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A preprint claims the planar Khavinson–Shapiro conjecture

Feng Shao and Weicheng Zhan claim that a bounded planar domain whose boundary is a finite union of disjoint Jordan curves has polynomial Dirichlet solutions for every polynomial boundary datum only when its boundary is an ellipse and the domain is the ellipse’s interior.

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Stable ID
math-planar-khavinson-shapiro-conjecture-proof-2026
Revision
math-planar-khavinson-shapiro-conjecture-proof-2026.v1
Field
Mathematics · Potential theory, partial differential equations and algebraic geometry
Evidence
Tier 1 · Peer reviewed: No
Record state
Provisional · Provisional claimed proof
Last checked

AI role

The manuscript does not disclose a substantive AI role in the proof or writing.

Record details

Problem or result
The planar Khavinson–Shapiro classification of domains with polynomial solutions to polynomial Dirichlet data
Authors
Feng Shao and Weicheng Zhan
Institutions
Xiamen University
Result date
First-version preprint submitted August 30, 2026

Why it matters

The conjecture asks whether exact polynomial solvability forces a domain’s geometry to be quadratic. The claimed theorem settles the planar case without assuming a smooth or connected boundary.

Limits

This is a first-version, non-peer-reviewed preprint with no located independent correctness assessment. Its scope is two-dimensional bounded domains with finitely many Jordan boundary components; it does not settle the higher-dimensional conjecture.

Sources

  1. Primary: Shao and Zhan, planar Khavinson–Shapiro proof claim
  2. Problem context: Tikaradze, planar polynomial Dirichlet formulation

Correction and revision history

  1. 2026-09-01 — Added after primary-source, scope, status, AI-role and limitation review.

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