On the Hilbert depth of the quotient ring of the edge ideal of a star graph

TL;DR

This paper investigates the Hilbert depth of the quotient ring of the edge ideal of a star graph, establishing bounds and asymptotic behavior as the number of vertices increases.

Contribution

It provides new bounds for the Hilbert depth of the quotient ring of star graph edge ideals and determines its limit ratio as the graph size grows.

Findings

Lower bound: hdepth ≥ ⌈n/2⌉ + ⌊√n⌋ - 2

Upper bound: hdepth ≤ ⌈n/2⌉ + ⌊εn⌋ + A - 2 for any ε>0

As n→∞, hdepth/n approaches 1/2

Abstract

Let $S_n=K[x_1,\ldots,x_n,y]$ and $I_n=(x_1y,x_2y,\ldots,x_ny)\subset S_n$ be the edge ideal of star graph. We prove that $\operatorname{hdepth}(S_n/I_n)\geq \left\lceil \frac{n}{2} \right\rceil + \left\lfloor \sqrt{n} \right\rfloor - 2$. Also, we show that for any $\varepsilon>0$, there exists some integer $A=A(\varepsilon)\geq 0$ such that $\operatorname{hdepth}(S_n/I_n)\leq \left\lceil \frac{n}{2} \right\rceil + \left\lfloor \varepsilon n \right\rfloor + A - 2$. We deduce that $\lim\limits_{n\to\infty} \frac{1}{n}\operatorname{hdepth}(S_n/I_n) = \frac{1}{2}$.