Introduction

      Representation

 
Polynomial Rings and Polynomials

      Creation of Polynomial Rings
            PolynomialRing(R, n) : Rng, RngIntElt -> RngMPol
            PolynomialRing(R, n, order) : Rng, RngIntElt, MonStgElt, ... -> RngMPol
            Example RngMPol_AssignNames (H25E1)
            Example RngMPol_Global (H25E2)

      Print Names
            AssignNames(~P, s) : RngMPol, [ MonStgElt ]) ->
            Name(P, i) : RngMPol, RngIntElt -> RngMPolElt

      Graded Polynomial Rings

      Creation of Polynomials
            P . i : RngMPol, RngInt -> RngMPolElt
            elt< R | a > : RngMPol, RngElt -> RngMPolElt
            MultivariatePolynomial(P, f, i) : RngMPol, RngUPolElt, RngIntElt -> RngMPolElt

 
Structure Operations

      Related Structures
            BaseRing(P) : RngMPol -> Rng

      Numerical Invariants
            Rank(P) : RngMPol -> RngIntElt

      Ring Predicates and Booleans

      Changing Coefficient Ring
            ChangeRing(P, S) : RngMPol, Rng -> RngMPol

      Homomorphisms
            hom< P -> S | f, y₁, ..., y_n > : RngMPol, Rng -> Map
            Example RngMPol_Homomorphism (H25E3)

 
Element Operations

      Arithmetic Operators

      Equality and Membership

      Predicates on Ring Elements
            IsDivisibleBy(a, b) : RngMPolElt, RngMPolElt -> BoolElt, RngMPolElt
            IsAlgebraicallyDependent(S) : RngMPolElt -> BoolElt

      Coefficients, Monomials and Terms
            Coefficients(f) : RngMPolElt -> [ RngElt ]
            Coefficients(f, i) : RngMPolElt, RngIntElt -> [ RngElt ]
            Coefficient(f, i, k) : RngMPolElt, RngIntElt, RngIntElt -> RngElt
            LeadingCoefficient(f) : RngMPolElt -> RngElt
            LeadingCoefficient(f, i) : RngMPolElt, RngIntElt -> RngElt
            Length(f) : RngMPolElt -> RngIntElt
            TrailingCoefficient(f) : RngMPolElt -> RngElt
            TrailingCoefficient(f, i) : RngMPolElt, RngIntElt -> RngElt
            CoefficientDenominator(f) : RngMPolElt -> RngElt
            MonomialCoefficient(f, m) : RngMPolElt, RngMPolElt -> RngElt
            Monomials(f) : RngMPolElt -> [ RngMPolElt ]
            CoefficientsAndMonomials(f) : RngMPolElt -> [ RngElt ], [ RngMPolElt ]
            LeadingMonomial(f) : RngMPolElt -> RngMPolElt
            Terms(f) : RngMPolElt -> [ RngMPolElt ]
            Terms(f, i) : RngMPolElt, RngIntElt -> [ RngMPolElt ]
            Term(f, i, k) : RngMPolElt, RngIntElt, RngIntElt -> RngMPolElt
            LeadingTerm(f) : RngMPolElt -> RngMPolElt
            LeadingTerm(f, i) : RngMPolElt, RngIntElt -> RngMPolElt
            TrailingTerm(f) : RngMPolElt -> RngElt
            TrailingTerm(f, i) : RngMPolElt, RngIntElt -> RngElt
            Exponents(f) : RngMPolElt -> [ RngIntElt ]
            Monomial(P, E) : RngMPol, [ RngIntElt ] -> RngMPolElt
            Polynomial(C, M) : [RngElt], [RngMPolElt] -> RngMPolElt
            Example RngMPol_Coefficients (H25E4)

      Degrees
            Degree(f, i) : RngMPolElt, RngIntElt -> RngIntElt
            TotalDegree(f) : RngMPolElt -> RngIntElt
            LeadingTotalDegree(f) : RngMPolElt -> RngIntElt

