Seigal's Conjecture on Gram Determinants of Real Binary Tensors Fails for n ≥ 4

For a real tensor of format 2×⋯×2 with n factors, let d_i be the determinant of the Gram matrix M_i M_i^T of its i-th flattening M_i. Seigal called the set of tuples (d_1, …, d_n) of tensors in the Frobenius unit ball the Gram locus, described it for n = 3, and conjectured that for n ≥ 4 it is cut out of the cube [0, 1/4]^n by a single polynomial inequality Q_1 ≥ Q_2 (Linear Algebra Appl. 2018, Conjecture 1.5; Oberwolfach Reports 2017). We point out that this conjecture is false for every n ≥ 4. For instance, (9/100, 9/100, 9/100, 6/25) satisfies Q_1 > 500 Q_2 but is not the Gram tuple of any real or complex tensor in the unit ball, and the counterexamples form a set with nonempty interior. This is a short corollary of known results: by theorems of Domanov, Stegeman and De Lathauwer (2017), and of Higuchi, Sudbery and Szulc (2003) for complex tensors, plus an elementary step from the unit sphere to the unit ball, the Gram locus is the set of d ∈ [0, 1/4]^n with Σ_{j≠i} √(1 − 4d_j) − √(1 − 4d_i) ≤ n − 2 for all i. From this we obtain the correct n-ary form of Seigal's theorem for n = 3, a union of two basic semialgebraic sets given by explicit integer polynomials, and we show that for no n ≥ 3 is the Gram locus the nonnegativity set of a single polynomial inside the cube (for n = 2 it is). We also correct a misprinted inequality in Seigal's theorem for n = 3. This is an unrefereed note.