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

Cogdell, James; Jorgenson, Jay; Smajlović, Lejla

doi:10.1090/tran/8432

Evaluating the Mahler measure of linear forms via the Kronecker limit formula on complex projective space
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by James Cogdell, Jay Jorgenson and Lejla Smajlović PDF

Trans. Amer. Math. Soc. 374 (2021), 6769-6796 Request permission

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 divisor $\mathcal D$ 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 $\mathbb {C}\mathbb {P}^n$ for $n \geq 2$ and $\mathcal D$ is a hyperplane, meaning the divisor of a linear form $P_D({z})$ for ${z} = (\mathcal {Z}_{j}) \in \mathbb {C}\mathbb {P}^n$. 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 $P_{D}$.

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