Introduction

 
Representation and Monomial Orders

      Lexicographical: lex

      Graded Lexicographical: glex

      Graded Reverse Lexicographical: grevlex

      Graded Reverse Lexicographical (Weighted): grevlexw

      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 ] ]
            SetSparseMonomialMinRank(R) : RngIntElt ->
            GetSparseMonomialMinRank() : -> RngIntElt
            Example GB_Order (H115E1)

      Creation of Graded Polynomial Rings
            PolynomialRing(R, Q) : Rng, [ RngIntElt ] -> RngMPol
            Grading(P) : RngMPol -> [ RngIntElt ]
            HomogeneousWeightsSearch(S) : [ RngMPol ] -> BoolElt, [ 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 (H115E2)

      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 ], [ RngIntElt ]
            GroebnerBasis(S: parameters) : [ RngMPolElt ] -> [ RngMPolElt ], [ RngIntElt ], []
            GroebnerBasisUnreduced(S: parameters) : [ RngMPolElt ] -> [ RngMPolElt ]
            GroebnerBasis(S, d: parameters) : [ RngMPol ], RngInt -> RngMPolElt
            SetGBGlobalModular(f) : BoolElt ->
            SetFaugereModular(f) : BoolElt ->

      The Dense Variant of the F₄ algorithm

      Related Functions
            HasGroebnerBasis(I) : RngMPol -> BoolElt
            EasyIdeal(I) : RngMPol -> RngMPol, [ RngIntElt ]
            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

      Construction of Input Systems
            MinRankSystem(K, n, k, r) : FldFin, RngIntElt, RngIntElt, RngIntElt -> [ RngMPolBool ]
            HFESystem(q, n, D) : RngIntElt, RngIntElt, RngIntElt -> [ RngMPolBool ]

      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 (H115E3)
            Example GB_RungeKutta2 (H115E4)
            Example GB_SolveOverGF2 (H115E5)
            Example GB_GBoverZ (H115E6)
            Example GB_FindingPrimes (H115E7)
            Example GB_QuadraticOrderGB (H115E8)
            Example GB_Coordinates (H115E9)
            Example GB_ValuationRing (H115E10)

      Degree-d Gröbner Bases
            GroebnerBasis(S, d : parameters) : [ RngMPolElt ], RngInt -> RngMPolElt
            Example GB_Degree-d (H115E11)

 
Changing Coefficient Ring
      ChangeRing(I, S) : RngMPol, Rng -> RngMPol
      Example GB_ChangeRing (H115E12)

 
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 (H115E13)

 
Hilbert-driven Gröbner Basis Construction
      HilbertGroebnerBasis(S, H) : [ RngMPolElt ], FldFunRatUElt -> BoolElt, [ RngMPolElt ]
      SetVerbose("HilbertGroebner", v) : MonStgElt, RngIntElt ->
      Example GB_HilbertGroebner (H115E14)

 
SAT solver
      SAT(B) : [ RngMPolBoolElt ] -> BoolElt, [ FldFinElt ]
      Example GB_SAT (H115E15)

 
Bibliography

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