On a New Type of Multiplier Sequences and Discriminant Amoebae

In the classical paper [PS14] I. Schur and G. Polýa introduced and characterized two types of multiplier sequences, i.e. linear operators acting diagonally in the monomial basis of R[x] and sending real-rooted (resp. with all real roots of the same sign) polynomials to real-rooted polynomials. Motivated by a fundamental property of amoebae and discriminants discovered in [GKZ94] we introduce below a new class of sign-independently real-rooted polynomials and describe diagonal operators preserving the latter set. A pleasant circumstance in our description is that this class of diagonal operators essentially coincides with the class of all log-concave sequences.