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- W2952161849 abstract "Let $I$ be a monomial ideal in the polynomial ring $S=mathbb{K}[x_1,...,x_n]$. We study the Stanley depth of the integral closure $bar{I}$ of $I$. We prove that for every integer $kgeq 1$, the inequalities ${rm sdepth} (S/bar{I^k}) leq {rm sdepth} (S/bar{I})$ and ${rm sdepth} (bar{I^k}) leq {rm sdepth} (bar{I})$ hold. We also prove that for every monomial ideal $Isubset S$ there exist integers $k_1,k_2geq 1$, such that for every $sgeq 1$, the inequalities ${rm sdepth} (S/I^{sk_1}) leq {rm sdepth} (S/bar{I})$ and ${rm sdepth} (I^{sk_2}) leq {rm sdepth} (bar{I})$ hold. In particular, $min_k {{rm sdepth} (S/I^k)} leq {rm sdepth} (S/bar{I})$ and $min_k {{rm sdepth} (I^k)} leq {rm sdepth} (bar{I})$. We conjecture that for every integrally closed monomial ideal $I$, the inequalities ${rm sdepth}(S/I)geq n-ell(I)$ and ${rm sdepth} (I)geq n-ell(I)+1$ 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 $kgg 0$, provided that $I$ is a normal ideal." @default.
- W2952161849 created "2019-06-27" @default.
- W2952161849 creator A5004484087 @default.
- W2952161849 date "2012-05-31" @default.
- W2952161849 modified "2023-09-27" @default.
- W2952161849 title "STANLEY DEPTH OF THE INTEGRAL CLOSURE OF MONOMIAL IDEALS" @default.
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