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TSpreadIdeals > isFTVector

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

Synopsis

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

See also

Ways to use isFTVector :

For the programmer

The object isFTVector is a method function.