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UNIVARIATE POLYNOMIAL RINGS

 
Acknowledgements
 
Introduction
      Representation
 
Creation Functions
      Creation of Structures
      Print Options
      Creation of Elements
 
Structure Operations
      Related Structures
      Changing Rings
      Numerical Invariants
      Ring Predicates and Booleans
      Homomorphisms
 
Element Operations
      Parent and Category
      Arithmetic Operators
      Equality and Membership
      Predicates on Ring Elements
      Coefficients and Terms
      Degree
      Roots
      Derivative, Integral
      Evaluation, Interpolation
      Quotient and Remainder
      Modular Arithmetic
      Other Operations
 
Common Divisors and Common Multiples
      Common Divisors and Common Multiples
      Content and Primitive Part
 
Polynomials over the Integers
 
Polynomials over Finite Fields
 
Factorization
      Factorization and Irreducibility
      Resultant and Discriminant
      Hensel Lifting
 
Ideals and Quotient Rings
      Creation of Ideals and Quotients
      Ideal Arithmetic
      Other Functions on Ideals
      Other Functions on Quotients
 
Special Families of Polynomials
      Orthogonal Polynomials
      Permutation Polynomials
      The Bernoulli Polynomial
      Swinnerton-Dyer Polynomials
 
Bibliography







DETAILS

 
Introduction

      Representation

 
Creation Functions

      Creation of Structures
            PolynomialAlgebra(R) : Rng -> RngUPol
            Example RngPol_Creation (H23E1)

      Print Options
            AssignNames(~P, s) : RngUPol, [ MonStgElt ]) ->
            Name(P, i) : RngUPol, RngIntElt -> RngUPolElt

      Creation of Elements
            P . 1 : RngUPol, RngInt -> RngPolElt
            elt< P | a0, ..., ad > : RngUPol, RngElt, ..., RngElt -> RngUPolElt
            P ! s : RngUPol, RngElt -> RngPolElt
            Polynomial(Q) : [ RngElt ] -> RngUPolElt
            Polynomial(R, Q) : Rng, [ RngElt] -> RngUPolElt
            Polynomial(R, f) : Rng, RngUPolElt -> RngUPolElt
            Example RngPol_Polynomials (H23E2)

 
Structure Operations

      Related Structures
            BaseRing(P) : RngUPol -> Rng

      Changing Rings
            ChangeRing(P, S) : RngUPol, Rng -> RngUPol, Map
            ChangeRing(P, S, f) : RngUPol, Rng, Map -> RngUPol, Map
            Example RngPol_ChangeRing (H23E3)

      Numerical Invariants
            Rank(P) : RngUPol -> RngIntElt
            # P : RngUPolRes -> RngIntElt

      Ring Predicates and Booleans

      Homomorphisms
            hom< P -> S | f, y > : RngUPol, Rng, Map, RngElt -> Map
            Example RngPol_Homomorphism (H23E4)

 
Element Operations

      Parent and Category

      Arithmetic Operators

      Equality and Membership

      Predicates on Ring Elements

      Coefficients and Terms
            Coefficients(p) : RngUPolElt -> [ RngElt ]
            Coefficient(p, i) : RngUPolElt, RngIntElt -> RngElt
            MonomialCoefficient(p, m) : RngUPolElt, RngUPolElt -> RngElt
            LeadingCoefficient(p) : RngUPolElt -> RngElt
            TrailingCoefficient(p) : RngUPolElt -> RngElt
            ConstantCoefficient(p) : RngUPolElt -> RngElt
            Terms(p) : RngUPolElt -> [ RngUPolElt ]
            LeadingTerm(p) : RngUPolElt -> RngUPolElt
            TrailingTerm(p) : RngUPolElt -> RngUPolElt
            Monomials(p) : RngUPolElt -> SeqEnum
            Support(p) : RngUPolElt -> [RngIntElt], [RngElt]
            Round(p) : RngUPolElt -> RngUPolElt
            Valuation(p) : RngUPolElt -> RngIntElt

