Hilbert polynomial of length functions

Let λ be a general length function for modules over a Noetherian ring R. We use λ to introduce Hilbert series and polynomials for R[X]-modules, measuring the growth rate of λ. We show that the leading term μ 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, μ 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.