MAGMA Computational Algebra System

IDEAL THEORY AND GRÖBNER BASES

DETAILS

 
Introduction

 
Representation and Monomial Orders

      Lexicographical: lex

      Graded Lexicographical: glex

      Graded Reverse Lexicographical: grev-lex

      Graded Reverse Lexicographical (Weighted): grev-lexw

      Elimination (k): elim

      Elimination List: elim

      Inverse Block: invblock

      Univariate: univ

      Weight: weight

 
Polynomial Rings and Ideals

      Creation of Polynomial Rings and Accessing their Monomial Orders
            PolynomialRing(R, n) : Rng, RngIntElt -> RngMPol
            PolynomialRing(R, n, order) : Rng, RngIntElt, MonStgElt, ... -> RngMPol
            PolynomialRing(R, n, T) : Rng, RngIntElt, Tup -> RngMPol
            MonomialOrder(P) : RngMPol -> Tup
            MonomialOrderWeightVectors(P) : RngMPol -> [ [ FldRatElt ] ]
            Example GB_Order (H98E1)

      Creation of Graded Polynomial Rings
            PolynomialRing(R, Q) : Rng, [ RngIntElt ] -> RngMPol
            Grading(P) : RngMPol -> [ RngIntElt ]

      Element Operations Using the Grading
            Degree(f) : RngMPolElt -> RngIntElt
            LeadingWeightedDegree(f) : RngMPolElt -> RngIntElt
            IsHomogeneous(f) : RngMPolElt -> BoolElt
            HomogeneousComponent(f, d) : RngMPolElt, RngIntElt -> RngMPolElt
            HomogeneousComponents(f) : RngMPolElt -> [ RngMPolElt ]
            MonomialsOfDegree(P, d) : RngMPolElt, RngIntElt -> {@ RngMPolElt @}
            MonomialsOfWeightedDegree(P, d) : RngMPolElt, RngIntElt -> {@ RngMPolElt @}
            Example GB_Graded (H98E2)

      Creation of Ideals and Accessing their Bases
            ideal<P | L> : RngMPol, List -> RngMPol
            Ideal(B) : [ RngMPolElt ] -> RngMPol
            Ideal(f) : RngMPolElt -> RngMPol
            IdealWithFixedBasis(B) : [ RngMPolElt ] -> RngMPol
            Basis(I) : RngMPol -> [ RngMPolElt ]
            BasisElement(I, i) : RngMPol, RngIntElt -> RngMPolElt

 
Gröbner Bases

      Gröbner Bases over Fields

      Gröbner Bases over Euclidean Rings

      Construction of Gröbner Bases
            Groebner(I: parameters) : RngMPol ->
            GroebnerBasis(I: parameters) : RngMPol -> RngMPolElt
            GroebnerBasis(S: parameters) : [ RngMPolElt ] -> [ RngMPolElt ]
            GroebnerBasisUnreduced(S: parameters) : [ RngMPolElt ] -> [ RngMPolElt ]
            GroebnerBasis(S, d: parameters) : [ RngMPol ], RngInt -> RngMPolElt

      Related Functions
            HasGroebnerBasis(I) : RngMPol -> BoolElt
            EasyIdeal(I) : RngMPol -> RngMPol
            EasyBasis(I) : RngMPol -> [ RngMPolElt ]
            SmallBasis(I) : RngMPol -> [ RngMPolElt ]
            MarkGroebner(I) : RngMPol ->
            IsGroebner(S) : { RngMPolElt } -> BoolElt
            Coordinates(I, f) : RngMPol, RngMPolElt -> [ RngMPolElt ]
            CoordinateMatrix(I) : RngMPol -> Matrix
            Reduce(S) : [ RngMPolElt ] -> [ RngMPolElt ]
            ReduceGroebnerBasis(S) : [ RngMPolElt ] -> [ RngMPolElt ]

      Gröbner Bases of Boolean Polynomial Rings
            BooleanPolynomialRing(n) : RngIntElt -> RngMPolBool
            BooleanPolynomialRing(n, order) : RngIntElt, MonStgElt -> RngMPolBool
            BooleanPolynomialRing(B, Q) : RngMPolBool, [RngIntElt] -> RngMPolBoolElt

      Verbosity
            SetVerbose("Groebner", v) : MonStgElt, RngIntElt ->
            SetVerbose("Buchberger", v) : MonStgElt, RngIntElt ->
            SetVerbose("Faugere", v) : MonStgElt, RngIntElt ->
            SetVerbose("FGLM", v) : MonStgElt, RngIntElt ->
            SetVerbose("GroebnerWalk", v) : MonStgElt, RngIntElt ->
            Example GB_Cyclic6 (H98E3)
            Example GB_RungeKutta2 (H98E4)
            Example GB_SolveOverGF2 (H98E5)
            Example GB_GBoverZ (H98E6)
            Example GB_FindingPrimes (H98E7)
            Example GB_QuadraticOrderGB (H98E8)
            Example GB_Coordinates (H98E9)
            Example GB_ValuationRing (H98E10)

      Degree-d Gröbner Bases
            GroebnerBasis(S, d : parameters) : [ RngMPolElt ], RngInt -> RngMPolElt
            Example GB_Degree-d (H98E11)

 
SAT solver
      SAT(B) : [ RngMPolBoolElt ] -> BoolElt, [ FldFinElt ]
      Example GB_SAT (H98E12)

