# fmpz_mpoly.h – multivariate polynomials over the integers

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

## Types, macros and constants

type fmpz_mpoly_struct

  A structure holding a multivariate integer polynomial.

type fmpz_mpoly_t

  An array of length \(1\) of `fmpz_mpoly_struct`.

type fmpz_mpoly_ctx_struct

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

type fmpz_mpoly_ctx_t

  An array of length \(1\) of `fmpz_mpoly_ctx_struct`.

## Context object

void fmpz_mpoly_ctx_init(fmpz_mpoly_ctx_t ctx, slong nvars, const ordering_t ord)

  Initialise a context object for a polynomial ring with the given number of variables and the given ordering. The possibilities for the ordering are `ORD_LEX`, `ORD_DEGLEX` and `ORD_DEGREVLEX`.

slong fmpz_mpoly_ctx_nvars(const fmpz_mpoly_ctx_t ctx)

  Return the number of variables used to initialize the context.

ordering_t fmpz_mpoly_ctx_ord(const fmpz_mpoly_ctx_t ctx)

  Return the ordering used to initialize the context.

void fmpz_mpoly_ctx_clear(fmpz_mpoly_ctx_t ctx)

  Release up any space allocated by ctx.

## Memory management

void fmpz_mpoly_init(fmpz_mpoly_t A, const fmpz_mpoly_ctx_t ctx)

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

void fmpz_mpoly_init2(fmpz_mpoly_t A, slong alloc, const fmpz_mpoly_ctx_t ctx)

  Initialise A for use with the given and 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 exponents.

void fmpz_mpoly_init3(fmpz_mpoly_t A, slong alloc, flint_bitcnt_t bits, const fmpz_mpoly_ctx_t ctx)

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

void fmpz_mpoly_fit_length(fmpz_mpoly_t A, slong len, const fmpz_mpoly_ctx_t ctx)

  Ensure that A has space for at least len terms.

void fmpz_mpoly_fit_bits(fmpz_mpoly_t A, flint_bitcnt_t bits, const fmpz_mpoly_ctx_t ctx)

  Ensure that the exponent fields of A have at least bits bits.

void fmpz_mpoly_realloc(fmpz_mpoly_t A, slong alloc, const fmpz_mpoly_ctx_t ctx)

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

void fmpz_mpoly_clear(fmpz_mpoly_t A, const fmpz_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.

char *fmpz_mpoly_get_str_pretty(const fmpz_mpoly_t A, const char **x, const fmpz_mpoly_ctx_t ctx)

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

int fmpz_mpoly_fprint_pretty(FILE *file, const fmpz_mpoly_t A, const char **x, const fmpz_mpoly_ctx_t ctx)

  Print a string representing A to file.

int fmpz_mpoly_print_pretty(const fmpz_mpoly_t A, const char **x, const fmpz_mpoly_ctx_t ctx)

  Print a string representing A to `stdout`.

int fmpz_mpoly_set_str_pretty(fmpz_mpoly_t A, const char *str, const char **x, const fmpz_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

void fmpz_mpoly_gen(fmpz_mpoly_t A, slong var, const fmpz_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.

int fmpz_mpoly_is_gen(const fmpz_mpoly_t A, slong var, const fmpz_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 \(0\).

void fmpz_mpoly_set(fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_mpoly_ctx_t ctx)

  Set A to B.

int fmpz_mpoly_equal(const fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_mpoly_ctx_t ctx)

  Return \(1\) if A is equal to B, else return \(0\).

void fmpz_mpoly_swap(fmpz_mpoly_t poly1, fmpz_mpoly_t poly2, const fmpz_mpoly_ctx_t ctx)

  Efficiently swap A and B.

int _fmpz_mpoly_fits_small(const fmpz *poly, slong len)

  Return 1 if the array of coefficients of length len consists entirely of values that are small `fmpz` values, i.e. of at most `FLINT_BITS - 2` bits plus a sign bit.

slong fmpz_mpoly_max_bits(const fmpz_mpoly_t A)

  Computes the maximum number of bits \(b\) required to represent the absolute values of the coefficients of A. If all of the coefficients are positive, \(b\) is returned, otherwise \(-b\) is returned.

## Constants

int fmpz_mpoly_is_fmpz(const fmpz_mpoly_t A, const fmpz_mpoly_ctx_t ctx)

  Return \(1\) if A is a constant, else return \(0\).

void fmpz_mpoly_get_fmpz(fmpz_t c, const fmpz_mpoly_t A, const fmpz_mpoly_ctx_t ctx)

  Assuming that A is a constant, set c to this constant. This function throws if A is not a constant.

void fmpz_mpoly_set_fmpz(fmpz_mpoly_t A, const fmpz_t c, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_set_ui(fmpz_mpoly_t A, ulong c, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_set_si(fmpz_mpoly_t A, slong c, const fmpz_mpoly_ctx_t ctx)

  Set A to the constant c.

void fmpz_mpoly_zero(fmpz_mpoly_t A, const fmpz_mpoly_ctx_t ctx)

  Set A to the constant \(0\).

void fmpz_mpoly_one(fmpz_mpoly_t A, const fmpz_mpoly_ctx_t ctx)

  Set A to the constant \(1\).

int fmpz_mpoly_equal_fmpz(const fmpz_mpoly_t A, const fmpz_t c, const fmpz_mpoly_ctx_t ctx)
int fmpz_mpoly_equal_ui(const fmpz_mpoly_t A, ulong c, const fmpz_mpoly_ctx_t ctx)
int fmpz_mpoly_equal_si(const fmpz_mpoly_t A, slong c, const fmpz_mpoly_ctx_t ctx)

  Return \(1\) if A is equal to the constant c, else return \(0\).

int fmpz_mpoly_is_zero(const fmpz_mpoly_t A, const fmpz_mpoly_ctx_t ctx)

  Return \(1\) if A is the constant \(0\), else return \(0\).

int fmpz_mpoly_is_one(const fmpz_mpoly_t A, const fmpz_mpoly_ctx_t ctx)

  Return \(1\) if A is the constant \(1\), else return \(0\).

