.. _nmod-mpoly:

**nmod_mpoly.h** -- multivariate polynomials over integers mod n (word-size n)
===============================================================================

    The exponents follow the ``mpoly`` interface.
    A coefficient may be referenced as a ``ulong *``.

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

.. type:: nmod_mpoly_struct

    A structure holding a multivariate polynomial over the integers modulo *n* for word-sized *n*.

.. type:: nmod_mpoly_t

    An array of length `1` of ``nmod_mpoly_struct``.

.. type:: nmod_mpoly_ctx_struct

    Context structure representing the parent ring of an ``nmod_mpoly``.

.. type:: nmod_mpoly_ctx_t

    An array of length `1` of ``nmod_mpoly_ctx_struct``.


Context object
--------------------------------------------------------------------------------


.. function:: void nmod_mpoly_ctx_init(nmod_mpoly_ctx_t ctx, slong nvars, const ordering_t ord, ulong n)

    Initialise a context object for a polynomial ring with the given number of variables and the given ordering.
    It will have coefficients modulo *n*. Setting `n = 0` will give undefined behavior.
    The possibilities for the ordering are ``ORD_LEX``, ``ORD_DEGLEX`` and ``ORD_DEGREVLEX``.

.. function:: slong nmod_mpoly_ctx_nvars(const nmod_mpoly_ctx_t ctx)

    Return the number of variables used to initialize the context.

.. function:: ordering_t nmod_mpoly_ctx_ord(const nmod_mpoly_ctx_t ctx)

    Return the ordering used to initialize the context.

.. function:: ulong nmod_mpoly_ctx_modulus(const nmod_mpoly_ctx_t ctx)

    Return the modulus used to initialize the context.

.. function:: void nmod_mpoly_ctx_clear(nmod_mpoly_ctx_t ctx)

    Release any space allocated by *ctx*.



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


.. function:: void nmod_mpoly_init(nmod_mpoly_t A, const nmod_mpoly_ctx_t ctx)

    Initialise *A* for use with the given an initialised context object. Its value is set to zero.

.. function:: void nmod_mpoly_init2(nmod_mpoly_t A, slong alloc, const nmod_mpoly_ctx_t ctx)

    Initialise *A* for use with the given an initialised context object. Its value is set to zero.
    It is allocated with space for *alloc* terms and at least ``MPOLY_MIN_BITS`` bits for the exponent widths.

.. function:: void nmod_mpoly_init3(nmod_mpoly_t A, slong alloc, flint_bitcnt_t bits, const nmod_mpoly_ctx_t ctx)

    Initialise *A* for use with the given an initialised context object. Its value is set to zero.
    It is allocated with space for *alloc* terms and *bits* bits for the exponents.

.. function:: void nmod_mpoly_fit_length(nmod_mpoly_t A, slong len, const nmod_mpoly_ctx_t ctx)

    Ensure that *A* has space for at least *len* terms.

.. function:: void nmod_mpoly_realloc(nmod_mpoly_t A, slong alloc, const nmod_mpoly_ctx_t ctx)

    Reallocate *A* to have space for *alloc* terms. 
    Assumes the current length of the polynomial is not greater than *alloc*.

.. function:: void nmod_mpoly_clear(nmod_mpoly_t A, const nmod_mpoly_ctx_t ctx)

    Release any space allocated for *A*.


Input/Output
--------------------------------------------------------------------------------

    The variable strings in *x* start with the variable of most significance at index `0`. If *x* is ``NULL``, the variables are named ``x1``, ``x2``, etc.

.. function:: char * nmod_mpoly_get_str_pretty(const nmod_mpoly_t A, const char ** x, const nmod_mpoly_ctx_t ctx)

    Return a string, which the user is responsible for cleaning up, representing *A*, given an array of variable strings *x*.

.. function:: int nmod_mpoly_fprint_pretty(FILE * file, const nmod_mpoly_t A, const char ** x, const nmod_mpoly_ctx_t ctx)

    Print a string representing *A* to *file*.

.. function:: int nmod_mpoly_print_pretty(const nmod_mpoly_t A, const char ** x, const nmod_mpoly_ctx_t ctx)

    Print a string representing *A* to ``stdout``.

.. function:: int nmod_mpoly_set_str_pretty(nmod_mpoly_t A, const char * str, const char ** x, const nmod_mpoly_ctx_t ctx)

    Set *A* to the polynomial in the null-terminates string *str* given an array *x* of variable strings.
    If parsing *str* fails, *A* is set to zero, and `-1` is returned. Otherwise, `0`  is returned.
    The operations ``+``, ``-``, ``*``, and ``/`` are permitted along with integers and the variables in *x*. The character ``^`` must be immediately followed by the (integer) exponent.
    If any division is not exact, parsing fails.


