On the zero sets of harmonic polynomials
Ioann Vasilyev
TL;DR
For every $d\ge 3$ the paper resolves Wavre's Problem 151 by constructing a nonzero harmonic polynomial vanishing on the $d-2$ skeleton $S_{d-2,d}$ of the unit cube, exemplified by $f_d(x)=\prod_{i<j}(x_i^2-x_j^2)$ with $\Delta f_d=0$. It then uses harmonic morphisms to generate infinite families of harmonic polynomials vanishing on the same set in the unit ball for all $n\ge 4$, notably via $P_k(x)=\mathrm{Re}(\phi_1+i\phi_2)^{2k+1}$ with $\phi_1$ and $\phi_2$ defined, and extends these ideas to odd dimensions; a general lifting principle ensures the construction persists under composition with harmonic morphisms. A parallel geometric result characterizes when a nonzero harmonic function can vanish on the boundary of a triangular prism, tying the existence to discrete reflection tilings of the cross-section and classifying the four tileable triangles. The work connects explicit polynomial constructions, harmonic morphisms, and reflection-group theory, extending prior results by Logunov and Malinnikova and highlighting open questions about zero sets on unions of affine codimension-two 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.