Hilbert series of modules over positively graded polynomials rings

TL;DR

This paper investigates the limitations of Hilbert series for finitely generated modules over positively graded polynomial rings, providing counterexamples and conditions under which they can be realized.

Contribution

It demonstrates that not all formal power series satisfying certain conditions are Hilbert series of finitely generated modules, answering a recent open question.

Findings

Counterexamples of formal power series not realizable as Hilbert series

Multiplying by a scalar can make some series realizable

Addresses an open question in the theory of Hilbert series

Abstract

In this note, we give examples of formal power series satisfying certain conditions that cannot be realized as Hilbert series of finitely generated modules. This answers to the negative a question raised in a recent article by the second and the third author. On the other hand, we show that the answer is positive after multiplication with a scalar.