fmpq_mpoly.h – multivariate polynomials over the rational numbers

Description.

Types, macros and constants

fmpq_mpoly_ctx_struct

fmpq_mpoly_ctx_t

Description.

fmpq_mpoly_struct

fmpq_mpoly_t

Description.

Context object

An array of type ulong * or fmpz ** is used to communicate exponent vectors. These exponent vectors must have length equal to the number of variables in the polynmial ring. The element of this exponent vector at index \(0\) corresponds to the most significant variable in the monomial ordering. For example, if the polynomial is \(7*x^2*y+8*y*z+9\) and the variables are ordered so that \(x>y>z\), the degree function will return \({2,1,1}\). Similarly, the exponent vector of the \(0\)-index term in this polynomial is \({2,1,0}\), while the \(2\)-index term has exponent vector \({0,0,0}\) and coefficient \(9\).

void fmpq_mpoly_ctx_init(fmpq_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 fmpq_mpoly_ctx_nvars(fmpq_mpoly_ctx_t ctx)

Return the number of variables used to initialize the context.

void fmpq_mpoly_ctx_clear(fmpq_mpoly_ctx_t ctx)

Release up any space allocated by an fmpq_mpoly_ctx_t.

Memory management

void fmpq_mpoly_init(fmpq_mpoly_t poly, const fmpq_mpoly_ctx_t ctx)

Initialise poly for use, given an initialised context object. Its value is set to 0.

void fmpq_mpoly_init2(fmpq_mpoly_t poly, slong alloc, const fmpq_mpoly_ctx_t ctx)

Initialise poly for use, with space for at least alloc terms, given an initialised context. Its value is set to 0.

void fmpq_mpoly_realloc(fmpq_mpoly_t poly, slong len, const fmpq_mpoly_ctx_t ctx)

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

void fmpq_mpoly_fit_length(fmpq_mpoly_t poly, slong len, const fmpq_mpoly_ctx_t ctx)

Reallocate poly to have space for at least len terms. No truncation is performed if len is less than the currently allocated number of terms; the allocated space can only grow.

void fmpq_mpoly_fit_bits(fmpq_mpoly_t poly, slong bits, const fmpq_mpoly_ctx_t ctx)

Reallocate the polynomial to have space for exponent fields of the given number of bits. This function can increase the number of bits only.

void fmpq_mpoly_clear(fmpq_mpoly_t poly, const fmpq_mpoly_ctx_t ctx)

Release any space allocated an fmpq_mpoly_t.

Degrees

int fmpq_mpoly_degrees_fit_si(const fmpq_mpoly_t poly, const fmpq_mpoly_ctx_t ctx)

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

void fmpq_mpoly_degrees_si(slong * degs, const fmpq_mpoly_t poly, const fmpq_mpoly_ctx_t ctx)

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

slong fmpq_mpoly_degree_si(const fmpq_mpoly_t poly, slong var, const fmpq_mpoly_ctx_t ctx)

Return the degree of poly with respect to the variable of index var. If poly is zero, the return is -1.

void fmpq_mpoly_degrees_fmpz(fmpz ** degs, const fmpq_mpoly_t poly, const fmpq_mpoly_ctx_t ctx)

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

void fmpq_mpoly_degree_fmpz(fmpz_t deg, const fmpq_mpoly_t poly, slong var, const fmpq_mpoly_ctx_t ctx)

Set deg to the degree of poly with respect to the variable of index var. If poly is zero, set deg to -1.

int fmpq_mpoly_totaldegree_fits_si(const fmpq_mpoly_t A, const fmpq_mpoly_ctx_t ctx)

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

slong fmpq_mpoly_totaldegree_si(const fmpq_mpoly_t A, const fmpq_mpoly_ctx_t ctx)

Return the total degree of A assuming it fits into an slong. If A is zero, the return is -1.

void fmpq_mpoly_totaldegree_fmpz(fmpz_t tdeg, const fmpq_mpoly_t A, const fmpq_mpoly_ctx_t ctx)

Set tdeg to the total degree of A. If A is zero, tdeg is set to -1.