      Univariate Polynomials
            IsUnivariate(f) : RngMPolElt -> BoolElt, RngUPolElt, RngIntElt
            IsUnivariate(f, i) : RngMPolElt, RngIntElt -> BoolElt, RngUPolElt
            UnivariatePolynomial(f) : RngMPolElt -> RngUPolElt
            Example RngMPol_UnivariatePolynomial (H25E5)

      Derivative, Integral
            Derivative(f, i) : RngMPolElt, RngIntElt -> RngMPolElt
            Derivative(f, k, i) : RngMPolElt, RngIntElt -> RngMPolElt
            Integral(f, i) : RngMPolElt, RngIntElt -> RngMPolElt
            JacobianMatrix( [ f ] ) : [ RngMPolElt ] -> RngMPol

      Evaluation, Interpolation
            Evaluate(f, s) : RngMPolElt, [ RngElt ] -> RngElt
            Evaluate(f, i, r) : RngMPolElt, RngMPolElt, RngElt -> RngMPolElt
            Interpolation(I, V, i) : [ RngElt ], [ RngMPolElt ], RngIntElt -> RngMPolElt
            Example RngMPol_Interpolate (H25E6)

      Quotient and Reductum
            f div g : RngMPolElt, RngMPolElt -> RngMPolElt
            Reductum(f) : RngMPolElt -> RngMPolElt
            Reductum(f, i) : RngMPolElt, RngIntElt -> RngMPolElt

      Diagonalizing a Polynomial of Degree 2
            SymmetricBilinearForm(f) : RngMPolElt -> ModMatRngElt
            DiagonalForm(f) : RngMPolElt -> RngMPolElt, ModMatRngElt
            Example RngMPol_Sym_Bi_Linear (H25E7)

 
Greatest Common Divisors

      Common Divisors and Common Multiples
            GreatestCommonDivisor(f, g) : RngMPolElt, RngMPolElt -> RngMPolElt
            GCD(Q) : [ RngMPolElt ] -> RngMPolElt
            LeastCommonMultiple(f, g) : RngMPolElt, RngMPolElt -> RngMPolElt
            LCM(Q) : [ RngMPolElt ] -> RngMPolElt
            Normalize(f) : RngMPolElt -> RngMPolElt
            ClearDenominators(f) : RngMPolElt -> RngMPolElt

      Content and Primitive Part
            Content(f) : RngMPolElt -> RngIntElt
            PrimitivePart(f) : RngMPolElt -> RngMPolElt
            ContentAndPrimitivePart(f) : RngMPolElt -> RngIntElt, RngMPolElt

 
Factorization and Irreducibility
      Factorization(f) : RngMPolElt -> [ < RngMPolElt, RngIntElt >], RngElt
      SquarefreeFactorization(f) : RngMPolElt -> [ <RngMPolElt, RngIntElt> ]
      SquarefreePart(f) : RngMPolElt -> RngMPolElt
      IsIrreducible(f) : RngMPolElt -> BoolElt
      SetVerbose("PolyFact", v) : MonStgElt, RngIntElt ->
      Example RngMPol_Trinomials (H25E8)
      Example RngMPol_Vandermonde (H25E9)
      Example RngMPol_Heron (H25E10)
      Example RngMPol_FiniteFieldFactorization (H25E11)

 
Resultants and Discriminants
      Resultant(f, g, i) : RngMPolElt, RngMPolElt, RngIntElt -> RngMPolElt
      Discriminant(f, i) : RngMPolElt, RngIntElt -> RngMPolElt

 
Polynomials over the Integers
      Sign(f) : RngMPolElt -> RngIntElt
      AbsoluteValue(f) : RngMPolElt -> RngMPolElt
      MaxNorm(f) : RngMPolElt -> RngIntElt
      SumNorm(f) : RngMPolElt -> RngIntElt

 
Bibliography

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