Graded Cohen–Macaulay Domains and Lattice Polytopes with Short h-Vector

Let P be a lattice polytope with the $$h^{*}$$ -vector $$(1, h^*_1, \ldots , h^*_s)$$ . In this note we show that if $$h_s^* \le h_1^*$$ then Ehrhart ring $${\mathbb {k}}[P]$$ is generated in degrees at most $$s-1$$ as {k}}$$ -algebra. particular, $$s=2$$ and $$h_2^* IDP. To see this, corresponding statement for semi-standard graded Cohen–Macaulay domains over algebraically closed fields.