2019/07/19 by Felipe Rincón, Cynthia Vinzant, Rincón, Felipe +3 · 1 citation
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Affine transformation #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Codimension #Combinatorics #Combinatorics (math.CO) #Dimension (graph theory) #FOS: Mathematics #Hyperbolic equilibrium point #Hyperbolic function #Hyperbolic manifold #Hyperplane #Mathematical analysis #Mathematics #Polynomial and algebraic computation #Pure mathematics #Sign (mathematics) #Subspace topology #Variety (cybernetics) #math.AG #math.CO
paper · pdf · doi:10.48550/arxiv.1907.08545
published in arXiv (Cornell University) (Cornell University) · Final version to appear in Advances in Mathematics. Corollary 4.9 from version 2 is replaced with stronger Theorem 4.10 and Remark 4.11, on M-convexity of tropicalization of coefficients of stable polynomials and generalizations
openalex publication_date 2019/07/19 · arxiv created 2021/03/04 · arxiv updated 2021/03/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
A variety of codimension c in complex affine space is called positively hyperbolic if the imaginary part of any point in it does not lie in any positive linear subspace of dimension c. Positively hyperbolic hypersurfaces are defined by stable polynomials. We give a new characterization of positively hyperbolic varieties using sign variations, and show that they are equivalently defined by being hyperbolic with respect to the positive part of the Grassmannian, in the sense of Shamovich and Vinnikov. We prove that positively hyperbolic projective varieties have tropicalizations that are locally subfans of the type A hyperplane arrangement defined by xi = xj, in which the maximal cones satisfy a non-crossing condition. This gives new proofs of some results of Choe--Oxley--Sokal--Wagner and Brändén on Newton polytopes and tropicalizations of stable polynomials. We settle the question of which tropical varieties can be obtained as tropicalizations of positively hyperbolic varieties in the case of tropical toric varieties, constant-coefficient tropical curves, and Bergman fans. Along the way, we also give a new characterization of positroids in terms of a non-crossing condition on their Bergman fans.