Representation and Monomial Orders
Graded Reverse Lexicographical: grev-lex
Graded Reverse Lexicographical (Weighted): grev-lexw
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 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
Wed Apr 24 15:09:57 EST 2013