fq_nmod_mpoly.h – multivariate polynomials over finite fields of word-sized characteristic¶
The exponents follow the
mpoly
interface. No references to the coefficients are available.
Types, macros and constants¶
-
type fq_nmod_mpoly_struct¶
A structure holding a multivariate polynomial over a finite field of word-sized characteristic.
-
type fq_nmod_mpoly_t¶
An array of length \(1\) of
fq_nmod_mpoly_struct
.
-
type fq_nmod_mpoly_ctx_struct¶
Context structure representing the parent ring of an
fq_nmod_mpoly
.
-
type fq_nmod_mpoly_ctx_t¶
An array of length \(1\) of
fq_nmod_mpoly_ctx_struct
.
Context object¶
-
void fq_nmod_mpoly_ctx_init(fq_nmod_mpoly_ctx_t ctx, slong nvars, const ordering_t ord, const fq_nmod_ctx_t fqctx)¶
Initialise a context object for a polynomial ring with the given number of variables and the given ordering. It will have coefficients in the finite field fqctx. The possibilities for the ordering are
ORD_LEX
,ORD_DEGLEX
andORD_DEGREVLEX
.
-
slong fq_nmod_mpoly_ctx_nvars(const fq_nmod_mpoly_ctx_t ctx)¶
Return the number of variables used to initialize the context.
-
ordering_t fq_nmod_mpoly_ctx_ord(const fq_nmod_mpoly_ctx_t ctx)¶
Return the ordering used to initialize the context.
-
void fq_nmod_mpoly_ctx_clear(fq_nmod_mpoly_ctx_t ctx)¶
Release any space allocated by an ctx.
Memory management¶
-
void fq_nmod_mpoly_init(fq_nmod_mpoly_t A, const fq_nmod_mpoly_ctx_t ctx)¶
Initialise A for use with the given an initialised context object. Its value is set to zero.
-
void fq_nmod_mpoly_init2(fq_nmod_mpoly_t A, slong alloc, const fq_nmod_mpoly_ctx_t 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.
-
void fq_nmod_mpoly_init3(fq_nmod_mpoly_t A, slong alloc, flint_bitcnt_t bits, const fq_nmod_mpoly_ctx_t 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.
-
void fq_nmod_mpoly_fit_length(fq_nmod_mpoly_t A, slong len, const fq_nmod_mpoly_ctx_t ctx)¶
Ensure that A has space for at least len terms.
-
void fq_nmod_mpoly_realloc(fq_nmod_mpoly_t A, slong alloc, const fq_nmod_mpoly_ctx_t ctx)¶
Reallocate A to have space for alloc terms. Assumes the current length of the polynomial is not greater than alloc.
-
void fq_nmod_mpoly_clear(fq_nmod_mpoly_t A, const fq_nmod_mpoly_ctx_t ctx)¶
Release any space allocated for A.
Input/Output¶
The variable strings in x start with the variable of most significance at index \(0\). If x is
NULL
, the variables are namedx1
,x2
, etc.
-
char *fq_nmod_mpoly_get_str_pretty(const fq_nmod_mpoly_t A, const char **x, const fq_nmod_mpoly_ctx_t ctx)¶
Return a string, which the user is responsible for cleaning up, representing A, given an array of variable strings x.
-
int fq_nmod_mpoly_fprint_pretty(FILE *file, const fq_nmod_mpoly_t A, const char **x, const fq_nmod_mpoly_ctx_t ctx)¶
Print a string representing A to file.
-
int fq_nmod_mpoly_print_pretty(const fq_nmod_mpoly_t A, const char **x, const fq_nmod_mpoly_ctx_t ctx)¶
Print a string representing A to
stdout
.
-
int fq_nmod_mpoly_set_str_pretty(fq_nmod_mpoly_t A, const char *str, const char **x, const fq_nmod_mpoly_ctx_t 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.
Basic manipulation¶
-
void fq_nmod_mpoly_gen(fq_nmod_mpoly_t A, slong var, const fq_nmod_mpoly_ctx_t 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.
-
int fq_nmod_mpoly_is_gen(const fq_nmod_mpoly_t A, slong var, const fq_nmod_mpoly_ctx_t 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 \(0\).
