Schinzel-type bounds for the Mahler measure on lemniscates

Abstract

We study the generalized Mahler measure on lemniscates, and prove a sharp lower bound for the measure of totally real integer polynomials that includes the classical result of Schinzel expressed in terms of the golden ratio. Moreover, we completely characterize many cases when this lower bound is attained. For example, we explicitly describe all lemniscates and the corresponding quadratic polynomials that achieve our lower bound for the generalized Mahler measure. It turns out that the extremal polynomials attaining the bound must have even degree. The main computational part of this work is related to finding many extremals of degree four and higher, which is a new feature compared to the original Schinzel’s theorem where only quadratic irreducible extremals are possible.