Proof of the planar Khavinson-Shapiro conjecture

Feng Shao; Weicheng Zhan

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arXiv · 2608.29909

Proof of the planar Khavinson-Shapiro conjecture

Feng Shao·Weicheng Zhan

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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.

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Feng Shao, Weicheng Zhan. 2026-08-30. Proof of the planar Khavinson-Shapiro conjecture. https://arxiv.org/abs/2608.29909

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