A generalization of a theorem of Boyd and Lawton (Q2862957)

The authors generalize a result that the reviewer conjectured in [J. Number Theory 13, 116--121 (1981; Zbl 0447.12003)] that was completely proved by \textit{W. M. Lawton} [J. Number Theory 16, 356--362 (1983; Zbl 0516.12018)]. This relates the convergence of certain sequences of one variable Mahler measures to a multi-variable Mahler measure. Here the (logarithmic) Mahler measure of a multivariable Laurent polynomial \(P\) in \(n\) variables is the average over the \(n\)-torus of \(\log|P|\). The generalization the authors provide is to replace the usual Mahler measure by one of three other possibilities: (i) the \textit{generalized Mahler measure} which replaces \(\log|P|\) by the maximum of \(\log|P_k|\) where \(\{P_k, k=1...s\}\) is a finite set of \(n\)-variable polynomials, (ii) the \textit{multiple Mahler measure} which replaces \(\log|P|\) by the product of \(\log|P_k|\) with \(P_k\) as in (i), or (iii) the \textit{higher Mahler measure} which is the special case of (ii) when all \(P_k\) are the same. The proofs require some delicate hard analysis and make use of an important Lemma of Lawton from his paper of 1983.