-
void fq_nmod_mpoly_set(fq_nmod_mpoly_t A, const fq_nmod_mpoly_t B, const fq_nmod_mpoly_ctx_t ctx)¶
Set A to B.
-
int fq_nmod_mpoly_equal(const fq_nmod_mpoly_t A, const fq_nmod_mpoly_t B, const fq_nmod_mpoly_ctx_t ctx)¶
Return \(1\) if A is equal to B, else return \(0\).
-
void fq_nmod_mpoly_swap(fq_nmod_mpoly_t A, fq_nmod_mpoly_t B, const fq_nmod_mpoly_ctx_t ctx)¶
Efficiently swap A and B.
Constants¶
-
int fq_nmod_mpoly_is_fq_nmod(const fq_nmod_mpoly_t A, const fq_nmod_mpoly_ctx_t ctx)¶
Return \(1\) if A is a constant, else return \(0\).
-
void fq_nmod_mpoly_get_fq_nmod(fq_nmod_t c, const fq_nmod_mpoly_t A, const fq_nmod_mpoly_ctx_t ctx)¶
Assuming that A is a constant, set c to this constant. This function throws if A is not a constant.
-
void fq_nmod_mpoly_set_fq_nmod(fq_nmod_mpoly_t A, const fq_nmod_t c, const fq_nmod_mpoly_ctx_t ctx)¶
-
void fq_nmod_mpoly_set_ui(fq_nmod_mpoly_t A, ulong c, const fq_nmod_mpoly_ctx_t ctx)¶
Set A to the constant c.
-
void fq_nmod_mpoly_set_fq_nmod_gen(fq_nmod_mpoly_t A, const fq_nmod_mpoly_ctx_t ctx)¶
Set A to the constant given by
fq_nmod_gen()
.
-
void fq_nmod_mpoly_zero(fq_nmod_mpoly_t A, const fq_nmod_mpoly_ctx_t ctx)¶
Set A to the constant \(0\).
-
void fq_nmod_mpoly_one(fq_nmod_mpoly_t A, const fq_nmod_mpoly_ctx_t ctx)¶
Set A to the constant \(1\).
-
int fq_nmod_mpoly_equal_fq_nmod(const fq_nmod_mpoly_t A, const fq_nmod_t c, const fq_nmod_mpoly_ctx_t ctx)¶
Return \(1\) if A is equal to the constant c, else return \(0\).
-
int fq_nmod_mpoly_is_zero(const fq_nmod_mpoly_t A, const fq_nmod_mpoly_ctx_t ctx)¶
Return \(1\) if A is the constant \(0\), else return \(0\).
-
int fq_nmod_mpoly_is_one(const fq_nmod_mpoly_t A, const fq_nmod_mpoly_ctx_t ctx)¶
Return \(1\) if A is the constant \(1\), else return \(0\).
Degrees¶
-
int fq_nmod_mpoly_degrees_fit_si(const fq_nmod_mpoly_t A, const fq_nmod_mpoly_ctx_t ctx)¶
Return \(1\) if the degrees of A with respect to each variable fit into an
slong
, otherwise return \(0\).
-
void fq_nmod_mpoly_degrees_fmpz(fmpz **degs, const fq_nmod_mpoly_t A, const fq_nmod_mpoly_ctx_t ctx)¶
-
void fq_nmod_mpoly_degrees_si(slong *degs, const fq_nmod_mpoly_t A, const fq_nmod_mpoly_ctx_t ctx)¶
Set degs to the degrees of A with respect to each variable. If A is zero, all degrees are set to \(-1\).
-
void fq_nmod_mpoly_degree_fmpz(fmpz_t deg, const fq_nmod_mpoly_t A, slong var, const fq_nmod_mpoly_ctx_t ctx)¶
-
slong fq_nmod_mpoly_degree_si(const fq_nmod_mpoly_t A, slong var, const fq_nmod_mpoly_ctx_t 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 \(-1\).
-
int fq_nmod_mpoly_total_degree_fits_si(const fq_nmod_mpoly_t A, const fq_nmod_mpoly_ctx_t ctx)¶
Return \(1\) if the total degree of A fits into an
slong
, otherwise return \(0\).
