Evaluating the Mahler measure of linear forms via Kronecker limit formulas on complex projective space

TL;DR

This paper derives an explicit series formula for the Mahler measure of linear forms on complex projective space using Kronecker limit formulas, connecting complex geometry and number theory.

Contribution

It applies Kronecker limit formulas to evaluate Mahler measures of hyperplane divisors in complex projective space, providing explicit series expressions.

Findings

Explicit series formula for Mahler measure of linear forms

Connection between Mahler measure and Kronecker limit formulas

Series terms involve rational numbers, multinomial coefficients, and coefficient norms

Abstract

In Cogdell et al., \it LMS Lecture Notes Series \bf 459, \rm 393--427 (2020), \rm the authors proved an analogue of Kronecker's limit formula associated to any divisor $\mathcal D$ which is smooth in codimension one on any smooth K\"ahler manifold $X$. In the present article, we apply the aforementioned Kronecker limit formula in the case when $X$ is complex projective space $\CC\PP^n$ for $n \geq 2$ and $\mathcal D$ is a hyperplane, meaning the divisor of a linear form $P_D({z})$ for ${z} = (\mathcal{Z}_{j}) \in \CC\PP^n$. Our main result is an explicit evaluation of the Mahler measure of $P_{D}$ as a convergent series whose each term is given in terms of rational numbers, multinomial coefficients, and the $L^{2}$-norm of the vector of coefficients of $P_{D}$.