2017/06/19 by Jonathan Leake, Leake, Jonathan, Jonathan R. Leake
Computer Science · Mathematics · #Advanced Topics in Algebra #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Mathematics and Applications #Matrix Theory and Algorithms #Polynomial and algebraic computation #Representation Theory (math.RT) #math.CV #math.RT
paper · pdf · doi:10.48550/arxiv.1706.06168
openalex publication_date 2017/06/19 · arxiv created 2021/08/06 · arxiv updated 2021/08/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In 2009, Borcea and Brändén 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-Brändén result to characterize a certain subclass of the linear operators which preserve such polynomials.