Coefficients

void fmpq_mpoly_denominator(fmpz_t d, const fmpq_mpoly_t poly, const fmpq_mpoly_ctx_t ctx)

Set \(d\) to the denominator of \(poly\), the smallest positive integer \(d\) such that \(d`*``poly`\) has integer coefficients.

void fmpq_mpoly_get_coeff_fmpq_monomial(fmpq_t c, const fmpq_mpoly_t poly, const fmpq_mpoly_t poly2, const fmpq_mpoly_ctx_t ctx)

Assuming that poly2 is a monomial, set \(c\) to the coefficient of the corresponding monomial in poly. This function thows if poly2 is not a monomial.

void fmpq_mpoly_set_coeff_fmpq_monomial(fmpq_mpoly_t poly, const fmpq_t c, const fmpq_mpoly_t poly2, const fmpq_mpoly_ctx_t ctx)

Assuming that poly2 is a monomial, set the coefficient of the corresponding monomial in poly to \(c\). This function thows if poly2 is not a monomial.

void fmpq_mpoly_get_coeff_fmpq_fmpz(fmpq_t c, const fmpq_mpoly_t poly, fmpz * const * exp, const fmpq_mpoly_ctx_t ctx)

Set \(c\) to the coefficient of the monomial with exponent exp.

void fmpq_mpoly_set_coeff_fmpq_fmpz(fmpq_mpoly_t poly, const fmpq_t c, fmpz * const * exp, fmpq_mpoly_ctx_t ctx)

Set the coefficient of the monomial with exponent exp to \(c\).

void fmpq_mpoly_get_coeff_fmpq_ui(fmpq_t c, const fmpq_mpoly_t poly, ulong const * exp, const fmpq_mpoly_ctx_t ctx)

Set \(c\) to the coefficient of the monomial with exponent exp.

void fmpq_mpoly_set_coeff_fmpq_ui(fmpq_mpoly_t poly, const fmpq_t c, ulong const * exp, fmpq_mpoly_ctx_t ctx)

Set the coefficient of the monomial with exponent exp to \(c\).

Internal operations

An fmpq_mpoly_t poly is an fmpz_mpoly_t zpoly together with an fmpq_t content representing any content. In order to be in canonical form, either zpoly and content must both be zero, or zcode must be a polynomial with content 1 and positive leading term. In either the case the representation poly = content * zpoly holds.

void fmpq_mpoly_canonicalise(fmpq_mpoly_t poly, const fmpz_mpoly_ctx_t ctx)

Factor out content from the internal fmpz_mpoly_t. If this fmpz_mpoly_t was in canonical form, the resulting poly will be. This function expects the internal fmpz_mpoly_t to be in canonical form. If this is not the case, the container operations of fmpq_mpoly_sort and fmpq_mpoly_combine_like_terms may be useful. All operations other than the container operations produce canonical output and expect their inputs to be canonical.

void fmpq_mpoly_assert_canonical(const fmpq_mpoly_t poly, const fmpq_mpoly_ctx_t ctx)

Throw if poly is not in canonical form.

Container operations

Some of these functions deal with violations of the internal canonical representation. A call to fmpq_mpoly_sort followed by a call to fmpq_mpoly_combine_like_terms should leave a polynomial in canonical form. The pushterm functions run in constant average time if the terms pushed have bounded denominators, and a term is appened even if the specified coefficient is zero.

slong fmpq_mpoly_length(const fmpq_mpoly_t poly, const fmpq_mpoly_ctx_t ctx)

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

void fmpq_mpoly_get_termcoeff_fmpq(fmpq_t x, const fmpq_mpoly_t poly, slong i, const fmpq_mpoly_ctx_t ctx)

Set \(x\) to coefficient of index \(i\), starting with \(i = 0\) for the term with most significance.

