Polynomial Ring Ideal Operations

• Introduction
• Creation of Polynomial Rings and their Ideals
• First Operations on 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
    • IdealQuotient(I, J): RngMPol, RngMPol → RngMPol
    • ColonIdeal(I, f): RngMPol, RngMPolElt → RngMPol, RngIntElt
    • IdealQuotient(I, f): RngMPol, RngMPolElt → RngMPol, RngIntElt
    • ColonIdealEquivalent(I, f): RngMPol, RngMPolElt → RngMPol, RngMPolElt
    • Saturation(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
    • ReesIdeal(P, J, I): RngMPol, RngMPol, RngMPol → RngMPol, Map
    • ReesIdeal(R, I): RngMPolRes, 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 Arithmetic
  • 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: Element Operations with Ideals
• Computation of Varieties
  • Variety(I): RngMPol → [ ModTupFldElt ]
  • Variety(I, L): RngMPol, Rng → [ ModTupFldElt ]
  • VarietySequence(I): RngMPol → [ [ RngElt ] ]
  • VarietySequence(I, L): RngMPol, Rng → [ [ RngElt ] ]
  • VarietySize(I): RngMPol → RngIntElt
  • Example: Variety over a Finite Field
• Multiplicities
  • MilnorNumber(f): RngMPolElt → RngElt
  • TjurinaNumber(f): RngMPolElt → RngElt
  • Example: Milnor and Tjurina Numbers
• Elimination
  • Construction of Elimination Ideals
    • EliminationIdeal(I, k: parameters): RngMPol, RngIntElt → RngMPol
    • EliminationIdeal(I, S): RngMPol, { RngIntElt } → RngMPol
    • EliminationIdeal(I, S): RngMPol, { RngMPolElt } → RngMPol
    • Example: Elimination over a Quadratic Order
  • Univariate Elimination Ideal Generators
    • UnivariateEliminationIdealGenerator(I, i): RngMPol, RngIntElt → RngMPolElt
    • UnivariateEliminationIdealGenerators(I): RngMPol → [ RngMPolElt ]
    • Example: Elimination Ideals from Neural Network Theory
    • Example: Radical of a Zero-Dimensional Ideal
  • Relation Ideals
    • RelationIdeal(Q): [ RngMPol ] → RngMPol
    • RelationIdeal(Q, T): [ RngMPol ], RngMPol → RngMPol
    • Example: Expressing One in Terms of Generators
• Variable Extension of Ideals
  • VariableExtension(I, k, b): RngMPol, RngIntElt, BoolElt → RngMPol, Map
  • VariableExtension(I, k, b, order): RngMPol, RngIntElt, BoolElt, ... → RngMPol, Map
• Homogenization of Ideals
  • Homogenization(I, b): RngMPol, RngIntElt, BoolElt → RngMPol, Map
  • Homogenization(I, b, order): RngMPol, RngIntElt, BoolElt, ... → RngMPol, Map
  • Homogenization(I): RngMPol, RngIntElt, BoolElt → RngMPol, Map
  • Homogenization(I, order): 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: Radical of an Ideal
  • 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: Primary Decomposition
  • Triangular Decomposition
    • TriangularDecomposition(I): RngMPol → [ RngMPol ], BoolElt
    • Example: Triangular Decomposition of Cyclic-5
  • Equidimensional Decomposition
    • EquidimensionalPart(I): RngMPol → RngMPol
    • EquidimensionalDecomposition(I): RngMPol → [ RngMPol ]
    • FineEquidimensionalDecomposition(I): RngMPol → SeqEnum
    • Example: Equidimensional Decomposition
• Normalisation and Noether Normalisation
  • Noether Normalisation
    • NoetherNormalisation(I): RngMPol → [RngMPolElt], Map, Map
    • NoetherNormalization(I): RngMPol → [RngMPolElt], Map, Map
    • Example: Noether Normalisation
  • Normalisation
    • Normalisation(I): RngMPol → List
    • Normalization(I): RngMPol → List
    • Example: Normalisation of Plane Curves
• 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: Hilbert Series of a Matrix Square Ideal
• Syzygies
  • SyzygyMatrix(Q): [ RngMPolElt ] → ModMatRngElt
  • Example: Syzygy Matrix
• 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: Implicitization of a Monomial Map
• Symmetric Polynomials
  • ElementarySymmetricPolynomial(P, k): RngMPol, RngIntElt → RngMPolElt
  • IsSymmetric(f): RngMPolElt → BoolElt, RngMPolElt
  • IsSymmetric(f, S): RngMPolElt, RngMPol → BoolElt, RngMPolElt
  • Example: Symmetric Polynomial in Elementary Terms
• Functions for Polynomial Algebra and Module Generators
  • MinimalAlgebraGenerators(L): [ RngMPol ] → [ RngMPol ]
  • 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: Homogeneous Module Test
• Bibliography