      Degree
            Degree(p) : RngUPolElt -> RngIntElt

      Roots
            Roots(p) : RngUPolElt -> [ < RngElt, RngIntElt> ]
            Roots(p, S) : RngUPolElt -> [ < RngElt, RngIntElt> ]
            HasRoot(p) : RngUPolElt -> BoolElt, RngElt
            HasRoot(p, S) : RngUPolElt, Rng -> BoolElt, RngElt
            SmallRoots(p, N, X) : RngUPolElt, RngElt, RngElt -> [RngElt]
            Example RngPol_SmallRootsUsage (H23E5)
            SetVerbose("SmallRoots", v) : MonStgElt, RngIntElt ->

      Derivative, Integral
            Derivative(p) : RngUPolElt -> RngUPolElt
            Derivative(p, n) : RngUPolElt, RngIntElt -> RngUPolElt
            Integral(p) : RngUPolElt -> RngUPolElt

      Evaluation, Interpolation
            Evaluate(p, r) : RngUPolElt, RngElt -> RngElt
            Interpolation(I, V) : [ RngElt ], [ RngElt ] -> RngUPolElt

      Quotient and Remainder
            Quotrem(f, g) : RngUPolElt, RngUPolElt -> RngUPolElt, RngUPolElt
            f div g : RngUPolElt, RngUPolElt -> RngUPolElt
            IsDivisibleBy(a, b) : RngUPolElt, RngUPolElt -> BoolElt, RngUPolElt
            ExactQuotient(f, g) : RngUPolElt, RngUPolElt -> RngUPolElt
            f mod g : RngUPolElt, RngUPolElt -> RngUPolElt
            Valuation(f, g) : RngUPolElt, RngUPolElt -> RngIntElt
            Reductum(f) : RngUPolElt -> RngUPolElt
            PseudoRemainder(f, g) : RngUPolElt, RngUPolElt -> RngUPolElt
            EuclideanNorm(p) : RngUPol -> RngIntElt

      Modular Arithmetic
            Modexp(f, n, g) : RngUPolElt, RngIntElt, RngUPolElt -> RngUPolElt
            ChineseRemainderTheorem(X, M) : [RngUPolElt], [RngUPolElt] -> RngUPolElt

      Other Operations
            ReciprocalPolynomial(f) : RngUPolElt -> RngUPolElt
            PowerPolynomial(f,n) : RngUPolElt, RngIntElt -> RngUPolElt
            f ^ M : RngUPolElt, Mtrx -> RngUPolElt

 
Common Divisors and Common Multiples

      Common Divisors and Common Multiples
            GreatestCommonDivisor(f, g) : RngUPolElt, RngUPolElt -> RngUPolElt
            ExtendedGreatestCommonDivisor(f, g) : RngUPolElt, RngUPolElt -> RngUPolElt, RngUPolElt, RngUPolElt
            LeastCommonMultiple(f, g) : RngUPolElt, RngUPolElt -> RngUPolElt
            Normalize(f) : RngUPolElt -> RngUPolElt

      Content and Primitive Part
            Content(p) : RngUPolElt -> RngIntElt
            PrimitivePart(p) : RngUPolElt -> RngUPolElt
            ContentAndPrimitivePart(p) : RngUPolElt -> RngIntElt, RngUPolElt

 
Polynomials over the Integers
      Sign(p) : RngUPolElt -> RngIntElt
      AbsoluteValue(p) : RngUPolElt -> RngUPolElt
      MaxNorm(p) : RngUPolElt -> RngIntElt
      SumNorm(p) : RngUPolElt -> RngIntElt
      DedekindTest(p, m) : RngUPolElt, RngIntElt -> Boolelt

