A note on $p$-adic higher Mahler measures

TL;DR

This paper introduces $p$-adic higher Mahler measures, extending classical concepts and establishing $p$-adic analogues of existing results, thus broadening the understanding of Mahler measures in number theory.

Contribution

It defines $p$-adic higher Mahler measures and proves their properties, providing the first $p$-adic analogues of known results in the classical setting.

Findings

Introduction of $p$-adic higher Mahler measures

Proof of $p$-adic analogues of Akatsuka's results

Extension of Mahler measure theory to $p$-adic context

Abstract

Kurokawa, Lal\'{\i}n and Ochiai introduced and studied the higher Mahler measures, which are generalization of the classical Mahler measure. In this article, we introduce $p$-adic higher Mahler measures and prove $p$-adic analogues of Akatsuka's results.