On the zero sets of harmonic polynomials

2025/06/30 by Ioann Vasilyev, Vasilyev, Ioann · 1 citation
Mathematics · #31B05 #58E20 #Analysis of PDEs (math.AP) #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2506.24116

openalex publication_date 2025/06/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we consider nonzero harmonic functions vanishing on some subsets of \mathbb Rⁿ. We give a positive solution to Problem 151 from the Scottish Book posed by R. Wavre in 1936. In more detail, we construct a nonzero harmonic polynomial that vanishes on the edges of the unit cube. Moreover, using harmonic morphisms we build new nontrivial families of harmonic polynomials that vanish at the same set in the unit ball in \mathbb Rⁿ for all n ≥ 4. This extends certain results by Logunov and Malinnikova. We also present new results on harmonic functions in the space whose zero sets are unions of affine codimension two subspaces.

Citations

• Twistor construction of some multivalued harmonic functions on \bf R³
• Quadratic cones on which few harmonic functions vanish
• Upper bounds for the number of isolated critical points via Thom-Milnor theorem
• Weighted gradient inequalities and unique continuation problems
• Quadratic harmonic morphisms and O-systems

Cited by

• 90+ years of the Scottish Book

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