-
void fq_nmod_mpoly_total_degree_fmpz(fmpz_t tdeg, const fq_nmod_mpoly_t A, const fq_nmod_mpoly_ctx_t ctx)¶
-
slong fq_nmod_mpoly_total_degree_si(const fq_nmod_mpoly_t A, const fq_nmod_mpoly_ctx_t ctx)¶
Either return or set tdeg to the total degree of A. If A is zero, the total degree is defined to be \(-1\).
-
void fq_nmod_mpoly_used_vars(int *used, const fq_nmod_mpoly_t A, const fq_nmod_mpoly_ctx_t ctx)¶
For each variable index \(i\), set
used[i]
to nonzero if the variable of index \(i\) appears in A and to zero otherwise.
Coefficients¶
-
void fq_nmod_mpoly_get_coeff_fq_nmod_monomial(fq_nmod_t c, const fq_nmod_mpoly_t A, const fq_nmod_mpoly_t M, const fq_nmod_mpoly_ctx_t 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.
-
void fq_nmod_mpoly_set_coeff_fq_nmod_monomial(fq_nmod_mpoly_t A, const fq_nmod_t c, const fq_nmod_mpoly_t M, const fq_nmod_mpoly_ctx_t 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.
-
void fq_nmod_mpoly_get_coeff_fq_nmod_fmpz(fq_nmod_t c, const fq_nmod_mpoly_t A, fmpz *const *exp, const fq_nmod_mpoly_ctx_t ctx)¶
-
void fq_nmod_mpoly_get_coeff_fq_nmod_ui(fq_nmod_t c, const fq_nmod_mpoly_t A, const ulong *exp, const fq_nmod_mpoly_ctx_t ctx)¶
Set c to the coefficient of the monomial with exponent vector exp.
-
void fq_nmod_mpoly_set_coeff_fq_nmod_fmpz(fq_nmod_mpoly_t A, const fq_nmod_t c, fmpz *const *exp, const fq_nmod_mpoly_ctx_t ctx)¶
-
void fq_nmod_mpoly_set_coeff_fq_nmod_ui(fq_nmod_mpoly_t A, const fq_nmod_t c, const ulong *exp, const fq_nmod_mpoly_ctx_t ctx)¶
Set the coefficient of the monomial with exponent exp to c.
-
void fq_nmod_mpoly_get_coeff_vars_ui(fq_nmod_mpoly_t C, const fq_nmod_mpoly_t A, const slong *vars, const ulong *exps, slong length, const fq_nmod_mpoly_ctx_t 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.
Comparison¶
-
int fq_nmod_mpoly_cmp(const fq_nmod_mpoly_t A, const fq_nmod_mpoly_t B, const fq_nmod_mpoly_ctx_t 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.
Container operations¶
These functions deal with violations of the internal canonical representation. If a term index is negative or not strictly less than the length of the polynomial, the function will throw.
-
int fq_nmod_mpoly_is_canonical(const fq_nmod_mpoly_t A, const fq_nmod_mpoly_ctx_t ctx)¶
Return \(1\) if A is in canonical form. Otherwise, return \(0\). To be in canonical form, all of the terms must have nonzero coefficients, and the terms must be sorted from greatest to least.
-
slong fq_nmod_mpoly_length(const fq_nmod_mpoly_t A, const fq_nmod_mpoly_ctx_t ctx)¶
Return the number of terms in A. If the polynomial is in canonical form, this will be the number of nonzero coefficients.
-
void fq_nmod_mpoly_resize(fq_nmod_mpoly_t A, slong new_length, const fq_nmod_mpoly_ctx_t ctx)¶
Set the length of A to
new_length
. Terms are either deleted from the end, or new zero terms are appended.
-
void fq_nmod_mpoly_get_term_coeff_fq_nmod(fq_nmod_t c, const fq_nmod_mpoly_t A, slong i, const fq_nmod_mpoly_ctx_t ctx)¶
Set c to the coefficient of the term of index i.
-
void fq_nmod_mpoly_set_term_coeff_ui(fq_nmod_mpoly_t A, slong i, ulong c, const fq_nmod_mpoly_ctx_t ctx)¶
Set the coefficient of the term of index i to c.
