Multivariate Pólya–Schur classification problems in the Weyl algebra

By Julius Borcea, Petter Brand'en. 2006

105 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/acc8645eaf02c793dab94e4852d1f560aca90c22

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Citations in the corpus

Interlacing families II: Mixed characteristic polynomials and the Kadison{Singer problem (2013, methodology): “One can use general results, such as Theorem 1.3 of [16] or Proposition 2.2 of [39], to prove that these operators preserve real stability.”
The Lee-Yang and Pólya-Schur programs. I. Linear operators preserving stability (2008, methodology): “Progress on Problem 2 was recently made in [ 9 ] where a complete solution was given for finite order partial differential operators when � = � n and � is an open half-plane.”
Spectral graph theory (2019, background): “Rather than dive into the deep theory of real-stable polynomials, we will instead point the interested reader to [4] for a proper treatment of the subject, and we will simply note the following facts without proof.”
Symmetric Decompositions and Real-Rootedness (2018, unknown): “6 in [7]).”
Applications of stable polynomials to mixed determinants: Johnson's conjectures, unimodality, and symmetrized Fischer products (2006, background): “A univariate polynomial f(z) with complex coefficients is called stable if f(z) 6= 0 for all z ∈ C with Im(z) > 0.”
Polya-Schur master theorems for circular domains and their boundaries (2006, unknown): “C V ] 2 7 Ju l 2”
Negative dependence and the geometry of polynomials (2007, methodology): “Moreover, we define a partial order on the set of strongly Rayleigh measures (by means of the notion of proper position for multivariate stable polynomials studied in [7, 8, 9 , 15]), and use it to settle Pemantle’s questions and conjectures on stochastic domination for truncations of “negatively d”
Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees (2013, background): “In [5], Borcea and Brändén characterize a class of differential operators that preserve real stability.”
Multivariate stable polynomials: theory and applications (2009, background): “The introduction of [4], in particular, gives a good account of the genesis of the theory.”