## Degrees

int fmpz_mpoly_degrees_fit_si(const fmpz_mpoly_t A, const fmpz_mpoly_ctx_t ctx)

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

void fmpz_mpoly_degrees_fmpz(fmpz **degs, const fmpz_mpoly_t A, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_degrees_si(slong *degs, const fmpz_mpoly_t A, const fmpz_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 \(-1\).

void fmpz_mpoly_degree_fmpz(fmpz_t deg, const fmpz_mpoly_t A, slong var, const fmpz_mpoly_ctx_t ctx)
slong fmpz_mpoly_degree_si(const fmpz_mpoly_t A, slong var, const fmpz_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 \(-1\).

int fmpz_mpoly_total_degree_fits_si(const fmpz_mpoly_t A, const fmpz_mpoly_ctx_t ctx)

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

void fmpz_mpoly_total_degree_fmpz(fmpz_t tdeg, const fmpz_mpoly_t A, const fmpz_mpoly_ctx_t ctx)
slong fmpz_mpoly_total_degree_si(const fmpz_mpoly_t A, const fmpz_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 \(-1\).

void fmpz_mpoly_used_vars(int *used, const fmpz_mpoly_t A, const fmpz_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

void fmpz_mpoly_get_coeff_fmpz_monomial(fmpz_t c, const fmpz_mpoly_t A, const fmpz_mpoly_t M, const fmpz_mpoly_ctx_t ctx)

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

void fmpz_mpoly_set_coeff_fmpz_monomial(fmpz_mpoly_t poly, const fmpz_t c, const fmpz_mpoly_t poly2, const fmpz_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.

void fmpz_mpoly_get_coeff_fmpz_fmpz(fmpz_t c, const fmpz_mpoly_t A, fmpz *const *exp, const fmpz_mpoly_ctx_t ctx)
ulong fmpz_mpoly_get_coeff_ui_fmpz(const fmpz_mpoly_t A, fmpz *const *exp, const fmpz_mpoly_ctx_t ctx)
slong fmpz_mpoly_get_coeff_si_fmpz(const fmpz_mpoly_t A, fmpz *const *exp, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_get_coeff_fmpz_ui(fmpz_t c, const fmpz_mpoly_t A, const ulong *exp, const fmpz_mpoly_ctx_t ctx)
ulong fmpz_mpoly_get_coeff_ui_ui(const fmpz_mpoly_t A, const ulong *exp, const fmpz_mpoly_ctx_t ctx)
slong fmpz_mpoly_get_coeff_si_ui(const fmpz_mpoly_t A, const ulong *exp, const fmpz_mpoly_ctx_t ctx)

  Either return or set c to the coefficient of the monomial with exponent vector exp.

void fmpz_mpoly_set_coeff_fmpz_fmpz(fmpz_mpoly_t A, const fmpz_t c, fmpz *const *exp, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_set_coeff_ui_fmpz(fmpz_mpoly_t A, ulong c, fmpz *const *exp, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_set_coeff_si_fmpz(fmpz_mpoly_t A, slong c, fmpz *const *exp, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_set_coeff_fmpz_ui(fmpz_mpoly_t A, const fmpz_t c, const ulong *exp, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_set_coeff_ui_ui(fmpz_mpoly_t A, ulong c, const ulong *exp, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_set_coeff_si_ui(fmpz_mpoly_t A, slong c, const ulong *exp, const fmpz_mpoly_ctx_t ctx)

  Set the coefficient of the monomial with exponent vector exp to c.

void fmpz_mpoly_get_coeff_vars_ui(fmpz_mpoly_t C, const fmpz_mpoly_t A, const slong *vars, const ulong *exps, slong length, const fmpz_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

int fmpz_mpoly_cmp(const fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_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.

## Conversion

int fmpz_mpoly_is_fmpz_poly(const fmpz_mpoly_t A, slong var, const fmpz_mpoly_ctx_t ctx)

  Return whether A is a univariate polynomial in the variable with index var.

int fmpz_mpoly_get_fmpz_poly(fmpz_poly_t A, const fmpz_mpoly_t B, slong var, const fmpz_mpoly_ctx_t ctx)

  If B is a univariate polynomial in the variable with index var, set A to this polynomial and return 1; otherwise return 0.

void fmpz_mpoly_set_fmpz_poly(fmpz_mpoly_t A, const fmpz_poly_t B, slong var, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_set_gen_fmpz_poly(fmpz_mpoly_t A, slong var, const fmpz_poly_t B, const fmpz_mpoly_ctx_t ctx)

  Set A to the univariate polynomial B in the variable with index var.

## 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.

fmpz *fmpz_mpoly_term_coeff_ref(fmpz_mpoly_t A, slong i, const fmpz_mpoly_ctx_t ctx)

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

int fmpz_mpoly_is_canonical(const fmpz_mpoly_t A, const fmpz_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 coefficient, and the terms must be sorted from greatest to least.

slong fmpz_mpoly_length(const fmpz_mpoly_t A, const fmpz_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.

void fmpz_mpoly_resize(fmpz_mpoly_t A, slong new_length, const fmpz_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.

void fmpz_mpoly_get_term_coeff_fmpz(fmpz_t c, const fmpz_mpoly_t A, slong i, const fmpz_mpoly_ctx_t ctx)
ulong fmpz_mpoly_get_term_coeff_ui(const fmpz_mpoly_t A, slong i, const fmpz_mpoly_ctx_t ctx)
slong fmpz_mpoly_get_term_coeff_si(const fmpz_mpoly_t poly, slong i, const fmpz_mpoly_ctx_t ctx)

