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

TL;DR

This work tackles the logarithmic Fekete problem on the unit sphere by converting the real optimization into a finite algebraic problem via a dot-product (x_ij) formulation and symmetry reduction. Using Gröbner bases and the Autonne-Takagi factorization, it enumerates all critical configurations for up to six points in all dimensions, recovering known results in a unified framework and revealing new real and complex configurations for six points. The authors classify these critical points (saddles and minima) and provide detailed energies, ranks, and, for six points, a complete account including multiplicities of certain solutions. The methodology provides a rigorous, computer-assisted enumeration of all critical configurations, offering a comprehensive inventory of saddle points and global minima with clear geometric and algebraic interpretations, and it sets the stage for extending the classification to seven points.

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.

Theorems & Definitions (12)

• Definition 1: Dot product matrix
• Definition 2: Cartesian coordinates matrix
• Proposition 1: Autonne-Takagi factorization horn2013
• Corollary 1
• proof
• Proposition 2: horn2013
• Theorem 1: cox2005
• Proposition 3
• Corollary 2
• Proposition 4: hartshorne2013algebraic