EVALUATING THE MAHLER MEASURE OF LINEAR FORMS VIA THE KRONECKER LIMIT FORMULA ON COMPLEX PROJECTIVE SPACE

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 divisor D which is smooth in codimension one on any smooth Kahler manifold X. In the present article, we apply the aforementioned Kronecker limit formula in the case when X is complex projective space CPn for n >= 2 and D is a hyperplane, meaning the divisor of a linear form P-D (z) for z = (Z(j)) is an element of CPn 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.