Lexsegment ideals and their h-polynomials

Abstract

Let $S=K[x_1,\dots ,x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $\mathrm{deg}{x}_i=1$ and $I\subset S$ a homogeneous ideal of $S$ with $\dim S/I=d$. The Hilbert series of $S/I$ is of the form $h_{S/I}(\lambda )/(1-\lambda {)}^d$, where $h_{S/I}(\lambda )=h_0+h_1\lambda +h_2{\lambda }^2+\cdots +h_s{\lambda }^s$ with $h_s\ne 0$ is the $h$-polynomial of $S/I$. Given arbitrary integers $r\ge 1$ and $s\ge 1$, a lexsegment ideal $I$ of $S=K[x_1,\dots ,x_n]$, where $n\le \max \{r,s\}+2$, satisfying $\mathrm{reg}(S/I)=r$ and $\mathrm{deg}{h}_{S/I}(\lambda )=s$ will be constructed.

4 pages

References in corpus (1)

• Castelnuovo-Mumford regularity of products of ideals

Cited by in corpus (1)

• Induced matching numbers of finite graphs and edge ideals