New Multiplier Sequences via Discriminant Amoebae

In their classic 1914 paper, Polýa and Schur introduced and characterized two types of linear operators acting diagonally on the monomial basis of R[x], sending real-rooted polynomials (resp. polynomials with all nonzero roots of the same sign) to real-rooted polynomials. Motivated by fundamental properties of amoebae and discriminants discovered by Gelfand, Kapranov, and Zelevinsky, we introduce two new natural classes of polynomials and describe diagonal operators preserving these new classes. A pleasant circumstance in our description is that these classes have a simple explicit description, one of them coinciding with the class of log-concave sequences.