Proof of the planar Khavinson-Shapiro conjecture

Source abstract

Let $Ω\subset\mathbb{R}^2$ 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 $\partialΩ$ with a harmonic polynomial, then $\partialΩ$ is an ellipse and $Ω$ is its bounded interior.