Heights of polynomials over lemniscates

TL;DR

This paper introduces a family of polynomial heights based on $L_p$ norms over lemniscates, providing explicit formulas, minimal height polynomials, and exploring connections to classical number theory conjectures.

Contribution

It defines a new class of polynomial heights over lemniscates, derives explicit minimal height polynomials, and relates these to classical Mahler measure and Lehmer's conjecture.

Findings

Explicit form for the geometric mean of polynomials over lemniscates.

Existence and uniqueness of polynomials with minimal height.

Connections to classical results on Mahler measure and algebraic integers.

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.