# Harmonic functions vanishing on a cone

## TL;DR

This paper investigates the dimensionality of harmonic functions vanishing on a specific quadratic harmonic cone, providing evidence that for the right circular case, this family is finite dimensional, using an innovative arithmetic approach.

## Contribution

The paper introduces a novel arithmetic method to analyze harmonic functions on quadratic cones, extending previous ideas and addressing a longstanding open question.

## Key findings

- Harmonic functions vanishing on the right circular harmonic cone are likely finite dimensional.
- The new arithmetic method extends Holt and Ille's ideas and resembles Hensel's Lemma.
- No known nondegenerate quadratic harmonic cone has a proven infinite-dimensional family of vanishing harmonic functions.

## Abstract

Let $Z$ be a quadratic harmonic cone in $\mathbb{R}^{3}$. We consider the family $\mathcal{H}(Z)$ of all harmonic functions vanishing on $Z$. Is $\mathcal{H}(Z)$ finite or infinite dimensional? Some aspects of this question go back to as early as the 19th century. To the best of our knowledge, no nondegenerate quadratic harmonic cone exists for which the answer to this question is known. In this paper we study the right circular harmonic cone and give evidence that the family of harmonic functions vanishing on it is, maybe surprisingly, finite dimensional. We introduce an arithmetic method to handle this question which extends ideas of Holt and Ille and is reminiscent of Hensel's Lemma.