrence. See ``fmpq_poly``
    for the Legendre polynomials.

.. function:: void _fmpz_poly_hermite_h(fmpz * coeffs, ulong n)

    Sets ``coeffs`` to the coefficient array of the Hermite
    polynomial `H_n(x)`, defined by `H'_n(x) = 2nH_{n-1}(x)`, for `n\ge0`.
    The coefficients are calculated using a hypergeometric recurrence.
    The length of the array will be ``n+1``.

.. function:: void fmpz_poly_hermite_h(fmpz_poly_t poly, ulong n)

    Sets ``poly`` to the Hermite polynomial `H_n(x)`,
    defined by `H'_n(x) = 2nH_{n-1}(x)`, for `n\ge0`. The coefficients are
    calculated using a hypergeometric recurrence.

.. function:: void _fmpz_poly_hermite_he(fmpz * coeffs, ulong n)

    Sets ``coeffs`` to the coefficient array of the Hermite
    polynomial `He_n(x)`, defined by
    `He_n(x) = 2^{-\tfrac{n}{2}}H_n\left(\frac{x}{\sqrt2}\right)`, for `n\ge0`.
    The coefficients are calculated using a hypergeometric recurrence.
    The length of the array will be ``n+1``.

.. function:: void fmpz_poly_hermite_he(fmpz_poly_t poly, ulong n)

    Sets ``poly`` to the Hermite polynomial `He_n(x)`,
    defined by `He_n(x) = 2^{-\tfrac{n}{2}}H_n\left(\frac{x}{\sqrt2}\right)`, for `n\ge0`.
    The coefficients are calculated using a hypergeometric recurrence.


Fibonacci polynomials
--------------------------------------------------------------------------------


.. function:: void _fmpz_poly_fibonacci(fmpz * coeffs, ulong n)

    Sets ``coeffs`` to the coefficient array of the `n`-th Fibonacci polynomial.
    The coefficients are calculated using a hypergeometric recurrence.

.. function:: void fmpz_poly_fibonacci(fmpz_poly_t poly, ulong n)

    Sets ``poly`` to the `n`-th Fibonacci polynomial.
    The coefficients are calculated using a hypergeometric recurrence.


Eulerian numbers and polynomials
--------------------------------------------------------------------------------

Eulerian numbers are the coefficients to the Eulerian polynomials

.. math::

    A_n(x) = \sum_{m = 0}^{n} A(n, m) x^m,

where the Eulerian polynomials are defined by the exponential generating
function

.. math::

    \frac{x - 1}{x - e^{(x - 1) t}}
    = \sum_{n = 0}^{\infty} A_n(x) \frac{t^n}{n!}.

The Eulerian numbers can be expressed explicitly via the formula

.. math::

    A(n, m) = \sum_{k = 0}^{m + 1} (-1)^k \binom{n + 1}{k} (m + 1 - k)^n.

Note: Not to be confused with Euler numbers and polynomials.

.. function:: void fmpz_eulerian_polynomial(fmpz_poly_t res, ulong n)

    Sets ``res`` to the Eulerian polynomial `A_n(x)`, where we define
    `A_0(x) = 1`. The polynomial is calculated via a recursive relation.


Modular forms and q-series
--------------------------------------------------------------------------------


.. function:: void _fmpz_poly_eta_qexp(fmpz * f, slong r, slong len)
              void fmpz_poly_eta_qexp(fmpz_poly_t f, slong r, slong n)

    Sets `f` to the `q`-expansion to length `n` of the
    Dedekind eta function (without the leading factor
    `q^{1/24}`) raised to the power `r`, i.e.
    `(q^{-1/24} \eta(q))^r = \prod_{k=1}^{\infty} (1 - q^k)^r`.

    In particular, `r = -1` gives the generating function
    of the partition function `p(k)`, and `r = 24` gives,
    after multiplication by `q`,
    the modular discriminant `\Delta(q)` which generates
    the Ramanujan tau function `\tau(k)`.

    This function uses sparse formulas for `r = 1, 2, 3, 4, 6`
    and otherwise reduces to one of those cases using power series arithmetic.

.. function:: void _fmpz_poly_theta_qexp(fmpz * f, slong r, slong len)
              void fmpz_poly_theta_qexp(fmpz_poly_t f, slong r, slong n)

    Sets `f` to the `q`-expansion to length `n` of the
    Jacobi theta function raised to the power `r`, i.e. `\vartheta(q)^r`
    where `\vartheta(q) = 1 + 2 \sum_{k=1}^{\infty} q^{k^2}`.

    This function uses sparse formulas for `r = 1, 2`
    and otherwise reduces to those cases using power series arithmetic.


CLD bounds
--------------------------------------------------------------------------------


.. function:: void fmpz_poly_CLD_bound(fmpz_t res, const fmpz_poly_t f, slong n)

    Compute a bound on the `n` coefficient of `fg'/g` where `g` is any
    factor of `f`.
