Totally Nonnegative Toeplitz Matrices and Hodge-Riemann Relations in Codimension Two

It is shown that in the Euclidean space of real rectangular matrices of a fixed size, the closure of the subset of totally positive Toeplitz matrices is equal to the set of totally nonnegative Toeplitz matrices. The proof appeals to an interpretation of a Toeplitz matrix as the coefficient matrix of the higher mixed Hessian of a homogeneous polynomial in two variables. Regarding the homogeneous polynomial as the Macaulay dual generator of a certain graded Artinian Gorenstein algebra, total positivity of its Toeplitz matrix then corresponds to the mixed Hodge-Riemann relations on its algebra. This short note is based on results from the longer paper “Higher Lorentzian polynomials, Higher Hessians, and Hodge-Riemann relations on Artinian Gorenstein algebras in codimension two” by P. Macias Marques, C. McDaniel, A. Seceleanu, and J. Watanabe ( arXiv:2208.05653v3 ).