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

Abstract

In Cogdell et al., LMS Lecture Notes Series 459, 393–427 (2020), the authors proved an analogue of Kronecker’s limit formula associated to any divisorD which is smooth in codimension one on any smooth Kähler manifold X . In the present article, we apply the aforementioned Kronecker limit formula in the case when X is complex projective space CP for n ≥ 2 and D is a hyperplane, meaning the divisor of a linear form PD(z) for z = (Zj) ∈ CP. Our main result is an explicit evaluation of the Mahler measure of PD as a convergent series whose each term is given in terms of rational numbers, multinomial coefficients, and the L-norm of the vector of coefficients of PD.