Graded Cohen-Macaulay domains and lattice polytopes with short $h$-vector

TL;DR

This paper establishes a condition on the $h^*$-vector of lattice polytopes that guarantees the integer decomposition property (IDP), extending the result to Cohen-Macaulay domains.

Contribution

It proves that for lattice polytopes with a short $h^*$-vector, specifically when $h_2^* \,\leq\, h_1^*$, the polytope has the IDP, and generalizes this to Cohen-Macaulay domains.

Findings

Polytopes with $h_2^* \,\leq\, h_1^*$ are IDP.

Extension of results to Cohen-Macaulay domains.

Short $h^*$-vector condition implies IDP.

Abstract

Let P be a lattice polytope with $h^*$-vector $(1, h^*_1, h^*_2)$. In this note we show that if $h_2^* \leq h_1^*$, then $P$ is IDP. More generally, we show the corresponding statements for semi-standard graded Cohen-Macaulay domains over algebraically closed fields.