Create a new FmpzModMPoly

Create a new FmpzModMPolyCtx

Use a FmpzModMPolyCtx

fmpz_mod_mpoly_ctx_init ctx nvars ord p

Initialise a context object for a polynomial ring modulo n with nvars variables and ordering ord. The possibilities for the ordering are ORD_LEX, ORD_DEGLEX and ORD_DEGREVLEX.

fmpz_mod_mpoly_ctx_nvars ctx

Return the number of variables used to initialize the context.

fmpz_mod_mpoly_ctx_ord ctx

Return the ordering used to initialize the context.

fmpz_mod_mpoly_ctx_get_modulus n ctx

Set n to the modulus used to initialize the context.

fmpz_mod_mpoly_ctx_clear ctx

Release up any space allocated by an ctx.

fmpz_mod_mpoly_init A ctx

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

fmpz_mod_mpoly_init2 A alloc ctx

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

fmpz_mod_mpoly_init3 A alloc bits ctx

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

fmpz_mod_mpoly_clear A ctx

Release any space allocated for A.

fmpz_mod_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.

fmpz_mod_mpoly_fprint_pretty file A x ctx

Print a string representing A to file.

fmpz_mod_mpoly_print_pretty A x ctx

Print a string representing A to stdout.

fmpz_mod_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.

fmpz_mod_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.

fmpz_mod_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 \(0\).

fmpz_mod_mpoly_set A B ctx

Set A to B.

fmpz_mod_mpoly_equal A B ctx

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

fmpz_mod_mpoly_swap poly1 poly2 ctx

Efficiently swap A and B.

fmpz_mod_mpoly_is_fmpz A ctx

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

fmpz_mod_mpoly_get_fmpz c A ctx

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

fmpz_mod_mpoly_set_fmpz A c ctx

fmpz_mod_mpoly_set_ui A c ctx

fmpz_mod_mpoly_set_si A c ctx

Set A to the constant c.

fmpz_mod_mpoly_zero A ctx

Set A to the constant \(0\).

fmpz_mod_mpoly_one A ctx

Set A to the constant \(1\).

fmpz_mod_mpoly_equal_fmpz A c ctx

fmpz_mod_mpoly_equal_ui A c ctx

fmpz_mod_mpoly_equal_si A c ctx

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

fmpz_mod_mpoly_is_zero A ctx

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

fmpz_mod_mpoly_is_one A ctx

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

fmpz_mod_mpoly_degrees_fit_si A ctx

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

fmpz_mod_mpoly_degrees_fmpz degs A ctx

fmpz_mod_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 \(-1\).

fmpz_mod_mpoly_degree_fmpz deg A var ctx

fmpz_mod_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 \(-1\).

fmpz_mod_mpoly_total_degree_fits_si A ctx

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

fmpz_mod_mpoly_total_degree_fmpz tdeg A ctx

fmpz_mod_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 \(-1\).

fmpz_mod_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.

fmpz_mod_mpoly_get_coeff_fmpz_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.

fmpz_mod_mpoly_set_coeff_fmpz_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.

fmpz_mod_mpoly_get_coeff_fmpz_fmpz c A exp ctx

fmpz_mod_mpoly_get_coeff_fmpz_ui c A exp ctx

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

fmpz_mod_mpoly_set_coeff_fmpz_fmpz A c exp ctx

fmpz_mod_mpoly_set_coeff_ui_fmpz A c exp ctx

fmpz_mod_mpoly_set_coeff_si_fmpz A c exp ctx

fmpz_mod_mpoly_set_coeff_fmpz_ui A c exp ctx

fmpz_mod_mpoly_set_coeff_ui_ui A c exp ctx

fmpz_mod_mpoly_set_coeff_si_ui A c exp ctx

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

fmpz_mod_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.

fmpz_mod_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.

fmpz_mod_mpoly_is_canonical A 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.

fmpz_mod_mpoly_length A ctx

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

fmpz_mod_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.

fmpz_mod_mpoly_get_term_coeff_fmpz c A i ctx

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

fmpz_mod_mpoly_set_term_coeff_fmpz A i c ctx

fmpz_mod_mpoly_set_term_coeff_ui A i c ctx

fmpz_mod_mpoly_set_term_coeff_si A i c ctx

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

fmpz_mod_mpoly_term_exp_fits_si poly i ctx

fmpz_mod_mpoly_term_exp_fits_ui poly 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 \(0\).

fmpz_mod_mpoly_get_term_exp_fmpz exp A i ctx

fmpz_mod_mpoly_get_term_exp_ui exp A i ctx

fmpz_mod_mpoly_get_term_exp_si exp 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).

fmpz_mod_mpoly_get_term_var_exp_ui A i var ctx

fmpz_mod_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).

fmpz_mod_mpoly_set_term_exp_fmpz A i exp ctx

fmpz_mod_mpoly_set_term_exp_ui A i exp ctx

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

fmpz_mod_mpoly_get_term M A i ctx

Set M to the term of index i in A.

fmpz_mod_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.

