# tStronglyStableMon -- give the t-strongly stable set generated by a given monomial

## Synopsis

• Usage:
tStronglyStableMon(u,t)
• Inputs:
• u, a t-spread monomial of a polynomial ring
• t, a positive integer that idenfies the t-spread contest
• Outputs:
• a list, the list of all the t-spread monomials of the t-strongly stable set generated by u

## Description

the function tStronglyStableMon(u,t) gives the list of all the monomials belonging to the t-strongly stable set generated by u, that is, $B_t\{u\}.$
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_9] o1 = S o1 : PolynomialRing i2 : tStronglyStableMon(x_2*x_5*x_8,2) 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 , x x x , 1 3 5 1 3 6 1 3 7 1 3 8 1 4 6 1 4 7 1 4 8 1 5 7 1 5 8 ------------------------------------------------------------------------ x x x , x x x , x x x , x x x , x x x } 2 4 6 2 4 7 2 4 8 2 5 7 2 5 8 o2 : List i3 : tStronglyStableMon(x_2*x_5*x_8,3) o3 = {x x x , x x x , x x x , x x x } 1 4 7 1 4 8 1 5 8 2 5 8 o3 : List