An areal analog of Mahler's measure

TL;DR

This paper introduces an areal analog of Mahler's measure based on area integrals over the unit disk, exploring its properties, relationship with Mahler's measure, and implications for extremal problems in Bergman spaces.

Contribution

It defines a new height function on polynomial spaces using area measure, analyzing its properties and comparing it to classical Mahler's measure.

Findings

The areal measure is multiplicative and lower bounds Mahler's measure.

It can be substantially lower than Mahler's measure.

The measure has connections to extremal problems in Bergman spaces.

Abstract

We consider a version of height on polynomial spaces defined by the integral over the normalized area measure on the unit disk. This natural analog of Mahler's measure arises in connection with extremal problems for Bergman spaces. It inherits many nice properties such as the multiplicative one. However, this height is a lower bound for Mahler's measure, and it can be substantially lower. We discuss some similarities and differences between the two.