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

• tLastMon -- give the last t-spread monomial of the Borel shadow a given t-spread monomial
• tPascalIdeal -- give the Pascal ideal of t-spread monomials of a given polynomial ring
• tSpreadList -- give the set of all t-spread monomials of a given list
• tSpreadIdeal -- give the ideal generated by the t-spread monomials which are among the generators of a given ideal
• isTSpread -- whether a monomial, a list of monomials or a monomial ideal is t-spread
• tShadow -- give the t-spread shadow of a given t-spread monomial or a given list of t-spread monomials

T-lex structures

• tNextMon -- give the t-lex successor of a given t-spread monomial
• tLexSeg -- give the t-lex segment with given extremes
• isTLexSeg -- whether a set of t-spread monomials is a t-lex segment
• tLexMon -- give the smallest initial t-lex segment containing a given monomial
• countTLexMon -- give the cardinality of the smallest initial t-lex segment containing a given monomial
• tVeroneseSet -- give the Veronese set of t-spread monomials of a given degree
• tVeroneseIdeal -- give the Veronese ideal of t-spread monomials of a given degree
• isTLexIdeal -- whether a given t-spread ideal is t-spread lex

T-strongly stable structures

• tStronglyStableSeg -- give the t-strongly stable segment with the given extremes
• isTStronglyStableSeg -- whether the given list of t-spread monomials is a t-strongly stable segment
• tStronglyStableMon -- give the t-strongly stable set generated by a given monomial
• countTStronglyStableMon -- give the cardinality of the t-strongly stable set generated by a given monomial
• tStronglyStableIdeal -- give the smallest t-strongly stable ideal containing a given t-spread ideal
• isTStronglyStableIdeal -- whether a given t-spread ideal is t-strongly stable

Kruskal-Katona's Theorem

• tMacaulayExpansion -- compute the t-Macaulay expansion of a positive integer
• fTVector -- compute the ft-vector of a given t-spread ideal of a polynomial ring
• isFTVector -- whether a given list of nonnegative integers is the ft-vector of a t-strongly stable ideal of a given polynomial ring
• 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

Extremal Betti numbers

• 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

Services

• solveBinomialExpansion -- compute the sum of a binomial expansion
• initialDegree -- return the initial degree of a given graded ideal
• initialIdeal -- return the initial ideal of a given ideal
• 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

Options

• Shift -- optional boolean argument for tMacaulayExpansion
• FixedMax -- optional boolean argument for tNextMon
• MaxInd -- optional integer argument for tLastMon