Acta Mathematica Vietnamica
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Lexsegment Ideals and Their $h$-Polynomials
Takayuki Hibi
,
Kazunori Matsuda
Abstract
Let $S = K[x_1,\dots , x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $\deg x_i = 1$ 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 $h_{S/I}(\lambda) = h_0 + h_1\lambda + h_2\lambda^2 + \dots + h_s\lambda^s$ with $h_s\not= 0$ is the $h$-polynomial of $S/I$. Given arbitrary integers $r \geq 1$ and $s \geq 1$, a lexsegment ideal $I$ of $S = K[x_1,\dots,x_n]$, where $n \leq \max\{r,s\} + 2$, satisfying $\mathrm{reg\,}(S/I) = r$ and $\deg h_{S/I}(\lambda) = s$ will be constructed.