Hilbert polynomial of length functions

TL;DR

This paper introduces Hilbert series and polynomials for modules over Noetherian rings using a general length function, establishing an invariant that refines algebraic entropy and exploring multivariate extensions.

Contribution

It defines Hilbert polynomials for R[X]-modules based on a general length function, linking the leading coefficient to refined entropy invariants and module dimension.

Findings

The leading term of the Hilbert polynomial is an invariant of the module.

The invariant refines algebraic entropy and its receptive version.

Multivariate Hilbert polynomials are also considered.

Abstract

Let $\lambda$ be a general length function for modules over a Noetherian ring R. We use $\lambda$ to introduce Hilbert series and polynomials for R[X]-modules, measuring the growth rate of~$\lambda$. We show that the leading term $\mu$ of the Hilbert polynomial is an invariant of the module, which refines both the algebraic entropy and the receptive algebraic entropy; its degree is a suitable notion of dimension for $R[X]$-modules. Similar to algebraic entropy, $\mu$ in general is not additive for exact sequence of $R[X]$-modules: we demonstrate how to adapt of certain entropy constructions to this new invariant. We also consider multi-variate versions of the Hilbert polynomial.