Two-variable polynomials with dynamical Mahler measure zero

Abstract

We discuss several aspects of the dynamical Mahler measure for multivariate polynomials. We prove a weak dynamical version of Boyd-Lawton formula and we characterize the polynomials with integer coefficients having dynamical Mahler measure zero both for the case of one variable (Kronecker's lemma) and for the case of two variables, under the assumption that the dynamical version of Lehmer's question is true.

Limits of Mahler measures in multiple variables

“…However, if no explicit estimates of the constants are needed, our method with Dimitrov-Habegger's bound readily gives an exponent 1 2(k−1) in Theorem 4.1, as discussed in Remark 4.15. The quality of this last exponent in the bound (14) is better than the one obtained in [20,Theorem A.1] for m = 1, noting that the authors explicitly remark that they do not strive to get optimal exponent. The crucial property of the bound provided by (27) is that the constants involved remain bounded if we replace P by P A , for any matrix A ∈ Z m×n .…”