Positively Hyperbolic Varieties, Tropicalization, and Positroids

TL;DR

This paper introduces positively hyperbolic varieties, characterizes them via sign variations, and explores their tropicalizations, linking stability, hyperbolicity, and combinatorial structures like positroids and hyperplane arrangements.

Contribution

It provides a new characterization of positively hyperbolic varieties through sign variations and establishes their tropicalizations as non-crossing hyperplane arrangements, connecting stability with tropical geometry.

Findings

Positively hyperbolic varieties are characterized by sign variations.

Tropicalizations of these varieties form non-crossing hyperplane arrangements.

The work links stability, hyperbolicity, and positroids through tropical geometry.

Abstract

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 $x_i = x_j$, in which the maximal cones satisfy a non-crossing condition. This gives new proofs of some results of Choe--Oxley--Sokal--Wagner and Br\"and\'en 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.