 
Polynomials over Finite Fields
      PrimePolynomials(R, d) : RngUPol, RngIntElt -> SeqEnum[ RngUPolElt ]
      RandomPrimePolynomial(R, d) : RngUPol, RngIntElt -> RngUPolElt
      NumberOfPrimePolynomials(q, d) : RngIntElt, RngIntElt -> RngIntElt
      JacobiSymbol(a,b) : RngUPol, RngUPol -> RngIntElt

 
Factorization

      Factorization and Irreducibility
            Factorization(f) : RngUPolElt -> [ < RngUPolElt, RngIntElt >], RngElt
            HasPolynomialFactorization(R) : Rng -> BoolElt
            SetVerbose("PolyFact", v) : MonStgElt, RngIntElt ->
            FactorisationToPolynomial(f) : [Tup] -> BoolElt
            Example RngPol_SwinnertonDyerPolynomial (H23E6)
            SquarefreeFactorization(f) : RngUPolElt -> [ <RngUPolElt, RngIntElt> ]
            DistinctDegreeFactorization(f) : RngUPolElt -> [ <RngIntElt, RngUPolElt> ]
            EqualDegreeFactorization(f, d, g) : RngUPolElt, RngIntElt, RngUPolElt -> [ RngUPolElt ]
            IsIrreducible(f) : RngUPolElt -> BoolElt
            IsSeparable(f) : RngUPolElt -> BoolElt
            QMatrix(f) : RngUPolElt -> AlgMatElt

      Resultant and Discriminant
            Discriminant(f) : RngUPolElt -> RngIntElt
            Resultant(f, g) : RngUPolElt, RngUPolElt -> RngElt
            CompanionMatrix(f) : RngUPolElt -> AlgMatElt

      Hensel Lifting
            HenselLift(f, s, P) : RngUPolElt, [ RngUPolElt ], RngUPol -> [ RngUPolElt ]
            Example RngPol_Hensel (H23E7)

 
Ideals and Quotient Rings

      Creation of Ideals and Quotients
            ideal< R | a1, ..., ar > : RngUPol, RngUPolElt, ..., RngUPolElt -> RngUPol
            quo< R | I > : RngUPol, RngUPol -> RngUPolRes

      Ideal Arithmetic
            I + J : RngUPol, RngUPol -> RngUPol
            I * J : RngUPol, RngUPol -> RngUPol
            I meet J : RngUPol, RngUPol -> RngUPol
            a in I : RngUPolElt, RngUPol -> BoolElt
            a notin I : RngUPolElt, RngUPol -> BoolElt
            I eq J : RngUPol, RngUPol -> BoolElt
            I ne J : RngUPol, RngUPol -> BoolElt
            I subset J : RngUPol, RngUPol -> BoolElt
            I notsubset J : RngUPol, RngUPol -> BoolElt

      Other Functions on Ideals
            I . 1 : RngUPol -> RngUPolElt

      Other Functions on Quotients
            Modulus(Q) : RngUPolRes -> RngUPolElt
            PreimageRing(Q) : RngUPolRes -> RngUPol

 
Special Families of Polynomials

      Orthogonal Polynomials
            ChebyshevFirst(n) : RngIntElt -> RngUPolElt
            ChebyshevSecond(n) : RngIntElt -> RngUPolElt
            LegendrePolynomial(n) : RngIntElt -> RngUPolElt
            LaguerrePolynomial(n) : RngIntElt -> RngUPolElt
            HermitePolynomial(n) : RngIntElt -> RngUPolElt
            GegenbauerPolynomial(n, m) : RngIntElt, RngElt ->RngUPolElt

      Permutation Polynomials
            DicksonFirst(n, a) : RngIntElt, RngElt -> RngUPolElt
            DicksonSecond(n, a) : RngIntElt, RngElt -> RngUPolElt

      The Bernoulli Polynomial
            BernoulliPolynomial(n) : RngIntElt -> RngUPolElt

      Swinnerton-Dyer Polynomials
            SwinnertonDyerPolynomial(n) : RngIntElt -> RngUPolElt

 
Bibliography

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