Characterization of Logarithmic Fekete Critical Configurations of at Most Six Points in All Dimensions

TL;DR

This paper classifies all critical point configurations for the logarithmic Fekete problem with up to six points in any dimension, providing new proofs and insights into their structure and global minimizers.

Contribution

It offers a comprehensive classification of critical configurations for up to six points across all dimensions using computational algebraic geometry, and investigates the global minimizer for seven points on the sphere.

Findings

All critical configurations for up to six points are classified.

New proofs of existing key results are provided using a unified approach.

Numerical evidence suggests the global minimizer for seven points on S^2 is unique among critical configurations.

Abstract

We consider the logarithmic Fekete problem, which consists of placing a fixed number of points on the unit sphere in $\mathbb{R}^d$, in such a way that the product of all pairs of mutual Euclidean distances is maximized or, equivalently, so that their logarithmic energy is minimized. Using tools from Computational Algebraic Geometry, we find and classify all critical configurations for this problem when considering at most six points in every dimension $d$. In particular, our approach gives new proofs of several key results appearing in the literature, with the benefit of using a unified approach. Furthermore, for seven points in $S^2$, we characterize the global minimizer among critical configurations having at least one pair of antipodal points, and give numerical evidence to support the conjecture that this configuration is also the unrestricted global minimizer.