  Either return or set c to the coefficient of the term of index i.

void fmpz_mpoly_set_term_coeff_fmpz(fmpz_mpoly_t A, slong i, const fmpz_t c, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_set_term_coeff_ui(fmpz_mpoly_t A, slong i, ulong c, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_set_term_coeff_si(fmpz_mpoly_t A, slong i, slong c, const fmpz_mpoly_ctx_t ctx)

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

int fmpz_mpoly_term_exp_fits_si(const fmpz_mpoly_t poly, slong i, const fmpz_mpoly_ctx_t ctx)
int fmpz_mpoly_term_exp_fits_ui(const fmpz_mpoly_t poly, slong i, const fmpz_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 \(0\).

void fmpz_mpoly_get_term_exp_fmpz(fmpz **exp, const fmpz_mpoly_t A, slong i, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_get_term_exp_ui(ulong *exp, const fmpz_mpoly_t A, slong i, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_get_term_exp_si(slong *exp, const fmpz_mpoly_t A, slong i, const fmpz_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`).

ulong fmpz_mpoly_get_term_var_exp_ui(const fmpz_mpoly_t A, slong i, slong var, const fmpz_mpoly_ctx_t ctx)
slong fmpz_mpoly_get_term_var_exp_si(const fmpz_mpoly_t A, slong i, slong var, const fmpz_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`).

void fmpz_mpoly_set_term_exp_fmpz(fmpz_mpoly_t A, slong i, fmpz *const *exp, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_set_term_exp_ui(fmpz_mpoly_t A, slong i, const ulong *exp, const fmpz_mpoly_ctx_t ctx)

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

void fmpz_mpoly_get_term(fmpz_mpoly_t M, const fmpz_mpoly_t A, slong i, const fmpz_mpoly_ctx_t ctx)

  Set \(M\) to the term of index i in A.

void fmpz_mpoly_get_term_monomial(fmpz_mpoly_t M, const fmpz_mpoly_t A, slong i, const fmpz_mpoly_ctx_t ctx)

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

void fmpz_mpoly_push_term_fmpz_fmpz(fmpz_mpoly_t A, const fmpz_t c, fmpz *const *exp, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_push_term_fmpz_ffmpz(fmpz_mpoly_t A, const fmpz_t c, const fmpz *exp, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_push_term_ui_fmpz(fmpz_mpoly_t A, ulong c, fmpz *const *exp, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_push_term_ui_ffmpz(fmpz_mpoly_t A, ulong c, const fmpz *exp, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_push_term_si_fmpz(fmpz_mpoly_t A, slong c, fmpz *const *exp, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_push_term_si_ffmpz(fmpz_mpoly_t A, slong c, const fmpz *exp, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_push_term_fmpz_ui(fmpz_mpoly_t A, const fmpz_t c, const ulong *exp, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_push_term_ui_ui(fmpz_mpoly_t A, ulong c, const ulong *exp, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_push_term_si_ui(fmpz_mpoly_t A, slong c, const ulong *exp, const fmpz_mpoly_ctx_t ctx)

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

void fmpz_mpoly_sort_terms(fmpz_mpoly_t A, const fmpz_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 size of A.

void fmpz_mpoly_combine_like_terms(fmpz_mpoly_t A, const fmpz_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 size of A.

void fmpz_mpoly_reverse(fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_mpoly_ctx_t ctx)

  Set A to the reversal of B.

## Random generation

void fmpz_mpoly_randtest_bound(fmpz_mpoly_t A, flint_rand_t state, slong length, ulong coeff_bits, ulong exp_bound, const fmpz_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)`.

void fmpz_mpoly_randtest_bounds(fmpz_mpoly_t A, flint_rand_t state, slong length, ulong coeff_bits, ulong *exp_bounds, const fmpz_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])`.

void fmpz_mpoly_randtest_bits(fmpz_mpoly_t A, flint_rand_t state, slong length, ulong coeff_bits, ulong exp_bits, const fmpz_mpoly_ctx_t ctx)

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

  The parameter `coeff_bits` to the three functions `fmpz_mpoly_randtest_{bound|bounds|bits}` is merely a suggestion for the approximate bit count of the resulting signed coefficients. The function `fmpz_mpoly_max_bits()` will give the exact bit count of the result.

## Addition/Subtraction

void fmpz_mpoly_add_fmpz(fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_t c, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_add_ui(fmpz_mpoly_t A, const fmpz_mpoly_t B, ulong c, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_add_si(fmpz_mpoly_t A, const fmpz_mpoly_t B, slong c, const fmpz_mpoly_ctx_t ctx)

  Set A to \(B + c\). If A and B are aliased, this function will probably run quickly.

void fmpz_mpoly_sub_fmpz(fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_t c, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_sub_ui(fmpz_mpoly_t A, const fmpz_mpoly_t B, ulong c, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_sub_si(fmpz_mpoly_t A, const fmpz_mpoly_t B, slong c, const fmpz_mpoly_ctx_t ctx)

  Set A to \(B - c\). If A and B are aliased, this function will probably run quickly.

void fmpz_mpoly_add(fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_mpoly_t C, const fmpz_mpoly_ctx_t ctx)

  Set A to \(B + C\). If A and B are aliased, this function might run in time proportional to the size of \(C\).

void fmpz_mpoly_sub(fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_mpoly_t C, const fmpz_mpoly_ctx_t ctx)

  Set A to \(B - C\). If A and B are aliased, this function might run in time proportional to the size of \(C\).