void fmpq_mpoly_set_termcoeff_fmpq(fmpq_mpoly_t poly, slong i, const fmpq_t x, const fmpq_mpoly_ctx_t ctx)

Set the coefficient of index \(i\) to \(x\), starting with \(i = 0\) for the term with most significance.

int fmpq_mpoly_termexp_fits_si(const fmpq_mpoly_t poly, slong i, const fmpq_mpoly_ctx_t ctx)

Return 1 if all entries of the exponent vector of the term of index \(i\) fit into an slong. Otherwise, return 0.

int fmpq_mpoly_termexp_fits_ui(const fmpq_mpoly_t poly, slong i, const fmpq_mpoly_ctx_t ctx)

Return 1 if all entries of the exponent vector of the term of index \(i\) fit into a ulong. Otherwise, return 0.

void fmpq_mpoly_get_termexp_ui(ulong * exps, const fmpq_mpoly_t poly, slong i, const fmpq_mpoly_ctx_t ctx)

Get the exponent vector of the given polynomial with index \(i\).

void fmpq_mpoly_get_termexp_fmpz(fmpz ** exps, const fmpq_mpoly_t poly, slong i, const fmpq_mpoly_ctx_t ctx)

Get the exponent vector of the given polynomial with index \(i\).

void fmpq_mpoly_set_termexp_ui(fmpq_mpoly_t poly, slong i, const ulong * exps, const fmpq_mpoly_ctx_t ctx)

Set the exponent vector of the given polynomial with index \(i\).

void fmpq_mpoly_set_termexp_fmpz(fmpq_mpoly_t poly, slong i, fmpz * const * exps, const const fmpq_mpoly_ctx_t ctx)

Set the exponent vector of the given polynomial with index \(i\).

void fmpq_mpoly_pushterm_fmpq_fmpz(fmpz_mpoly_t poly, const fmpq_t c, fmpz * const * exp, const fmpq_mpoly_ctx_t ctx)

Append a term to poly with the given coefficient and exponents.

void fmpq_mpoly_pushterm_fmpz_fmpz(fmpz_mpoly_t poly, const fmpq_t c, fmpz * const * exp, const fmpq_mpoly_ctx_t ctx)

Append a term to poly with the given coefficient and exponents.

void fmpq_mpoly_pushterm_ui_fmpz(fmpz_mpoly_t poly, ulong c, fmpz * const * exp, const fmpq_mpoly_ctx_t ctx)

Append a term to poly with the given coefficient and exponents.

void fmpq_mpoly_pushterm_si_fmpz(fmpz_mpoly_t poly, slong c, fmpz * const * exp, const fmpq_mpoly_ctx_t ctx)

Append a term to poly with the given coefficient and exponents.

void fmpq_mpoly_pushterm_fmpq_ui(fmpz_mpoly_t poly, const fmpq_t c, const ulong * exp, const fmpq_mpoly_ctx_t ctx)

Append a term to poly with the given coefficient and exponents.

void fmpq_mpoly_pushterm_fmpz_ui(fmpz_mpoly_t poly, const fmpq_t c, const ulong * exp, const fmpq_mpoly_ctx_t ctx)

Append a term to poly with the given coefficient and exponents.

void fmpq_mpoly_pushterm_ui_ui(fmpz_mpoly_t poly, ulong c, const ulong * exp, const fmpq_mpoly_ctx_t ctx)

Append a term to poly with the given coefficient and exponents.

void fmpq_mpoly_pushterm_si_ui(fmpz_mpoly_t poly, slong c, const ulong * exp, const fmpq_mpoly_ctx_t ctx)

Append a term to poly with the given coefficient and exponents.

void fmpq_mpoly_sort_terms(fmpq_mpoly_t poly, const fmpq_mpoly_ctx_t ctx)

Sort the internal fmpz_mpoly_t into the canonical ordering dictated by the ordering in ctx. This function does not combine like terms, nor does it delete terms with coefficient zero. Even if all terms have distinct exponents, the result may still not be in canonical form, because content may not be factored out.

void fmpq_mpoly_combine_like_terms(fmpq_mpoly_t poly, const fmpz_mpoly_ctx_t ctx)

Combine adjacent like terms in the internal fmpz_poly and then factor out content via fmpq_mpoly_canonicalise. If the terms of poly were sorted to begin with, the result will be in canonical form.

