Skip to content
DraconiticResearch
Search Sources API
203,234 studies31 sourcesUpdated just now
PreprintarXiv licencearXiv

Proof of the planar Khavinson-Shapiro conjecture

  • Feng Shao
  • Weicheng Zhan
Open at sourcePDFDOICite

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.

Related

  1. A proof of Wang's planar D-coloring conjecture
    Harshit Verma
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  2. Rayleigh's conjecture for vibrating plates: simpler planar proof and nonlinear extensions
    Dorin Bucur, Richard S. Laugesen
    arXivPreprintarXiv licenceFull textAnalysis & PDEsarXiv
  3. An elementary proof of a tangent-cone identity for rank-one planar Lipschitz maps
    Igor Buchowiec
    arXivPreprintarXiv licenceFull textGeometry & topologyarXiv
  4. Degreewise Cut-Semigroup Saturation for K5-Minor-Free Graphs and Seymour's Planar Edge-Colouring Conjecture
    SeungJu Lee
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
  5. The Borowiecki-Broere Generalized Total Coloring Conjecture for Planar Graphs
    James Alexander Schreib
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{shao2026proof,
  title = {{Proof of the planar Khavinson-Shapiro conjecture}},
  author = {Feng Shao and Weicheng Zhan},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.29909},
  eprint = {2608.29909},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.29909},
}
RIS
TY  - UNPB
TI  - Proof of the planar Khavinson-Shapiro conjecture
AU  - Feng Shao
AU  - Weicheng Zhan
PY  - 2026
DA  - 2026/08/30
PB  - arXiv
DO  - 10.48550/arxiv.2608.29909
UR  - https://arxiv.org/abs/2608.29909
AB  - Let Ω⊂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 ℝ^2 agrees on ∂Ω with a harmonic polynomial, then ∂Ω is an ellipse and Ω is its bounded interior.
ER  -