Generalized Mahler measures of Laurent polynomials
The paper deals with the following generalization of Mahler measure introduced by \textit{M. Lalín} and \textit{T. Mittal} [Res. Number Theory 4, No. 2, Paper No. 16, 23 p. (2018; Zbl 1454.11195)]: If \(\mathfrak a =(a_1,a_2,\dots,a_n)\) is an \(n\)-tuple of positive numbers and \(P(x_1,\dots,x_n)\in \mathcal{C}(x_1,\dots,x_n)^*\), then\N\[\N\text{m}_\mathfrak a (P) = \frac1{(2\pi i)^n}\int_{T^n_\mathfrak a}\log(|P(x_1,\dots,x_n)|)\frac{dx_1}{x_1}\cdots \frac{dx_n}{x_n},\N\]\Nwhere\N\[\NT^n_\mathfrak a = \{(x_1,\dots,x_n):\ |x_1|=a_1,\dots,|x_n|=a_n\}.\N\]\NIn the case \(a_1=\dots=a_n=1\) one gets the usual logarithmic Mahler measure \(\text{m}(P)\).\N\NThe author shows (Theorem 1.2) that if \(P(x_1,\dots,x_n)\) is a Laurent polynomial without constant term and \(P_k(x_1,\dots,x_n) = k - P(x_1,\dots,x_n)\), then for suitably chosen values of \(k\in\mathcal{C}\) the difference \( \text{m}_\mathfrak a(P) - \text{m}(P)\) can be determined, and in some cases makes this result more precise (Theorems 1.3--1.5). The last theorem determines m\(_{a,b}(x+1/x+y+1/y+4)\).
scientific article; zbMATH DE number 7887387
scientific article