## Scalar operations

void fmpz_mpoly_neg(fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_mpoly_ctx_t ctx)

  Set A to \(-B\).

void fmpz_mpoly_scalar_mul_fmpz(fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_t c, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_scalar_mul_ui(fmpz_mpoly_t A, const fmpz_mpoly_t B, ulong c, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_scalar_mul_si(fmpz_mpoly_t A, const fmpz_mpoly_t B, slong c, const fmpz_mpoly_ctx_t ctx)

  Set A to \(B \times c\).

void fmpz_mpoly_scalar_fmma(fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_t c, const fmpz_mpoly_t D, const fmpz_t e, const fmpz_mpoly_ctx_t ctx)

  Sets A to \(B \times c + D \times e\).

void fmpz_mpoly_scalar_divexact_fmpz(fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_t c, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_scalar_divexact_ui(fmpz_mpoly_t A, const fmpz_mpoly_t B, ulong c, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_scalar_divexact_si(fmpz_mpoly_t A, const fmpz_mpoly_t B, slong c, const fmpz_mpoly_ctx_t ctx)

  Set A to B divided by c. The division is assumed to be exact.

int fmpz_mpoly_scalar_divides_fmpz(fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_t c, const fmpz_mpoly_ctx_t ctx)
int fmpz_mpoly_scalar_divides_ui(fmpz_mpoly_t A, const fmpz_mpoly_t B, ulong c, const fmpz_mpoly_ctx_t ctx)
int fmpz_mpoly_scalar_divides_si(fmpz_mpoly_t A, const fmpz_mpoly_t B, slong c, const fmpz_mpoly_ctx_t ctx)

  If B is divisible by c, set A to the exact quotient and return \(1\), otherwise set A to zero and return \(0\).

## Differentiation/Integration

void fmpz_mpoly_derivative(fmpz_mpoly_t A, const fmpz_mpoly_t B, slong var, const fmpz_mpoly_ctx_t ctx)

  Set A to the derivative of B with respect to the variable of index \(var\).

void fmpz_mpoly_integral(fmpz_mpoly_t A, fmpz_t scale, const fmpz_mpoly_t B, slong var, const fmpz_mpoly_ctx_t ctx)

  Set A and scale so that A is an integral of \(scale \times B\) with respect to the variable of index var, where scale is positive and as small as possible.

## Evaluation

> These functions return \(0\) when the operation would imply unreasonable arithmetic.

int fmpz_mpoly_evaluate_all_fmpz(fmpz_t ev, const fmpz_mpoly_t A, fmpz *const *vals, const fmpz_mpoly_ctx_t ctx)

  Set ev to the evaluation of A where the variables are replaced by the corresponding elements of the array vals. Return \(1\) for success and \(0\) for failure.

int fmpz_mpoly_evaluate_one_fmpz(fmpz_mpoly_t A, const fmpz_mpoly_t B, slong var, const fmpz_t val, const fmpz_mpoly_ctx_t ctx)

  Set A to the evaluation of B where the variable of index var is replaced by `val`. Return \(1\) for success and \(0\) for failure.

int fmpz_mpoly_compose_fmpz_poly(fmpz_poly_t A, const fmpz_mpoly_t B, fmpz_poly_struct *const *C, const fmpz_mpoly_ctx_t ctxB)

  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.

int fmpz_mpoly_compose_fmpz_mpoly_geobucket(fmpz_mpoly_t A, const fmpz_mpoly_t B, fmpz_mpoly_struct *const *C, const fmpz_mpoly_ctx_t ctxB, const fmpz_mpoly_ctx_t ctxAC)
int fmpz_mpoly_compose_fmpz_mpoly_horner(fmpz_mpoly_t A, const fmpz_mpoly_t B, fmpz_mpoly_struct *const *C, const fmpz_mpoly_ctx_t ctxB, const fmpz_mpoly_ctx_t ctxAC)
int fmpz_mpoly_compose_fmpz_mpoly(fmpz_mpoly_t A, const fmpz_mpoly_t B, fmpz_mpoly_struct *const *C, const fmpz_mpoly_ctx_t ctxB, const fmpz_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. The length of the array C is the number of variables in ctxB. Neither A nor 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.

void fmpz_mpoly_compose_fmpz_mpoly_gen(fmpz_mpoly_t A, const fmpz_mpoly_t B, const slong *c, const fmpz_mpoly_ctx_t ctxB, const fmpz_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

void fmpz_mpoly_mul(fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_mpoly_t C, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_mul_threaded(fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_mpoly_t C, const fmpz_mpoly_ctx_t ctx, slong thread_limit)

  Set A to \(B \times C\).

void fmpz_mpoly_mul_johnson(fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_mpoly_t C, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_mul_heap_threaded(fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_mpoly_t C, const fmpz_mpoly_ctx_t ctx)

  Set A to \(B \times C\) using Johnson’s heap-based method. The first version always uses one thread.

int fmpz_mpoly_mul_array(fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_mpoly_t C, const fmpz_mpoly_ctx_t ctx)
int fmpz_mpoly_mul_array_threaded(fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_mpoly_t C, const fmpz_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.

int fmpz_mpoly_mul_dense(fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_mpoly_t C, const fmpz_mpoly_ctx_t ctx)

  Try to set A to \(B \times C\) using dense 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.

int fmpz_mpoly_pow_fmpz(fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_t k, const fmpz_mpoly_ctx_t ctx)

  Set A to B raised to the k-th power. Return \(1\) for success and \(0\) for failure.

int fmpz_mpoly_pow_ui(fmpz_mpoly_t A, const fmpz_mpoly_t B, ulong k, const fmpz_mpoly_ctx_t ctx)

  Set A to B raised to the k-th power. Return \(1\) for success and \(0\) for failure.

