Ruminations on Hejhal's theorem about the Bergman and Szego kernels

TL;DR

This paper presents a new proof of Hejhal's theorem on the nondegeneracy of a matrix linking Bergman and Szego kernels in planar domains, using Mergelyan's theorem, and explores related properties of kernel zeroes.

Contribution

It introduces a novel proof of Hejhal's theorem and investigates the connections between kernel properties and zero distributions.

Findings

New proof of Hejhal's theorem established

Connections between kernel zeroes and theorem properties explored

Proposed ideas for further understanding of the theorem

Abstract

We give a new proof of Dennis Hejhal's theorem on the nondegeneracy of the matrix that appears in the identity relating the Bergman and Szego kernels of a smoothly bounded finitely connected domain in the plane. Mergelyan's theorem is at the heart of the argument. We explore connections of Hejhal's theorem to properties of the zeroes of the Szego kernel and propose some ideas to better understand Hejhal's original theorem.