Basic manipulation
--------------------------------------------------------------------------------

.. function:: void nmod_mpoly_gen(nmod_mpoly_t A, slong var, const nmod_mpoly_ctx_t ctx)

    Set *A* to the variable of index *var*, where `var = 0` corresponds to the variable with the most significance with respect to the ordering. 

.. function:: int nmod_mpoly_is_gen(const nmod_mpoly_t A, slong var, const nmod_mpoly_ctx_t ctx)

    If `var \ge 0`, return `1` if *A* is equal to the `var`-th generator, otherwise return `0`.
    If `var < 0`, return `1` if the polynomial is equal to any generator, otherwise return `0`.

.. function:: void nmod_mpoly_set(nmod_mpoly_t A, const nmod_mpoly_t B, const nmod_mpoly_ctx_t ctx)
    
    Set *A* to *B*.

.. function:: int nmod_mpoly_equal(const nmod_mpoly_t A, const nmod_mpoly_t B, const nmod_mpoly_ctx_t ctx)

    Return `1` if *A* is equal to *B*, else return `0`.

.. function:: void nmod_mpoly_swap(nmod_mpoly_t A, nmod_mpoly_t B, const nmod_mpoly_ctx_t ctx)

    Efficiently swap *A* and *B*.


Constants
--------------------------------------------------------------------------------


.. function:: int nmod_mpoly_is_ui(const nmod_mpoly_t A, const nmod_mpoly_ctx_t ctx)

    Return `1` if *A* is a constant, else return `0`.

.. function:: ulong nmod_mpoly_get_ui(const nmod_mpoly_t A, const nmod_mpoly_ctx_t ctx)

    Assuming that *A* is a constant, return this constant.
    This function throws if *A* is not a constant.

.. function:: void nmod_mpoly_set_ui(nmod_mpoly_t A, ulong c, const nmod_mpoly_ctx_t ctx)

    Set *A* to the constant *c*.

.. function:: void nmod_mpoly_zero(nmod_mpoly_t A, const nmod_mpoly_ctx_t ctx)

    Set *A* to the constant `0`.

.. function:: void nmod_mpoly_one(nmod_mpoly_t A, const nmod_mpoly_ctx_t ctx)

    Set *A* to the constant `1`.

.. function:: int nmod_mpoly_equal_ui(const nmod_mpoly_t A, ulong c, const nmod_mpoly_ctx_t ctx)

    Return `1` if *A* is equal to the constant *c*, else return `0`.

.. function:: int nmod_mpoly_is_zero(const nmod_mpoly_t A, const nmod_mpoly_ctx_t ctx)

    Return `1` if *A* is the constant `0`, else return `0`.

.. function:: int nmod_mpoly_is_one(const nmod_mpoly_t A, const nmod_mpoly_ctx_t ctx)

    Return `1` if *A* is the constant `1`, else return `0`.


Degrees
--------------------------------------------------------------------------------


.. function:: int nmod_mpoly_degrees_fit_si(const nmod_mpoly_t A, const nmod_mpoly_ctx_t ctx)

    Return `1` if the degrees of *A* with respect to each variable fit into an ``slong``, otherwise return `0`.

.. function:: void nmod_mpoly_degrees_fmpz(fmpz ** degs, const nmod_mpoly_t A, const nmod_mpoly_ctx_t ctx)
              void nmod_mpoly_degrees_si(slong * degs, const nmod_mpoly_t A, const nmod_mpoly_ctx_t ctx)

    Set *degs* to the degrees of *A* with respect to each variable.
    If *A* is zero, all degrees are set to `-1`.

.. function:: void nmod_mpoly_degree_fmpz(fmpz_t deg, const nmod_mpoly_t A, slong var, const nmod_mpoly_ctx_t ctx)
              slong nmod_mpoly_degree_si(const nmod_mpoly_t A, slong var, const nmod_mpoly_ctx_t ctx)

    Either return or set *deg* to the degree of *A* with respect to the variable of index *var*.
    If *A* is zero, the degree is defined to be `-1`.

.. function:: int nmod_mpoly_total_degree_fits_si(const nmod_mpoly_t A, const nmod_mpoly_ctx_t ctx)

    Return `1` if the total degree of *A* fits into an ``slong``, otherwise return `0`.

.. function:: void nmod_mpoly_total_degree_fmpz(fmpz_t tdeg, const nmod_mpoly_t A, const nmod_mpoly_ctx_t ctx)
              slong nmod_mpoly_total_degree_si(const nmod_mpoly_t A, const nmod_mpoly_ctx_t ctx)

    Either return or set *tdeg* to the total degree of *A*.
    If *A* is zero, the total degree is defined to be `-1`.

.. function:: void nmod_mpoly_used_vars(int * used, const nmod_mpoly_t A, const nmod_mpoly_ctx_t ctx)

    For each variable index *i*, set ``used[i]`` to nonzero if the variable of index *i* appears in *A* and to zero otherwise.


