# Extreme value statistics for the roots of a complex Kac polynomial

## TL;DR

This paper studies the statistical behavior of the largest root in complex Gaussian Kac polynomials, focusing on fluctuations and large deviations, with results involving Fredholm determinants and integral series.

## Contribution

It provides a detailed analysis of the fluctuations and large deviations of the largest root, extending previous work with explicit formulas involving Fredholm determinants.

## Key findings

- Fluctuations characterized by Fredholm determinants.
- Large deviations described by a series of multiple integrals.
- Results deepen understanding of root distribution in random polynomials.

## Abstract

We investigate the fluctuations and large deviations of the root of largest modulus in a model of random polynomial with independent complex Gaussian coefficients (Kac polynomials). The fluctuations were recently computed by R. Butez (arxiv 1704.02761) and involve a Fredholm determinant. The precise large deviations show a particular function defined by a series of mutiple integrals in the same vein.