-
int fq_nmod_mpoly_term_exp_fits_si(const fq_nmod_mpoly_t A, slong i, const fq_nmod_mpoly_ctx_t ctx)¶
-
int fq_nmod_mpoly_term_exp_fits_ui(const fq_nmod_mpoly_t A, slong i, const fq_nmod_mpoly_ctx_t ctx)¶
Return \(1\) if all entries of the exponent vector of the term of index \(i\) fit into an
slong
(resp. aulong
). Otherwise, return \(0\).
-
void fq_nmod_mpoly_get_term_exp_fmpz(fmpz **exp, const fq_nmod_mpoly_t A, slong i, const fq_nmod_mpoly_ctx_t ctx)¶
-
void fq_nmod_mpoly_get_term_exp_ui(ulong *exp, const fq_nmod_mpoly_t A, slong i, const fq_nmod_mpoly_ctx_t ctx)¶
-
void fq_nmod_mpoly_get_term_exp_si(slong *exp, const fq_nmod_mpoly_t A, slong i, const fq_nmod_mpoly_ctx_t 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 aulong
(resp.slong
).
-
ulong fq_nmod_mpoly_get_term_var_exp_ui(const fq_nmod_mpoly_t A, slong i, slong var, const fq_nmod_mpoly_ctx_t ctx)¶
-
slong fq_nmod_mpoly_get_term_var_exp_si(const fq_nmod_mpoly_t A, slong i, slong var, const fq_nmod_mpoly_ctx_t 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
).
-
void fq_nmod_mpoly_set_term_exp_fmpz(fq_nmod_mpoly_t A, slong i, fmpz *const *exp, const fq_nmod_mpoly_ctx_t ctx)¶
-
void fq_nmod_mpoly_set_term_exp_ui(fq_nmod_mpoly_t A, slong i, const ulong *exp, const fq_nmod_mpoly_ctx_t ctx)¶
Set the exponent of the term of index i to exp.
-
void fq_nmod_mpoly_get_term(fq_nmod_mpoly_t M, const fq_nmod_mpoly_t A, slong i, const fq_nmod_mpoly_ctx_t ctx)¶
Set M to the term of index i in A.
-
void fq_nmod_mpoly_get_term_monomial(fq_nmod_mpoly_t M, const fq_nmod_mpoly_t A, slong i, const fq_nmod_mpoly_ctx_t ctx)¶
Set M to the monomial of the term of index i in A. The coefficient of M will be one.
-
void fq_nmod_mpoly_push_term_fq_nmod_fmpz(fq_nmod_mpoly_t A, const fq_nmod_t c, fmpz *const *exp, const fq_nmod_mpoly_ctx_t ctx)¶
-
void fq_nmod_mpoly_push_term_fq_nmod_ffmpz(fq_nmod_mpoly_t A, const fq_nmod_t c, const fmpz *exp, const fq_nmod_mpoly_ctx_t ctx)¶
-
void fq_nmod_mpoly_push_term_fq_nmod_ui(fq_nmod_mpoly_t A, const fq_nmod_t c, const ulong *exp, const fq_nmod_mpoly_ctx_t ctx)¶
Append a term to A with coefficient c and exponent vector exp. This function runs in constant average time.
-
void fq_nmod_mpoly_sort_terms(fq_nmod_mpoly_t A, const fq_nmod_mpoly_ctx_t 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 bit size of A.
-
void fq_nmod_mpoly_combine_like_terms(fq_nmod_mpoly_t A, const fq_nmod_mpoly_ctx_t 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 bit size of A.
-
void fq_nmod_mpoly_reverse(fq_nmod_mpoly_t A, const fq_nmod_mpoly_t B, const fq_nmod_mpoly_ctx_t ctx)¶
Set A to the reversal of B.
Random generation¶
-
void fq_nmod_mpoly_randtest_bound(fq_nmod_mpoly_t A, flint_rand_t state, slong length, ulong exp_bound, const fq_nmod_mpoly_ctx_t 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 ton_randint(state, exp_bound)
.
-
void fq_nmod_mpoly_randtest_bounds(fq_nmod_mpoly_t A, flint_rand_t state, slong length, ulong *exp_bounds, const fq_nmod_mpoly_ctx_t 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 ton_randint(state, exp_bounds[i])
.
