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POLYNOMIAL RING IDEAL OPERATIONS

 
Acknowledgements
 
Introduction
 
Creation of Polynomial Rings and their Ideals
 
First Operations on Ideals
      Simple Ideal Constructions
      Basic Commutative Algebra Operations
      Ideal Predicates
      Element Operations with Ideals
 
Computation of Varieties
 
Multiplicities
 
Elimination
      Construction of Elimination Ideals
      Univariate Elimination Ideal Generators
      Relation Ideals
 
Variable Extension of Ideals
 
Homogenization of Ideals
 
Extension and Contraction of Ideals
 
Dimension of Ideals
 
Radical and Decomposition of Ideals
      Radical
      Primary Decomposition
      Triangular Decomposition
      Equidimensional Decomposition
 
Normalisation and Noether Normalisation
      Noether Normalisation
      Normalisation
 
Hilbert Series and Hilbert Polynomial
 
Syzygies
 
Maps between Rings
 
Symmetric Polynomials
 
Functions for Polynomial Algebra and Module Generators
 
Bibliography







DETAILS

 
Introduction

 
Creation of Polynomial Rings and their Ideals

 
First Operations on Ideals

      Simple Ideal Constructions
            I + J : RngMPol, RngMPol -> RngMPol
            I * J : RngMPol, RngMPol -> RngMPol
            I ^ k : RngMPol, RngIntElt -> RngMPol
            I / J : RngMPol, RngMPol -> RngMPolRes

      Basic Commutative Algebra Operations
            QuotientDimension(I) : RngMPol -> RngIntElt
            ColonIdeal(I, J) : RngMPol, RngMPol -> RngMPol
            ColonIdeal(I, f) : RngMPol, RngMPolElt -> RngMPol, RngIntElt
            ColonIdealEquivalent(I, f) : RngMPol, RngMPolElt -> RngMPol, RngMPolElt
            Saturation(I, J) : RngMPol, RngMPol -> RngMPol
            Saturation(I): RngMPol -> RngMPol
            Generic(I) : RngMPol -> RngMPol
            LeadingMonomialIdeal(I) : RngMPol -> RngMPol
            I meet J : RngMPol, RngMPol -> RngMPol
            &meet S : [ RngMPol ] -> RngMPol
            RegularSequence(I): RngMPol -> SeqEnum
            ReesIdeal(P, I): RngMPol, RngMPol -> RngMPol, Map

      Ideal Predicates
            I eq J : RngMPol, RngMPol -> BoolElt
            I ne J : RngMPol, RngMPol -> BoolElt
            I notsubset J : RngMPol, RngMPol -> BoolElt
            I subset J : RngMPol, RngMPol -> BoolElt
            IsZero(I) : RngMPol -> BoolElt
            IsProper(I) : RngMPol -> BoolElt
            IsHomogeneous(I) : RngMPol -> BoolElt
            IsPrincipal(I) : RngMPol -> BoolElt, RngMPolElt
            IsPrimary(I) : RngMPol -> BoolElt
            IsPrime(I) : RngMPol -> BoolElt
            IsMaximal(I) : RngMPol -> BoolElt
            IsRadical(I) : RngMPol -> BoolElt
            IsZeroDimensional(I) : RngMPol -> BoolElt
            HasGrevlexOrder(I) : RngMPol -> BoolElt
            Example Ideal_IdealArithmetic (H106E1)

      Element Operations with Ideals
            f in I : RngMPolElt, RngMPol -> BoolElt
            f notin I : RngMPolElt, RngMPol -> BoolElt
            IsInRadical(f, I) : RngMPolElt, RngMPol -> BoolElt
            JacobianIdeal(f) : RngMPolElt -> RngMPol
            Example Ideal_ElementOperations (H106E2)

 
Computation of Varieties
      Variety(I) : RngMPol -> [ ModTupFldElt ]
      VarietySequence(I) : RngMPol -> [ [ RngElt ] ]
      VarietySizeOverAlgebraicClosure(I) : RngMPol -> RngIntElt
      Example Ideal_Variety (H106E3)

 
Multiplicities
      MilnorNumber(f) : RngMPolElt -> RngElt
      TjurinaNumber(f) : RngMPolElt -> RngElt
      Example Ideal_Variety (H106E4)

 
Elimination

      Construction of Elimination Ideals
            EliminationIdeal(I, k: parameters) : RngMPol, RngIntElt -> RngMPol
            EliminationIdeal(I, S) : RngMPol, { RngIntElt } -> RngMPol
            Example Ideal_QuadraticOrderElim (H106E5)

