Creation of Polynomial Rings and their Ideals
Simple Ideal Constructions
I + J : RngMPol, RngMPol -> RngMPol
I * J : RngMPol, RngMPol -> RngMPol
I ^ k : RngMPol, RngIntElt -> RngMPol
I / J : RngMPol, RngMPol -> RngMPolRes
Basic Commutative Algebra Operations
QuotientDimension(I) : RngMPol -> RngIntElt
ColonIdeal(I, J) : RngMPol, RngMPol -> RngMPol
ColonIdeal(I, f) : RngMPol, RngMPolElt -> RngMPol, RngIntElt
ColonIdealEquivalent(I, f) : RngMPol, RngMPolElt -> RngMPol, RngMPolElt
Saturation(I, J) : RngMPol, RngMPol -> RngMPol
Saturation(I): RngMPol -> RngMPol
Generic(I) : RngMPol -> RngMPol
LeadingMonomialIdeal(I) : RngMPol -> RngMPol
I meet J : RngMPol, RngMPol -> RngMPol
&meet S : [ RngMPol ] -> RngMPol
RegularSequence(I): RngMPol -> SeqEnum
ReesIdeal(P, I): RngMPol, RngMPol -> RngMPol, Map
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
IsHomogeneous(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
HasGrevlexOrder(I) : RngMPol -> BoolElt
Example Ideal_IdealArithmetic (H106E1)
Element Operations with Ideals
f in I : RngMPolElt, RngMPol -> BoolElt
f notin I : RngMPolElt, RngMPol -> BoolElt
IsInRadical(f, I) : RngMPolElt, RngMPol -> BoolElt
JacobianIdeal(f) : RngMPolElt -> RngMPol
Example Ideal_ElementOperations (H106E2)
Computation of Varieties
Variety(I) : RngMPol -> [ ModTupFldElt ]
VarietySequence(I) : RngMPol -> [ [ RngElt ] ]
VarietySizeOverAlgebraicClosure(I) : RngMPol -> RngIntElt
Example Ideal_Variety (H106E3)
Multiplicities
MilnorNumber(f) : RngMPolElt -> RngElt
TjurinaNumber(f) : RngMPolElt -> RngElt
Example Ideal_Variety (H106E4)
Construction of Elimination Ideals
EliminationIdeal(I, k: parameters) : RngMPol, RngIntElt -> RngMPol
EliminationIdeal(I, S) : RngMPol, { RngIntElt } -> RngMPol
Example Ideal_QuadraticOrderElim (H106E5)
Univariate Elimination Ideal Generators
UnivariateEliminationIdealGenerator(I, i) : RngMPol, RngIntElt -> RngMPolElt
UnivariateEliminationIdealGenerators(I) : RngMPol -> [ RngMPolElt ]
Example Ideal_EliminationIdeal (H106E6)
Example Ideal_ZRadical (H106E7)
Relation Ideals
RelationIdeal(Q) : [ RngMPol ] -> RngMPol
Example Ideal_RelationIdeal (H106E8)
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 Ideal_Radical (H106E9)
Primary Decomposition
PrimaryDecomposition(I) : RngMPol -> [ RngMPol ], [ RngMPol ]
RadicalDecomposition(I) : RngMPol -> [ RngMPol ]
ProbableRadicalDecomposition(I) : RngMPol -> [ RngMPol ]
MinimalDecomposition(S) : [ RngMPol ] -> [ RngMPol ]
SetVerbose("Decomposition", v) : MonStgElt, RngIntElt ->
Example Ideal_PrimaryDecomposition (H106E10)
Triangular Decomposition
TriangularDecomposition(I) : RngMPol -> [ RngMPol ], BoolElt
Example Ideal_TriangularDecomposition (H106E11)
Equidimensional Decomposition
EquidimensionalPart(I) : RngMPol -> RngMPol
Example Ideal_EquidimensionalDecomposition (H106E12)
Normalisation and Noether Normalisation
Noether Normalisation
NoetherNormalisation(I) : RngMPol -> [RngMPolElt],Map,Map
Example Ideal_NoetherNormalisation (H106E13)
Normalisation
Normalisation(I) : RngMPol -> List
Example Ideal_Normalisation (H106E14)
Hilbert Series and Hilbert Polynomial
HilbertSeries(I) : RngMPol -> FldFunUElt
HilbertSeries(I, p) : RngMPol, RngIntElt -> RngSerLaurElt
HilbertDenominator(I) : RngMPol -> RngUPol
HilbertNumerator(I) : RngMPol -> RngUPol
HilbertPolynomial(I) : RngMPol -> RngUPolElt, RngIntElt
Example Ideal_Hilbert (H106E15)
Syzygies
SyzygyMatrix(Q) : [ RngMPolElt ] -> ModMatRngElt
Example Ideal_SyzygyMatrix (H106E16)
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 Ideal_Map1 (H106E17)
Symmetric Polynomials
ElementarySymmetricPolynomial(P, k) : RngMPol, RngIntElt -> RngMPolElt
IsSymmetric(f) : RngMPolElt -> BoolElt, RngMPolElt
Example Ideal_IsSymmetric (H106E18)
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 ] ]
Example Ideal_HomogeneousModuleTest1 (H106E19)
Bibliography
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Version: V2.19 of
Wed Apr 24 15:09:57 EST 2013