-
void fq_nmod_mpoly_randtest_bits(fq_nmod_mpoly_t A, flint_rand_t state, slong length, ulong exp_bits, const fq_nmod_mpoly_ctx_t ctx)¶
Generate a random polynomial with length up to length and exponents whose packed form does not exceed the given bit count.
Addition/Subtraction¶
-
void fq_nmod_mpoly_add_fq_nmod(fq_nmod_mpoly_t A, const fq_nmod_mpoly_t B, const fq_nmod_t C, const fq_nmod_mpoly_ctx_t ctx)¶
Set A to \(B + c\).
-
void fq_nmod_mpoly_sub_fq_nmod(fq_nmod_mpoly_t A, const fq_nmod_mpoly_t B, const fq_nmod_t C, const fq_nmod_mpoly_ctx_t ctx)¶
Set A to \(B - c\).
-
void fq_nmod_mpoly_add(fq_nmod_mpoly_t A, const fq_nmod_mpoly_t B, const fq_nmod_mpoly_t C, const fq_nmod_mpoly_ctx_t ctx)¶
Set A to \(B + C\).
-
void fq_nmod_mpoly_sub(fq_nmod_mpoly_t A, const fq_nmod_mpoly_t B, const fq_nmod_mpoly_t C, const fq_nmod_mpoly_ctx_t ctx)¶
Set A to \(B - C\).
Scalar operations¶
-
void fq_nmod_mpoly_neg(fq_nmod_mpoly_t A, const fq_nmod_mpoly_t B, const fq_nmod_mpoly_ctx_t ctx)¶
Set A to \(-B\).
-
void fq_nmod_mpoly_scalar_mul_fq_nmod(fq_nmod_mpoly_t A, const fq_nmod_mpoly_t B, const fq_nmod_t c, const fq_nmod_mpoly_ctx_t ctx)¶
Set A to \(B \times c\).
-
void fq_nmod_mpoly_make_monic(fq_nmod_mpoly_t A, const fq_nmod_mpoly_t B, const fq_nmod_mpoly_ctx_t ctx)¶
Set A to B divided by the leading coefficient of B. This throws if B is zero.
Differentiation¶
-
void fq_nmod_mpoly_derivative(fq_nmod_mpoly_t A, const fq_nmod_mpoly_t B, slong var, const fq_nmod_mpoly_ctx_t ctx)¶
Set A to the derivative of B with respect to the variable of index var.
Evaluation¶
These functions return \(0\) when the operation would imply unreasonable arithmetic.
-
void fq_nmod_mpoly_evaluate_all_fq_nmod(fq_nmod_t ev, const fq_nmod_mpoly_t A, fq_nmod_struct *const *vals, const fq_nmod_mpoly_ctx_t ctx)¶
Set ev the evaluation of A where the variables are replaced by the corresponding elements of the array vals.
-
void fq_nmod_mpoly_evaluate_one_fq_nmod(fq_nmod_mpoly_t A, const fq_nmod_mpoly_t B, slong var, const fq_nmod_t val, const fq_nmod_mpoly_ctx_t ctx)¶
Set A to the evaluation of B where the variable of index var is replaced by val.
-
int fq_nmod_mpoly_compose_fq_nmod_poly(fq_nmod_poly_t A, const fq_nmod_mpoly_t B, fq_nmod_poly_struct *const *C, const fq_nmod_mpoly_ctx_t ctx)¶
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.
-
int fq_nmod_mpoly_compose_fq_nmod_mpoly(fq_nmod_mpoly_t A, const fq_nmod_mpoly_t B, fq_nmod_mpoly_struct *const *C, const fq_nmod_mpoly_ctx_t ctxB, const fq_nmod_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 A nor B is allowed to alias any other polynomial. Return \(1\) for success and \(0\) for failure.
-
void fq_nmod_mpoly_compose_fq_nmod_mpoly_gen(fq_nmod_mpoly_t A, const fq_nmod_mpoly_t B, const slong *c, const fq_nmod_mpoly_ctx_t ctxB, const fq_nmod_mpoly_ctx_t 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 anyc[i]
is negative, the corresponding variable of B is replaced by zero. Otherwise, it is expected thatc[i]
is less than the number of variables in ctxAC.
