# 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.

## 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 .