      Univariate Elimination Ideal Generators
            UnivariateEliminationIdealGenerator(I, i) : RngMPol, RngIntElt -> RngMPolElt
            UnivariateEliminationIdealGenerators(I) : RngMPol -> [ RngMPolElt ]
            Example Ideal_EliminationIdeal (H106E6)
            Example Ideal_ZRadical (H106E7)

      Relation Ideals
            RelationIdeal(Q) : [ RngMPol ] -> RngMPol
            Example Ideal_RelationIdeal (H106E8)

 
Variable Extension of Ideals
      VariableExtension(I, k, b) : RngMPol, RngIntElt, BoolElt -> RngMPol, Map

 
Homogenization of Ideals
      Homogenization(I, b) : RngMPol, RngIntElt, BoolElt -> RngMPol, Map

 
Extension and Contraction of Ideals
      Extension(I, U) : RngMPol, [ RngIntElt ] -> RngMPol, Map

 
Dimension of Ideals
      Dimension(I) : RngMPol -> RngIntElt, [ RngIntElt ]

 
Radical and Decomposition of Ideals

      Radical
            Radical(I) : RngMPol -> RngMPol
            Example Ideal_Radical (H106E9)

      Primary Decomposition
            PrimaryDecomposition(I) : RngMPol -> [ RngMPol ], [ RngMPol ]
            RadicalDecomposition(I) : RngMPol -> [ RngMPol ]
            ProbableRadicalDecomposition(I) : RngMPol -> [ RngMPol ]
            MinimalDecomposition(S) : [ RngMPol ] -> [ RngMPol ]
            SetVerbose("Decomposition", v) : MonStgElt, RngIntElt ->
            Example Ideal_PrimaryDecomposition (H106E10)

      Triangular Decomposition
            TriangularDecomposition(I) : RngMPol -> [ RngMPol ], BoolElt
            Example Ideal_TriangularDecomposition (H106E11)

      Equidimensional Decomposition
            EquidimensionalPart(I) : RngMPol -> RngMPol
            Example Ideal_EquidimensionalDecomposition (H106E12)

 
Normalisation and Noether Normalisation

      Noether Normalisation
            NoetherNormalisation(I) : RngMPol -> [RngMPolElt],Map,Map
            Example Ideal_NoetherNormalisation (H106E13)

      Normalisation
            Normalisation(I) : RngMPol -> List
            Example Ideal_Normalisation (H106E14)

 
Hilbert Series and Hilbert Polynomial
      HilbertSeries(I) : RngMPol -> FldFunUElt
      HilbertSeries(I, p) : RngMPol, RngIntElt -> RngSerLaurElt
      HilbertDenominator(I) : RngMPol -> RngUPol
      HilbertNumerator(I) : RngMPol -> RngUPol
      HilbertPolynomial(I) : RngMPol -> RngUPolElt, RngIntElt
      Example Ideal_Hilbert (H106E15)

 
Syzygies
      SyzygyMatrix(Q) : [ RngMPolElt ] -> ModMatRngElt
      Example Ideal_SyzygyMatrix (H106E16)

 
Maps between Rings
      PolyMapKernel(f) : Map -> RngMPol
      IsInImage(f, p) : Map, RngMPolElt -> [ BoolElt ]
      IsSurjective(f) : Map -> [ BoolElt ]
      Extension(phi, I): Map, RngMPol -> RngMPol
      Implicitization(phi) : Map -> RngMPol
      Example Ideal_Map1 (H106E17)

 
Symmetric Polynomials
      ElementarySymmetricPolynomial(P, k) : RngMPol, RngIntElt -> RngMPolElt
      IsSymmetric(f) : RngMPolElt -> BoolElt, RngMPolElt
      Example Ideal_IsSymmetric (H106E18)

 
Functions for Polynomial Algebra and Module Generators
      MinimalAlgebraGenerators(L) : [ RngMPol ] -> [ RngMPol ]
      HomogeneousModuleTest(P, S, F) : [ RngMPol ], [ RngMPol ], RngMPol -> BoolElt, [ RngMPol ]
      HomogeneousModuleTest(P, S, L) : [ RngMPol ], [ RngMPol ], [ RngMPol ] -> [ BoolElt ], [ [ RngMPol ] ]
      HomogeneousModuleTestBasis(P, S, L) : [ RngMPol ], [ RngMPol ], [ RngMPol ] -> [ BoolElt ], [ [ RngMPol ] ]
      Example Ideal_HomogeneousModuleTest1 (H106E19)

 
Bibliography

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