Winter 2024 AN INCREASING NORMALIZED DEPTH FUNCTION
S. A. Seyed Fakhari
J. Commut. Algebra 16(4): 497-499 (Winter 2024). DOI: 10.1216/jca.2024.16.497

Abstract

Let 𝕂 be a field and S=𝕂[x1,,xn] be the polynomial ring in n variables over 𝕂. Assume that I is a squarefree monomial ideal of S. For every integer k1, we denote the k-th squarefree power of I by I[k]. The normalized depth function of I is defined as gI(k)=depth(SI[k])(dk1), where dk denotes the minimum degree of monomials belonging to I[k]. Erey, Herzog, Hibi and Saeedi Madani conjectured that for any squarefree monomial ideal I, the function gI(k) is nonincreasing. In this short note, we provide a counterexample for this conjecture. Our example in fact shows that gI(2)gI(1) can be arbitrarily large.

Citation

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S. A. Seyed Fakhari. "AN INCREASING NORMALIZED DEPTH FUNCTION." J. Commut. Algebra 16 (4) 497 - 499, Winter 2024. https://doi.org/10.1216/jca.2024.16.497

Information

Received: 9 November 2023; Revised: 24 March 2024; Accepted: 27 March 2024; Published: Winter 2024
First available in Project Euclid: 6 January 2025

MathSciNet: MR3294858
Digital Object Identifier: 10.1216/jca.2024.16.497

Subjects:
Primary: 05E40 , 13C15

Keywords: normalized depth function , squarefree power

Rights: Copyright © 2024 Rocky Mountain Mathematics Consortium

Vol.16 • No. 4 • Winter 2024
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