Lexsegment ideals and their h-polynomials

TL;DR

This paper constructs lexsegment ideals in polynomial rings with specified regularity and h-polynomial degree, advancing understanding of their algebraic and combinatorial properties.

Contribution

It introduces a method to explicitly construct lexsegment ideals with prescribed regularity and h-polynomial degree, expanding the class of known ideals with controlled invariants.

Findings

Constructed lexsegment ideals with given regularity and h-polynomial degree.

Established bounds on the number of variables needed for such ideals.

Provided explicit examples illustrating the construction.

Abstract

Let $S = K[x_1, \ldots, x_n]$ 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 $h_s \neq 0$ is the $h$-polynomial of $S/I$. Given arbitrary integers $r \geq 1$ and $s \geq 1$, a lexsegment ideal $I$ of $S = K[x_1, \ldots, x_n]$, where $n \leq \max\{r, s\} + 2$, satisfying $\mathrm{reg}(S/I) = r$ and $ \mathrm{deg}\ h_{S/I}(\lambda) = s$ will be constructed.