Integral closure and Hilbert series of a special monomial ideal

TL;DR

This paper investigates the integral closure properties, Hilbert series, and Freiman property of a specific class of monomial ideals in polynomial rings, providing new insights into their algebraic and combinatorial structure.

Contribution

It characterizes the unmixedness of the integral closure, computes the Hilbert series, and proves the ideal is Freiman, advancing understanding of these monomial ideals.

Findings

Integral closure is unmixed under certain conditions.

Explicit Hilbert series formula derived.

The ideal is shown to be Freiman.

Abstract

Let $R=K[x_1,\ldots, x_n]$ be the polynomial ring in $n$ variables over a field $K$ and let $M_{n,t}=(x^{e_1},\ldots, x^{e_n})$ be a monomial ideal of $R$, where $x^{e_i}=x_1^t\ldots x_{i-1}^tx_{i+1}^t\ldots x_n^t$. We study the unmixedness of its integral closure. Furthermore, we compute the Hilbert series of this ideal and we show that this ideal is Freiman.