Universal large deviations for Kac polynomials

TL;DR

This paper establishes the universal large deviations principle for the empirical measures of zeros of Kac polynomials with i.i.d. coefficients, under broad conditions on the coefficient distribution.

Contribution

It proves the universality of large deviations for zeros of Kac polynomials with general coefficient distributions satisfying mild decay and non-vanishing conditions.

Findings

Large deviations principle holds universally for zeros of Kac polynomials.

Results apply to coefficients with densities on C, R, or R+ that do not vanish too fast.

The framework covers broad classes of coefficient distributions.

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 C, R or R + , under the assumption that the density does not vanish too fast at zero and decays at least as exp --|x| $\rho$ , $\rho$ \textgreater{} 0, at infinity.