## Division

int fmpz_mpoly_divides(fmpz_mpoly_t Q, const fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_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 \(0\).

void fmpz_mpoly_divrem(fmpz_mpoly_t Q, fmpz_mpoly_t R, const fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_mpoly_ctx_t ctx)

  Set \(Q\) and \(R\) to the quotient and remainder of A divided by B. The monomials in R divisible by the leading monomial of B will have coefficients reduced modulo the absolute value of the leading coefficient of B. Note that this function is not very useful if the leading coefficient B is not a unit.

void fmpz_mpoly_quasidivrem(fmpz_t scale, fmpz_mpoly_t Q, fmpz_mpoly_t R, const fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_mpoly_ctx_t ctx)

  Set scale, Q and R so that Q and R are the quotient and remainder of \(scale \times A\) divided by B. No monomials in R will be divisible by the leading monomial of B.

void fmpz_mpoly_div(fmpz_mpoly_t Q, const fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_mpoly_ctx_t ctx)

  Perform the operation of `fmpz_mpoly_divrem()` and discard R. Note that this function is not very useful if the division is not exact and the leading coefficient B is not a unit.

void fmpz_mpoly_quasidiv(fmpz_t scale, fmpz_mpoly_t Q, const fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_mpoly_ctx_t ctx)

  Perform the operation of `fmpz_mpoly_quasidivrem()` and discard R.

void fmpz_mpoly_divrem_ideal(fmpz_mpoly_struct **Q, fmpz_mpoly_t R, const fmpz_mpoly_t A, fmpz_mpoly_struct *const *B, slong len, const fmpz_mpoly_ctx_t ctx)

  This function is as per `fmpz_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. Note that this function is not very useful if there is no unit among the leading coefficients in the array B.

void fmpz_mpoly_quasidivrem_ideal(fmpz_t scale, fmpz_mpoly_struct **Q, fmpz_mpoly_t R, const fmpz_mpoly_t A, fmpz_mpoly_struct *const *B, slong len, const fmpz_mpoly_ctx_t ctx)

  This function is as per `fmpz_mpoly_quasidivrem()` 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.

## Greatest Common Divisor

void fmpz_mpoly_term_content(fmpz_mpoly_t M, const fmpz_mpoly_t A, const fmpz_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 positive coefficient.

int fmpz_mpoly_content_vars(fmpz_mpoly_t g, const fmpz_mpoly_t A, slong *vars, slong vars_length, const fmpz_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.

int fmpz_mpoly_gcd(fmpz_mpoly_t G, const fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_mpoly_ctx_t ctx)

  Try to set G to the GCD of A and B with positive leading coefficient. 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.

int fmpz_mpoly_gcd_cofactors(fmpz_mpoly_t G, fmpz_mpoly_t Abar, fmpz_mpoly_t Bbar, const fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_mpoly_ctx_t ctx)

  Do the operation of `fmpz_mpoly_gcd()` and also compute \(Abar = A/G\) and \(Bbar = B/G\) if successful.

int fmpz_mpoly_gcd_brown(fmpz_mpoly_t G, const fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_mpoly_ctx_t ctx)
int fmpz_mpoly_gcd_hensel(fmpz_mpoly_t G, const fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_mpoly_ctx_t ctx)
int fmpz_mpoly_gcd_subresultant(fmpz_mpoly_t G, const fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_mpoly_ctx_t ctx)
int fmpz_mpoly_gcd_zippel(fmpz_mpoly_t G, const fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_mpoly_ctx_t ctx)
int fmpz_mpoly_gcd_zippel2(fmpz_mpoly_t G, const fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_mpoly_ctx_t ctx)

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

int fmpz_mpoly_resultant(fmpz_mpoly_t R, const fmpz_mpoly_t A, const fmpz_mpoly_t B, slong var, const fmpz_mpoly_ctx_t ctx)

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

int fmpz_mpoly_discriminant(fmpz_mpoly_t D, const fmpz_mpoly_t A, slong var, const fmpz_mpoly_ctx_t ctx)

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

void fmpz_mpoly_primitive_part(fmpz_mpoly_t res, const fmpz_mpoly_t f, const fmpz_mpoly_ctx_t ctx)

  Sets res to the primitive part of f, obtained by dividing out the content of all coefficients and normalizing the leading coefficient to be positive. The zero polynomial is unchanged.

## Square Root

int fmpz_mpoly_sqrt_heap(fmpz_mpoly_t Q, const fmpz_mpoly_t A, const fmpz_mpoly_ctx_t ctx, int check)

  If A is a perfect square return \(1\) and set Q to the square root with positive leading coefficient. Otherwise return \(0\) and set Q to the zero polynomial. If \(check = 0\) the polynomial is assumed to be a perfect square. This can be significantly faster, but it will not detect non-squares with any reliability, and in the event of being passed a non-square the result is meaningless.

int fmpz_mpoly_sqrt(fmpz_mpoly_t q, const fmpz_mpoly_t A, const fmpz_mpoly_ctx_t ctx)

  If A is a perfect square return \(1\) and set Q to the square root with positive leading coefficient. Otherwise return \(0\) and set Q to zero.

int fmpz_mpoly_is_square(const fmpz_mpoly_t A, const fmpz_mpoly_ctx_t ctx)

  Return \(1\) if A is a perfect square, otherwise return \(0\).

