Create a new FmpqMPolyCtx

Use a FmpqMPolyCtx

fmpq_mpoly_ctx_init ctx nvars 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.

fmpq_mpoly_ctx_nvars ctx

Return the number of variables used to initialize the context.

fmpq_mpoly_ctx_ord ctx

Return the ordering used to initialize the context.

fmpq_mpoly_ctx_clear ctx

Release up any space allocated by ctx.

fmpq_mpoly_init A ctx

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

fmpq_mpoly_init2 A alloc 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.

fmpq_mpoly_init3 A alloc bits 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.

fmpq_mpoly_fit_length A len ctx

Ensure that A has space for at least len terms.

fmpq_mpoly_fit_bits A bits ctx

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

fmpq_mpoly_realloc A alloc ctx

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

fmpq_mpoly_clear A ctx

Release any space allocated for A.

fmpq_mpoly_get_str_pretty A x ctx

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

fmpq_mpoly_fprint_pretty file A x ctx

Print a string representing A to file.

fmpq_mpoly_print_pretty A x ctx

Print a string representing A to stdout. foreign import ccall "fmpq_mpoly.h fmpq_mpoly_print_pretty"

fmpq_mpoly_set_str_pretty A str x 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.

fmpq_mpoly_gen A var 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.

fmpq_mpoly_is_gen A var 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 \(0\).

fmpq_mpoly_set A B ctx

Set A to B.

fmpq_mpoly_equal A B ctx

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

fmpq_mpoly_swap A B ctx

Efficiently swap A and B.

fmpq_mpoly_is_fmpq A ctx

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

fmpq_mpoly_get_fmpq c A ctx

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

fmpq_mpoly_set_fmpq A c ctx

fmpq_mpoly_set_fmpz A c ctx

fmpq_mpoly_set_ui A c ctx

fmpq_mpoly_set_si A c ctx

Set A to the constant c.

fmpq_mpoly_zero A ctx

Set A to the constant \(0\).

fmpq_mpoly_one A ctx

Set A to the constant \(1\).

fmpq_mpoly_equal_fmpq A c ctx

fmpq_mpoly_equal_fmpz A c ctx

fmpq_mpoly_equal_ui A c ctx

fmpq_mpoly_equal_si A c ctx

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

fmpq_mpoly_is_zero A ctx

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

fmpq_mpoly_is_one A ctx

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

fmpq_mpoly_degrees_fit_si A ctx

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

fmpq_mpoly_degrees_fmpz degs A ctx

fmpq_mpoly_degrees_si degs A ctx

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

fmpq_mpoly_degree_fmpz deg A var ctx

fmpq_mpoly_degree_si A var 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 \(-1\).

fmpq_mpoly_total_degree_fits_si A ctx

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

fmpq_mpoly_total_degree_fmpz tdeg A ctx

fmpq_mpoly_total_degree_si A ctx

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

fmpq_mpoly_used_vars used A ctx

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

fmpq_mpoly_get_denominator d A ctx

Set d to the denominator of A, the smallest positive integer \(d\) such that \(d \times A\) has integer coefficients.

fmpq_mpoly_get_coeff_fmpq_monomial c A M 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.

fmpq_mpoly_set_coeff_fmpq_monomial A c M 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.

fmpq_mpoly_get_coeff_fmpq_fmpz c A exp ctx

fmpq_mpoly_get_coeff_fmpq_ui c A exp ctx

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

fmpq_mpoly_set_coeff_fmpq_fmpz A c exp ctx

fmpq_mpoly_set_coeff_fmpq_ui A c exp ctx

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

fmpq_mpoly_get_coeff_vars_ui C A vars exps length 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.

fmpq_mpoly_cmp A B 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.

fmpq_mpoly_content_ref A ctx

Return a reference to the content of A.

fmpq_mpoly_zpoly_ref A ctx

Return a reference to the integer polynomial of A.

fmpq_mpoly_zpoly_term_coeff_ref A i ctx

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

fmpq_mpoly_is_canonical A ctx

Return \(1\) if A is in canonical form. Otherwise, return \(0\). An fmpq_mpoly_t is represented as the product of an fmpq_t content and an fmpz_mpoly_t zpoly. The representation is considered canonical when either (1) both content and zpoly are zero, or (2) both content and zpoly are nonzero and canonical and zpoly is reduced. A nonzero zpoly is considered reduced when the coefficients have GCD one and the leading coefficient is positive.

fmpq_mpoly_length A ctx

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

fmpq_mpoly_resize A new_length ctx

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

fmpq_mpoly_get_term_coeff_fmpq c A i ctx

Set c to coefficient of index i

fmpq_mpoly_set_term_coeff_fmpq A i c ctx

Set the coefficient of index i to c.

fmpq_mpoly_term_exp_fits_si A i ctx

fmpq_mpoly_term_exp_fits_ui A i 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 \(0\).

fmpq_mpoly_get_term_exp_fmpz exps A i ctx

fmpq_mpoly_get_term_exp_ui exps A i ctx

fmpq_mpoly_get_term_exp_si exps A i 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).

fmpq_mpoly_get_term_var_exp_ui A i var ctx

fmpq_mpoly_get_term_var_exp_si A i var 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).

fmpq_mpoly_set_term_exp_fmpz A i exps ctx

fmpq_mpoly_set_term_exp_ui A i exps ctx

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

fmpq_mpoly_get_term M A i ctx

Set M to the term of index i in A.