Set/Negate

void fmpq_mpoly_set(fmpq_mpoly_t poly1, const fmpq_mpoly_t poly2, const fmpq_mpoly_ctx_t ctx)

Set poly1 to poly2.

void fmpq_mpoly_swap(fmpq_mpoly_t poly1, fmpq_mpoly_t poly2, const fmpq_mpoly_ctx_t ctx)

Efficiently swap the contents of the two given polynomials. No copying is performed; the swap is accomplished by swapping pointers.

void fmpq_mpoly_gen(fmpq_mpoly_t poly, slong i, const fmpq_mpoly_ctx_t ctx)

Set poly to the \(i\)-th generator (variable), where \(i = 0\) corresponds to the variable with the most significance with respect to the ordering.

void fmpq_mpoly_neg(fmpq_mpoly_t poly1, const fmpq_mpoly_t poly2, const fmpq_mpoly_ctx_t ctx)

Set poly1 to \(-```poly2`\).

Constants

int fmpq_mpoly_is_fmpq(const fmpq_mpoly_t poly, const fmpq_mpoly_ctx_t ctx)

Return 1 if poly is a constant, else return 0.

void fmpq_mpoly_get_fmpq(fmpq_t c, const fmpq_mpoly_t poly, const fmpq_mpoly_ctx_t ctx)

Assuming that poly is a constant, set \(c\) to this constant. This function throws if poly is not a constant.

void fmpq_mpoly_set_fmpq(fmpq_mpoly_t poly, const fmpq_t c, const fmpq_mpoly_ctx_t ctx);

Set poly to the constant \(c\).

void fmpq_mpoly_set_fmpz(fmpq_mpoly_t poly, const fmpz_t c, const fmpq_mpoly_ctx_t ctx);

Set poly to the constant \(c\).

void fmpq_mpoly_set_ui(fmpq_mpoly_t poly, ulong c, const fmpq_mpoly_ctx_t ctx);

Set poly to the constant \(c\).

void fmpq_mpoly_set_si(fmpq_mpoly_t poly, slong c, const fmpq_mpoly_ctx_t ctx);

Set poly to the constant \(c\).

void fmpq_mpoly_zero(fmpq_mpoly_t poly, const fmpq_mpoly_ctx_t ctx)

Set poly to the constant 0.

void fmpq_mpoly_one(fmpq_mpoly_t poly, const fmpq_mpoly_ctx_t ctx)

Set poly to the constant 1.

Comparison

int fmpq_mpoly_equal(fmpq_mpoly_t poly1, const fmpq_mpoly_t poly2, const fmpq_mpoly_ctx_t ctx)

Return 1 if poly1 is equal to poly2, else return 0.

int fmpq_mpoly_equal_fmpq(const fmpq_mpoly_t poly, fmpq_t c, const fmpq_mpoly_ctx_t ctx)

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

int fmpq_mpoly_equal_fmpz(const fmpq_mpoly_t poly, fmpz_t c, const fmpq_mpoly_ctx_t ctx)

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

int fmpq_mpoly_equal_ui(const fmpq_mpoly_t poly, ulong  c, const fmpq_mpoly_ctx_t ctx)

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

int fmpq_mpoly_equal_si(const fmpq_mpoly_t poly, slong  c, const fmpq_mpoly_ctx_t ctx)

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

int fmpq_mpoly_is_zero(const fmpq_mpoly_t poly, const fmpq_mpoly_ctx_t ctx)

Return 1 if poly is equal to the constant 0, else return 0.

int fmpq_mpoly_is_one(const fmpq_mpoly_t poly, const fmpq_mpoly_ctx_t ctx)

Return 1 if poly is equal to the constant 1, else return 0.

int fmpq_mpoly_is_gen(const fmpq_mpoly_t poly, slong i, const fmpq_mpoly_ctx_t ctx)

If \(i \ge 0\), return 1 if poly is equal to the \(i\)-th generator, otherwise return 0. If \(i < 0\), return 1 if the polynomial is equal to any generator, otherwise return 0.

