# fmpz_mpoly_q.h – multivariate rational functions over Q

An `fmpz_mpoly_q_t` represents an element of \(\mathbb{Q}(x_1,\ldots,x_n)\) for fixed n as a pair of FLINT multivariate polynomials (`fmpz_mpoly_t`). Instances are always kept in canonical form by ensuring that the GCD of numerator and denominator is 1 and that the coefficient of the leading term of the denominator is positive.

The user must create a multivariate polynomial context (`fmpz_mpoly_ctx_t`) specifying the number of variables n and the monomial ordering.

## Types and macros

type fmpz_mpoly_q_struct

type fmpz_mpoly_q_t

  An fmpz_mpoly_q_struct consists of a pair of fmpz_mpoly_struct:s. An fmpz_mpoly_q_t is defined as an array of length one of type fmpz_mpoly_q_struct, permitting an fmpz_mpoly_q_t to be passed by reference.

fmpz_mpoly_q_numref(x)

  Macro returning a pointer to the numerator of x which can be used as an fmpz_mpoly_t.

fmpz_mpoly_q_denref(x)

  Macro returning a pointer to the denominator of x which can be used as an fmpz_mpoly_t.

## Memory management

void fmpz_mpoly_q_init(fmpz_mpoly_q_t res, const fmpz_mpoly_ctx_t ctx)

  Initializes res for use, and sets its value to zero.

void fmpz_mpoly_q_clear(fmpz_mpoly_q_t res, const fmpz_mpoly_ctx_t ctx)

  Clears res, freeing or recycling its allocated memory.

## Assignment

void fmpz_mpoly_q_swap(fmpz_mpoly_q_t x, fmpz_mpoly_q_t y, const fmpz_mpoly_ctx_t ctx)

  Swaps the values of x and y efficiently.

void fmpz_mpoly_q_set(fmpz_mpoly_q_t res, const fmpz_mpoly_q_t x, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_q_set_fmpq(fmpz_mpoly_q_t res, const fmpq_t x, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_q_set_fmpz(fmpz_mpoly_q_t res, const fmpz_t x, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_q_set_si(fmpz_mpoly_q_t res, slong x, const fmpz_mpoly_ctx_t ctx)

  Sets res to the value x.

## Canonicalisation

void fmpz_mpoly_q_canonicalise(fmpz_mpoly_q_t x, const fmpz_mpoly_ctx_t ctx)

  Puts the numerator and denominator of x in canonical form by removing common content and making the leading term of the denominator positive.

int fmpz_mpoly_q_is_canonical(const fmpz_mpoly_q_t x, const fmpz_mpoly_ctx_t ctx)

  Returns whether x is in canonical form.

  In addition to verifying that the numerator and denominator have no common content and that the leading term of the denominator is positive, this function checks that the denominator is nonzero and that the numerator and denominator have correctly sorted terms (these properties should normally hold; verifying them provides an extra consistency check for test code).

## Properties

int fmpz_mpoly_q_is_zero(const fmpz_mpoly_q_t x, const fmpz_mpoly_ctx_t ctx)

  Returns whether x is the constant 0.

int fmpz_mpoly_q_is_one(const fmpz_mpoly_q_t x, const fmpz_mpoly_ctx_t ctx)

  Returns whether x is the constant 1.

void fmpz_mpoly_q_used_vars(int *used, const fmpz_mpoly_q_t f, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_q_used_vars_num(int *used, const fmpz_mpoly_q_t f, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_q_used_vars_den(int *used, const fmpz_mpoly_q_t f, const fmpz_mpoly_ctx_t ctx)

  For each variable, sets the corresponding entry in used to the boolean flag indicating whether that variable appears in the rational function (respectively its numerator or denominator).

## Special values

void fmpz_mpoly_q_zero(fmpz_mpoly_q_t res, const fmpz_mpoly_ctx_t ctx)

  Sets res to the constant 0.

void fmpz_mpoly_q_one(fmpz_mpoly_q_t res, const fmpz_mpoly_ctx_t ctx)

  Sets res to the constant 1.

void fmpz_mpoly_q_gen(fmpz_mpoly_q_t res, slong i, const fmpz_mpoly_ctx_t ctx)

  Sets res to the generator \(x_{i+1}\). Requires \(0 \le i < n\) where n is the number of variables of ctx.

