fmpq_mpoly.h

Initialises a context for polynomials in nvars variables with the monomial ordering ord, one of ORD_LEX, ORD_DEGLEX and ORD_DEGREVLEX.

Initialises a context with a random number of variables between 0 and max_nvars and a random ordering, for testing.

Return the number of variables and the ordering of the context.

Frees the memory used by the context.

Initialise $A$ to zero, in the last two versions with room for alloc terms, and in the last with exponent fields of bits bits (otherwise at least MPOLY_MIN_BITS).

Ensure room for len terms, ensure exponent fields of at least bits bits (repacking the terms), and reallocate the integer part to exactly alloc terms (its length must not exceed alloc).

Frees the memory used by $A$.

Returns a string representation of $A$, to be freed with flint_free.

Print $A$ to file or to stdout, with coefficients such as 1/2*x^2.

Sets $A$ to the polynomial given by the null-terminated string str and returns 0, or returns $-1$ if parsing fails, in which case the value of $A$ is unspecified. The syntax is that of fmpz_mpoly_set_str_pretty: integers, the names in x, parentheses, +, -, *, / and ^ followed by a nonnegative integer; division by a polynomial must be exact, and division by integers gives rational coefficients.

Sets $A$ to the variable of index var.

If $\mathit{var} \ge 0$, returns whether $A$ is the variable of index var; if $\mathit{var} < 0$, returns whether $A$ is any variable.

Set $A$ to $B$, and swap $A$ and $B$ in constant time.

Returns whether $A = B$.

Returns whether $A$ is a constant.

Sets $c$ to the constant $A$; throws if $A$ is not constant.

Set $A$ to the constant $c$, to 0 or to 1.

Return whether $A$ is the constant $c$, 0 or 1.

Returns whether the degree of $A$ in each variable fits an slong.

Set degs[i] to the degree of $A$ in the variable of index $i$.

Set or return the degree of $A$ in the variable of index var.

Return whether the total degree of $A$ fits an slong, and set or return the total degree.

Sets used[i] to a nonzero value if the variable of index $i$ occurs in $A$, and to zero otherwise.

Sets $d$ to the smallest positive integer such that $dA$ has integer coefficients (the denominator of the content).

Returns whether the leading coefficient of $A$ is 1 (false for zero).

Get or set the coefficient in $A$ of the monomial of the single-term polynomial $M$ (whose coefficient is ignored). Throw if $M$ does not have exactly one term.

Set $c$ to the coefficient of the monomial with exponent vector exp (zero if it does not occur). The underscore version takes the exponents as a plain array of fmpz.

Set the coefficient of the monomial with exponent vector exp to $c$, inserting or removing a term as needed.

Sets $C$ to the coefficient of $\prod_k x_{\mathit{vars}[k]}^{\mathit{exps}[k]}$ in $A$ viewed as a polynomial in the variables vars (a polynomial in the other variables). The length variables in vars must be distinct, with $0 < \mathit{length} \le n$.

Returns whether $A$ involves no variable other than the one of index var.

Sets $A$ to $B$ as a univariate polynomial in the variable of index var and returns 1. Returns 0 if $B$ involves another variable or the degree does not fit an slong.

Sets $A$ to the univariate polynomial $B$ in the variable of index var.

Returns $1$, $-1$ or $0$ according to a fixed total order on polynomials that agrees with the monomial ordering for monomials with equal coefficients.

Return pointers to the content, to the integer part, and to the integer coefficient of the term of index $i$ of the integer part.

Return whether $A$ is in the canonical form described in the section on types, respectively throw an exception describing the defect if poly is not.

Returns the number of terms of $A$.

Sets the length of $A$ to new_length, deleting terms at the end or appending terms with zero coefficient and zero exponents.

Get or set the coefficient of the term of index $i$.

Return whether every exponent of the term of index $i$ fits an slong, respectively a ulong.

Set exps to the exponent vector of the term of index $i$. The ui and si versions throw if an exponent does not fit.

Return the exponent of the variable of index var in the term of index $i$; throw if it does not fit.

Set the exponent vector of the term of index $i$, which may break the ordering of the terms.

Set $M$ to the term of index $i$, respectively its monomial with coefficient 1.

Append a term with coefficient $c$ and exponent vector exp (an array of pointers to fmpz, an array of fmpz, or an array of ulong). The integer part and the denominator of the content are rescaled as needed, so the cost per term is constant on average when the denominators of the pushed coefficients are bounded. The result is not reduced; finish with fmpq_mpoly_reduce (or sort and combine first).

Moves the content and sign of the integer part into the content, so that the integer part is primitive with positive leading coefficient. If the terms of $A$ are nonzero, sorted and have distinct exponents, the result is canonical.

A shortcut for fmpq_mpoly_reduce after an addition: if the integer part has easy_length terms (no cancellation happened), it is assumed to be primitive and only its sign is normalised; otherwise fmpq_mpoly_reduce is called. The integer part must have at most easy_length terms.

Sorts the terms of the integer part into decreasing monomial order, without combining like terms, removing zero terms or reducing.

Combines adjacent terms with equal exponents, removes zero terms and calls fmpq_mpoly_reduce. If the terms were sorted, the result is canonical.

Set $A$ to a random polynomial with at most length terms and exponents below exp_bound, below exp_bounds[i] for variable $i$, or of at most exp_bits bits.

Set $A$ to $B + c$, respectively $B - c$.