Basic arithmetic

void fmpq_mpoly_add_fmpq(fmpq_mpoly_t poly1, const fmpq_mpoly_t poly2, const fmpq_t c, const fmpq_mpoly_ctx_t ctx);

Set poly1 to poly2 plus \(c\).

void fmpq_mpoly_add_fmpz(fmpq_mpoly_t poly1, const fmpq_mpoly_t poly2, const fmpz_t c, const fmpq_mpoly_ctx_t ctx);

Set poly1 to poly2 plus \(c\).

void fmpq_mpoly_add_ui(fmpq_mpoly_t poly1, const fmpq_mpoly_t poly2, ulong c, const fmpq_mpoly_ctx_t ctx);

Set poly1 to poly2 plus \(c\).

void fmpq_mpoly_add_si(fmpq_mpoly_t poly1, const fmpq_mpoly_t poly2, slong c, const fmpq_mpoly_ctx_t ctx);

Set poly1 to poly2 plus \(c\).

void fmpq_mpoly_sub_fmpq(fmpq_mpoly_t poly1, const fmpq_mpoly_t poly2, const fmpq_t c, const fmpq_mpoly_ctx_t ctx);

Set poly1 to poly2 minus \(c\).

void fmpq_mpoly_sub_fmpz(fmpq_mpoly_t poly1, const fmpq_mpoly_t poly2, const fmpz_t c, const fmpq_mpoly_ctx_t ctx);

Set poly1 to poly2 minus \(c\).

void fmpq_mpoly_sub_ui(fmpq_mpoly_t poly1, const fmpq_mpoly_t poly2, ulong c, const fmpq_mpoly_ctx_t ctx);

Set poly1 to poly2 minus \(c\).

void fmpq_mpoly_sub_si(fmpq_mpoly_t poly1, const fmpq_mpoly_t poly2, slong c, const fmpq_mpoly_ctx_t ctx);

Set poly1 to poly2 minus \(c\).

void fmpq_mpoly_add(fmpq_mpoly_t poly1, const fmpq_mpoly_t poly2, const fmpq_mpoly_t poly3, const fmpq_mpoly_ctx_t ctx)

Set poly1 to poly2 plus poly3.

void fmpq_mpoly_sub(fmpq_mpoly_t poly1, const fmpq_mpoly_t poly2, const fmpq_mpoly_t poly3, const fmpq_mpoly_ctx_t ctx)

Set poly1 to poly2 minus poly3.

Scalar operations

void fmpq_mpoly_scalar_mul_fmpq(fmpq_mpoly_t poly1, const fmpq_mpoly_t poly2, const fmpq_t c, const fmpq_mpoly_ctx_t ctx)

Set poly1 to poly2 times \(c\).

void fmpq_mpoly_scalar_mul_fmpz(fmpq_mpoly_t poly1, const fmpq_mpoly_t poly2, const fmpz_t c, const fmpq_mpoly_ctx_t ctx)

Set poly1 to poly2 times \(c\).

void fmpq_mpoly_scalar_mul_ui(fmpq_mpoly_t poly1, const fmpq_mpoly_t poly2, ulong  c, const fmpq_mpoly_ctx_t ctx)

Set poly1 to poly2 times \(c\).

void fmpq_mpoly_scalar_mul_si(fmpq_mpoly_t poly1, const fmpq_mpoly_t poly2, slong  c, const fmpq_mpoly_ctx_t ctx)

