Alexandrov’s inequality and conjectures on some Toeplitz matrices

We study determinant inequalities for certain Toeplitz-like matrices over C. For fixed n and N ⩾ 1, let Q be the n × (n + N − 1) zero–one Toeplitz matrix with Qij = 1 for 0 ⩽ j − i ⩽ N − 1 and Qij = 0 otherwise. We prove that det(QQ∗) is the minimum of det(RR∗) over all complex matrices R with the same dimensions as Q satisfying ∣Rij∣ ⩾ 1 whenever Qij = 1 and Rij = 0 otherwise. Although R has a Toeplitz-like band structure, it is not required to be actually Toeplitz. Our proof involves Alexandrov’s inequality for polarized determinants and its generalizations. This problem is motivated by Littlewood’s conjecture on the minimum 1-norm of N-term exponential sums on the unit circle. We also discuss polarized Bazin–Reiss–Picquet identities, some connections with k-tree enumeration, and analogous conjectured inequalities for the elementary symmetric functions of QQ∗.