Proof of the planar Khavinson-Shapiro conjecture
Feng Shao, Weicheng Zhan
Abstract
Let Ω⊂R2 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 R2 agrees on ∂Ω with a harmonic polynomial, then ∂Ω is an ellipse and Ω is its bounded interior.
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