In 2008, Borcea and Brändén characterized the set of linear operators preserving the set of polynomials which are non-vanishing (stable) when all inputs are in a given circular region (usually some half-plane). The theory of stable polynomials has recently found many applications in combinatorics and optimization, and in particular was used in the solution to the Kadison-Singer problem. In this talk, we present a few extensions of Borcea-Brändén characterization, which follow nicely from an intuitive and unifying simplification of their proof. First we discuss immediate corollaries, including similar results for polynomials which have all their roots in a given interval or ray. Next we draw a connection to Gurvits' notion of capacity, giving a theory of capacity-preserving operators. Capacity has been used to give a simple proof of the permanent lower bound for doubly stochastic matrices, and more recently as a progress measure in a polynomial time algorithm for the noncommutative symbolic matrix singularity problem. Finally we further discuss the relation of capacity-preserving operators to this and other recent work.
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