On the product of the singular values of a binary tensor

A real binary tensor consists of 2d real entries arranged into hypercube format 2×d. For d = 2, a real binary tensor is a 2 × 2 matrix with two singular values. Their product is the determinant. We generalize this formula to d ≥ 2. Given a partition μ ⊢ d and a μ-symmetric real binary tensor t, we study the distance function from t to the variety Xμℝ of μ-symmetric real binary tensors of rank one. The study of the local minima of this function is related to the computation of the singular values of t. Denoting with the complexification of Xμℝ, the Euclidean Distance polynomial EDpoly Xμ∨,t(ε2) of the dual variety of Xμ at t has among its roots the singular values of t. On one hand, the lowest coefficient of EDpoly Xμ∨,t(ε2) is the square of the μ-discriminant of t times a product of sum of squares polynomials. On the other hand, we describe the variety of μ-symmetric binary tensors that do not admit the maximum number of singular values, counted with multiplicity. Finally, we compute symbolically all the coefficients of EDpoly Xμ∨,t(ε2) for tensors of format 2 × 2 × 2.