LAUSR

Abstract We study E-eigenvalues of a symmetric tensor f of degree d on a finite-dimensional Euclidean vector space V, and their relation with the E-characteristic polynomial of f. We show that the leading coefficient of the E-characteristic polynomial of f, when it has maximum degree, is the ( d − 2 ) -th power (respectively the ( ( d − 2 ) / 2 ) -th power) when d is odd (respectively when d is even) of the Q ˜ -discriminant, where Q ˜ is the d-th Veronese embedding of the isotropic quadric Q ⊆ P ( V ) . This fact, together with a known formula for the constant term of the E-characteristic polynomial of f, leads to a closed formula for the product of the E-eigenvalues of f, which generalizes the fact that the determinant of a symmetric matrix is equal to the product of its eigenvalues.