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 (H116E1)

      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 (H116E2)

 
Computation of Varieties
      Variety(I) : RngMPol -> [ ModTupFldElt ]
      VarietySequence(I) : RngMPol -> [ [ RngElt ] ]
      VarietySize(I) : RngMPol -> RngIntElt
      Example Ideal_Variety (H116E3)

 
Multiplicities
      MilnorNumber(f) : RngMPolElt -> RngElt
      TjurinaNumber(f) : RngMPolElt -> RngElt
      Example Ideal_Variety (H116E4)

 
Elimination

      Construction of Elimination Ideals
            EliminationIdeal(I, k: parameters) : RngMPol, RngIntElt -> RngMPol
            EliminationIdeal(I, S) : RngMPol, { RngIntElt } -> RngMPol
            Example Ideal_QuadraticOrderElim (H116E5)

      Univariate Elimination Ideal Generators
            UnivariateEliminationIdealGenerator(I, i) : RngMPol, RngIntElt -> RngMPolElt
            UnivariateEliminationIdealGenerators(I) : RngMPol -> [ RngMPolElt ]
            Example Ideal_EliminationIdeal (H116E6)
            Example Ideal_ZRadical (H116E7)

      Relation Ideals
            RelationIdeal(Q) : [ RngMPol ] -> RngMPol
            Example Ideal_RelationIdeal (H116E8)

 
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 (H116E9)

      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 (H116E10)

      Triangular Decomposition
            TriangularDecomposition(I) : RngMPol -> [ RngMPol ], BoolElt
            Example Ideal_TriangularDecomposition (H116E11)

      Equidimensional Decomposition
            EquidimensionalPart(I) : RngMPol -> RngMPol
            Example Ideal_EquidimensionalDecomposition (H116E12)

 
Normalisation and Noether Normalisation

      Noether Normalisation
            NoetherNormalisation(I) : RngMPol -> [RngMPolElt],Map,Map
            Example Ideal_NoetherNormalisation (H116E13)

      Normalisation
            Normalisation(I) : RngMPol -> List
            Example Ideal_Normalisation (H116E14)

 
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 (H116E15)

 
Syzygies
      SyzygyMatrix(Q) : [ RngMPolElt ] -> ModMatRngElt
      Example Ideal_SyzygyMatrix (H116E16)

 
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 (H116E17)

 
Symmetric Polynomials
      ElementarySymmetricPolynomial(P, k) : RngMPol, RngIntElt -> RngMPolElt
      IsSymmetric(f) : RngMPolElt -> BoolElt, RngMPolElt
      Example Ideal_IsSymmetric (H116E18)

 
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 (H116E19)

 
Bibliography

[Next][Prev] [Right] [____] [Up] [Index] [Root]