newFmpzMPoly ctx

Construct a new FmpzMPoly with context ctx.

withNewFmpzMPoly ctx

Execute computation f on a new FmpzMPoly with context ctx.

Create a new FmpzMPolyCtx

Use a FmpzMPolyCtx

fmpz_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.

fmpz_mpoly_ctx_nvars ctx

Return the number of variables used to initialize the context.

fmpz_mpoly_ctx_ord ctx

Return the ordering used to initialize the context.

fmpz_mpoly_ctx_clear ctx

Release up any space allocated by ctx.

fmpz_mpoly_init A ctx

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

fmpz_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.

fmpz_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.

fmpz_mpoly_fit_length A len ctx

Ensure that A has space for at least len terms.

fmpz_mpoly_fit_bits A bits ctx

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

fmpz_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.

fmpz_mpoly_clear A ctx

Release any space allocated for A.

fmpz_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_mpoly_fprint_pretty file A x ctx

Print a string representing A to file.

fmpz_mpoly_print_pretty A x ctx

Print a string representing A to stdout. foreign import ccall "fmpz_mpoly.h fmpz_mpoly_print_pretty" fmpz_mpoly_print_pretty :: Ptr CFmpzMPoly -> Ptr (Ptr CChar) -> Ptr CFmpzMPolyCtx -> IO CInt

fmpz_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_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_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_mpoly_set A B ctx

Set A to B.

fmpz_mpoly_equal A B ctx

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

fmpz_mpoly_swap poly1 poly2 ctx

Efficiently swap A and B.

_fmpz_mpoly_fits_small poly 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.

fmpz_mpoly_max_bits 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.

fmpz_mpoly_is_fmpz A ctx

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

fmpz_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_mpoly_set_fmpz A c ctx

Set A to the constant c.

fmpz_mpoly_zero A ctx

Set A to the constant \(0\).

fmpz_mpoly_one A ctx

Set A to the constant \(1\).

fmpz_mpoly_equal_fmpz A c ctx

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

fmpz_mpoly_is_zero A ctx

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

fmpz_mpoly_is_one A ctx

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

fmpz_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_mpoly_degrees_fmpz 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_mpoly_degree_fmpz deg 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_mpoly_total_degree_fits_si A ctx

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

fmpz_mpoly_total_degree_fmpz tdeg 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_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_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_mpoly_set_coeff_fmpz_monomial poly c poly2 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_mpoly_get_coeff_fmpz_fmpz c A exp ctx

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

fmpz_mpoly_set_coeff_fmpz_fmpz A c exp ctx

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

fmpz_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_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_mpoly_is_fmpz_poly A var ctx

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

fmpz_mpoly_get_fmpz_poly A B var ctx

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

fmpz_mpoly_set_fmpz_poly A B var ctx

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

fmpz_mpoly_term_coeff_ref A i ctx

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

fmpz_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_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_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_mpoly_get_term_coeff_fmpz c A i ctx

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

fmpz_mpoly_set_term_coeff_fmpz A i c ctx

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

fmpz_mpoly_term_exp_fits_si 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_mpoly_get_term_exp_fmpz 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_mpoly_get_term_var_exp_ui 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_mpoly_set_term_exp_fmpz A i exp ctx

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

fmpz_mpoly_get_term M A i ctx

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

fmpz_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_mpoly_push_term_fmpz_fmpz A c exp ctx

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

fmpz_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_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_mpoly_reverse A B ctx

Set A to the reversal of B.

fmpz_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).

fmpz_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]).

fmpz_mpoly_randtest_bits A state length coeff_bits exp_bits 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.

fmpz_mpoly_add_fmpz A B c ctx

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

fmpz_mpoly_sub_fmpz A B c ctx

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

fmpz_mpoly_add A B C ctx

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

fmpz_mpoly_sub A B C ctx

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

fmpz_mpoly_neg A B ctx

Set A to \(-B\).

fmpz_mpoly_scalar_mul_fmpz A B c ctx

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

fmpz_mpoly_scalar_fmma A B c D e ctx

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

fmpz_mpoly_scalar_divexact_fmpz A B c ctx

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

fmpz_mpoly_scalar_divides_fmpz A B c ctx

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

fmpz_mpoly_derivative A B var ctx

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

fmpz_mpoly_integral A scale B var 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.

fmpz_mpoly_evaluate_all_fmpz 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.

fmpz_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_mpoly_compose_fmpz_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.

fmpz_mpoly_compose_fmpz_mpoly_geobucket 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_mpoly_compose_fmpz_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.

fmpz_mpoly_mul A B C ctx

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

fmpz_mpoly_mul_johnson A B C ctx

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

fmpz_mpoly_mul_array A B C 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.