Set poly1 to poly2 times \(c\).

void fmpq_mpoly_scalar_div_fmpq(fmpq_mpoly_t poly1, const fmpq_mpoly_t poly2, const fmpq_t c, const fmpq_mpoly_ctx_t ctx)

Set poly1 to poly2 divided by \(c\).

void fmpq_mpoly_scalar_div_fmpz(fmpq_mpoly_t poly1, const fmpq_mpoly_t poly2, const fmpz_t c, const fmpq_mpoly_ctx_t ctx)

Set poly1 to poly2 divided by \(c\).

void fmpq_mpoly_scalar_div_ui(fmpq_mpoly_t poly1, const fmpq_mpoly_t poly2, ulong  c, const fmpq_mpoly_ctx_t ctx)

Set poly1 to poly2 divided by \(c\).

void fmpq_mpoly_scalar_div_si(fmpq_mpoly_t poly1, const fmpq_mpoly_t poly2, slong  c, const fmpq_mpoly_ctx_t ctx)

Set poly1 to poly2 divided by \(c\).

void fmpq_mpoly_make_monic_inplace(fmpq_mpoly_t poly1, const fmpq_mpoly_ctx_t ctx)

Divide poly1 by its leading coefficient. An expection is raised if poly1 is zero.

Multiplication

void fmpq_mpoly_mul(fmpq_mpoly_t poly1, const fmpq_mpoly_t poly2, const fmpq_mpoly_t poly3, const fmpq_mpoly_ctx_t ctx)

Set poly1 to poly2 times poly3.

void fmpq_mpoly_mul_threaded(fmpq_mpoly_t poly1, const fmpq_mpoly_t poly2, const fmpq_mpoly_t poly3, const fmpq_mpoly_ctx_t ctx)

Set poly1 to poly2 times poly3 using multiple threads.

Powering

void fmpq_mpoly_pow_fmpz(fmpq_mpoly_t poly1, const fmpq_mpoly_t poly2, const fmpz_t k, const fmpq_mpoly_ctx_t ctx)

Set poly1 to poly2 raised to the \(k\)-th power. An expection is raised if \(k < 0\) or if \(k\) is large and the polynomial is not a monomial with coefficient \(\pm1\).

void fmpq_mpoly_pow_si(fmpq_mpoly_t poly1, const fmpq_mpoly_t poly2, slong k, const fmpq_mpoly_ctx_t ctx)

Set poly1 to poly2 raised to the \(k\)-th power. An expection is raised if \(k < 0\).

Divisibility testing

int fmpq_mpoly_divides(fmpq_mpoly_t poly1, const fmpq_mpoly_t poly2, const fmpq_mpoly_t poly3, const fmpq_mpoly_ctx_t ctx)

Set poly1 to poly2 divided by poly3 and return 1 if the quotient is exact. Otherwise return 0.

Division

void fmpq_mpoly_div(fmpq_mpoly_t polyq, const fmpq_mpoly_t poly2, const fmpq_mpoly_t poly3, const fmpq_mpoly_ctx_t ctx)

Set polyq to the quotient of poly2 by poly3, discarding the remainder. An expection is currently raised if poly2 or poly3 have bit counts greater than FLINT_BITS.

void fmpq_mpoly_divrem(fmpq_mpoly_t q, fmpq_mpoly_t r, const fmpq_mpoly_t poly2, const fmpq_mpoly_t poly3, const fmpq_mpoly_ctx_t ctx)

Set polyq and polyr to the quotient and remainder of poly2 divided by poly3. An expection is currently raised if poly2 or poly3 have bit counts greater than FLINT_BITS.

Reduction

void fmpq_mpoly_divrem_ideal(fmpq_mpoly_struct ** q, fmpq_mpoly_t r, const fmpq_mpoly_t poly2, fmpq_mpoly_struct * const * poly3, slong len, const fmpq_mpoly_ctx_t ctx)

This function is as per fmpq_mpoly_divrem 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]. An expection is currently raised if any input polynomials have bit counts greater than FLINT_BITS.

