Multivariate Pólya-Schur classification problems in the Weyl algebra
J. Borcea, P. Brändén · Proceedings of the London Mathematical Society · 2010
A multivariate polynomial is stable if it is nonvanishing whenever all variables have positive imaginary parts. We classify all linear partial differential operators in the Weyl algebra 𝒜n that preserve stability. An important tool that we develop in the process is the higher-dimensional generalization of Pólya–Schur's notion of multiplier sequence. We characterize all multivariate multiplier sequences as well as those of finite order. Next, we establish a multivariate extension of the Cauchy–Poincaré interlacing theorem and prove a natural analog of the Lax conjecture for real stable polynomials in two variables. Using the latter we describe all operators in 𝒜1 that preserve univariate hyperbolic polynomials by means of determinants and homogenized symbols. Our methods also yield homotopical properties for symbols of linear stability preservers and a duality theorem showing that an operator in 𝒜n preserves stability if and only if its Fischer–Fock adjoint does. These are powerful multivariate extensions of the classical Hermite–Poulain–Jensen theorem, Pólya's curve theorem and Schur–Maló–Szegőcomposition theorems. Examples and applications to strict stability preservers are also discussed.