Universal large deviations for Kac polynomials

Abstract

We prove the universality of the large deviations principle for the empirical measures of zeros of random polynomials whose coefficients are i.i.d. random variables possessing a density with respect to the Lebesgue measure on $\mathbb{C} $, $\mathbb{R} $ or $\mathbb{R} ^+$, under the assumption that the density does not vanish too fast at zero and decays at least as $\exp -|x|^{\rho }$, $\rho >0$, at infinity.