On the zero sets of harmonic polynomials

TL;DR

This paper constructs explicit harmonic polynomials with prescribed zero sets, solving a classical problem and extending previous results by creating new families of such polynomials in higher dimensions.

Contribution

It provides a positive solution to a longstanding problem by explicitly constructing harmonic polynomials vanishing on specific sets, and introduces methods to generate families of such polynomials in multiple dimensions.

Findings

Constructed a harmonic polynomial vanishing on the edges of a unit cube.

Extended results to higher dimensions using harmonic morphisms.

Presented new insights into harmonic functions with zero sets as unions of affine subspaces.

Abstract

In this paper we consider nonzero harmonic functions vanishing on some subsets of $\mathbb R^n$. 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 $\mathbb R^n$ for all $n \geq 4$. 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.