Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the depth and Stanley depth of integral closure of powers of monomial
  ideals   Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be the
polynomial ring in $n$ variables over $\mathbb{K}$. Assume that $G$ is a graph
with edge ideal $I(G)$. We prove that the modules $S/\overline{I(G)^k}$ and
$\overline{I(G)^k}/\overline{I(G)^{k+1}}$ satisfy Stanley's inequality for
every integer $k\gg 0$. If $G$ is a non-bipartite graph, we show that the
ideals $\overline{I(G)^k}$ satisfy Stanley's inequality for all $k\gg 0$. For
every connected bipartite graph $G$ (with at least one edge), we prove that
${\rm sdepth}(I(G)^k)\geq 2$, for any positive integer $k\leq {\rm
girth}(G)/2+1$. This result partially answers a question asked in [20]. For any
proper monomial ideal $I$ of $S$, it is shown that the sequence $\{{\rm
depth}(\overline{I^k}/\overline{I^{k+1}})\}_{k=0}^{\infty}$ is convergent and
$\lim_{k\rightarrow\infty}{\rm
depth}(\overline{I^k}/\overline{I^{k+1}})=n-\ell(I)$, where $\ell(I)$ denotes
the analytic spread of $I$. Furthermore, it is proved that for any monomial
ideal $I$, there exists an integer $s$ such that $${\rm depth} (S/I^{sm}) \leq
{\rm depth} (S/\overline{I}),$$for every integer $m\geq 1$. We also determine a
value $s$ for which the above inequality holds. If $I$ is an integrally closed
ideal, we show that ${\rm depth}(S/I^m)\leq {\rm depth}(S/I)$, for every
integer $m\geq 1$. As a consequence, we obtain that for any integrally closed
monomial ideal $I$ and any integer $m\geq 1$, we have ${\rm Ass}(S/I)\subseteq
{\rm Ass}(S/I^m)$. \end{abstract}

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The categories of this paper are: ao-sci
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The categories of this paper are: patt-sol
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The categories of this paper are: supr-con
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The categories of this paper are: gr-qc
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The categories of this paper are: atom-ph
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The categories of this paper are: cs
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The categories of this paper are: nlin
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The categories of this paper are: nucl-th
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The categories of this paper are: math-ph
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The categories of this paper are: alg-geom
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The categories of this paper are: chao-dyn
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The categories of this paper are: comp-gas
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The categories of this paper are: nucl-ex
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The categories of this paper are: physics
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The categories of this paper are: acc-phys
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The categories of this paper are: adap-org