Coefficients
--------------------------------------------------------------------------------


.. function:: ulong nmod_mpoly_get_coeff_ui_monomial(const nmod_mpoly_t A, const nmod_mpoly_t M, const nmod_mpoly_ctx_t ctx)

    Assuming that *M* is a monomial, return the coefficient of the corresponding monomial in *A*.
    This function throws if *M* is not a monomial.

.. function:: void nmod_mpoly_set_coeff_ui_monomial(nmod_mpoly_t A, ulong c, const nmod_mpoly_t M, const nmod_mpoly_ctx_t ctx)

    Assuming that *M* is a monomial, set the coefficient of the corresponding monomial in *A* to *c*.
    This function throws if *M* is not a monomial.

.. function:: ulong nmod_mpoly_get_coeff_ui_fmpz(const nmod_mpoly_t A, fmpz * const * exp, const nmod_mpoly_ctx_t ctx)
              ulong nmod_mpoly_get_coeff_ui_ui(const nmod_mpoly_t A, const ulong * exp, const nmod_mpoly_ctx_t ctx)

    Return the coefficient of the monomial with exponent *exp*.

.. function:: void nmod_mpoly_set_coeff_ui_fmpz(nmod_mpoly_t A, ulong c, fmpz * const * exp, const nmod_mpoly_ctx_t ctx)
              void nmod_mpoly_set_coeff_ui_ui(nmod_mpoly_t A, ulong c, const ulong * exp, const nmod_mpoly_ctx_t ctx)

    Set the coefficient of the monomial with exponent *exp* to `c`.

.. function:: void nmod_mpoly_get_coeff_vars_ui(nmod_mpoly_t C, const nmod_mpoly_t A, const slong * vars, const ulong * exps, slong length, const nmod_mpoly_ctx_t ctx)

    Set *C* to the coefficient of *A* with respect to the variables in *vars* with powers in the corresponding array *exps*.
    Both *vars* and *exps* point to array of length *length*. It is assumed that ``0 < length \le nvars(A)`` and that the variables in *vars* are distinct. 


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


.. function:: int nmod_mpoly_cmp(const nmod_mpoly_t A, const nmod_mpoly_t B, const nmod_mpoly_ctx_t ctx)

    Return `1` (resp. `-1`, or `0`) if *A* is after (resp. before, same as) *B* in some arbitrary but fixed total ordering of the polynomials.
    This ordering agrees with the usual ordering of monomials when *A* and *B* are both monomials.


Container operations
--------------------------------------------------------------------------------

    These functions deal with violations of the internal canonical representation.
    If a term index is negative or not strictly less than the length of the polynomial, the function will throw.

.. function:: ulong * nmod_mpoly_term_coeff_ref(nmod_mpoly_t A, slong i, const nmod_mpoly_ctx_t ctx)

    Return a reference to the coefficient of index *i* of *A*.

.. function:: int nmod_mpoly_is_canonical(const nmod_mpoly_t A, const nmod_mpoly_ctx_t ctx)

    Return `1` if *A* is in canonical form. Otherwise, return `0`.
    To be in canonical form, all of the terms must have nonzero coefficients, and the terms must be sorted from greatest to least.

.. function:: slong nmod_mpoly_length(const nmod_mpoly_t A, const nmod_mpoly_ctx_t ctx)

    Return the number of terms in *A*.
    If the polynomial is in canonical form, this will be the number of nonzero coefficients.

.. function:: void nmod_mpoly_resize(nmod_mpoly_t A, slong new_length, const nmod_mpoly_ctx_t ctx)

    Set the length of *A* to ``new_length``.
    Terms are either deleted from the end, or new zero terms are appended.

.. function:: ulong nmod_mpoly_get_term_coeff_ui(const nmod_mpoly_t A, slong i, const nmod_mpoly_ctx_t ctx)

    Return the coefficient of the term of index *i*.

.. function:: void nmod_mpoly_set_term_coeff_ui(nmod_mpoly_t A, slong i, ulong c, const nmod_mpoly_ctx_t ctx)

    Set the coefficient of the term of index *i* to *c*.

.. function:: int nmod_mpoly_term_exp_fits_si(const nmod_mpoly_t A, slong i, const nmod_mpoly_ctx_t ctx)
              int nmod_mpoly_term_exp_fits_ui(const nmod_mpoly_t A, slong i, const nmod_mpoly_ctx_t ctx)

    Return `1` if all entries of the exponent vector of the term of index *i* fit into an ``slong`` (resp. a ``ulong``). Otherwise, return `0`.

