Positively hyperbolic varieties, tropicalization, and positroids

A multivariate complex polynomial is called stable if any line in any positive direction meets its hypersurface only at real points. Stable polynomials have close relations to matroids and hyperbolic programming. We will discuss a generalization of stability to algebraic varieties of codimension larger than one. They are varieties which are hyperbolic with respect to the nonnegative Grassmannian, following the notion of hyperbolicity studied by Shamovich, Vinnikov, Kummer, and Vinzant. We show that their tropicalization and Chow polytopes have nice combinatorial structures related to braid arrangements and positroids, generalizing some results of Choe, Oxley, Sokal, Wagner, and Brändén on Newton polytopes and tropicalizations of stable polynomials. This is based on joint work with Felipe Rincón and Cynthia Vinzant.