A Hessian criterion for totally positive Toeplitz matrices and a new proof of Cattani's theorem in two variables

TL;DR

This paper introduces a Hessian criterion for totally positive Toeplitz matrices, providing a new proof of Cattani's theorem in two variables and connecting algebraic relations with matrix positivity.

Contribution

It offers a novel Hessian criterion for Toeplitz matrices and a new proof of Cattani's theorem specifically for codimension two algebras.

Findings

Hessian criterion characterizes total positivity in Toeplitz matrices

New proof of Cattani's theorem in two variables

Connection between algebraic Hodge-Riemann relations and matrix positivity

Abstract

Cattani's theorem for graded Artinian Gorenstein algebras states that the ordinary Hodge-Riemann relations imply the mixed Hodge-Riemann relations under certain conditions. We give a new proof of this result for codimension two algebras. Our proof leads to a Hessian criterion for totally positive Toeplitz matrices, which is analogous to the Wronskian criterion for totally positive flags discovered recently by S. Karp.