Differentiation/Integration

void fmpq_mpoly_derivative(fmpq_mpoly_t poly1, const fmpq_mpoly_t poly2, slong var, const fmpq_mpoly_ctx_t ctx)

Set poly1 to the derivative of poly2 with respect to the variable of index var.

void fmpq_mpoly_integral(fmpq_mpoly_t poly1, const fmpq_mpoly_t poly2, slong var, const fmpq_mpoly_ctx_t ctx)

Set poly1 to the integral with the fewest number of terms of poly2 with respect to the variable of index var.

Evaluation

void fmpq_mpoly_evaluate_all_fmpq(fmpq_t ev, const fmpq_mpoly_t A, fmpq * const * vals, const fmpq_mpoly_ctx_t ctx)

Set ev the evaluation of A where the variables are replaced by the corresponding elements of the array vals.

void fmpq_mpoly_evaluate_one_fmpq(fmpq_mpoly_t A, const fmpq_mpoly_t B, slong var, const fmpq_t val, const fmpq_mpoly_ctx_t ctx)

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

void fmpq_mpoly_compose_fmpq_poly(fmpq_poly_t A, const fmpq_mpoly_t B, fmpq_poly_struct * const * C, const fmpq_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.

void fmpq_mpoly_compose_fmpq_mpoly(fmpq_mpoly_t A, const fmpq_mpoly_t B, fmpq_mpoly_struct * const * C, const fmpq_mpoly_ctx_t ctxB, const fmpq_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.

Greatest Common Divisor

int fmpq_mpoly_gcd(fmpq_mpoly_t poly1, const fmpq_mpoly_t poly2, const fmpq_mpoly_t poly3, const fmpq_mpoly_ctx_t ctx)

Set poly1 to the monic GCD of poly2 and poly3, assuming the return value is 1. If the return value is 0, the GCD was unable to be computed.

Input/Output

char * fmpq_mpoly_get_str_pretty(const fmpq_mpoly_t poly, const char ** x, const fmpq_mpoly_ctx_t ctx)

Return a string (which the user is responsible for cleaning up), representing poly, given an array of variable strings, starting with the variable of most significance with respect to the ordering.

int fmpq_mpoly_fprint_pretty(FILE * file, const fmpq_mpoly_t poly, const char ** x, const fmpq_mpoly_ctx_t ctx)

Print to the given stream a string representing poly, given an array of variable strings, starting with the variable of most significance with respect to the ordering. The number of characters written is returned.

int fmpq_mpoly_print_pretty(const fmpq_mpoly_t poly, const char ** x, const fmpq_mpoly_ctx_t ctx)

Print to stdout a string representing poly, given an array of variable strings, starting with the variable of most significance with respect to the ordering. The number of characters written is returned.

int fmpq_mpoly_set_str_pretty(fmpq_mpoly_t poly, const char * str, const char ** x, const fmpq_mpoly_ctx_t ctx)

Sets poly to the polynomial in the null-terminated string str given an array x of variable strings. If parsing str fails, poly 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.

Random generation

void fmpq_mpoly_randtest_bound(fmpq_mpoly_t poly, flint_rand_t state, slong length, mp_limb_t coeff_bits, slong exp_bound, const fmpq_mpoly_ctx_t ctx)

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

void fmpq_mpoly_randtest_bound(fmpq_mpoly_t poly, flint_rand_t state, slong length, mp_limb_t coeff_bits, slong exp_bound, const fmpq_mpoly_ctx_t ctx)

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

void fmpq_mpoly_randtest_bits(fmpq_mpoly_t poly, flint_rand_t state, slong length, mp_limb_t coeff_bits, mp_limb_t exp_bits, const fmpq_mpoly_ctx_t ctx)

Generate a random polynomial with length up to the given length, exponents whose packed form does not exceed the given bit count, and with rational coefficients of the given number of bits.