# isTStronglyStableIdeal -- whether a given t-spread ideal is t-strongly stable

## Synopsis

• Usage:
isTStronglyStableIdeal(I,t)
• Inputs:
• I, a t-spread ideal of a polynomial ring
• t, a positive integer that idenfies the t-spread contest
• Outputs:
• , whether the ideal I is t-spread lex

## Description

Let $\texttt{S}=K[x_1,\ldots,x_n]$, $\texttt{t}\geq 1$ and a t-spread ideal I. Then I is called a t-strongly stable ideal, if $[I_j]_t$ is a t-strongly stable set for all $j$.
We recall that $[I_j]_t$ is the t-spread part of the $j$-th graded component of IMoreover, a subset $N\subset M_{n,d,t}$ is called a t-strongly stable set if taking a t-spread monomial $u\in N$, for all $j\in \mathrm{supp}(u)$ and all $i,\ 1\leq i\leq j$, such that $x_i(u/x_j)$ is a t-spread monomial, then it follows that $x_i(u/x_j)\in N$.

Examples:

 i1 : S=QQ[x_1..x_6] o1 = S o1 : PolynomialRing i2 : isTStronglyStableIdeal(ideal {x_1*x_3,x_1*x_5},2) o2 = false i3 : isTStronglyStableIdeal(ideal {x_1*x_3,x_1*x_4,x_1*x_5,x_2*x_4,x_2*x_5},2) o3 = true

• tStronglyStableIdeal -- give the smallest t-strongly stable ideal containing a given t-spread ideal

## Ways to use isTStronglyStableIdeal :

• "isTStronglyStableIdeal(Ideal,ZZ)"

## For the programmer

The object isTStronglyStableIdeal is .