The recently posted MO-Q "Positivity of the coefficients of Taylor series associated to the Riemann hypothesis" (see also this MO-Q) has re-kindled my interest in hyperbolic polynomials--essentially those with real zeros only. I'm somewhat familiar with single variable case, in which my initial interests revolved around the relation of associated theorems to the calculus of Sheffer Appell polynomials, but a cursory look at the lit shows a much broader definition and, naturally, context for these polynomials. I was led by Zach Harris's effusive answer to the MSE-Q "Intuition behind the relevance of hyperbolic polynomials" to "Lecture 5: Hyperbolic Polynomials" by Pemantle which begins with a quote of G. C. Rota
The one contribution of mine that I hope will be remembered has consisted in pointing out that all sorts of problems of combinatorics can be viewed as problems of the location of the zeros of certain polynomials...
but goes on to give applications outside of combinatorics (see the chart and references therein). An MO search on hyperbolic polynomials gave more than two hundred hits. I think it might be worthwhile to have a compilation of applications related to the work and/or interests of MO users, so
What are the applications of hyperbolic polynomials?
(Not restricted to the single variable case. I see no issue with repeating any of the observations noted above or in the linked references, but please provide references in your response. For me, there is also the historical question of the synergy between work on the Riemann hypothesis, hyperbolic polynomials, symmetric function theory, and the finite operator calculus of Sheffer polynomial sequences--certainly many famous researchers in the mid to late 1800s and early 1900s did work on all of these subjects.)
