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arXiv · 2507.15043

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

Chris McDaniel
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Abstract

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.

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Chris McDaniel. 2025-07-20. A Hessian criterion for totally nonnegative Toeplitz matrices and a theorem of Cattani. https://arxiv.org/abs/2507.15043

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