fmpq_mpoly_get_term_monomial M A i ctx

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

fmpq_mpoly_push_term_fmpq_fmpz A c exp ctx

fmpq_mpoly_push_term_fmpz_fmpz A c exp ctx

fmpq_mpoly_push_term_ui_fmpz A c exp ctx

fmpq_mpoly_push_term_si_fmpz A c exp ctx

fmpq_mpoly_push_term_fmpq_ui A c exp ctx

fmpq_mpoly_push_term_fmpz_ui A c exp ctx

fmpq_mpoly_push_term_ui_ui A c exp ctx

fmpq_mpoly_push_term_si_ui A c exp ctx

Append a term to A with coefficient c and exponent vector exp. This function should run in constant average time if the terms pushed have bounded denominator.

fmpq_mpoly_reduce A ctx

Factor out necessary content from A->zpoly so that it is reduced. If the terms of A were nonzero and sorted with distinct exponents to begin with, the result will be in canonical form.

fmpq_mpoly_sort_terms A ctx

Sort the internal A->zpoly 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, nor does it reduce.

fmpq_mpoly_combine_like_terms A ctx

Combine adjacent like terms in the internal A->zpoly and then factor out content via a call to fmpq_mpoly_reduce. If the terms of A were sorted to begin with, the result will be in canonical form.

fmpq_mpoly_randtest_bound A state length coeff_bits exp_bound 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).

fmpq_mpoly_randtest_bounds A state length coeff_bits exp_bounds 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]).

fmpq_mpoly_randtest_bits A state length coeff_bits exp_bits ctx

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

The parameter coeff_bits to the three functions fmpq_mpoly_randtest_{bound|bounds|bits} is merely a suggestion for the approximate bit count of the resulting coefficients.

fmpq_mpoly_add_fmpq A B c ctx

fmpq_mpoly_add_fmpz A B c ctx

fmpq_mpoly_add_ui A B c ctx

fmpq_mpoly_add_si A B c ctx

Set A to \(B + c\).

fmpq_mpoly_sub_fmpq A B c ctx

fmpq_mpoly_sub_fmpz A B c ctx

fmpq_mpoly_sub_ui A B c ctx

fmpq_mpoly_sub_si A B c ctx

Set A to \(B - c\).

fmpq_mpoly_add A B C ctx

Set A to \(B + C\).

fmpq_mpoly_sub A B C ctx

Set A to \(B - C\).

fmpq_mpoly_neg A B ctx

Set A to \(-B\).

fmpq_mpoly_scalar_mul_fmpq A B c ctx

fmpq_mpoly_scalar_mul_fmpz A B c ctx

fmpq_mpoly_scalar_mul_ui A B c ctx

fmpq_mpoly_scalar_mul_si A B c ctx

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

fmpq_mpoly_scalar_div_fmpq A B c ctx

fmpq_mpoly_scalar_div_fmpz A B c ctx

fmpq_mpoly_scalar_div_ui A B c ctx

fmpq_mpoly_scalar_div_si A B c ctx

Set A to \(B/c\).

fmpq_mpoly_make_monic A B ctx

Set A to B divided by the leading coefficient of B. This throws if B is zero.

All of these functions run quickly if A and B are aliased.

fmpq_mpoly_derivative A B var ctx

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

fmpq_mpoly_integral A B var ctx

Set A to the integral with the fewest number of terms of B with respect to the variable of index var.

fmpq_mpoly_evaluate_all_fmpq ev A vals 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.

fmpq_mpoly_evaluate_one_fmpq A B var val 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.

fmpq_mpoly_compose_fmpq_poly A B C 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.

fmpq_mpoly_compose_fmpq_mpoly A B C ctxB 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 A nor B is allowed to alias any other polynomial. Return \(1\) for success and \(0\) for failure.

fmpq_mpoly_compose_fmpq_mpoly_gen A B c ctxB 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.

fmpq_mpoly_mul A B C ctx

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

fmpq_mpoly_pow_fmpz A B k ctx

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

fmpq_mpoly_pow_ui A B k ctx

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

fmpq_mpoly_divides Q A B 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 fmpq_mpoly_div may be faster if the quotient is known to be exact.

fmpq_mpoly_div Q A B ctx

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

fmpq_mpoly_divrem Q R A B ctx

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

fmpq_mpoly_divrem_ideal Q R A B len ctx

This function is as per fmpq_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.

fmpq_mpoly_content g A ctx

Set g to the (nonnegative) gcd of the coefficients of A.

fmpq_mpoly_term_content M A 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.

fmpq_mpoly_content_vars g A vars vars_length 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.

fmpq_mpoly_gcd G A B 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.

fmpq_mpoly_gcd_cofactors G Abar Bbar A B ctx

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

fmpq_mpoly_gcd_brown G A B ctx

fmpq_mpoly_gcd_hensel G A B ctx

fmpq_mpoly_gcd_subresultant G A B ctx

fmpq_mpoly_gcd_zippel G A B ctx

fmpq_mpoly_gcd_zippel2 G A B ctx

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

fmpq_mpoly_resultant R A B var ctx

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

fmpq_mpoly_discriminant D A var ctx

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

fmpq_mpoly_sqrt Q A 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.

fmpq_mpoly_is_square A ctx

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

fmpq_mpoly_univar_init A ctx

Initialize A.

fmpq_mpoly_univar_clear A ctx

Clear A.

fmpq_mpoly_univar_swap A B ctx

Swap A and B.

fmpq_mpoly_to_univar A B var 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.

fmpq_mpoly_from_univar A B var 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.

fmpq_mpoly_univar_degree_fits_si A ctx

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

fmpq_mpoly_univar_length A ctx

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

fmpq_mpoly_univar_get_term_exp_si A i ctx

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

fmpq_mpoly_univar_get_term_coeff c A i ctx

fmpq_mpoly_univar_swap_term_coeff c A i ctx

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