The finiteness of computing the ultrametric Mahler measure

TL;DR

The paper proves that computing the ultrametric Mahler measure $M_\infty(\alpha)$ can be reduced to a finite search by introducing a modified Mahler measure $\bar{M}$ on the quotient space $V$ and a finite projected set $\bar{\mathcal{B}}(\alpha)$. It shows $M_\infty(\alpha)=\bar{M}_\infty(\pi(\alpha))$ and, in practice, reduces to identifying the smallest index $J$ such that $\pi(\alpha)$ lies in the span of the first $J$ elements of $\bar{\mathcal{B}}(\alpha)$, giving $M_\infty(\alpha)=\bar{M}(\bar{b}_J)$. The paper develops a quadratic-case algorithm via $B2List$ to generate a usable finite set of candidates, analyzes when $\bar{M}(\gamma)=M(\gamma)$ (with exceptions only in $\mathbb{Q}(i)$ and $\mathbb{Q}(i\sqrt{3})$), and presents a worked example showing how to compute $M_\infty$ for a nontrivial quadratic algebraic number, thereby illustrating the approach and its limitations. The results bridge finiteness properties, Galois-theoretic reductions, and explicit computations to enable practical evaluation of ultrametric Mahler measures in low-degree cases.

Abstract

Recent work of Fili and the author examines an ultrametric version of the Mahler measure, denoted $M_\infty(α)$ for an algebraic number $α$. We show that the computation of $M_\infty(α)$ 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 $M_\infty(α)$.

Theorems & Definitions (29)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Theorem 2.1
• Theorem 2.2
• Example
• Lemma 3.1
• proof
• Lemma 3.2
• proof