On the exponential transform of multi-sheeted algebraic domains

TL;DR

This paper introduces multi-sheeted algebraic and quadrature domains as branched coverings over the Riemann sphere, showing their equivalence and relating their exponential transform to an elimination function, with applications to algebraic curves like the Neumann oval.

Contribution

It establishes the equivalence of multi-sheeted algebraic and quadrature domains and links their exponential transform to the elimination function, expanding the theory of algebraic domains.

Findings

Extended exponential transform matches the elimination function up to simple factors.

Multi-sheeted algebraic domains are characterized as branched coverings over the Riemann sphere.

Certain algebraic curves from the Neumann oval produce multi-sheeted algebraic domains.

Abstract

We introduce multi-sheeted versions of algebraic domains and quadrature domains, allowing them to be branched covering surfaces over the Riemann sphere. The two classes of domains turn out to be the same, and the main result states that the extended exponential transform of such a domain agrees, apart from some simple factors, with the extended elimination function for a generating pair of functions. In an example we discuss the algebraic curves associated to level curves of the Neumann oval, and determine which of these give rise to multi-sheeted algebraic domains.