On the Hilbert depth of the quotient ring of the edge ideal of a complete bipartite graph

TL;DR

This paper investigates the Hilbert depth of the quotient ring of the edge ideal of a complete bipartite graph, establishing bounds and conditions for equality based on the parameters n and m.

Contribution

It provides new lower bounds and characterizations for the Hilbert depth of these quotient rings, including exact values under specific conditions.

Findings

Proves that h(n,m) ≥ ⌈n/2⌉ for all n ≥ m.

Identifies when equality holds for h(n,m) based on m's interval.

Establishes bounds and inequalities for h(n,n) and between different m values.

Abstract

Let $n\geq m$ be two positive integers, $S_{n,m}=K[x_1,\ldots,x_n,y_1,\ldots,y_m]$ and $I_{n,m}=(x_iy_j\;:\;1\leq i\leq n,1\leq j\leq m)\subset S_{n,m}$ the edge ideal of a complete bipartite graph. Denote $h(n,m)=\operatorname{hdepth}(S_{n,m}/I_{n,m})$. We prove that $h(n,m)\geq \left\lceil \frac{n}{2} \right\rceil$ and the equality holds if $m$ belong to a certain interval centered in $\left\lceil \frac{n} {2} \right\rceil$. Also, we find some tight bounds for $h(n,n)$ and we prove several inequalities between $h(n,m)$ and $h(n,m')$.