How Many Eigenvalues of a Random Symmetric Tensor are Real?

Abstract

This article answers a question posed by Draisma and Horobet, who asked for a closed formula for the expected number of real eigenvalues of a random real symmetric tensor drawn from the Gaussian distribution relative to the Bombieri norm. This expected value is equal to the expected number of real critical points on the unit sphere of a Kostlan polynomial. We also derive an exact formula for the expected absolute value of the determinant of a matrix from the Gaussian Orthogonal Ensemble.