Polynomials having small heights on lemniscates

Abstract

The Mahler measure of a complex polynomial is the geometric mean of that polynomial over the unit circle. By a result of Kronecker, for nonconstant integer polynomials, the Mahler measure is equal to 1 if and only if all roots of the polynomial are 0 or roots of unity. In 1933, Lehmer asked if the Mahler measure for all other nonconstant integer polynomials had a lower bound greater than 1. In fact, Lehmer noted that the smallest such measure he had found belonged to a polynomial having degree 10 and to this day, no polynomial has been found which lowers this bound. We explore a generalization of the Mahler measure to lemniscates and investigate which properties of the classical Mahler measure are preserved by this generalization. In particular, we are interested in the analogues of Lehmer's question, and we investigate this matter both analytically and computationally. Our work is largely restricted to the classical Bernoulli lemniscate and its variations, but many of our results have applicability to a broad range of lemniscates.