.. function:: void nmod_mpoly_get_term_exp_fmpz(fmpz ** exp, const nmod_mpoly_t A, slong i, const nmod_mpoly_ctx_t ctx)
              void nmod_mpoly_get_term_exp_ui(ulong * exp, const nmod_mpoly_t A, slong i, const nmod_mpoly_ctx_t ctx)

.. function:: void nmod_mpoly_get_term_exp_si(slong * exp, const nmod_mpoly_t A, slong i, const nmod_mpoly_ctx_t ctx)

    Set *exp* to the exponent vector of the term of index *i*.
    The ``_ui`` (resp. ``_si``) version throws if any entry does not fit into a ``ulong`` (resp. ``slong``).

.. function:: ulong nmod_mpoly_get_term_var_exp_ui(const nmod_mpoly_t A, slong i, slong var, const nmod_mpoly_ctx_t ctx)
              slong nmod_mpoly_get_term_var_exp_si(const nmod_mpoly_t A, slong i, slong var, const nmod_mpoly_ctx_t ctx)

    Return the exponent of the variable *var* of the term of index *i*.
    This function throws if the exponent does not fit into a ``ulong`` (resp. ``slong``).

.. function:: void nmod_mpoly_set_term_exp_fmpz(nmod_mpoly_t A, slong i, fmpz * const * exp, const nmod_mpoly_ctx_t ctx)
              void nmod_mpoly_set_term_exp_ui(nmod_mpoly_t A, slong i, const ulong * exp, const nmod_mpoly_ctx_t ctx)

    Set the exponent of the term of index *i* to *exp*.

.. function:: void nmod_mpoly_get_term(nmod_mpoly_t M, const nmod_mpoly_t A, slong i, const nmod_mpoly_ctx_t ctx)

    Set *M* to the term of index *i* in *A*.

.. function:: void nmod_mpoly_get_term_monomial(nmod_mpoly_t M, const nmod_mpoly_t A, slong i, const nmod_mpoly_ctx_t ctx)

    Set *M* to the monomial of the term of index *i* in *A*. The coefficient of *M* will be one.

.. function:: void nmod_mpoly_push_term_ui_fmpz(nmod_mpoly_t A, ulong c, fmpz * const * exp, const nmod_mpoly_ctx_t ctx)
              void nmod_mpoly_push_term_ui_ffmpz(nmod_mpoly_t A, ulong c, const fmpz * exp, const nmod_mpoly_ctx_t ctx)
              void nmod_mpoly_push_term_ui_ui(nmod_mpoly_t A, ulong c, const ulong * exp, const nmod_mpoly_ctx_t ctx)

    Append a term to *A* with coefficient *c* and exponent vector *exp*.
    This function runs in constant average time.

.. function:: void nmod_mpoly_sort_terms(nmod_mpoly_t A, const nmod_mpoly_ctx_t ctx)

    Sort the terms of *A* into the canonical ordering dictated by the ordering in *ctx*.
    This function simply reorders the terms: It does not combine like terms, nor does it delete terms with coefficient zero.
    This function runs in linear time in the bit size of *A*.

.. function:: void nmod_mpoly_combine_like_terms(nmod_mpoly_t A, const nmod_mpoly_ctx_t ctx)

    Combine adjacent like terms in *A* and delete terms with coefficient zero.
    If the terms of *A* were sorted to begin with, the result will be in canonical form.
    This function runs in linear time in the bit size of *A*.

.. function:: void nmod_mpoly_reverse(nmod_mpoly_t A, const nmod_mpoly_t B, const nmod_mpoly_ctx_t ctx)

    Set *A* to the reversal of *B*.


Random generation
--------------------------------------------------------------------------------


.. function:: void nmod_mpoly_randtest_bound(nmod_mpoly_t A, flint_rand_t state, slong length, ulong exp_bound, const nmod_mpoly_ctx_t ctx)

    Generate a random polynomial with length up to *length* and exponents in the range ``[0, exp_bound - 1]``.
    The exponents of each variable are generated by calls to  ``n_randint(state, exp_bound)``.

.. function:: void nmod_mpoly_randtest_bounds(nmod_mpoly_t A, flint_rand_t state, slong length, ulong * exp_bounds, const nmod_mpoly_ctx_t ctx)

    Generate a random polynomial with length up to *length* and exponents in the range ``[0, exp_bounds[i] - 1]``.
    The exponents of the variable of index *i* are generated by calls to ``n_randint(state, exp_bounds[i])``.

.. function:: void nmod_mpoly_randtest_bits(nmod_mpoly_t A, flint_rand_t state, slong length, ulong exp_bits, const nmod_mpoly_ctx_t ctx)

    Generate a random polynomial with length up to *length* and exponents whose packed form does not exceed the given bit count.


Addition/Subtraction
--------------------------------------------------------------------------------


.. function:: void nmod_mpoly_add_ui(nmod_mpoly_t A, const nmod_mpoly_t B, ulong c, const nmod_mpoly_ctx_t ctx)

    Set *A* to `B + c`.

