Stanley depth of powers of the edge ideal of a forest

Abstract:

Let $\mathbb {K}$ be a field and $S=\mathbb {K}[x_1,\dots ,x_n]$ be the polynomial ring in $n$ variables over the field $\mathbb {K}$. Let $G$ be a forest with $p$ connected components $G_1,\ldots ,G_p$ and let $I=I(G)$ be its edge ideal in $S$. Suppose that $d_i$ is the diameter of $G_i$, $1\leq i\leq p$, and consider $d =\max \hspace {0.04cm}\{d_i\mid 1\leq i\leq p\}$. Morey has shown that for every $t\geq 1$, the quantity $\max \{\lceil \frac {d-t+2}{3}\rceil +p-1,p\}$ is a lower bound for $\textrm {depth}(S/I^t)$. In this paper, we show that for every $t\geq 1$, the mentioned quantity is also a lower bound for $\textrm {sdepth}(S/I^t)$. By combining this inequality with Burch’s inequality, we show that any sufficiently large powers of edge ideals of forests are Stanley. Finally, we state and prove a generalization of our main theorem.