Some spherical codes in S2 and their algebraic numbers

TL;DR

This paper explores spherical codes on S2, analyzing their algebraic properties and structures for minimal energy configurations, revealing complex algebraic numbers and polynomial relations that challenge current mathematical methods.

Contribution

It introduces a detailed analysis of spherical codes' algebraic structures and reports the recovery of numerous algebraic number sets from minimal energy configurations.

Findings

Identification of embedded polygonal structures in spherical codes

Recovery of 49 algebraic number sets from spherical codes

High algebraic degrees of minimal polynomials pose significant challenges

Abstract

The first 195 spherical codes for the global minima of 1 to 65 points on S2 have been obtained for 3 types of potentials: logarithmic, Coulomb, called the Thomson problem, and the inverse square law, with 77, 38, and 38 digits precision respectively. It was discovered that certain point sets have embedded polygonal structures, constraining the points, enabling them to be parameterized and to successfully recover the algebraic polynomial. So far 49 algebraic number sets have been recovered, but 109 more remain to be recovered from their 1,622 parameters, 983 known to 50,014 digit precision. The very high algebraic degree of these minimal polynomials eludes finding the algebraic numbers from the spherical codes and requires new mathematical tools to meet this challenge.