# tStronglyStableSeg -- give the t-strongly stable segment with the given extremes

## Synopsis

• Usage:
tStronglyStableSeg(v,u,t)
• Inputs:
• v, a t-spread monomial of a polynomial ring
• u, a t-spread monomial of a polynomial ring
• t, a positive integer that idenfies the t-spread contest
• Outputs:
• a list, the set of all the t-spread monomials of the t-stromgly stable set generated by u and smaller than u, with respect to lexcografic order

## Description

the function tStronglyStableSeg(v,u,t) gives the set of t-spread monomials belonging to the strongly stable set generated by u and smaller than v, that is, $B_t[v,u]=\{w\in B_t\{u\}\ :\ v\geq_\mathrm{slex} w\}.$
We recall that if $u\in M_{n,d,t}\subset S=K[x_1,\ldots,x_n]$ then $B_t\{u\}$ is the smallest t-strongly stable set of monomials of $M_{n,d,t}$ containing $u.$
Moreover, 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_14] o1 = S o1 : PolynomialRing i2 : tStronglyStableSeg(x_2*x_5*x_9*x_12,x_2*x_6*x_10*x_13,2) o2 = {x x x x , x x x x , x x x x , x x x x , x x x x , x x x x , 2 5 9 12 2 5 9 13 2 5 10 12 2 5 10 13 2 6 8 10 2 6 8 11 ------------------------------------------------------------------------ x x x x , x x x x , x x x x , x x x x , x x x x , x x x x , 2 6 8 12 2 6 8 13 2 6 9 11 2 6 9 12 2 6 9 13 2 6 10 12 ------------------------------------------------------------------------ x x x x } 2 6 10 13 o2 : List i3 : tStronglyStableSeg(x_2*x_5*x_9*x_12,x_2*x_6*x_10*x_13,3) o3 = {x x x x , x x x x , x x x x , x x x x , x x x x , x x x x } 2 5 9 12 2 5 9 13 2 5 10 13 2 6 9 12 2 6 9 13 2 6 10 13 o3 : List

## Ways to use tStronglyStableSeg :

• "tStronglyStableSeg(RingElement,RingElement,ZZ)"

## For the programmer

The object tStronglyStableSeg is .