fmpz_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_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_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_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_mpoly_divrem Q R A B 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.

fmpz_mpoly_quasidivrem scale Q R A B 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.

fmpz_mpoly_div Q A B 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.

fmpz_mpoly_quasidiv scale Q A B ctx

Perform the operation of fmpz_mpoly_quasidivrem and discard R.

fmpz_mpoly_divrem_ideal Q R A B len 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.

fmpz_mpoly_quasidivrem_ideal scale Q R A B len 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.

fmpz_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 positive coefficient.

fmpz_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_mpoly_gcd G A B 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.

fmpz_mpoly_gcd_cofactors G Abar Bbar A B ctx

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

fmpz_mpoly_gcd_brown G A B ctx

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

fmpz_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_mpoly_discriminant D A var ctx

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

fmpz_mpoly_primitive_part res f 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.

fmpz_mpoly_sqrt_heap Q A ctx 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.

fmpz_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.

fmpz_mpoly_is_square A ctx

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

fmpz_mpoly_univar_init A ctx

Initialize A.

fmpz_mpoly_univar_clear A ctx

Clear A.

fmpz_mpoly_univar_swap A B ctx

Swap A and B.

fmpz_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_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_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_mpoly_univar_length A ctx

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

fmpz_mpoly_univar_get_term_exp_si A i ctx

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

fmpz_mpoly_univar_get_term_coeff c A i ctx

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

fmpz_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_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_mpoly_inflate when possible.

fmpz_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.

fmpz_mpoly_pow_fps A B k 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 \(k \geq 2\).

_fmpz_mpoly_divides_array poly1 exp1 alloc poly2 exp2 len2 poly3 exp3 len3 mults num 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.

fmpz_mpoly_divides_array poly1 poly2 poly3 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.

_fmpz_mpoly_divides_monagan_pearce poly1 exp1 alloc poly2 exp2 len2 poly3 exp3 len3 bits N

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.

fmpz_mpoly_divides_heap_threaded Q A B ctx thread_limit

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.

The threaded version takes an upper limit on the number of threads to use, while the first version always uses one thread.

_fmpz_mpoly_div_monagan_pearce polyq expq allocq poly2 exp2 len2 poly3 exp3 len3 bits N

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.

fmpz_mpoly_div_monagan_pearce polyq poly2 poly3 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.

_fmpz_mpoly_divrem_monagan_pearce lenr polyq expq allocq polyr expr allocr poly2 exp2 len2 poly3 exp3 len3 bits N

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.

fmpz_mpoly_divrem_monagan_pearce q r poly2 poly3 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.

_fmpz_mpoly_divrem_array lenr polyq expq allocq polyr expr allocr poly2 exp2 len2 poly3 exp3 len3 mults num 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.

fmpz_mpoly_divrem_array q r poly2 poly3 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.

fmpz_mpoly_quasidivrem_heap scale q r poly2 poly3 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.

_fmpz_mpoly_divrem_ideal_monagan_pearce polyq polyr expr allocr poly2 exp2 len2 poly3 exp3 len N bits ctx

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.

fmpz_mpoly_divrem_ideal_monagan_pearce q r poly2 poly3 len 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].

fmpz_mpoly_vec_init vec len ctx

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

fmpz_mpoly_vec_clear vec ctx

Clears vec, freeing its allocated memory.

fmpz_mpoly_vec_print vec ctx

Prints vec to standard output.

fmpz_mpoly_vec_swap x y ctx

Swaps x and y efficiently.

fmpz_mpoly_vec_fit_length vec len ctx

Allocates room for len entries in vec.

fmpz_mpoly_vec_set dest src ctx

Sets dest to a copy of src.

fmpz_mpoly_vec_append vec f ctx

Appends f to the end of vec.

fmpz_mpoly_vec_insert_unique vec f 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.

fmpz_mpoly_vec_set_length vec len ctx

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

fmpz_mpoly_vec_randtest_not_zero vec state len poly_len bits exp_bound ctx

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

fmpz_mpoly_vec_set_primitive_unique res src 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.

fmpz_mpoly_spoly res f g ctx

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

fmpz_mpoly_reduction_primitive_part res f vec ctx

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

fmpz_mpoly_vec_is_groebner G F 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.

fmpz_mpoly_vec_is_autoreduced F ctx

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

fmpz_mpoly_vec_autoreduction H F ctx

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

fmpz_mpoly_vec_autoreduction_groebner H G 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).

fmpz_mpoly_buchberger_naive G F ctx

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

fmpz_mpoly_buchberger_naive_with_limits G F ideal_len_limit poly_len_limit poly_bits_limit 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.

fmpz_mpoly_symmetric res k 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.