The resultant on compact Riemann surfaces

TL;DR

This paper introduces a new concept of the resultant for meromorphic functions on compact Riemann surfaces, linking it to potential theory and algebraic dependence, with applications to quadrature domains.

Contribution

It defines the resultant for meromorphic functions on Riemann surfaces and explores its properties, including integral formulas and relations to potential theory and algebraic dependence.

Findings

Derived integral formulas for the resultant

Connected the resultant to potential theory

Expressed the exponential transform of quadrature domains via the resultant

Abstract

We introduce a notion of resultant of two meromorphic functions on a compact Riemann surface and demonstrate its usefulness in several respects. For example, we exhibit several integral formulas for the resultant, relate it to potential theory and give explicit formulas for the algebraic dependence between two meromorphic functions on a compact Riemann surface. As a particular application, the exponential transform of a quadrature domain in the complex plane is expressed in terms of the resultant of two meromorphic functions on the Schottky double of the domain.