## Univariate Functions

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

void fmpz_mpoly_univar_init(fmpz_mpoly_univar_t A, const fmpz_mpoly_ctx_t ctx)

  Initialize A.

void fmpz_mpoly_univar_clear(fmpz_mpoly_univar_t A, const fmpz_mpoly_ctx_t ctx)

  Clear A.

void fmpz_mpoly_univar_swap(fmpz_mpoly_univar_t A, fmpz_mpoly_univar_t B, const fmpz_mpoly_ctx_t ctx)

  Swap A and B.

void fmpz_mpoly_to_univar(fmpz_mpoly_univar_t A, const fmpz_mpoly_t B, slong var, const fmpz_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.

void fmpz_mpoly_from_univar(fmpz_mpoly_t A, const fmpz_mpoly_univar_t B, slong var, const fmpz_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.

int fmpz_mpoly_univar_degree_fits_si(const fmpz_mpoly_univar_t A, const fmpz_mpoly_ctx_t ctx)

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

slong fmpz_mpoly_univar_length(const fmpz_mpoly_univar_t A, const fmpz_mpoly_ctx_t ctx)

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

slong fmpz_mpoly_univar_get_term_exp_si(fmpz_mpoly_univar_t A, slong i, const fmpz_mpoly_ctx_t ctx)

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

void fmpz_mpoly_univar_get_term_coeff(fmpz_mpoly_t c, const fmpz_mpoly_univar_t A, slong i, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_univar_swap_term_coeff(fmpz_mpoly_t c, fmpz_mpoly_univar_t A, slong i, const fmpz_mpoly_ctx_t ctx)

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

## Internal Functions

void fmpz_mpoly_inflate(fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz *shift, const fmpz *stride, const fmpz_mpoly_ctx_t ctx)

  Apply the function `e -> shift[v] + stride[v]*e` to each exponent `e` corresponding to the variable `v`. It is assumed that each shift and stride is not negative.

void fmpz_mpoly_deflate(fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz *shift, const fmpz *stride, const fmpz_mpoly_ctx_t ctx)

  Apply the function `e -> (e - shift[v])/stride[v]` to each exponent `e` corresponding to the variable `v`. If any `stride[v]` is zero, the corresponding numerator `e - shift[v]` is assumed to be zero, and the quotient is defined as zero. This allows the function to undo the operation performed by `fmpz_mpoly_inflate()` when possible.

void fmpz_mpoly_deflation(fmpz *shift, fmpz *stride, const fmpz_mpoly_t A, const fmpz_mpoly_ctx_t ctx)

  For each variable \(v\) let \(S_v\) be the set of exponents appearing on \(v\). Set `shift[v]` to \(\operatorname{min}(S_v)\) and set `stride[v]` to \(\operatorname{gcd}(S-\operatorname{min}(S_v))\). If A is zero, all shifts and strides are set to zero.

void fmpz_mpoly_pow_fps(fmpz_mpoly_t A, const fmpz_mpoly_t B, ulong k, const fmpz_mpoly_ctx_t ctx)

  Set A to B raised to the k-th power, using the Monagan and Pearce FPS algorithm. It is assumed that B is not zero and \(k \geq 2\).

slong _fmpz_mpoly_divides_array(fmpz **poly1, ulong **exp1, slong *alloc, const fmpz *poly2, const ulong *exp2, slong len2, const fmpz *poly3, const ulong *exp3, slong len3, slong *mults, slong num, slong bits)

  Use dense array exact division to set `(poly1, exp1, alloc)` to `(poly2, exp3, len2)` divided by `(poly3, exp3, len3)` in `num` variables, given a list of multipliers to tightly pack exponents and a number of bits for the fields of the exponents of the result. The array “mults” is a list of bases to be used in encoding the array indices from the exponents. The function reallocates its output, hence the double indirection, and returns the length of its output if the quotient is exact, or zero if not. It is assumed that `poly2` is not zero. No aliasing is allowed.

int fmpz_mpoly_divides_array(fmpz_mpoly_t poly1, const fmpz_mpoly_t poly2, const fmpz_mpoly_t poly3, const fmpz_mpoly_ctx_t ctx)

  Set `poly1` to `poly2` divided by `poly3`, using a big dense array to accumulate coefficients, and return 1 if the quotient is exact. Otherwise, return 0 if the quotient is not exact. If the array will be larger than some internally set parameter, the function fails silently and returns \(-1\) so that some other method may be called. This function is most efficient on dense inputs. Note that the function `fmpz_mpoly_div_monagan_pearce` below may be much faster if the quotient is known to be exact.

slong _fmpz_mpoly_divides_monagan_pearce(fmpz **poly1, ulong **exp1, slong *alloc, const fmpz *poly2, const ulong *exp2, slong len2, const fmpz *poly3, const ulong *exp3, slong len3, ulong bits, slong N, const ulong *cmpmask)

  Set `(poly1, exp1, alloc)` to `(poly2, exp3, len2)` divided by `(poly3, exp3, len3)` and return 1 if the quotient is exact. Otherwise return 0. The function assumes exponent vectors that each fit in \(N\) words, and are packed into fields of the given number of bits. Assumes input polys are nonzero. Implements “Polynomial division using dynamic arrays, heaps and packed exponents” by Michael Monagan and Roman Pearce. No aliasing is allowed.

int fmpz_mpoly_divides_monagan_pearce(fmpz_mpoly_t poly1, const fmpz_mpoly_t poly2, const fmpz_mpoly_t poly3, const fmpz_mpoly_ctx_t ctx)

  Set `poly1` to `poly2` divided by `poly3` and return 1 if the quotient is exact. Otherwise return 0. The function uses the algorithm of Michael Monagan and Roman Pearce. Note that the function `fmpz_mpoly_div_monagan_pearce` below may be much faster if the quotient is known to be exact.

int fmpz_mpoly_divides_heap_threaded(fmpz_mpoly_t Q, const fmpz_mpoly_t A, const fmpz_mpoly_t B, const fmpz_mpoly_ctx_t ctx)

  The same method as used as in `fmpz_mpoly_divides_monagan_pearce()`, but is also multi-threaded.

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 fmpz_mpoly_divides_heap_threaded`.

