Intrinsic valuation entropy

TL;DR

This paper generalizes the concept of intrinsic entropy from Abelian groups to modules over certain valuation domains, establishing it as a length function and proving its key properties and uniqueness.

Contribution

It introduces a new intrinsic entropy for modules over Archimedean valuation domains and proves its fundamental properties and uniqueness.

Findings

Entropy is a length function for $R[X]$-modules.

Satisfies an adapted Intrinsic Algebraic Yuzvinski Formula.

Is essentially the unique invariant with these properties.

Abstract

We extend the notion of intrinsic entropy for endomorphisms of Abelian groups to endomorphisms of modules over an Archimedean non-discrete valuation domain $R$, using the natural non-discrete length function introduced by Northcott and Reufel for such a category of modules. We prove that this notion of entropy is a length function for the category of $R[X]$-modules, it satisfies (a suitably adapted version of) the Intrinsic Algebraic Yuzvinski Formula and that it is essentially the unique invariant for $Mod(R[X])$ with these properties.