Proof of the planar Khavinson-Shapiro conjecture

Abstract

Let $\Omega \subset {\mathbb{R}}^2$ be a bounded domain whose boundary is a finite union of pairwise disjoint Jordan curves. We prove the planar Khavinson--Shapiro conjecture in this setting: if every polynomial on ${\mathbb{R}}^2$ agrees on $\partial \Omega$ with a harmonic polynomial, then $\partial \Omega$ is an ellipse and $\Omega$ is its bounded interior.

arXiv comment21 pages