Hilbert polynomial of length functions

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 sequences of R[X]-modules: we demonstrate how to adapt certain entropy constructions to this new invariant. We also consider multi-variate versions of the Hilbert polynomial.