Odd cycles and Hilbert functions of their toric rings

Abstract

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

4 pages