.. function:: void nmod_mpoly_sub_ui(nmod_mpoly_t A, const nmod_mpoly_t B, ulong c, const nmod_mpoly_ctx_t ctx)

    Set *A* to `B - c`.

.. function:: void nmod_mpoly_add(nmod_mpoly_t A, const nmod_mpoly_t B, const nmod_mpoly_t C, const nmod_mpoly_ctx_t ctx)

    Set *A* to `B + C`.

.. function:: void nmod_mpoly_sub(nmod_mpoly_t A, const nmod_mpoly_t B, const nmod_mpoly_t C, const nmod_mpoly_ctx_t ctx)

    Set *A* to `B - C`.


Scalar operations
--------------------------------------------------------------------------------

.. function:: void nmod_mpoly_neg(nmod_mpoly_t A, const nmod_mpoly_t B, const nmod_mpoly_ctx_t ctx)
    
    Set *A* to `-B`.

.. function:: void nmod_mpoly_scalar_mul_ui(nmod_mpoly_t A, const nmod_mpoly_t B, ulong c, const nmod_mpoly_ctx_t ctx)

    Set *A* to `B \times c`.

.. function:: void nmod_mpoly_make_monic(nmod_mpoly_t A, const nmod_mpoly_t B, const nmod_mpoly_ctx_t ctx)

    Set *A* to *B* divided by the leading coefficient of *B*.
    This throws if *B* is zero or the leading coefficient is not invertible.


Differentiation
--------------------------------------------------------------------------------


.. function:: void nmod_mpoly_derivative(nmod_mpoly_t A, const nmod_mpoly_t B, slong var, const nmod_mpoly_ctx_t ctx)

    Set *A* to the derivative of *B* with respect to the variable of index *var*.


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

    These functions return `0` when the operation would imply unreasonable arithmetic.

.. function:: ulong nmod_mpoly_evaluate_all_ui(const nmod_mpoly_t A, const ulong * vals, const nmod_mpoly_ctx_t ctx)

    Return the evaluation of *A* where the variables are replaced by the corresponding elements of the array *vals*.

.. function:: void nmod_mpoly_evaluate_one_ui(nmod_mpoly_t A, const nmod_mpoly_t B, slong var, ulong val, const nmod_mpoly_ctx_t ctx)

    Set *A* to the evaluation of *B* where the variable of index *var* is replaced by *val*.

.. function:: int nmod_mpoly_compose_nmod_poly(nmod_poly_t A, const nmod_mpoly_t B, nmod_poly_struct * const * C, const nmod_mpoly_ctx_t ctx)

    Set *A* to the evaluation of *B* where the variables are replaced by the corresponding elements of the array *C*.
    The context object of *B* is *ctxB*.
    Return `1` for success and `0` for failure.

.. function:: int nmod_mpoly_compose_nmod_mpoly_geobucket(nmod_mpoly_t A, const nmod_mpoly_t B, nmod_mpoly_struct * const * C, const nmod_mpoly_ctx_t ctxB, const nmod_mpoly_ctx_t ctxAC)
              int nmod_mpoly_compose_nmod_mpoly_horner(nmod_mpoly_t A, const nmod_mpoly_t B, nmod_mpoly_struct * const * C, const nmod_mpoly_ctx_t ctxB, const nmod_mpoly_ctx_t ctxAC)
              int nmod_mpoly_compose_nmod_mpoly(nmod_mpoly_t A, const nmod_mpoly_t B, nmod_mpoly_struct * const * C, const nmod_mpoly_ctx_t ctxB, const nmod_mpoly_ctx_t ctxAC)

    Set *A* to the evaluation of *B* where the variables are replaced by the corresponding elements of the array *C*.
    Both *A* and the elements of *C* have context object *ctxAC*, while *B* has context object *ctxB*.
    Neither of *A* and *B* is allowed to alias any other polynomial.
    Return `1` for success and `0` for failure.
    The main method attempts to perform the calculation using matrices and chooses heuristically between the ``geobucket`` and ``horner`` methods if needed.

.. function:: void nmod_mpoly_compose_nmod_mpoly_gen(nmod_mpoly_t A, const nmod_mpoly_t B, const slong * c, const nmod_mpoly_ctx_t ctxB, const nmod_mpoly_ctx_t ctxAC)

    Set *A* to the evaluation of *B* where the variable of index *i* in *ctxB* is replaced by the variable of index ``c[i]`` in *ctxAC*.
    The length of the array *C* is the number of variables in *ctxB*.
    If any ``c[i]`` is negative, the corresponding variable of *B* is replaced by zero. Otherwise, it is expected that ``c[i]`` is less than the number of variables in *ctxAC*.


