The product of the eigenvalues of a symmetric tensor

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.