Real roots of non-centered random polynomials

Abstract

We study the fluctuations of the number of real roots of random polynomials with independent, nonzero-mean coefficients. Such non-centered ensembles arise naturally in signal-plus-noise models and in random perturbations of deterministic polynomials. While Ibragimov and Maslova (1971) established the leading asymptotics of the expected number of real roots for non-centered polynomials with i.i.d. coefficients, the corresponding variance asymptotics and central limit theorem have remained open for more than fifty years. This stands in sharp contrast to the centered case, where the fluctuation theory is now well understood across a wide range of ensembles. We resolve this gap by developing novel comparison principles that reduce the fluctuation theory of a non-centered ensemble to that of its centered counterpart. These principles yield sharp variance asymptotics and central limit theorems for broad classes of ensembles, including Kac and hyperbolic polynomials, their derivatives, and related extensions. In particular, for both Kac and hyperbolic polynomials, the leading variance constant equals exactly one-half of that in the centered case, reflecting asymmetric suppression of fluctuations across the two regions where roots concentrate. Our results provide the first comprehensive fluctuation theory for the number of real roots of non-centered random polynomials.