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TSpreadIdeals -- A package for working with t-spread ideals of polynomial rings

Description

TSpreadIdeals is a package for creating and manipulating t-spread ideals of polynomial rings. Let $u=x_{i_1}x_{i_2}\cdots x_{i_d}$ a monomial of $S=K[x_1,\ldots,x_n]$, with $1\le i_1\le i_2\le\dots\le i_d\le n$. The monomial u is called $t$-spread if $i_{j+1}-i_j\ge t$ for all $j\in [d-1].$
A list of monomials is t-spread if all the monomials of the list are t-spread. A monomial ideal is called t-spread if it is generated by t-spread monomials.
Some references can be found in the papers Computational methods for t-spread monomial ideals and A numerical characterization of the extremal Betti numbers of t-spread strongly stable ideals.

Menu

T-spread managing

T-lex structures

T-strongly stable structures

Kruskal-Katona's Theorem

Extremal Betti numbers

Services

Options

Author

Version

This documentation describes version 1.0 of TSpreadIdeals.

Citation

If you have used this package in your research, please cite it as follows:

@misc{TSpreadIdealsSource,
  title = {{TSpreadIdeals: t-spread ideals of a polynomial ring. Version~1.0}},
  author = {Luca Amata},
  howpublished = {A \emph{Macaulay2} package available at
    \url{https://github.com/Macaulay2/M2/tree/master/M2/Macaulay2/packages}}
}

