Stanley depth of the integral closure of monomial ideals

Abstract

Let $I$ be a monomial ideal in the polynomial ring $S=\mathbb K [x_1,\dots ,x_n]$. We study the Stanley depth of the integral closure $\overline{I}$ of $I$. We prove that for every integer $k\ge 1$, the inequalities $\text{ sdepth} (S/\overline{I^k}) \le \text{ sdepth} (S/\overline{I})$ and $\text{ sdepth} (\overline{I^k}) \le \text{ sdepth} (\overline{I})$ hold. We also prove that for every monomial ideal $I\subset S$ there exist integers $k_1,k_2\ge 1$, such that for every $s\ge 1$, the inequalities $\text{ sdepth} (S/I^{sk_1}) \le \text{ sdepth} (S/\overline{I})$ and $\text{ sdepth} (I^{sk_2}) \le \text{ sdepth} (\overline{I})$ hold. In particular, $\min _k \{\text{ sdepth} (S/I^k)\} \le \text{ sdepth} (S/\overline{I})$ and $\min _k \{\text{ sdepth} (I^k)\} \le \text{ sdepth} (\overline{I})$. We conjecture that for every integrally closed monomial ideal $I$, the inequalities $\text{ sdepth}(S/I)\ge n-\ell (I)$ and $\text{ sdepth} (I)\ge n-\ell (I)+1$ hold, where $\ell (I)$ is the analytic spread of $I$. Assuming the conjecture is true, it follows together with the Burch’s inequality that Stanley’s conjecture holds for $I^k$ and $S/I^k$ for $k\gg 0$, provided that $I$ is a normal ideal.