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

slong _fmpz_mpoly_div_monagan_pearce(fmpz **polyq, ulong **expq, slong *allocq, const fmpz *poly2, const ulong *exp2, slong len2, const fmpz *poly3, const ulong *exp3, slong len3, slong bits, slong N, const ulong *cmpmask)

  Set `(polyq, expq, allocq)` to the quotient of `(poly2, exp2, len2)` by `(poly3, exp3, len3)` discarding remainder (with notional remainder coefficients reduced modulo the leading coefficient of `(poly3, exp3, len3)`), and return the length of the quotient. The function reallocates its output, hence the double indirection. The function assumes the exponent vectors all fit in \(N\) words. The exponent vectors are assumed to have fields with the given number of bits. Assumes input polynomials are nonzero. Implements “Polynomial division using dynamic arrays, heaps and packed exponents” by Michael Monagan and Roman Pearce. No aliasing is allowed.

void fmpz_mpoly_div_monagan_pearce(fmpz_mpoly_t polyq, const fmpz_mpoly_t poly2, const fmpz_mpoly_t poly3, const fmpz_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.

slong _fmpz_mpoly_divrem_monagan_pearce(slong *lenr, fmpz **polyq, ulong **expq, slong *allocq, fmpz **polyr, ulong **expr, slong *allocr, const fmpz *poly2, const ulong *exp2, slong len2, const fmpz *poly3, const ulong *exp3, slong len3, slong bits, slong N, const ulong *cmpmask)

  Set `(polyq, expq, allocq)` and `(polyr, expr, allocr)` to the quotient and remainder of `(poly2, exp2, len2)` by `(poly3, exp3, len3)` (with remainder coefficients reduced modulo the leading coefficient of `(poly3, exp3, len3)`), and return the length of the quotient. The function reallocates its outputs, hence the double indirection. The function assumes the exponent vectors all fit in \(N\) words. The exponent vectors are assumed to have fields with the given number of bits. Assumes input polynomials are nonzero. Implements “Polynomial division using dynamic arrays, heaps and packed exponents” by Michael Monagan and Roman Pearce. No aliasing is allowed.

void fmpz_mpoly_divrem_monagan_pearce(fmpz_mpoly_t q, fmpz_mpoly_t r, const fmpz_mpoly_t poly2, const fmpz_mpoly_t poly3, const fmpz_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.

slong _fmpz_mpoly_divrem_array(slong *lenr, fmpz **polyq, ulong **expq, slong *allocq, fmpz **polyr, ulong **expr, slong *allocr, const fmpz *poly2, const ulong *exp2, slong len2, const fmpz *poly3, const ulong *exp3, slong len3, slong *mults, slong num, slong bits)

  Use dense array division to set `(polyq, expq, allocq)` and `(polyr, expr, allocr)` to the quotient and remainder of `(poly2, exp2, len2)` divided by `(poly3, exp3, len3)` in `num` variables, given a list of multipliers to tightly pack exponents and a number of bits for the fields of the exponents of the result. The function reallocates its outputs, hence the double indirection. The array `mults` is a list of bases to be used in encoding the array indices from the exponents. The function returns the length of the quotient. It is assumed that the input polynomials are not zero. No aliasing is allowed.

int fmpz_mpoly_divrem_array(fmpz_mpoly_t q, fmpz_mpoly_t r, const fmpz_mpoly_t poly2, const fmpz_mpoly_t poly3, const fmpz_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`). The function is implemented using dense arrays, and is efficient when the inputs are fairly dense. If the array will be larger than some internally set parameter, the function silently returns 0 so that another function can be called, otherwise it returns 1.

void fmpz_mpoly_quasidivrem_heap(fmpz_t scale, fmpz_mpoly_t q, fmpz_mpoly_t r, const fmpz_mpoly_t poly2, const fmpz_mpoly_t poly3, const fmpz_mpoly_ctx_t ctx)

  Set `scale`, `q` and `r` so that `scale*poly2 = q*poly3 + r` and no monomial in `r` is divisible by the leading monomial of `poly3`, where `scale` is positive and as small as possible. This function throws an exception if `poly3` is zero or if an exponent overflow occurs.

slong _fmpz_mpoly_divrem_ideal_monagan_pearce(fmpz_mpoly_struct **polyq, fmpz **polyr, ulong **expr, slong *allocr, const fmpz *poly2, const ulong *exp2, slong len2, fmpz_mpoly_struct *const *poly3, ulong *const *exp3, slong len, slong N, slong bits, const fmpz_mpoly_ctx_t ctx, const ulong *cmpmask)

  This function is as per `_fmpz_mpoly_divrem_monagan_pearce` except that it takes an array of divisor polynomials `poly3` and an array of repacked exponent arrays `exp3`, which may alias the exponent arrays of `poly3`, and it returns an array of quotient polynomials `polyq`. The number of divisor (and hence quotient) polynomials is given by `len`. The function computes polynomials \(q_i\) such that \(r = a - \sum_{i=0}^{\mbox{len - 1}} q_ib_i\), where the \(q_i\) are the quotient polynomials and the \(b_i\) are the divisor polynomials.

void fmpz_mpoly_divrem_ideal_monagan_pearce(fmpz_mpoly_struct **q, fmpz_mpoly_t r, const fmpz_mpoly_t poly2, fmpz_mpoly_struct *const *poly3, slong len, const fmpz_mpoly_ctx_t ctx)

  This function is as per `fmpz_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]`.

