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LexIdeals :: LexIdeals

LexIdeals -- a package for working with lex ideals

Description

LexIdeals is a package for creating lexicographic ideals and lex-plus-powers (LPP) ideals. There are also several functions for use with the multiplicity conjectures of Herzog, Huneke, and Srinivasan.

Author

Version

This documentation describes version 1.2 of LexIdeals.

Source code

The source code from which this documentation is derived is in the file LexIdeals.m2.

Exports

  • Functions and commands
    • cancelAll -- make all potentially possible cancellations in the graded free resolution of an ideal
    • generateLPPs -- return all LPP ideals corresponding to a given Hilbert function
    • hilbertFunct -- return the Hilbert function of a polynomial ring mod a homogeneous ideal as a list
    • isCM -- test whether a polynomial ring modulo a homogeneous ideal is Cohen-Macaulay
    • isHF -- is a finite list a Hilbert function of a polynomial ring mod a homogeneous ideal
    • isLexIdeal -- determine whether an ideal is a lexicographic ideal
    • isLPP -- determine whether an ideal is an LPP ideal
    • isPurePower -- determine whether a ring element is a pure power of a variable
    • lexIdeal -- produce a lexicographic ideal
    • LPP -- return the lex-plus-powers (LPP) ideal corresponding to a given Hilbert function and power sequence
    • macaulayBound -- the bound on the growth of a Hilbert function from Macaulay's Theorem
    • macaulayLowerOperator -- the a_<d> operator used in Green's proof of Macaulay's Theorem
    • macaulayRep -- the Macaulay representation of an integer
    • multBounds -- determine whether an ideal satisfies the upper and lower bounds of the multiplicity conjecture
    • multLowerBound -- determine whether an ideal satisfies the lower bound of the multiplicity conjecture
    • multUpperBound -- determine whether an ideal satisfies the upper bound of the multiplicity conjecture
    • multUpperHF -- test a sufficient condition for the upper bound of the multiplicity conjecture
  • Methods
    • "cancelAll(Ideal)" -- see cancelAll -- make all potentially possible cancellations in the graded free resolution of an ideal
    • "generateLPPs(PolynomialRing,List)" -- see generateLPPs -- return all LPP ideals corresponding to a given Hilbert function
    • "hilbertFunct(Ideal)" -- see hilbertFunct -- return the Hilbert function of a polynomial ring mod a homogeneous ideal as a list
    • "isCM(Ideal)" -- see isCM -- test whether a polynomial ring modulo a homogeneous ideal is Cohen-Macaulay
    • "isHF(List)" -- see isHF -- is a finite list a Hilbert function of a polynomial ring mod a homogeneous ideal
    • "isLexIdeal(Ideal)" -- see isLexIdeal -- determine whether an ideal is a lexicographic ideal
    • "isLPP(Ideal)" -- see isLPP -- determine whether an ideal is an LPP ideal
    • "isPurePower(RingElement)" -- see isPurePower -- determine whether a ring element is a pure power of a variable
    • "lexIdeal(Ideal)" -- see lexIdeal -- produce a lexicographic ideal
    • "lexIdeal(PolynomialRing,List)" -- see lexIdeal -- produce a lexicographic ideal
    • "lexIdeal(QuotientRing,List)" -- see lexIdeal -- produce a lexicographic ideal
    • "LPP(PolynomialRing,List,List)" -- see LPP -- return the lex-plus-powers (LPP) ideal corresponding to a given Hilbert function and power sequence
    • "macaulayBound(ZZ,ZZ)" -- see macaulayBound -- the bound on the growth of a Hilbert function from Macaulay's Theorem
    • "macaulayLowerOperator(ZZ,ZZ)" -- see macaulayLowerOperator -- the a_<d> operator used in Green's proof of Macaulay's Theorem
    • "macaulayRep(ZZ,ZZ)" -- see macaulayRep -- the Macaulay representation of an integer
    • "multBounds(Ideal)" -- see multBounds -- determine whether an ideal satisfies the upper and lower bounds of the multiplicity conjecture
    • "multLowerBound(Ideal)" -- see multLowerBound -- determine whether an ideal satisfies the lower bound of the multiplicity conjecture
    • "multUpperBound(Ideal)" -- see multUpperBound -- determine whether an ideal satisfies the upper bound of the multiplicity conjecture
    • "multUpperHF(PolynomialRing,List)" -- see multUpperHF -- test a sufficient condition for the upper bound of the multiplicity conjecture
  • Symbols

For the programmer

The object LexIdeals is a package.