A Representation Theoretic Explanation of the Borcea-Br\"and\'en Characterization

In 2009, Borcea and Br\"and\'en characterize all linear operators on multivariate polynomials which preserve the property of being non-vanishing (stable) on products of prescribed open circular regions. We give a representation theoretic interpretation of their findings, which generalizes and simplifies their result and leads to a conceptual unification of many related results in polynomial stability theory. At the heart of this unification is a generalized Grace's theorem which addresses polynomials whose roots are all contained in some real interval or ray. This generalization allows us to extend the Borcea-Br\"and\'en result to characterize a certain subclass of the linear operators which preserve such polynomials.