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


.. function:: void nmod_mpoly_mul(nmod_mpoly_t A, const nmod_mpoly_t B, const nmod_mpoly_t C, const nmod_mpoly_ctx_t ctx)

    Set *A* to `B \times C`.

.. function:: void nmod_mpoly_mul_johnson(nmod_mpoly_t A, const nmod_mpoly_t B, const nmod_mpoly_t C, const nmod_mpoly_ctx_t ctx)
              void nmod_mpoly_mul_heap_threaded(nmod_mpoly_t A, const nmod_mpoly_t B, const nmod_mpoly_t C, const nmod_mpoly_ctx_t ctx)

    Set *A* to `B \times C` using Johnson's heap-based method.
    The first version always uses one thread.

.. function:: int nmod_mpoly_mul_array(nmod_mpoly_t A, const nmod_mpoly_t B, const nmod_mpoly_t C, const nmod_mpoly_ctx_t ctx)
              int nmod_mpoly_mul_array_threaded(nmod_mpoly_t A, const nmod_mpoly_t B, const nmod_mpoly_t C, const nmod_mpoly_ctx_t ctx)

    Try to set *A* to `B \times C` using arrays.
    If the return is `0`, the operation was unsuccessful. Otherwise, it was successful, and the return is `1`.
    The first version always uses one thread.

.. function:: int nmod_mpoly_mul_dense(nmod_mpoly_t A, const nmod_mpoly_t B, const nmod_mpoly_t C, const nmod_mpoly_ctx_t ctx)

    Try to set *A* to `B \times C` using univariate arithmetic.
    If the return is `0`, the operation was unsuccessful. Otherwise, it was successful and the return is `1`.


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

    These functions return `0` when the operation would imply unreasonable arithmetic.

.. function:: int nmod_mpoly_pow_fmpz(nmod_mpoly_t A, const nmod_mpoly_t B, const fmpz_t k, const nmod_mpoly_ctx_t ctx)

    Set *A* to *B* raised to the *k*-th power.
    Return `1` for success and `0` for failure.

.. function:: int nmod_mpoly_pow_ui(nmod_mpoly_t A, const nmod_mpoly_t B, ulong k, const nmod_mpoly_ctx_t ctx)

    Set *A* to *B* raised to the *k*-th power.
    Return `1` for success and `0` for failure.


Division
--------------------------------------------------------------------------------

The division functions assume that the modulus is prime.

.. function:: int nmod_mpoly_divides(nmod_mpoly_t Q, const nmod_mpoly_t A, const nmod_mpoly_t B, const nmod_mpoly_ctx_t ctx)

    If *A* is divisible by *B*, set *Q* to the exact quotient and return `1`. Otherwise, set *Q* to zero and return `0`.
    Note that the function ``nmod_mpoly_div`` below may be faster if the quotient is known to be exact.

.. function:: void nmod_mpoly_div(nmod_mpoly_t Q, const nmod_mpoly_t A, const nmod_mpoly_t B, const nmod_mpoly_ctx_t ctx)

    Set *Q* to the quotient of *A* by *B*, discarding the remainder.

.. function:: void nmod_mpoly_divrem(nmod_mpoly_t Q, nmod_mpoly_t R, const nmod_mpoly_t A, const nmod_mpoly_t B, const nmod_mpoly_ctx_t ctx)

    Set *Q* and *R* to the quotient and remainder of *A* divided by *B*.

.. function:: void nmod_mpoly_divrem_ideal(nmod_mpoly_struct ** Q, nmod_mpoly_t R, const nmod_mpoly_t A, nmod_mpoly_struct * const * B, slong len, const nmod_mpoly_ctx_t ctx)

    This function is as per :func:`nmod_mpoly_divrem` except that it takes an array of divisor polynomials *B* and it returns an array of quotient polynomials *Q*.
    The number of divisor (and hence quotient) polynomials, is given by *len*.


.. function:: int nmod_mpoly_divides_dense(nmod_mpoly_t Q, const nmod_mpoly_t A, const nmod_mpoly_t B, const nmod_mpoly_ctx_t ctx)

    Try to do the operation of ``nmod_mpoly_divides`` using univariate arithmetic.
    If the return is `-1`, the operation was unsuccessful. Otherwise, it was successful and the return is `0` or `1`.

.. function:: int nmod_mpoly_divides_monagan_pearce(nmod_mpoly_t Q, const nmod_mpoly_t A, const nmod_mpoly_t B, const nmod_mpoly_ctx_t ctx)

    Do the operation of ``nmod_mpoly_divides`` using the algorithm of Michael Monagan and Roman Pearce.