Multiplication¶
-
void fq_nmod_mpoly_mul(fq_nmod_mpoly_t A, const fq_nmod_mpoly_t B, const fq_nmod_mpoly_t C, const fq_nmod_mpoly_ctx_t ctx)¶
Set A to B times C.
Powering¶
These functions return \(0\) when the operation would imply unreasonable arithmetic.
-
int fq_nmod_mpoly_pow_fmpz(fq_nmod_mpoly_t A, const fq_nmod_mpoly_t B, const fmpz_t k, const fq_nmod_mpoly_ctx_t ctx)¶
Set A to \(B\) raised to the k-th power. Return \(1\) for success and \(0\) for failure.
-
int fq_nmod_mpoly_pow_ui(fq_nmod_mpoly_t A, const fq_nmod_mpoly_t B, ulong k, const fq_nmod_mpoly_ctx_t ctx)¶
Set A to \(B\) raised to the k-th power. Return \(1\) for success and \(0\) for failure.
Division¶
-
int fq_nmod_mpoly_divides(fq_nmod_mpoly_t Q, const fq_nmod_mpoly_t A, const fq_nmod_mpoly_t B, const fq_nmod_mpoly_ctx_t ctx)¶
If A is divisible by B, set Q to the exact quotient and return \(1\). Otherwise, set Q to zero and return \(0\).
-
void fq_nmod_mpoly_div(fq_nmod_mpoly_t Q, const fq_nmod_mpoly_t A, const fq_nmod_mpoly_t B, const fq_nmod_mpoly_ctx_t ctx)¶
Set Q to the quotient of A by B, discarding the remainder.
-
void fq_nmod_mpoly_divrem(fq_nmod_mpoly_t Q, fq_nmod_mpoly_t R, const fq_nmod_mpoly_t A, const fq_nmod_mpoly_t B, const fq_nmod_mpoly_ctx_t ctx)¶
Set Q and R to the quotient and remainder of A divided by B.
-
void fq_nmod_mpoly_divrem_ideal(fq_nmod_mpoly_struct **Q, fq_nmod_mpoly_t R, const fq_nmod_mpoly_t A, fq_nmod_mpoly_struct *const *B, slong len, const fq_nmod_mpoly_ctx_t ctx)¶
This function is as per
fq_nmod_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.
Greatest Common Divisor¶
-
void fq_nmod_mpoly_term_content(fq_nmod_mpoly_t M, const fq_nmod_mpoly_t A, const fq_nmod_mpoly_ctx_t 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.
-
int fq_nmod_mpoly_content_vars(fq_nmod_mpoly_t g, const fq_nmod_mpoly_t A, slong *vars, slong vars_length, const fq_nmod_mpoly_ctx_t 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.
-
int fq_nmod_mpoly_gcd(fq_nmod_mpoly_t G, const fq_nmod_mpoly_t A, const fq_nmod_mpoly_t B, const fq_nmod_mpoly_ctx_t 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.
-
int fq_nmod_mpoly_gcd_cofactors(fq_nmod_mpoly_t G, fq_nmod_mpoly_t Abar, fq_nmod_mpoly_t Bbar, const fq_nmod_mpoly_t A, const fq_nmod_mpoly_t B, const fq_nmod_mpoly_ctx_t ctx)¶
Do the operation of
fq_nmod_mpoly_gcd()
and also compute \(Abar = A/G\) and \(Bbar = B/G\) if successful.
-
int fq_nmod_mpoly_gcd_brown(fq_nmod_mpoly_t G, const fq_nmod_mpoly_t A, const fq_nmod_mpoly_t B, const fq_nmod_mpoly_ctx_t ctx)¶
-
int fq_nmod_mpoly_gcd_hensel(fq_nmod_mpoly_t G, const fq_nmod_mpoly_t A, const fq_nmod_mpoly_t B, const fq_nmod_mpoly_ctx_t ctx)¶
-
int fq_nmod_mpoly_gcd_zippel(fq_nmod_mpoly_t G, const fq_nmod_mpoly_t A, const fq_nmod_mpoly_t B, const fq_nmod_mpoly_ctx_t ctx)¶
Try to set G to the GCD of A and B using various algorithms.
