Odd cycles and Hilbert functions of their toric rings

TL;DR

This paper investigates the Hilbert functions of specific normal toric rings, demonstrating that certain symmetric sequences satisfying particular properties can be realized as h-vectors of Cohen–Macaulay standard G-domains for small sizes.

Contribution

It characterizes new classes of Hilbert functions that can occur for Cohen–Macaulay toric rings, focusing on sequences with odd cycle structures.

Findings

Constructs explicit examples of Hilbert functions for small cases

Shows these sequences satisfy Cohen–Macaulay conditions

Provides criteria for possible h-vectors in this setting

Abstract

Studying Hilbert functions of concrete examples of normal toric rings, it is demonstrated that, for each $1 \leq s \leq 5$, an $O$-sequence $(h_0, h_1, \ldots, h_{2s-1}) \in \mathbb{Z}_{\geq 0}^{2s}$ satisfying the properties that (i) $h_0 \leq h_1 \leq \cdots \leq h_{s-1}$, (ii) $h_{2s-1} = h_0$, $h_{2s-2} = h_1$ and (iii) $h_{2s - 1 - i} = h_i + (-1)^{i}$, $2 \leq i \leq s - 1$, can be the $h$-vector of a Cohen--Macaulay standard $G$-domain.