# isFTVector -- wheter a given list of nonnegative intengers is the ft-vector of a t-strongly stable ideal of a given polynomial ring

## Synopsis

• Usage:
isFTVector(S,f,t)
• Inputs:
• S, a polynomial ring
• f, a list of nonnegative integers
• t, a positive integer that idenfies the t-spread contest
• Outputs:
• , wheter the list f represents the ft-vector of a t-strongly stable ideal of S

## Description

Let $\texttt{S}=K[x_1,\ldots,x_n]$, $\texttt{t}\geq 1$ and let $\texttt{f}=(f(0),f(1),\ldots,f(d),\ldots)$ be a sequence of nonnegative integers. The sequence f is the $\texttt{f}_\texttt{t}$-vector of a t-spread ideal of S if the following conditions hold:
$f(0)\leq 1,\ f(1)\leq n \text{ and } f(d+1)\leq \texttt{tMacaulayExpansion(f(d),n,d,t,Shift=>true)}$ for all $d>1.$
Let I be a t-spread ideal of the polynomial ring $S=K[x_1,\ldots,x_n]$ One can define the $f_\texttt{t}$-vector of I as $f_\texttt{t}(\texttt{I})=\left( f_{\texttt{t},-1}(\texttt{I}), f_{\texttt{t},0}(\texttt{I}), \ldots, f_{\texttt{t},j}(\texttt{I}), \ldots \right),$
where $f_{\texttt{t},j-1}(\texttt{I})=|[S_j]_t|-|[I_j]_t|$ and $[I_j]_t$ is the t-spread part of the $j$-th graded component of I.

Example:

 i1 : S=QQ[x_1..x_8] o1 = S o1 : PolynomialRing i2 : f={1,8,20,10,0} o2 = {1, 8, 20, 10, 0} o2 : List i3 : isFTVector(S,f,2) o3 = true i4 : S=QQ[x_1..x_7] o4 = S o4 : PolynomialRing i5 : isFTVector(S,f,2) o5 = false