On the Hilbert depth of the quotient ring of the edge ideal of a complete bipartite graph
Andreea I. Bordianu, Mircea Cimpoeas
TL;DR
This paper analyzes the Hilbert depth of the quotient $S_{n,m}/I_{n,m}$ where $I_{n,m}$ is the edge ideal of the complete bipartite graph $K_{n,m}$ and $S_{n,m}=K[x_1,\dots,x_n,y_1,\dots,y_m]$. Building on a squarefree monomial criterion for Hilbert depth, the authors establish a universal lower bound $h(n,m)\ge\left\lceil\frac{n}{2}\right\rceil$, derive exact values in parity-driven regimes, and prove several monotonicity and symmetry relations among different $m$-values, including $h(n,\left\lfloor n/2\right\rfloor)=h(n,\left\lceil n/2\right\rceil)=\left\lceil\frac{n}{2}\right\rceil$ and $\lim_{n\to\infty}\frac{1}{n}h(n,m(n))=\tfrac{1}{2}$. They also provide sharp bounds for $h(n,n)$ and formulate conjectures that would yield sharper lower bounds for the diagonal case $h(n,n)$. The results illuminate how the Hilbert depth of the quotient behaves with respect to the bipartite sizes and offer a framework for further refinement via proposed conjectures.
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 $h(n,m')$.