Heights of polynomials over lemniscates

Abstract

We consider a family of heights defined by the $L_p$ norms of polynomials with respect to the equilibrium measure of a lemniscate for $0 \le p \le \infty$, where $p=0$ corresponds to the geometric mean (the generalized Mahler measure) and $p=\infty$ corresponds to the standard supremum norm. This special choice of the measure allows to find an explicit form for the geometric mean of a polynomial, and estimate it via certain resultant. For lemniscates satisfying appropriate hypotheses, we establish explicit polynomials of lowest height, and also show their uniqueness. We discuss relations between the standard results on the Mahler measure and their analogues for lemniscates that include generalizations of Kronecker's theorem on algebraic integers in the unit disk, as well as of Lehmer's conjecture.