-
int fq_nmod_mpoly_resultant(fq_nmod_mpoly_t R, const fq_nmod_mpoly_t A, const fq_nmod_mpoly_t B, slong var, const fq_nmod_mpoly_ctx_t ctx)¶
Try to set R to the resultant of A and B with respect to the variable of index var.
-
int fq_nmod_mpoly_discriminant(fq_nmod_mpoly_t D, const fq_nmod_mpoly_t A, slong var, const fq_nmod_mpoly_ctx_t ctx)¶
Try to set D to the discriminant of A with respect to the variable of index var.
Square Root¶
-
int fq_nmod_mpoly_sqrt(fq_nmod_mpoly_t Q, const fq_nmod_mpoly_t A, const fq_nmod_mpoly_ctx_t ctx)¶
If \(Q^2=A\) has a solution, set \(Q\) to a solution and return \(1\), otherwise return \(0\) and set \(Q\) to zero.
-
int fq_nmod_mpoly_is_square(const fq_nmod_mpoly_t A, const fq_nmod_mpoly_ctx_t ctx)¶
Return \(1\) if A is a perfect square, otherwise return \(0\).
-
int fq_nmod_mpoly_quadratic_root(fq_nmod_mpoly_t Q, const fq_nmod_mpoly_t A, const fq_nmod_mpoly_t B, const fq_nmod_mpoly_ctx_t ctx)¶
If \(Q^2+AQ=B\) has a solution, set \(Q\) to a solution and return \(1\), otherwise return \(0\).
Univariate Functions¶
An
fq_nmod_mpoly_univar_t
holds a univariate polynomial in some main variable withfq_nmod_mpoly_t
coefficients in the remaining variables. These functions are useful when one wants to rewrite an element of \(\mathbb{F}_q[x_1, \dots, x_m]\) as an element of \((\mathbb{F}_q[x_1, \dots, x_{v-1}, x_{v+1}, \dots, x_m])[x_v]\) and vice versa.
-
void fq_nmod_mpoly_univar_init(fq_nmod_mpoly_univar_t A, const fq_nmod_mpoly_ctx_t ctx)¶
Initialize A.
-
void fq_nmod_mpoly_univar_clear(fq_nmod_mpoly_univar_t A, const fq_nmod_mpoly_ctx_t ctx)¶
Clear A.
-
void fq_nmod_mpoly_univar_swap(fq_nmod_mpoly_univar_t A, fq_nmod_mpoly_univar_t B, const fq_nmod_mpoly_ctx_t ctx)¶
Swap A and \(B\).
-
void fq_nmod_mpoly_to_univar(fq_nmod_mpoly_univar_t A, const fq_nmod_mpoly_t B, slong var, const fq_nmod_mpoly_ctx_t 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.
-
void fq_nmod_mpoly_from_univar(fq_nmod_mpoly_t A, const fq_nmod_mpoly_univar_t B, slong var, const fq_nmod_mpoly_ctx_t 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.
-
int fq_nmod_mpoly_univar_degree_fits_si(const fq_nmod_mpoly_univar_t A, const fq_nmod_mpoly_ctx_t ctx)¶
Return \(1\) if the degree of A with respect to the main variable fits an
slong
. Otherwise, return \(0\).
-
slong fq_nmod_mpoly_univar_length(const fq_nmod_mpoly_univar_t A, const fq_nmod_mpoly_ctx_t ctx)¶
Return the number of terms in A with respect to the main variable.
-
slong fq_nmod_mpoly_univar_get_term_exp_si(fq_nmod_mpoly_univar_t A, slong i, const fq_nmod_mpoly_ctx_t ctx)¶
Return the exponent of the term of index i of A.
-
void fq_nmod_mpoly_univar_get_term_coeff(fq_nmod_mpoly_t c, const fq_nmod_mpoly_univar_t A, slong i, const fq_nmod_mpoly_ctx_t ctx)¶
-
void fq_nmod_mpoly_univar_swap_term_coeff(fq_nmod_mpoly_t c, fq_nmod_mpoly_univar_t A, slong i, const fq_nmod_mpoly_ctx_t ctx)¶
Set (resp. swap) c to (resp. with) the coefficient of the term of index i of A.