A note on isoparametric polynomials

TL;DR

This paper characterizes homogeneous polynomial solutions to a specific gradient magnitude equation, revealing they are either radially symmetric or composed of Chebyshev and Cartan-Münzner polynomials, thus advancing understanding of isoparametric polynomials.

Contribution

It classifies all homogeneous polynomial solutions to a gradient magnitude equation, linking them to well-known polynomial families and providing a complete characterization.

Findings

Solutions are either radially symmetric or composed of Chebyshev and Cartan-Münzner polynomials.

Provides a complete classification of solutions to the gradient magnitude equation.

Connects solutions to classical polynomial families in algebra and geometry.

Abstract

We show that any homogeneous polynomial solution of |\nabla F(x)|^2=m^2|x|^(2m-2), m>1, is either a radially symmetric polynomial F(x)=\pm |x|^m (for even m's) or it is a composition of a Chebychev polynomial and a Cartan-M\"unzner polynomial.