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- W4387075992 abstract "Let $mathbb{K}$ be a field and $S=mathbb{K}[x_1,ldots,x_n]$ be the polynomial ring in $n$ variables over $mathbb{K}$. Assume that $I$ is a squarefree monomial ideal of $S$. For every integer $kgeq 1$, we denote the $k$-th squarefree power of $I$ by $I^{[k]}$. The normalized depth function of $I$ is the defined as $g_I(k)={rm depth}(S/I^{[k]})-(d_k-1)$, where $d_k$ denotes the minimum degree of monomials belonging to $I^{[k]}$. Erey, Herzo, Hibi and Saeedi Madani conjectured that for any squarefree monomial ideal $I$, the function $g_I(k)$ is nonincreasing. In this short note, we provide a counterexample for this conjecture. Our example in fact shows that $g_I(2)-g_I(1)$ can be arbitrarily large." @default.
- W4387075992 created "2023-09-27" @default.
- W4387075992 creator A5004484087 @default.
- W4387075992 date "2023-09-25" @default.
- W4387075992 modified "2023-09-28" @default.
- W4387075992 title "An Increasing normalized depth function" @default.
- W4387075992 doi "https://doi.org/10.48550/arxiv.2309.13892" @default.
- W4387075992 hasPublicationYear "2023" @default.
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