.. function:: int nmod_mpoly_divides_heap_threaded(nmod_mpoly_t Q, const nmod_mpoly_t A, const nmod_mpoly_t B, const nmod_mpoly_ctx_t ctx)

    Do the operation of ``nmod_mpoly_divides`` using the heap and multiple
    threads. This function should only be called once ``global_thread_pool`` has
    been initialized.

.. note::

    This function is only defined if the machine is known to be strongly ordered
    during the configuration. To check whether this function is defined during
    compilation-time, use the C preprocessor macro
    ``#ifdef nmod_mpoly_divides_heap_threaded``.

    Note that, if the system is known to be strongly ordered, the underlying
    algorithm for this function is utilized in :func:`nmod_mpoly_divides`.
    Hence, you may find it easier to use this function instead if the C
    preprocessor is not available.


Greatest Common Divisor
--------------------------------------------------------------------------------

The greatest common divisor functions assume that the modulus is prime.

.. function:: void nmod_mpoly_term_content(nmod_mpoly_t M, const nmod_mpoly_t A, const nmod_mpoly_ctx_t ctx)

    Set *M* to the GCD of the terms of *A*.
    If *A* is zero, *M* will be zero. Otherwise, *M* will be a monomial with coefficient one.

.. function:: int nmod_mpoly_content_vars(nmod_mpoly_t g, const nmod_mpoly_t A, slong * vars, slong vars_length, const nmod_mpoly_ctx_t ctx)

    Set *g* to the GCD of the coefficients of *A* when viewed as a polynomial in the variables *vars*.
    Return `1` for success and `0` for failure. Upon success, *g* will be independent of the variables *vars*.

.. function:: int nmod_mpoly_gcd(nmod_mpoly_t G, const nmod_mpoly_t A, const nmod_mpoly_t B, const nmod_mpoly_ctx_t ctx)

    Try to set *G* to the monic GCD of *A* and *B*. The GCD of zero and zero is defined to be zero.
    If the return is `1` the function was successful. Otherwise the return is  `0` and *G* is left untouched.

.. function:: int nmod_mpoly_gcd_cofactors(nmod_mpoly_t G, nmod_mpoly_t Abar, nmod_mpoly_t Bbar, const nmod_mpoly_t A, const nmod_mpoly_t B, const nmod_mpoly_ctx_t ctx)

    Do the operation of :func:`nmod_mpoly_gcd` and also compute `Abar = A/G` and `Bbar = B/G` if successful.

.. function:: int nmod_mpoly_gcd_brown(nmod_mpoly_t G, const nmod_mpoly_t A, const nmod_mpoly_t B, const nmod_mpoly_ctx_t ctx)
              int nmod_mpoly_gcd_hensel(nmod_mpoly_t G, const nmod_mpoly_t A, const nmod_mpoly_t B, const nmod_mpoly_ctx_t ctx)
              int nmod_mpoly_gcd_zippel(nmod_mpoly_t G, const nmod_mpoly_t A, const nmod_mpoly_t B, const nmod_mpoly_ctx_t ctx)

    Try to set *G* to the GCD of *A* and *B* using various algorithms.

.. function:: int nmod_mpoly_resultant(nmod_mpoly_t R, const nmod_mpoly_t A, const nmod_mpoly_t B, slong var, const nmod_mpoly_ctx_t ctx)

    Try to set *R* to the resultant of *A* and *B* with respect to the variable of index *var*.

.. function:: int nmod_mpoly_discriminant(nmod_mpoly_t D, const nmod_mpoly_t A, slong var, const nmod_mpoly_ctx_t ctx)

    Try to set *D* to the discriminant of *A* with respect to the variable of index *var*.


Square Root
--------------------------------------------------------------------------------

The square root functions assume that the modulus is prime for correct operation.

.. function:: int nmod_mpoly_sqrt(nmod_mpoly_t Q, const nmod_mpoly_t A, const nmod_mpoly_ctx_t ctx)

    If `Q^2=A` has a solution, set *Q* to a solution and return `1`, otherwise return `0` and set *Q* to zero.

.. function:: int nmod_mpoly_is_square(const nmod_mpoly_t A, const nmod_mpoly_ctx_t ctx)

    Return `1` if *A* is a perfect square, otherwise return `0`.

.. function:: int nmod_mpoly_quadratic_root(nmod_mpoly_t Q, const nmod_mpoly_t A, const nmod_mpoly_t B, const nmod_mpoly_ctx_t ctx)

    If `Q^2+AQ=B` has a solution, set *Q* to a solution and return `1`, otherwise return `0`.

Univariate Functions
--------------------------------------------------------------------------------

    An ``nmod_mpoly_univar_t`` holds a univariate polynomial in some main variable
    with ``nmod_mpoly_t`` coefficients in the remaining variables. These functions
    are useful when one wants to rewrite an element of `\mathbb{Z}/n\mathbb{Z}[x_1, \dots, x_m]`
    as an element of `(\mathbb{Z}/n\mathbb{Z}[x_1, \dots, x_{v-1}, x_{v+1}, \dots, x_m])[x_v]`
    and vice versa.

