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GRÖBNER BASES

 
Acknowledgements
 
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
      Creation of Graded Polynomial Rings
      Element Operations Using the Grading
      Creation of Ideals and Accessing their Bases
 
Gröbner Bases
      Gröbner Bases over Fields
      Gröbner Bases over Euclidean Rings
      Construction of Gröbner Bases
      Related Functions
      Gröbner Bases of Boolean Polynomial Rings
      Verbosity
      Degree-d Gröbner Bases
 
Changing Coefficient Ring
 
Changing Monomial Order
 
Hilbert-driven Gröbner Basis Construction
 
SAT solver
 
Bibliography







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 (H105E1)

      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 (H105E2)

      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
            NormalForm(f, I) : RngMPolElt, RngMPol -> RngMPolElt
            NormalForm(f, S) : RngMPolElt, [ RngMPolElt ] -> RngMPolElt, [ RngMPolElt ]
            SPolynomial(f, g) : RngMPolElt, RngMPolElt -> RngMPolElt
            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 (H105E3)
            Example GB_RungeKutta2 (H105E4)
            Example GB_SolveOverGF2 (H105E5)
            Example GB_GBoverZ (H105E6)
            Example GB_FindingPrimes (H105E7)
            Example GB_QuadraticOrderGB (H105E8)
            Example GB_Coordinates (H105E9)
            Example GB_ValuationRing (H105E10)

      Degree-d Gröbner Bases
            GroebnerBasis(S, d : parameters) : [ RngMPolElt ], RngInt -> RngMPolElt
            Example GB_Degree-d (H105E11)

 
Changing Coefficient Ring
      ChangeRing(I, S) : RngMPol, Rng -> RngMPol
      Example GB_ChangeRing (H105E12)

 
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 (H105E13)

 
Hilbert-driven Gröbner Basis Construction
      HilbertGroebnerBasis(S, H) : [ RngMPolElt ], FldFunRatUElt -> BoolElt, [ RngMPolElt ]
      SetVerbose("HilbertGroebner", v) : MonStgElt, RngIntElt ->
      Example GB_HilbertGroebner (H105E14)

 
SAT solver
      SAT(B) : [ RngMPolBoolElt ] -> BoolElt, [ FldFinElt ]
      Example GB_SAT (H105E15)

 
Bibliography

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