A generalization of Cartan's theorem on isoparametric cubics

TL;DR

This paper extends Cartan's classical result by classifying all homogeneous cubic solutions to a specific eiconal equation, revealing they are either standard or exceptional Cartan cubics in certain dimensions.

Contribution

It generalizes Cartan's theorem by identifying all cubic solutions to the eiconal equation as either standard or exceptional Cartan cubics in specified dimensions.

Findings

Homogeneous cubic solutions are rotationally equivalent to known forms.

Classifies solutions in dimensions 5, 8, 14, 26.

Includes four exceptional Cartan cubic polynomials.

Abstract

We give a generalization of the well-known result of E. Cartan on isoparametric cubics by showing that a homogeneous cubic polynomial solution of the eiconal equation $|\nabla f|^2=9|x|^4$ must be rotationally equivalent to either $x_n^3-3x_n(x_1^2+...+x_{n-1}^2)$, or to one of four exceptional Cartan cubic polynomials in dimensions $n=5,8,14,26$.