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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, released February 01, 2021.

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/stable/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 /build/reproducible-path/macaulay2-1.25.06+ds/M2/Macaulay2/packages/TSpreadIdeals.m2:725:0.