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MULTIVARIATE POLYNOMIAL RINGS

 
Acknowledgements
 
Introduction
      Representation
 
Polynomial Rings and Polynomials
      Creation of Polynomial Rings
      Print Names
      Graded Polynomial Rings
      Creation of Polynomials
 
Structure Operations
      Related Structures
      Numerical Invariants
      Ring Predicates and Booleans
      Changing Coefficient Ring
      Homomorphisms
 
Element Operations
      Arithmetic Operators
      Equality and Membership
      Predicates on Ring Elements
      Coefficients, Monomials and Terms
      Degrees
      Univariate Polynomials
      Derivative, Integral
      Evaluation, Interpolation
      Quotient and Reductum
      Diagonalizing a Polynomial of Degree 2
 
Greatest Common Divisors
      Common Divisors and Common Multiples
      Content and Primitive Part
 
Factorization and Irreducibility
 
Resultants and Discriminants
 
Polynomials over the Integers
 
Bibliography







DETAILS

 
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 (H24E1)
            Example RngMPol_Global (H24E2)

      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, ..., yn > : RngMPol, Rng -> Map
            Example RngMPol_Homomorphism (H24E3)

 
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
            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 (H24E4)

      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 (H24E5)

      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 (H24E6)

      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 (H24E7)

 
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 (H24E8)
      Example RngMPol_Vandermonde (H24E9)
      Example RngMPol_Heron (H24E10)
      Example RngMPol_FiniteFieldFactorization (H24E11)

 
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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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012