Parole chiave

**Abstract** Let $Q$ be an affine monoid and $k[Q]$ the associated monoid $k$-algebra, where we let $k$ be a field. In this talk, in the case where $k[Q]$ is standard graded, a difference of the Hilbert series of $k[Q]$ and its normalization is discussed. More precisely, we prove that if $k[Q]$ satisfies Serre’s condition (S2), then the degree of the $h$-polynomial of $k[Q]$ is always greater than or equal to that of its normalization. Moreover, we also show counterexamples of this statement if we drop the assumption (S2).