Real algebraic geometry of real algebraic Jordan curves in the plane and the Bergman kernel

TL;DR

This paper characterizes restrictions of real rational functions on algebraic Jordan curves using the Dirichlet-to-Neumann map and Bergman kernel, providing new decompositions and descriptions for these curves, including multiply connected cases.

Contribution

It introduces a novel characterization of rational functions on algebraic Jordan curves via the Dirichlet-to-Neumann map and Bergman kernel, with applications to curve description.

Findings

Partial fractions-like decomposition for rational functions

New descriptions of algebraic Jordan curves

Extension to multiply connected domains

Abstract

We characterize the space of restrictions of real rational functions to certain algebraic Jordan curves in the plane via the Dirichlet-to-Neumann map associated to the domain in the complex plane bounded by the curve and its Bergman kernel. The characterization leads to a partial fractions-like decomposition for such rational functions and new ways to describe such Jordan curves. The multiply connected case is also explored.