fmpz_mod_mpoly_push_term_fmpz_fmpz A c exp ctx

fmpz_mod_mpoly_push_term_ui_fmpz A c exp ctx

fmpz_mod_mpoly_push_term_si_fmpz A c exp ctx

fmpz_mod_mpoly_push_term_fmpz_ui A c exp ctx

fmpz_mod_mpoly_push_term_ui_ui A c exp ctx

fmpz_mod_mpoly_push_term_si_ui A c exp ctx

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

fmpz_mod_mpoly_sort_terms A 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.

fmpz_mod_mpoly_combine_like_terms A 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.

fmpz_mod_mpoly_randtest_bound A state length 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).

fmpz_mod_mpoly_randtest_bounds A state length 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]).

fmpz_mod_mpoly_randtest_bits A state length exp_bits ctx

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

fmpz_mod_mpoly_add_fmpz A B c ctx

fmpz_mod_mpoly_add_ui A B c ctx

fmpz_mod_mpoly_add_si A B c ctx

Set A to \(B + c\).

fmpz_mod_mpoly_sub_fmpz A B c ctx

fmpz_mod_mpoly_sub_ui A B c ctx

fmpz_mod_mpoly_sub_si A B c ctx

Set A to \(B - c\).

fmpz_mod_mpoly_add A B C ctx

Set A to \(B + C\).

fmpz_mod_mpoly_sub A B C ctx

Set A to \(B - C\).

fmpz_mod_mpoly_neg A B ctx

Set A to \(-B\).

fmpz_mod_mpoly_scalar_mul_fmpz A B c ctx

fmpz_mod_mpoly_scalar_mul_ui A B c ctx

fmpz_mod_mpoly_scalar_mul_si A B c ctx

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

fmpz_mod_mpoly_scalar_addmul_fmpz A B C d ctx

Sets A to \(B + C \times d\).

fmpz_mod_mpoly_make_monic A B ctx

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

fmpz_mod_mpoly_derivative A B var ctx

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

fmpz_mod_mpoly_evaluate_all_fmpz eval A vals ctx

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

fmpz_mod_mpoly_evaluate_one_fmpz 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.

fmpz_mod_mpoly_compose_fmpz_mod_mpoly_geobucket A B C ctxB ctxAC

fmpz_mod_mpoly_compose_fmpz_mod_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. 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.

fmpz_mod_mpoly_mul A B C ctx

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

fmpz_mod_mpoly_mul_johnson A B C ctx

Set A to \(B \times C\) using Johnson's heap-based method.

fmpz_mod_mpoly_mul_dense A B C 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 \(1\).

fmpz_mod_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.

fmpz_mod_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.

fmpz_mod_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 \(0\).

fmpz_mod_mpoly_div Q A B ctx

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

fmpz_mod_mpoly_divrem Q R A B ctx

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

fmpz_mod_mpoly_divrem_ideal Q R A B len ctx

This function is as per fmpz_mod_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.

fmpz_mod_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.

fmpz_mod_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.

fmpz_mod_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.

fmpz_mod_mpoly_gcd_cofactors G Abar Bbar A B ctx

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

fmpz_mod_mpoly_gcd_brown G A B ctx

fmpz_mod_mpoly_gcd_hensel G A B ctx

fmpz_mod_mpoly_gcd_subresultant G A B ctx

fmpz_mod_mpoly_gcd_zippel G A B ctx

fmpz_mod_mpoly_gcd_zippel2 G A B ctx

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

fmpz_mod_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.

fmpz_mod_mpoly_discriminant D A var ctx

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

fmpz_mod_mpoly_sqrt Q A ctx

If \(Q^2=A\) has a solution, set Q to a solution and return \(1\), otherwise return \(0\) and set Q to zero.

fmpz_mod_mpoly_is_square A ctx

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

fmpz_mod_mpoly_quadratic_root Q A B ctx

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

fmpz_mod_mpoly_univar_init A ctx

Initialize A.

fmpz_mod_mpoly_univar_clear A ctx

Clear A.

fmpz_mod_mpoly_univar_swap A B ctx

Swap A and B.

fmpz_mod_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.

fmpz_mod_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.

fmpz_mod_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 \(0\).

fmpz_mod_mpoly_univar_length A ctx

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

fmpz_mod_mpoly_univar_get_term_exp_si A i ctx

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

fmpz_mod_mpoly_univar_get_term_coeff c A i ctx

fmpz_mod_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.

fmpz_mod_mpoly_univar_set_coeff_ui Ax e c ctx

Set the coefficient of \(X^e\) in Ax to c.

fmpz_mod_mpoly_univar_resultant R Ax Bx ctx

Try to set R to the resultant of Ax and Bx.

fmpz_mod_mpoly_univar_discriminant D Ax ctx

Try to set D to the discriminant of Ax.

fmpz_mod_mpoly_inflate A B shift stride 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.

fmpz_mod_mpoly_deflate A B shift stride 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_mod_mpoly_inflate when possible.

fmpz_mod_mpoly_deflation shift stride A 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.