Extreme points of the Vandermonde determinant on the sphere and some limits involving the generalized Vandermonde determinant

TL;DR

This paper investigates the extreme points of Vandermonde determinants on the sphere, providing analytical solutions, visual analysis, and exploring limits involving generalized Vandermonde matrices across various dimensions.

Contribution

It offers explicit analytical expressions for the roots of Vandermonde determinants on the sphere and explores their relation to Hermite polynomials and generalized matrices.

Findings

Roots of Vandermonde determinants are roots of rescaled Hermite polynomials.

Analytical expressions are provided for dimensions three to seven.

Relations between ordinary and generalized Vandermonde matrices involving limits are discussed.

Abstract

The values of the determinant of Vandermonde matrices with real elements are analyzed both visually and analytically over the unit sphere in various dimensions. For three dimensions some generalized Vandermonde matrices are analyzed visually. The extreme points of the ordinary Vandermonde determinant on finite-dimensional unit spheres are given as the roots of rescaled Hermite polynomials and a recursion relation is provided for the polynomial coefficients. Analytical expressions for these roots are also given for dimension three to seven. A transformation of the optimization problem is provided and some relations between the ordinary and generalized Vandermonde matrices involving limits are discussed.