Set $A$ to $B + C$, respectively $B - C$. The integer parts are combined with the cofactors of the GCD of the contents.

Sets $A$ to $-B$.

Set $A$ to $cB$.

Set $A$ to $B/c$ for nonzero $c$.

Sets $A$ to $B$ divided by its leading coefficient; throws if $B$ is zero.

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

Sets $A$ to the antiderivative of $B$ with respect to the variable of index var that has no term free of that variable.

Sets ev to $A$ evaluated at $x_i = \mathit{vals}[i]$ for all $i$.

Sets $A$ to $B$ with the variable of index var replaced by val.

Sets $A$ to $B(C_0, \ldots, C_{n-1})$ for univariate polynomials $C_i$.

Sets $A$ to $B(C_0, \ldots, C_{n-1})$, where $B$ is in ctxB and $A$ and the $C_i$ are in ctxAC. $A$ must not alias $B$ or any $C_i$.

Sets $A$ to $B$ with the variable of index $i$ of ctxB replaced by the variable of index c[i] of ctxAC, or by zero if c[i] is negative. Each nonnegative c[i] must be less than the number of variables of ctxAC.

Sets $A$ to $BC$, multiplying the integer parts with fmpz_mpoly_mul (see Multiplication).

Set $A$ to $B^k$. Return 0 if $k$ is negative (in the fmpz version) or the result would be unreasonably large, and 1 otherwise.

If $B$ divides $A$, sets $Q = A/B$ and returns 1; otherwise sets $Q$ to zero and returns 0. See Division.

Set $Q$ (and $R$) to the quotient (and remainder) of $A$ by $B$, with $A = QB + R$ and no term of $R$ divisible by the leading monomial of $B$. They are computed with fmpz_mpoly_quasidiv and fmpz_mpoly_quasidivrem on the integer parts.

Generalised division by the len divisors $B_i$: sets the $Q_i$ and $R$ with $A = \sum_i Q_i B_i + R$ and no term of $R$ divisible by the leading monomial of any $B_i$.

If $A$ is the square of a polynomial over $\mathbb{Q}$, sets $Q$ to the square root with positive leading coefficient and returns 1; otherwise sets $Q$ to zero and returns 0.

Returns whether $A$ is the square of a polynomial over $\mathbb{Q}$.

Sets $g$ to the nonnegative content of $A$ (the absolute value of the stored content).

Sets $M$ to the GCD of the terms of $A$: zero if $A$ is zero, and otherwise the largest monomial dividing every term, with coefficient 1.

Sets $g$ to the GCD of the coefficients of $A$ viewed as a polynomial in the variables vars, so that $g$ does not involve these variables. Returns 1 for success and 0 for failure.

Sets $G$ to the monic GCD of $A$ and $B$.

Sets $G$ to the monic GCD of $A$ and $B$ and also $\bar A = A/G$ and $\bar B = B/G$.

Set $G$ to the monic GCD of $A$ and $B$ using one particular algorithm. The warning on fmpz_mpoly_gcd_zippel applies to fmpq_mpoly_gcd_zippel.

Sets $R$ to the resultant of $A$ and $B$ with respect to the variable of index var, computed with fmpz_mpoly_resultant on the integer parts and multiplied by the appropriate powers of the contents. Returns 1 for success and 0 for failure.

Sets $R$ to the discriminant of $A$ with respect to the variable of index var. Returns 1 for success and 0 for failure.

Sets $A$ to $B$ with each exponent $e$ of variable $i$ replaced by $\mathit{shift}[i] + \mathit{stride}[i] \cdot e$.

Initialise to zero, free, swap in constant time, and ensure room for length terms.

Prints $A$ as a sum of terms (c)*X^e.

Throws if the exponents of $A$ are not nonnegative and strictly decreasing or a coefficient is not canonical.

Sets $A$ to $B$ as a polynomial in the variable of index var.

Set $A$ to the polynomial represented by $B$, with the variable of index var as main variable. The coefficients of $B$ must not involve that variable. The second version packs the result into Abits bits, which must suffice.

Return whether the degree of $A$ fits an slong, the number of terms, and the exponent of the term of index $i$.

Set $c$ to, respectively swap $c$ with, the coefficient of the term of index $i$.

Prepares appending a term with coefficient $C$: multiplies the integer part by the numerator of the content (which becomes 1) and by the denominator of $C \cdot \mathrm{den}(c)$, so that the new term can be appended with the integer coefficient left in the numerator of $C$.

Makes the canonical nonzero $A$ monic by setting its content to the inverse of the leading coefficient of the integer part.

Writes $B(s_0 x_0, \ldots, s_{n-1} x_{n-1})$, for the rationals $s_i$ in scales, as Acontent times the integer coefficients Acoeff on the monomials of $B$, by clearing the denominators $b_i^{d_i}$ with $d_i$ the maximal exponent of $x_i$ and taking out $a_i^{f_i}$ with $f_i$ the minimal exponent. Returns 0 if the powers would be too large. Used by the composition functions.

Sets $A$ to $B$ with exponents packed into Abits bits. Returns 0 if the exponents do not fit, in which case $A$ is unspecified.

Fills the generic ring interface R (used by the expression parser) with the operations of this module.

Test helper: throws if some term of $r$ is divisible by the leading monomial of $g$, using fmpz_mpoly_remainder_strongtest on the integer parts.