The normalized depth function of squarefree powers

TL;DR

This paper introduces the normalized depth function for squarefree powers of monomial ideals, especially focusing on edge ideals of graphs, and explores its properties and conjectures about its behavior.

Contribution

It defines the normalized depth function for squarefree powers and investigates its properties, particularly for edge ideals of graphs, proposing a conjecture on its monotonicity.

Findings

Normalized depth function is strongly supported by computational evidence.

The paper conjectures that the normalized depth function is nonincreasing.

Deep study of the normalized depth function for edge ideals of finite simple graphs.

Abstract

The depth of squarefree powers of a squarefree monomial ideal is introduced. Let $I$ be a squarefree monomial ideal of the polynomial ring $S=K[x_1,\ldots,x_n]$. The $k$-th squarefree power $I^{[k]}$ of $I$ is the ideal of $S$ generated by those squarefree monomials $u_1\cdots u_k$ with each $u_i\in G(I)$, where $G(I)$ is the unique minimal system of monomial generators of $I$. Let $d_k$ denote the minimum degree of monomials belonging to $G(I^{[k]})$. One has $\operatorname{depth}(S/I^{[k]}) \geq d_k -1$. Setting $g_I(k) = \operatorname{depth}(S/I^{[k]}) - (d_k - 1)$, one calls $g_I(k)$ the normalized depth function of $I$. The computational experience strongly invites us to propose the conjecture that the normalized depth function is nonincreasing. In the present paper, especially the normalized depth function of the edge ideal of a finite simple graph is deeply studied.