On mother body measures with algebraic Cauchy transform

TL;DR

This paper investigates the existence and properties of motherbody measures for algebraic functions' Cauchy transforms, focusing on algebraic functions and quadratic equations with polynomial coefficients in potential theory.

Contribution

It identifies a large class of algebraic functions likely to have positive motherbody measures and explores the representability of algebraic functions as Cauchy transforms, including quadratic cases.

Findings

A large class of algebraic functions likely admits positive motherbody measures.

Detailed analysis of algebraic functions satisfying quadratic equations with polynomial coefficients.

Open problems and conjectures related to the representability of algebraic functions as Cauchy transforms.

Abstract

Below we discuss the existence of a motherbody measure for the exterior inverse problem in potential theory in the complex plane. More exactly, we study the question of representability almost everywhere (a.e.) in C of (a branch of) an irreducible algebraic function as the Cauchy transform of a signed measure supported on a finite number of compact semi-analytic curves and a finite number of isolated points. Firstly, we present a large class of algebraic functions for which there (conjecturally) always exists a positive measure with the above properties. This class was discovered in our earlier study %of the eigenpolynomials of exactly solvable linear differential operators. Secondly, we investigate in detail the representability problem in the case when the Cauchy transform satisfies a quadratic equation with polynomial coefficients a.e. in C. Several conjectures and open problems are posed.