2. Blackbox polynomials

This package provides a blackbox polynomial domain and is due to L. Lehmann. The verbose flag is BbpVerbose.

2.1 Creation of a blackbox polynomial algebra

intrinsic: BlackboxPolynomialAlgebra (R::Rng, n::RngIntElt) -> RngMPol

It returns a ring of multivariate blackbox polynomials over R in n variables.

2.2 Creation of elements

intrinsic: Var (x::RngMPolElt) -> Rec

It returns the blackbox polynomial element corresponding to x.

intrinsic: Cst (E::RngMPol, c) -> Rec

• E is a domain returned by BlackboxPolynomialAlgebra.
• c is an element of the base ring of R.

It returns the blackbox polynomial corresponding to c.

2.3 Arithmetic operations

All elementary ring operations are supported.

intrinsic: '+' (x::Rec) -> Rec

intrinsic: '+' (x::Rec, y) -> Rec

intrinsic: '+' (y, x::Rec) -> Rec

intrinsic: '-' (x::Rec) -> Rec

intrinsic: '-' (x::Rec, y) -> Rec

intrinsic: '-' (y, x::Rec) -> Rec

intrinsic: '*' (x::Rec, y) -> Rec

intrinsic: '*' (y, x::Rec) -> Rec

intrinsic: '/' (x::Rec, c) -> Rec

c must be invertible.

intrinsic: '^' (x::Rec, n::RngIntElt) -> Rec

n must be a non-negative integer.

intrinsic: Add (E::RngMPol, l::[Rec]) -> Rec

• E, returned by BlackboxPolynomialAlgebra.
• l, a sequence.

It returns the sum of the element of l.

intrinsic: Product (E::RngMPol, l::[Rec]) -> Rec

• E, returned by BlackboxPolynomialAlgebra.
• l, a sequence.

It returns the product of the element of l.

2.4 Equality tests

Equality test corresponds to the test on the representations: two blackbox polynomials are equal iff their address is the same.

intrinsic: IsZero (e::Rec) -> BoolElt

Tells whether the expression e is the constant 0 or not.

intrinsic: IsOne (e::Rec) -> BoolElt

Tells whether the expression e is the constant 1 or not.

2.5 Evaluation

intrinsic: ClearRememberTableValues (~E::RngMPol)

It clears the remember table of the current evaluation process.

intrinsic: Evaluate (F::Rec, x::[]) -> .

intrinsic: Evaluate (F::[Rec], x::[]) -> .

It returns the sequence of the values of the elements of F when the i-th variable is specialized to x[i], for i from 1 to #x. If F is empty then it returns [Universe(x)|].
Error conditions:

#x must equal the rank of the polynomial domain.

intrinsic: InitializeEvaluation (F::[Rec])

• F, a nonempty sequence of expressions.

It initializes the evaluation process that will serve to evaluate the element of F only.

intrinsic: ChangeUniverseValues (~E::RngMPol, R::Rng, f::UserProgram)

• E, a domain returned by BlackboxPolynomialAlgebra.
• R, a ring
• f, map into R.

It applies f onto the remember table of the current evaluation process.

intrinsic: Derivative (e::Rec, i::RngIntElt: algorithm:="backward") -> Rec

The derivative of e wrt its i-th variable.

intrinsic: Gradient (e::Rec: algorithm:="backward") -> []

intrinsic: Gradient (F::[Rec]: algorithm:="backward") -> []

Sequence of the partial derivatives. The parameter algorithm can be "backward" (for Baur-Strassen) or "forward".

intrinsic: ClearRememberTableGradients (~E::RngMPol)

It clears the remember tables used for gradient storage.

2.6 Degree, coefficients

intrinsic: ConvertToPolynomial (e::Rec) -> .

intrinsic: ConvertToPolynomial (e::[Rec]) -> .

• e is an expression or a sequence of expressions.

It returns the multivariate polynomial elements represented by e.

intrinsic: IsHomogeneous (e::Rec) -> BoolElt

It tells whether e is homogeneous (w.r.t. to grading on the variables of its polynomial ring). It is not probabilistic.

intrinsic: TotalDegree (e::Rec: Strategy:="UpperBound") -> RngIntElt

intrinsic: TotalDegree (F::[Rec]: Strategy:="UpperBound") -> .

It returns the total degrees of the mulivariate polynomial represented by F.
Parameter: Strategy can be either "Deterministic" or "Probabilistic". Strategy can also be "UpperBound" in order to compute a deterministic upper bound only.

2.7 Linear algebra

The following linear algebra functionalities are based on Berkowitz' algorithm.

intrinsic: CharacteristicPolynomialBerkowitz (A) -> .

• A, sequence of sequences of expressions, viewed as a square matrix.

It returns the sequence of coefficients of the characteristic polynomial of A, the last element being the coefficient of degree 0.

intrinsic: DeterminantBerkowitz (A) -> .

• A, sequence of sequences of Expressions, viewed as a square matrix.

It returns the determinant of A.

2.8 Printing

intrinsic: PrintBbp (s::[])

intrinsic: PrintBbp (e::Rec)

It prints the expression represented by e.