## Input and 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.

void fmpz_mpoly_q_print_pretty(const fmpz_mpoly_q_t f, const char **x, const fmpz_mpoly_ctx_t ctx)

  Prints res to standard output. If x is not NULL, the strings in x are used as the symbols for the variables.

char *fmpz_mpoly_q_get_str_pretty(const fmpz_mpoly_q_t f, const char **x, const fmpz_mpoly_ctx_t ctx)

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

int fmpz_mpoly_q_set_str_pretty(fmpz_mpoly_q_t res, const char *s, const char **x, fmpz_mpoly_ctx_t ctx)

  Set res to the fraction in the null-terminated string str given an array x of variable strings. If parsing str fails, res 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 division by zero occurs, parsing fails.

## Random generation

void fmpz_mpoly_q_randtest(fmpz_mpoly_q_t res, flint_rand_t state, slong length, ulong coeff_bits, slong exp_bound, const fmpz_mpoly_ctx_t ctx)

  Sets res to a random rational function where both numerator and denominator have up to length terms, coefficients up to size coeff_bits, and exponents strictly smaller than exp_bound.

## Comparisons

int fmpz_mpoly_q_equal(const fmpz_mpoly_q_t x, const fmpz_mpoly_q_t y, const fmpz_mpoly_ctx_t ctx)

  Returns whether x and y are equal.

## Arithmetic

void fmpz_mpoly_q_neg(fmpz_mpoly_q_t res, const fmpz_mpoly_q_t x, const fmpz_mpoly_ctx_t ctx)

  Sets res to the negation of x.

void fmpz_mpoly_q_add(fmpz_mpoly_q_t res, const fmpz_mpoly_q_t x, const fmpz_mpoly_q_t y, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_q_add_fmpq(fmpz_mpoly_q_t res, const fmpz_mpoly_q_t x, const fmpq_t y, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_q_add_fmpz(fmpz_mpoly_q_t res, const fmpz_mpoly_q_t x, const fmpz_t y, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_q_add_si(fmpz_mpoly_q_t res, const fmpz_mpoly_q_t x, slong y, const fmpz_mpoly_ctx_t ctx)

  Sets res to the sum of x and y.

void fmpz_mpoly_q_sub(fmpz_mpoly_q_t res, const fmpz_mpoly_q_t x, const fmpz_mpoly_q_t y, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_q_sub_fmpq(fmpz_mpoly_q_t res, const fmpz_mpoly_q_t x, const fmpq_t y, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_q_sub_fmpz(fmpz_mpoly_q_t res, const fmpz_mpoly_q_t x, const fmpz_t y, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_q_sub_si(fmpz_mpoly_q_t res, const fmpz_mpoly_q_t x, slong y, const fmpz_mpoly_ctx_t ctx)

  Sets res to the difference of x and y.

void fmpz_mpoly_q_mul(fmpz_mpoly_q_t res, const fmpz_mpoly_q_t x, const fmpz_mpoly_q_t y, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_q_mul_fmpq(fmpz_mpoly_q_t res, const fmpz_mpoly_q_t x, const fmpq_t y, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_q_mul_fmpz(fmpz_mpoly_q_t res, const fmpz_mpoly_q_t x, const fmpz_t y, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_q_mul_si(fmpz_mpoly_q_t res, const fmpz_mpoly_q_t x, slong y, const fmpz_mpoly_ctx_t ctx)

  Sets res to the product of x and y.

void fmpz_mpoly_q_div(fmpz_mpoly_q_t res, const fmpz_mpoly_q_t x, const fmpz_mpoly_q_t y, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_q_div_fmpq(fmpz_mpoly_q_t res, const fmpz_mpoly_q_t x, const fmpq_t y, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_q_div_fmpz(fmpz_mpoly_q_t res, const fmpz_mpoly_q_t x, const fmpz_t y, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_q_div_si(fmpz_mpoly_q_t res, const fmpz_mpoly_q_t x, slong y, const fmpz_mpoly_ctx_t ctx)

  Sets res to the quotient of x and y. Division by zero calls flint_abort.

void fmpz_mpoly_q_inv(fmpz_mpoly_q_t res, const fmpz_mpoly_q_t x, const fmpz_mpoly_ctx_t ctx)

  Sets res to the inverse of x. Division by zero calls flint_abort.

## Content

void _fmpz_mpoly_q_content(fmpz_t num, fmpz_t den, const fmpz_mpoly_t xnum, const fmpz_mpoly_t xden, const fmpz_mpoly_ctx_t ctx)
void fmpz_mpoly_q_content(fmpq_t res, const fmpz_mpoly_q_t x, const fmpz_mpoly_ctx_t ctx)

  Sets res to the content of the coefficients of x.