Real zero polynomials and Pólya-Schur type theorems

Abstract

Real zero preserving operators. Let P(C) denote the space of all polynomials with complex coefficients, regarded as functions on the complex plane. The differentiation operator D = d/dz acts on P(C); the action on the monomials is given by D z = n zn−1 for n = 1, 2, 3, . . .. We also have the operator D∗ of multiplicative differentiation, related toD viaD∗ = zD; the action on the monomials is given byD∗ z = n z for n = 0, 1, 2, . . .. Whereas D commutes with all translations, D∗ commutes with all dilations of the complex plane C. A third differentiation operator D] = zD is of interest; its action on the monomials is given by D] z = n z for n = 0, 1, 2, . . .. We get it by first inverting the plane (z 7→ 1/z), then applying minus differentiation −D, and by finally inverting back again. This means that studying D] on the polynomials is equivalent to studying the ordinary differentiation operator D on the space of all rational functions that are regular at all points of the extended plane with the exception of the origin. The Gauss-Lucas theorem states that if a polynomial p(z) has its zeros contained in some given convex set K, then its derivative Dp(z) = p′(z) has all its zeros in K as well (unless p(z) is constant, that is). In particular, if all the zeros are real, then so are the zeros of the derivative. Naturally, the same statement can be made for the multiplicative derivative D∗ as well, and for D], too. In this context, we should mention the classical theorem of Laguerre [2, p. 23], which extends the Gauss-Lucas theorem for the real zeros case to the more general setting of entire functions of genus 0 or 1. To simplify the later discussion, we introduce the notation P(C;R) for the collection of all polynomials with only real zeros, including all constants. This means that the zero polynomial is in P(C;R), although strictly speaking, it has plenty of non-real zeros. Clearly, P(C;R) constitutes a multiplicative semi-group. Let T : P(C)→ P(C) be a linear operator. Let us say that T is real zero preserving if T (P(C;R)) ⊂ P(C;R); it would be of interest to have a complete characterization of the real zero preserving operators. From the above remarks, we know that D, D∗, and D] are real zero preserving. To get some headway into this general problem, it is helpful to have some additional information regarding the given operator T .