Exports

  • Functions and commands
    • countTLexMon -- give the cardinality of the smallest initial t-lex segment containing a given monomial
    • countTStronglyStableMon -- give the cardinality of the t-strongly stable set generated by a given monomial
    • fTVector -- compute the ft-vector of a given t-spread ideal of a polynomial ring
    • initialDegree -- return the initial degree of a given graded ideal
    • initialIdeal -- return the initial ideal of a given ideal
    • isFTVector -- whether a given list of nonnegative integers is the ft-vector of a t-strongly stable ideal of a given polynomial ring
    • isTLexIdeal -- whether a given t-spread ideal is t-spread lex
    • isTLexSeg -- whether a set of t-spread monomials is a t-lex segment
    • isTSpread -- whether a monomial, a list of monomials or a monomial ideal is t-spread
    • isTStronglyStableIdeal -- whether a given t-spread ideal is t-strongly stable
    • isTStronglyStableSeg -- whether the given list of t-spread monomials is a t-strongly stable segment
    • minimalBettiNumbers -- return the minimal Betti numbers of the quotient ring corresponding to a given graded ideal
    • minimalBettiNumbersIdeal -- return the minimal Betti numbers of a given graded ideal
    • solveBinomialExpansion -- compute the sum of a binomial expansion
    • tExtremalBettiCorners -- give the corners of the extremal Betti numbers of a given t-strongly stable ideal
    • tExtremalBettiMonomials -- give the list of the t-spread basic monomials related to the given extremal Betti numbers configuration
    • tLastMon -- give the last t-spread monomial of the Borel shadow a given t-spread monomial
    • tLexIdeal -- returns the t-spread lex ideal with a given ft-vector or with the same ft-vector of a given t-strongly stable ideal
    • tLexMon -- give the smallest initial t-lex segment containing a given monomial
    • tLexSeg -- give the t-lex segment with given extremes
    • tMacaulayExpansion -- compute the t-Macaulay expansion of a positive integer
    • tNextMon -- give the t-lex successor of a given t-spread monomial
    • tPascalIdeal -- give the Pascal ideal of t-spread monomials of a given polynomial ring
    • tShadow -- give the t-spread shadow of a given t-spread monomial or a given list of t-spread monomials
    • tSpreadIdeal -- give the ideal generated by the t-spread monomials which are among the generators of a given ideal
    • tSpreadList -- give the set of all t-spread monomials of a given list
    • tStronglyStableIdeal -- give the smallest t-strongly stable ideal containing a given t-spread ideal
    • tStronglyStableMon -- give the t-strongly stable set generated by a given monomial
    • tStronglyStableSeg -- give the t-strongly stable segment with the given extremes
    • tVeroneseIdeal -- give the Veronese ideal of t-spread monomials of a given degree
    • tVeroneseSet -- give the Veronese set of t-spread monomials of a given degree
  • Methods
    • countTLexMon(RingElement,ZZ) -- see countTLexMon -- give the cardinality of the smallest initial t-lex segment containing a given monomial
    • countTStronglyStableMon(RingElement,ZZ) -- see countTStronglyStableMon -- give the cardinality of the t-strongly stable set generated by a given monomial
    • fTVector(Ideal,ZZ) -- see fTVector -- compute the ft-vector of a given t-spread ideal of a polynomial ring
    • initialDegree(Ideal) -- see initialDegree -- return the initial degree of a given graded ideal
    • initialIdeal(Ideal) -- see initialIdeal -- return the initial ideal of a given ideal
    • isFTVector(Ring,List,ZZ) -- see isFTVector -- whether a given list of nonnegative integers is the ft-vector of a t-strongly stable ideal of a given polynomial ring
    • isTLexIdeal(Ideal,ZZ) -- see isTLexIdeal -- whether a given t-spread ideal is t-spread lex
    • isTLexSeg(List,ZZ) -- see isTLexSeg -- whether a set of t-spread monomials is a t-lex segment
    • isTSpread(Ideal,ZZ) -- see isTSpread -- whether a monomial, a list of monomials or a monomial ideal is t-spread
    • isTSpread(List,ZZ) -- see isTSpread -- whether a monomial, a list of monomials or a monomial ideal is t-spread
    • isTSpread(RingElement,ZZ) -- see isTSpread -- whether a monomial, a list of monomials or a monomial ideal is t-spread
    • isTStronglyStableIdeal(Ideal,ZZ) -- see isTStronglyStableIdeal -- whether a given t-spread ideal is t-strongly stable
    • isTStronglyStableSeg(List,ZZ) -- see isTStronglyStableSeg -- whether the given list of t-spread monomials is a t-strongly stable segment
    • minimalBettiNumbers(Ideal) -- see minimalBettiNumbers -- return the minimal Betti numbers of the quotient ring corresponding to a given graded ideal
    • minimalBettiNumbersIdeal(Ideal) -- see minimalBettiNumbersIdeal -- return the minimal Betti numbers of a given graded ideal
    • solveBinomialExpansion(List) -- see solveBinomialExpansion -- compute the sum of a binomial expansion
    • tExtremalBettiCorners(Ideal,ZZ) -- see tExtremalBettiCorners -- give the corners of the extremal Betti numbers of a given t-strongly stable ideal
    • tExtremalBettiMonomials(Ring,List,List,ZZ) -- see tExtremalBettiMonomials -- give the list of the t-spread basic monomials related to the given extremal Betti numbers configuration
    • tLastMon(RingElement,ZZ,ZZ) -- see tLastMon -- give the last t-spread monomial of the Borel shadow a given t-spread monomial
    • tLexIdeal(Ideal,ZZ) -- see tLexIdeal -- returns the t-spread lex ideal with a given ft-vector or with the same ft-vector of a given t-strongly stable ideal
    • tLexIdeal(Ring,List,ZZ) -- see tLexIdeal -- returns the t-spread lex ideal with a given ft-vector or with the same ft-vector of a given t-strongly stable ideal
    • tLexMon(RingElement,ZZ) -- see tLexMon -- give the smallest initial t-lex segment containing a given monomial
    • tLexSeg(RingElement,RingElement,ZZ) -- see tLexSeg -- give the t-lex segment with given extremes
    • tMacaulayExpansion(ZZ,ZZ,ZZ,ZZ) -- see tMacaulayExpansion -- compute the t-Macaulay expansion of a positive integer
    • tNextMon(RingElement,ZZ) -- see tNextMon -- give the t-lex successor of a given t-spread monomial
    • tPascalIdeal(Ring,ZZ) -- see tPascalIdeal -- give the Pascal ideal of t-spread monomials of a given polynomial ring
    • tShadow(List,ZZ) -- see tShadow -- give the t-spread shadow of a given t-spread monomial or a given list of t-spread monomials
    • tShadow(RingElement,ZZ) -- see tShadow -- give the t-spread shadow of a given t-spread monomial or a given list of t-spread monomials
    • tSpreadIdeal(Ideal,ZZ) -- see tSpreadIdeal -- give the ideal generated by the t-spread monomials which are among the generators of a given ideal
    • tSpreadList(List,ZZ) -- see tSpreadList -- give the set of all t-spread monomials of a given list
    • tStronglyStableIdeal(Ideal,ZZ) -- see tStronglyStableIdeal -- give the smallest t-strongly stable ideal containing a given t-spread ideal
    • tStronglyStableMon(RingElement,ZZ) -- see tStronglyStableMon -- give the t-strongly stable set generated by a given monomial
    • tStronglyStableSeg(RingElement,RingElement,ZZ) -- see tStronglyStableSeg -- give the t-strongly stable segment with the given extremes
    • tVeroneseIdeal(Ring,ZZ,ZZ) -- see tVeroneseIdeal -- give the Veronese ideal of t-spread monomials of a given degree
    • tVeroneseSet(Ring,ZZ,ZZ) -- see tVeroneseSet -- give the Veronese set of t-spread monomials of a given degree
  • Symbols
    • FixedMax -- optional boolean argument for tNextMon
    • MaxInd -- optional integer argument for tLastMon
    • Shift -- optional boolean argument for tMacaulayExpansion

For the programmer

The object TSpreadIdeals is a package, defined in TSpreadIdeals.m2.

The source of this document is in TSpreadIdeals.m2:725:0.