The Borcea Branden characterization for stability preservers in the Weyl algebra, I

Let p be a polynomial in n variables with complex coefficients. The Weyl algebra consists of differential operators with associated polynomials called symbols. I shall prove at least one direction of a theorem due to Borcea and Branden that a differential operator in the Weyl algebra preserves stability if and only if its symbol is stable.

I shall describe a general strategy for using this to prove the real rootedness of families of polynomials and mention a few applications to spectral graph theory.

Josiah Sugarman