A Hessian criterion for totally nonnegative Toeplitz matrices and a theorem of Cattani

To a homogeneous real bivariate form we associate a family of Toeplitz matrices, as well as a family of auxiliary homogeneous real bivariate forms called higher Hessian polynomials.

We show that for a given form, certain positivity properties of its higher Hessian polynomials imply total nonnegativity of its Toeplitz matrices.

One of our main result turns out to be equivalent to a special case of a deep theorem from Hodge theory due to E.

Cattani.

Our other result uses a recent theorem of S.

Karp and K.

Purbhoo related to problems in real Schubert calculus.