Representation and Monomial Orders
Graded Reverse Lexicographical: grev-lex
Graded Reverse Lexicographical with Weights: grev-lexw
Creation of Polynomial Rings and Ideals
Creation of Polynomial Rings
PolynomialRing(R, n) : Rng, RngIntElt -> RngMPol
PolynomialRing(R, n, order) : Rng, RngIntElt, MonStgElt, ... -> RngMPol
Example GB_Order (H94E1)
Creation of Ideals and Accessing their Bases
ideal<P | L> : RngMPol, List -> 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
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 ->
Related Functions
HasGroebnerBasis(I) : RngMPol -> BoolElt
EasyIdeal(I) : RngMPol -> RngMPol
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 ]
Example GB_Cyclic6 (H94E2)
Example GB_RungeKutta2 (H94E3)
Example GB_SolveOverGF2 (H94E4)
Example GB_GBoverZ (H94E5)
Example GB_FindingPrimes (H94E6)
Example GB_QuadraticOrderGB (H94E7)
Example GB_Coordinates (H94E8)
Construction of New Ideals
I + J : RngMPol, RngMPol -> RngMPol
I * J : RngMPol, RngMPol -> RngMPol
I ^ k : RngMPol, RngIntElt -> RngMPol
I / J : RngMPol, RngMPol -> RngMPolRes
ColonIdeal(I, J) : RngMPol, RngMPol -> RngMPol
ColonIdeal(I, f) : RngMPol, RngMPolElt -> RngMPol, RngIntElt
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
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
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
Example GB_IdealArithmetic (H94E9)
Operations on 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
f notin I : RngMPolElt, RngMPol -> BoolElt
SPolynomial(f, g) : RngMPolElt, RngMPolElt -> RngMPolElt
Example GB_ElementOperations (H94E10)
Computation of Varieties
Variety(I) : RngMPol -> [ ModTupFldElt ]
VarietySequence(I) : RngMPol -> [ [ RngElt ] ]
VarietySizeOverAlgebraicClosure(I) : RngMPol -> RngIntElt
Example GB_Variety (H94E11)
Construction of Elimination Ideals
EliminationIdeal(I, k: parameters) : RngMPol, RngIntElt -> RngMPol
EliminationIdeal(I, S) : RngMPol, { RngIntElt } -> RngMPol
Example GB_QuadraticOrderElim (H94E12)
Univariate Elimination Ideal Generators
UnivariateEliminationIdealGenerator(I, i) : RngMPol, RngIntElt -> RngMPolElt
UnivariateEliminationIdealGenerators(I) : RngMPol -> [ RngMPolElt ]
Example GB_EliminationIdeal (H94E13)
Example GB_ZRadical (H94E14)
Relation Ideals
RelationIdeal(Q) : [ RngMPol ] -> RngMPol
Example GB_RelationIdeal (H94E15)
Changing Coefficient Ring
ChangeRing(I, S) : RngMPol, Rng -> RngMPol
Example GB_ChangeRing (H94E16)
Changing Monomial Order
ChangeOrder(I, Q) : RngMPol, RngMPol -> RngMPol, Map
ChangeOrder(I, order) : RngMPol, ..., -> RngMPol, Map
Example GB_ChangeOrder (H94E17)
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 (H94E18)
Primary Decomposition
PrimaryDecomposition(I) : RngMPol -> [ RngMPol ], [ RngMPol ]
RadicalDecomposition(I) : RngMPol -> [ RngMPol ]
ProbableRadicalDecomposition(I) : RngMPol -> [ RngMPol ]
SetVerbose("Decomposition", v) : MonStgElt, RngIntElt ->
Example GB_PrimaryDecomposition (H94E19)
Triangular Decomposition
TriangularDecomposition(I) : RngMPol -> [ RngMPol ]
Example GB_TriangularDecomposition (H94E20)
Equidimensional Decomposition
EquidimensionalPart(I) : RngMPol -> RngMPol
Example GB_EquidimensionalDecomposition (H94E21)
Normalisation and Noether Normalisation
Noether Normalisation
NoetherNormalisation(I) : RngMPol -> [RngMPolElt],Map,Map
Example GB_NoetherNormalisation (H94E22)
Normalisation
Normalisation(I) : RngMPol -> List
Example GB_Normalisation (H94E23)
Creation of Graded Polynomial Rings
PolynomialRing(R, Q) : Rng, [ RngIntElt ] -> RngMPol
VariableWeights(P) : RngMPol -> [ RngIntElt ]
Elements of Graded Polynomial Rings
WeightedDegree(f) : RngMPolElt -> RngIntElt
LeadingWeightedDegree(f) : RngMPolElt -> RngIntElt
IsHomogeneous(f) : RngMPolElt -> BoolElt
IsHomogeneous(I) : RngMPol -> BoolElt
HomogeneousComponent(f, d) : RngMPolElt, RngIntElt -> RngMPolElt
HomogeneousComponents(f) : RngMPolElt -> [ RngMPolElt ]
MonomialsOfDegree(P, d) : RngMPolElt, RngIntElt -> {@ RngMPolElt @}
MonomialsOfWeightedDegree(P, d) : RngMPolElt, RngIntElt -> {@ RngMPolElt @}
Degree-d Gröbner Bases
GroebnerBasis(S, d : parameters) : [ RngMPolElt ], RngInt -> RngMPolElt
Example GB_Graded (H94E24)
Example GB_Degree-d (H94E25)
Hilbert Series and Hilbert Polynomial
HilbertSeries(I) : RngMPol -> FldFunUElt
HilbertPolynomial(I) : RngMPol -> RngUPolElt, RngIntElt
Example GB_Hilbert (H94E26)
Hilbert-driven Gröbner Basis Construction
HilbertGroebnerBasis(S, H) : [ RngMPolElt ], FldFunRatUElt -> BoolElt, [ RngMPolElt ]
SetVerbose("HilbertGroebner", v) : MonStgElt, RngIntElt ->
Example GB_HilbertGroebner (H94E27)
Syzygy Modules
SyzygyModule(Q) : [ RngMPolElt ] -> ModTupRng
SyzygyMatrix(Q) : [ RngMPolElt ] -> ModMatRngElt
Example GB_SyzygyModule (H94E28)
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 (H94E29)
Symmetric Polynomials
ElementarySymmetricPolynomial(P, k) : RngMPol, RngIntElt -> RngMPolElt
IsSymmetric(f) : RngMPolElt -> BoolElt, RngMPolElt
Example GB_IsSymmetric (H94E30)
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 ] ]