.. function:: void nmod_mpoly_univar_init(nmod_mpoly_univar_t A, const nmod_mpoly_ctx_t ctx)

    Initialize *A*.

.. function:: void nmod_mpoly_univar_clear(nmod_mpoly_univar_t A, const nmod_mpoly_ctx_t ctx)

    Clear *A*.

.. function:: void nmod_mpoly_univar_swap(nmod_mpoly_univar_t A, nmod_mpoly_univar_t B, const nmod_mpoly_ctx_t ctx)

    Swap *A* and *B*.

.. function:: void nmod_mpoly_to_univar(nmod_mpoly_univar_t A, const nmod_mpoly_t B, slong var, const nmod_mpoly_ctx_t ctx)

    Set *A* to a univariate form of *B* by pulling out the variable of index *var*.
    The coefficients of *A* will still belong to the content *ctx* but will not depend on the variable of index *var*.

.. function:: void nmod_mpoly_from_univar(nmod_mpoly_t A, const nmod_mpoly_univar_t B, slong var, const nmod_mpoly_ctx_t ctx)

    Set *A* to the normal form of *B* by putting in the variable of index *var*.
    This function is undefined if the coefficients of *B* depend on the variable of index *var*.

.. function:: int nmod_mpoly_univar_degree_fits_si(const nmod_mpoly_univar_t A, const nmod_mpoly_ctx_t ctx)

    Return `1` if the degree of *A* with respect to the main variable fits an ``slong``. Otherwise, return `0`.

.. function:: slong nmod_mpoly_univar_length(const nmod_mpoly_univar_t A, const nmod_mpoly_ctx_t ctx)

    Return the number of terms in *A* with respect to the main variable.

.. function:: slong nmod_mpoly_univar_get_term_exp_si(nmod_mpoly_univar_t A, slong i, const nmod_mpoly_ctx_t ctx)

    Return the exponent of the term of index *i* of *A*.

.. function:: void nmod_mpoly_univar_get_term_coeff(nmod_mpoly_t c, const nmod_mpoly_univar_t A, slong i, const nmod_mpoly_ctx_t ctx)
              void nmod_mpoly_univar_swap_term_coeff(nmod_mpoly_t c, nmod_mpoly_univar_t A, slong i, const nmod_mpoly_ctx_t ctx)

    Set (resp. swap) *c* to (resp. with) the coefficient of the term of index *i* of *A*.


Internal Functions
--------------------------------------------------------------------------------

.. function:: void nmod_mpoly_pow_rmul(nmod_mpoly_t A, const nmod_mpoly_t B, ulong k, const nmod_mpoly_ctx_t ctx)

    Set *A* to *B* raised to the *k*-th power using repeated multiplications.

.. function:: void nmod_mpoly_div_monagan_pearce(nmod_mpoly_t polyq, const nmod_mpoly_t poly2, const nmod_mpoly_t poly3, const nmod_mpoly_ctx_t ctx)

    Set ``polyq`` to the quotient of ``poly2`` by ``poly3``,
    discarding the remainder (with notional remainder coefficients reduced
    modulo the leading coefficient of ``poly3``). Implements "Polynomial
    division using dynamic arrays, heaps and packed exponents" by Michael
    Monagan and Roman Pearce. This function is exceptionally efficient if the
    division is known to be exact.

.. function:: void nmod_mpoly_divrem_monagan_pearce(nmod_mpoly_t q, nmod_mpoly_t r, const nmod_mpoly_t poly2, const nmod_mpoly_t poly3, const nmod_mpoly_ctx_t ctx)

    Set ``polyq`` and ``polyr`` to the quotient and remainder of
    ``poly2`` divided by ``poly3``, (with remainder coefficients reduced
    modulo the leading coefficient of ``poly3``). Implements "Polynomial
    division using dynamic arrays, heaps and packed exponents" by Michael
    Monagan and Roman Pearce.


.. function:: void nmod_mpoly_divrem_ideal_monagan_pearce(nmod_mpoly_struct ** q, nmod_mpoly_t r, const nmod_mpoly_t poly2, nmod_mpoly_struct * const * poly3, slong len, const nmod_mpoly_ctx_t ctx)

    This function is as per ``nmod_mpoly_divrem_monagan_pearce`` except
    that it takes an array of divisor polynomials ``poly3``, and it returns
    an array of quotient polynomials ``q``. The number of divisor (and hence
    quotient) polynomials, is given by *len*. The function computes
    polynomials `q_i = q[i]` such that ``poly2`` is
    `r + \sum_{i=0}^{\mbox{len - 1}} q_ib_i`, where `b_i =` ``poly3[i]``.

