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

2025/07/20 by Chris McDaniel, McDaniel, Chris
Mathematics · #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #FOS: Mathematics #Holomorphic and Operator Theory #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.2507.15043

openalex publication_date 2025/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

Citations

• Universal Plücker coordinates for the Wronski map and positivity in real Schubert calculus
• Jeu de taquin and a monodromy problem for Wronskians of polynomials
• Mixed Lefschetz Theorems and Hodge-Riemann Bilinear Relations
• The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz

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