The finiteness of computing the ultrametric Mahler measure

TL;DR

This paper investigates the ultrametric Mahler measure for algebraic numbers, demonstrating that its computation can be reduced to a finite search, and provides examples where this approach is feasible.

Contribution

It introduces a method to compute the ultrametric Mahler measure by reducing it to a finite search, advancing understanding of its computational aspects.

Findings

Computation of $M_()$ reduces to a finite search.

Examples where the finite set can be explicitly determined.

Potential to compute $M_()$ in specific cases.

Abstract

Recent work of Fili and the author examines an ultrametric version of the Mahler measure, denoted $M_\infty(\alpha)$ for an algebraic number $\alpha$. We show that the computation of $M_\infty(\alpha)$ can be reduced to a certain search through a finite set. Although it is a open problem to record the points of this set in general, we provide some examples where it is reasonable to compute and our result can be used to determine $M_\infty(\alpha)$.