On polygonal measures with vanishing harmonic moments

TL;DR

This paper investigates conditions under which signed polygonal measures with vertices in a finite set have all harmonic moments vanish, revealing the dimension of the solution space and existence criteria for specific densities.

Contribution

It provides a formula for the dimension of the space of such measures and establishes existence or non-existence results based on the size of the vertex set.

Findings

Dimension of measure space is (|S|-3)(|S|-4)/2 for generic S.

No such measures exist for |S| ≤ 5.

Existence of measures with densities ±1 for |S| ≥ 6.

Abstract

A signed polygonal measure is the sum of finitely many real constant density measures supported on polygons. Given a finite set S in the plane, we study the existence of signed polygonal measures spanned by polygons with vertices in S, which have all harmonic moments vanishing. For S generic, we show that the dimension of the linear space of such measures is (|S|-3)(|S|-4)/2. We also investigate the situation where the resulting density is either 0, or 1, or -1, which corresponds to pairs of polygons of unit density having the same logarithmic potential at infinity. We show that such a signed measure does not exist if |S| is at most 5, but for each n at least 6 there exists an S, with |S|=n, giving rise to such a signed measure.