Graded algebras with cyclotomic Hilbert series

TL;DR

This paper characterizes graded algebras with Hilbert series roots on the unit circle, showing they are complete intersections under certain conditions like being Koszul or having an irreducible h-polynomial.

Contribution

It establishes a precise link between Hilbert-cyclotomic properties and complete intersection structures in graded algebras, extending classical results.

Findings

Hilbert-cyclotomic algebras coincide with complete intersections under specified conditions

In the Koszul case, classical results about deviations imply the main theorem

Roots of the numerator of the Hilbert series lie on the unit circle for these algebras

Abstract

Let $R$ be a positively graded algebra over a field. We say that $R$ is Hilbert-cyclotomic if the numerator of its reduced Hilbert series has all of its roots on the unit circle. Such rings arise naturally in commutative algebra, numerical semigroup theory and Ehrhart theory. If $R$ is standard graded, we prove that, under the additional hypothesis that $R$ is Koszul or has an irreducible $h$-polynomial, Hilbert-cyclotomic algebras coincide with complete intersections. In the Koszul case, this is a consequence of some classical results about the vanishing of deviations of a graded algebra.