## Vectors

type fmpz_mpoly_vec_struct

type fmpz_mpoly_vec_t

  A type holding a vector of `fmpz_mpoly_t`.

fmpz_mpoly_vec_entry(vec, i)

  Macro for accessing the entry at position i in vec.

void fmpz_mpoly_vec_init(fmpz_mpoly_vec_t vec, slong len, const fmpz_mpoly_ctx_t ctx)

  Initializes vec to a vector of length len, setting all entries to the zero polynomial.

void fmpz_mpoly_vec_clear(fmpz_mpoly_vec_t vec, const fmpz_mpoly_ctx_t ctx)

  Clears vec, freeing its allocated memory.

void fmpz_mpoly_vec_print(const fmpz_mpoly_vec_t vec, const fmpz_mpoly_ctx_t ctx)

  Prints vec to standard output.

void fmpz_mpoly_vec_swap(fmpz_mpoly_vec_t x, fmpz_mpoly_vec_t y, const fmpz_mpoly_ctx_t ctx)

  Swaps x and y efficiently.

void fmpz_mpoly_vec_fit_length(fmpz_mpoly_vec_t vec, slong len, const fmpz_mpoly_ctx_t ctx)

  Allocates room for len entries in vec.

void fmpz_mpoly_vec_set(fmpz_mpoly_vec_t dest, const fmpz_mpoly_vec_t src, const fmpz_mpoly_ctx_t ctx)

  Sets dest to a copy of src.

void fmpz_mpoly_vec_append(fmpz_mpoly_vec_t vec, const fmpz_mpoly_t f, const fmpz_mpoly_ctx_t ctx)

  Appends f to the end of vec.

slong fmpz_mpoly_vec_insert_unique(fmpz_mpoly_vec_t vec, const fmpz_mpoly_t f, const fmpz_mpoly_ctx_t ctx)

  Inserts f without duplication into vec and returns its index. If this polynomial already exists, vec is unchanged. If this polynomial does not exist in vec, it is appended.

void fmpz_mpoly_vec_set_length(fmpz_mpoly_vec_t vec, slong len, const fmpz_mpoly_ctx_t ctx)

  Sets the length of vec to len, truncating or zero-extending as needed.

void fmpz_mpoly_vec_randtest_not_zero(fmpz_mpoly_vec_t vec, flint_rand_t state, slong len, slong poly_len, slong bits, ulong exp_bound, fmpz_mpoly_ctx_t ctx)

  Sets vec to a random vector with exactly len entries, all nonzero, with random parameters defined by poly_len, bits and exp_bound.

void fmpz_mpoly_vec_set_primitive_unique(fmpz_mpoly_vec_t res, const fmpz_mpoly_vec_t src, const fmpz_mpoly_ctx_t ctx)

  Sets res to a vector containing all polynomials in src reduced to their primitive parts, without duplication. The zero polynomial is skipped if present. The output order is arbitrary.

## Ideals and Gröbner bases

The following methods deal with ideals in \(\mathbb{Q}[X_1,\ldots,X_n]\). We use primitive integer polynomials as normalised generators in place of monic rational polynomials.

void fmpz_mpoly_spoly(fmpz_mpoly_t res, const fmpz_mpoly_t f, const fmpz_mpoly_t g, const fmpz_mpoly_ctx_t ctx)

  Sets res to the S-polynomial of f and g, scaled to an integer polynomial by computing the LCM of the leading coefficients.

void fmpz_mpoly_reduction_primitive_part(fmpz_mpoly_t res, const fmpz_mpoly_t f, const fmpz_mpoly_vec_t vec, const fmpz_mpoly_ctx_t ctx)

  Sets res to the primitive part of the reduction (remainder of multivariate quasidivision with remainder) with respect to the polynomials vec.

int fmpz_mpoly_vec_is_groebner(const fmpz_mpoly_vec_t G, const fmpz_mpoly_vec_t F, const fmpz_mpoly_ctx_t ctx)

  If F is NULL, checks if G is a Gröbner basis. If F is not NULL, checks if G is a Gröbner basis for F.

int fmpz_mpoly_vec_is_autoreduced(const fmpz_mpoly_vec_t F, const fmpz_mpoly_ctx_t ctx)

  Checks whether the vector F is autoreduced (or inter-reduced).

void fmpz_mpoly_vec_autoreduction(fmpz_mpoly_vec_t H, const fmpz_mpoly_vec_t F, const fmpz_mpoly_ctx_t ctx)

  Sets H to the autoreduction (inter-reduction) of F.

void fmpz_mpoly_vec_autoreduction_groebner(fmpz_mpoly_vec_t H, const fmpz_mpoly_vec_t G, const fmpz_mpoly_ctx_t ctx)

  Sets H to the autoreduction (inter-reduction) of G. Assumes that G is a Gröbner basis. This produces a reduced Gröbner basis, which is unique (up to the sort order of the entries in the vector).

void fmpz_mpoly_buchberger_naive(fmpz_mpoly_vec_t G, const fmpz_mpoly_vec_t F, const fmpz_mpoly_ctx_t ctx)

  Sets G to a Gröbner basis for F, computed using a naive implementation of Buchberger’s algorithm.

int fmpz_mpoly_buchberger_naive_with_limits(fmpz_mpoly_vec_t G, const fmpz_mpoly_vec_t F, slong ideal_len_limit, slong poly_len_limit, slong poly_bits_limit, const fmpz_mpoly_ctx_t ctx)

  As `fmpz_mpoly_buchberger_naive()`, but halts if during the execution of Buchberger’s algorithm the length of the ideal basis set exceeds ideal_len_limit, the length of any polynomial exceeds poly_len_limit, or the size of the coefficients of any polynomial exceeds poly_bits_limit. Returns 1 for success and 0 for failure. On failure, G is a valid basis for F but it might not be a Gröbner basis.

## Special polynomials

void fmpz_mpoly_symmetric_gens(fmpz_mpoly_t res, ulong k, slong *vars, slong n, const fmpz_mpoly_ctx_t ctx)

void fmpz_mpoly_symmetric(fmpz_mpoly_t res, ulong k, const fmpz_mpoly_ctx_t ctx)

  Sets res to the elementary symmetric polynomial \(e_k(X_1,\ldots,X_n)\).

  The gens version takes \(X_1,\ldots,X_n\) to be the subset of generators given by vars and n. The indices in vars start from zero. Currently, the indices in vars must be distinct.