 
Operations on Ideals

      Basic Operations
            I + J : RngMPol, RngMPol -> RngMPol
            I * J : RngMPol, RngMPol -> RngMPol
            I ^ k : RngMPol, RngIntElt -> RngMPol
            I / J : RngMPol, RngMPol -> RngMPolRes
            QuotientDimension(I) : RngMPol -> RngIntElt
            ColonIdeal(I, J) : RngMPol, RngMPol -> RngMPol
            ColonIdeal(I, f) : RngMPol, RngMPolElt -> RngMPol, RngIntElt
            ColonIdealEquivalent(I, f) : RngMPol, RngMPolElt -> RngMPol, RngMPolElt
            Generic(I) : RngMPol -> RngMPol
            LeadingMonomialIdeal(I) : RngMPol -> RngMPol
            I meet J : RngMPol, RngMPol -> RngMPol
            &meet S : [ RngMPol ] -> RngMPol
            Saturation(I, J) : RngMPol, RngMPol -> RngMPol
            Saturation(I, x) : RngMPol, RngMPolElt -> RngMPol
            Saturation(I): RngMPol -> RngMPol
            RegularSequence(I): RngMPol -> SeqEnum

      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 GB_IdealArithmetic (H98E13)

      Elements of Ideals
            f in I : RngMPolElt, RngMPol -> BoolElt
            IsInRadical(f, I) : RngMPolElt, RngMPol -> BoolElt
            JacobianIdeal(f) : RngMPolElt -> RngMPol
            NormalForm(f, I) : RngMPolElt, RngMPol -> RngMPolElt
            NormalForm(f, S) : RngMPolElt, [ RngMPolElt ] -> RngMPolElt, [ RngMPolElt ]
            f notin I : RngMPolElt, RngMPol -> BoolElt
            SPolynomial(f, g) : RngMPolElt, RngMPolElt -> RngMPolElt
            Example GB_ElementOperations (H98E14)

 
Computation of Varieties
      Variety(I) : RngMPol -> [ ModTupFldElt ]
      VarietySequence(I) : RngMPol -> [ [ RngElt ] ]
      VarietySizeOverAlgebraicClosure(I) : RngMPol -> RngIntElt
      Example GB_Variety (H98E15)

 
Multiplicities
      MilnorNumber(f) : RngMPolElt -> RngElt
      TjurinaNumber(f) : RngMPolElt -> RngElt
      Example GB_Variety (H98E16)

 
Elimination

      Construction of Elimination Ideals
            EliminationIdeal(I, k: parameters) : RngMPol, RngIntElt -> RngMPol
            EliminationIdeal(I, S) : RngMPol, { RngIntElt } -> RngMPol
            Example GB_QuadraticOrderElim (H98E17)

      Univariate Elimination Ideal Generators
            UnivariateEliminationIdealGenerator(I, i) : RngMPol, RngIntElt -> RngMPolElt
            UnivariateEliminationIdealGenerators(I) : RngMPol -> [ RngMPolElt ]
            Example GB_EliminationIdeal (H98E18)
            Example GB_ZRadical (H98E19)

      Relation Ideals
            RelationIdeal(Q) : [ RngMPol ] -> RngMPol
            Example GB_RelationIdeal (H98E20)

 
Changing Coefficient Ring
      ChangeRing(I, S) : RngMPol, Rng -> RngMPol
      Example GB_ChangeRing (H98E21)

 
Changing Monomial Order
      ChangeOrder(I, Q) : RngMPol, RngMPol -> RngMPol, Map
      ChangeOrder(I, order) : RngMPol, ..., -> RngMPol, Map
      ChangeOrder(I, T) : RngMPol, Tup -> RngMPol
      Example GB_ChangeOrder (H98E22)

 
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 GB_Radical (H98E23)

      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 GB_PrimaryDecomposition (H98E24)

      Triangular Decomposition
            TriangularDecomposition(I) : RngMPol -> [ RngMPol ], BoolElt
            Example GB_TriangularDecomposition (H98E25)

      Equidimensional Decomposition
            EquidimensionalPart(I) : RngMPol -> RngMPol
            Example GB_EquidimensionalDecomposition (H98E26)

 
Normalisation and Noether Normalisation

      Noether Normalisation
            NoetherNormalisation(I) : RngMPol -> [RngMPolElt],Map,Map
            Example GB_NoetherNormalisation (H98E27)

      Normalisation
            Normalisation(I) : RngMPol -> List
            Example GB_Normalisation (H98E28)

 
Hilbert Series and Hilbert Polynomial
      HilbertSeries(I) : RngMPol -> FldFunUElt
      HilbertSeries(I, p) : RngMPol, RngIntElt -> RngSerLaurElt
      HilbertDenominator(M) : RngMPol -> RngUPol
      HilbertNumerator(M) : RngMPol -> RngUPol
      HilbertPolynomial(I) : RngMPol -> RngUPolElt, RngIntElt
      Example GB_Hilbert (H98E29)

 
Hilbert-driven Gröbner Basis Construction
      HilbertGroebnerBasis(S, H) : [ RngMPolElt ], FldFunRatUElt -> BoolElt, [ RngMPolElt ]
      SetVerbose("HilbertGroebner", v) : MonStgElt, RngIntElt ->
      Example GB_HilbertGroebner (H98E30)

 
Syzygy Modules
      SyzygyModule(Q) : [ RngMPolElt ] -> ModTupRng
      SyzygyMatrix(Q) : [ RngMPolElt ] -> ModMatRngElt
      Example GB_SyzygyModule (H98E31)

 
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 GB_Map1 (H98E32)

 
Symmetric Polynomials
      ElementarySymmetricPolynomial(P, k) : RngMPol, RngIntElt -> RngMPolElt
      IsSymmetric(f) : RngMPolElt -> BoolElt, RngMPolElt
      Example GB_IsSymmetric (H98E33)

 
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 ] ]

 
Bibliography

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