Representation and Monomial Orders
Graded Reverse Lexicographical: grevlex
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 (H50E1)
Creation of Ideals and Accessing their Bases
ideal<P | L> : RngMPol, List -> RngMPol
Ideal(Q) : [ 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 ]
Reduce(S) : [ RngMPolElt ] -> [ RngMPolElt ]
ReduceGroebnerBasis(S) : [ RngMPolElt ] -> [ RngMPolElt ]
Example GB_Cyclic6 (H50E2)
Example GB_RungeKutta2 (H50E3)
Example GB_GBoverZ (H50E4)
Example GB_FindingPrimes (H50E5)
Example GB_QuadraticOrderGB (H50E6)
Example GB_Coordinates (H50E7)
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
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 (H50E8)
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 (H50E9)
Computation of Varieties
Variety(I) : RngMPol -> [ ModTupFldElt ]
VarietySequence(I) : RngMPol -> [ [ RngElt ] ]
VarietySizeOverAlgebraicClosure(I) : RngMPol -> RngIntElt
Example GB_Variety (H50E10)
Construction of Elimination Ideals
EliminationIdeal(I, k: parameters) : RngMPol, RngIntElt -> RngMPol
EliminationIdeal(I, S) : RngMPol, { RngIntElt } -> RngMPol
Example GB_QuadraticOrderElim (H50E11)
Univariate Elimination Ideal Generators
UnivariateEliminationIdealGenerator(I, i) : RngMPol, RngIntElt -> RngMPolElt
UnivariateEliminationIdealGenerators(I) : RngMPol -> [ RngMPolElt ]
Example GB_EliminationIdeal (H50E12)
Example GB_ZRadical (H50E13)
Relation Ideals
RelationIdeal(Q) : [ RngMPol ] -> RngMPol
Example GB_RelationIdeal (H50E14)
Changing Coefficient Ring
ChangeRing(I, S) : RngMPol, Rng -> RngMPol
Example GB_ChangeRing (H50E15)
Changing Monomial Order
ChangeOrder(I, Q) : RngMPol, RngMPol -> RngMPol, Map
ChangeOrder(I, order) : RngMPol, ..., -> RngMPol, Map
Example GB_ChangeOrder (H50E16)
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 (H50E17)
Primary Decomposition
PrimaryDecomposition(I) : RngMPol -> [ RngMPol ], [ RngMPol ]
RadicalDecomposition(I) : RngMPol -> [ RngMPol ]
ProbableRadicalDecomposition(I) : RngMPol -> [ RngMPol ]
SetVerbose("Decomposition", v) : MonStgElt, RngIntElt ->
Example GB_PrimaryDecomposition (H50E18)
Triangular Decomposition
TriangularDecomposition(I) : RngMPol -> [ RngMPol ]
Example GB_TriangularDecomposition (H50E19)
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) : RngMPolElt -> {@ RngMPolElt @}
MonomialsOfWeightedDegree(P) : RngMPolElt -> {@ RngMPolElt @}
Degree-d Gröbner Bases
GroebnerBasis(S, d : parameters) : [ RngMPol ], RngInt -> RngMPolElt
Example GB_Graded (H50E20)
Example GB_Degree-d (H50E21)
Hilbert Series and Hilbert Polynomial
HilbertSeries(I) : RngMPol -> FldFunUElt
HilbertPolynomial(M) : RngMPol -> RngUPolElt, RngIntElt
Example GB_Hilbert (H50E22)
Hilbert-driven Gröbner Basis Construction
HilbertGroebnerBasis(S, H) : [ RngMPolElt ], FldFunRatUElt -> BoolElt, [ RngMPolElt ]
SetVerbose("HilbertGroebner", v) : MonStgElt, RngIntElt ->
Example GB_HilbertGroebner (H50E23)
Syzygy Modules
SyzygyModule(Q) : [ RngMPolElt ] -> ModTupRng
SyzygyMatrix(Q) : [ RngMPolElt ] -> ModMatRngElt
Example GB_SyzygyModule (H50E24)
Maps between Polynomial Rings
PolyMapKernel(f) : Map -> RngMPol
IsInImage(f, p) : Map, RngMPolElt -> [ BoolElt ]
IsSurjective(f) : Map -> [ BoolElt ]
Implicitization(f) : Map -> RngMPol
Example GB_Map1 (H50E25)
Symmetric Polynomials
ElementarySymmetricPolynomial(P, k) : RngMPol, RngIntElt -> RngMPolElt
IsSymmetric(f) : RngMPolElt -> BoolElt, RngMPolElt
Example GB_IsSymmetric (H50E26)
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 ] ]