Limiting value of higher Mahler measure

TL;DR

This paper derives an explicit formula for the asymptotic behavior of the higher Mahler measure of Laurent polynomials, providing insights into its limiting value as the order increases.

Contribution

It introduces a new explicit formula for the limit of the normalized higher Mahler measure as the order tends to infinity.

Findings

Explicit formula for the limit of |m_k(P)|/k! as k approaches infinity

Provides a deeper understanding of the asymptotic behavior of higher Mahler measures

Advances theoretical knowledge in Mahler measure analysis

Abstract

We consider the $k$-higher Mahler measure $m_k(P)$ of a Laurent polynomial $P$ as the integral of $\log ^k \left| P \right|$ over the complex unit circle. In this paper we derive an explicit formula for the value of $\left| m_k(P) \right|/k!$ as $k \to \infty.$