# 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

## Synopsis

• Usage:
tLexIdeal(S,f,t) or tLexIdeal(I,t)
• Inputs:
• S, a polynomial ring
• f, a sequence of nonnegative integers
• I, a t-strongly stable ideal of a polynomial ring
• t, a positive integer that idenfies the t-spread contest
• Outputs:
• an ideal, the t-spread lex ideal with ft-vector f or the t-spread lex ideal with the same ft-vector of I

## Description

It has been proved that if I is a t-strongly stable ideal then a unique t-lex ideal with the same $f_\texttt{t}$-vector of I exists.
Let $\texttt{S}=K[x_1,\ldots,x_n]$, $\texttt{t}\geq 1$. The method tLexIdeal(S,f,t) gives the t-lex ideal of S with f as $f_\texttt{t}$-vector, if exists. The overloading method tLexIdeal(I,t) gives the t-lex ideal with the same $f_\texttt{t}$-vector of the t-strongly stable ideal I.

Examples:

 i1 : S=QQ[x_1..x_8] o1 = S o1 : PolynomialRing i2 : f={1,8,2,0,0} o2 = {1, 8, 2, 0, 0} o2 : List i3 : I=tLexIdeal(S,f,2) o3 = ideal (x x , x x , x x , x x , x x , x x , x x , x x , x x , x x , x x , 1 3 1 4 2 4 1 5 2 5 3 5 1 6 2 6 3 6 4 6 1 7 ------------------------------------------------------------------------ x x , x x , x x , x x , x x , x x , x x , x x ) 2 7 3 7 4 7 5 7 1 8 2 8 3 8 4 8 o3 : Ideal of S i4 : fTVector(I,2)==f o4 = true i5 : isTLexIdeal(I,2) o5 = true i6 : J=tStronglyStableIdeal(ideal {x_1*x_4*x_6},2) o6 = ideal (x x x , x x x , x x x ) 1 3 5 1 3 6 1 4 6 o6 : Ideal of S i7 : K=tLexIdeal(J,2) o7 = ideal (x x x , x x x , x x x , x x x x ) 1 3 5 1 3 6 1 3 7 1 4 6 8 o7 : Ideal of S i8 : fTVector(J,2)==fTVector(K,2) o8 = true

• isTLexIdeal -- whether a given t-spread ideal is t-spread lex
• fTVector -- compute the ft-vector of a given t-spread ideal of a polynomial ring
• isFTVector -- wheter a given list of nonnegative intengers is the ft-vector of a t-strongly stable ideal of a given polynomial ring
• isTStronglyStableIdeal -- whether a given t-spread ideal is t-strongly stable

## Ways to use tLexIdeal :

• "tLexIdeal(Ideal,ZZ)"
• "tLexIdeal(Ring,List,ZZ)"

## For the programmer

The object tLexIdeal is .