Universality for low degree factors of random polynomials over finite fields

TL;DR

This paper demonstrates that the distribution of low degree irreducible factors of random polynomials over finite fields exhibits universality, similar to uniform polynomials, using Fourier analysis and recent equidistribution tools.

Contribution

It extends universality results to non-uniform coefficients and multiple roots, employing Fourier analysis and recent equidistribution techniques.

Findings

Distribution of low degree factors matches that of uniform polynomials

Universality holds under certain parameter conditions, including prime fields with size up to exponential in degree

Methods extend to handle multiple roots and Hasse derivatives

Abstract

We show that the counts of low degree irreducible factors of a random polynomial $f$ over $\mathbb{F}_q$ with independent but non-uniform coefficients behave like that of a uniform random polynomial, exhibiting a form of universality for random polynomials over finite fields. Our strongest results require various assumptions on the parameters, but we are able to obtain results requiring only $q=p$ a prime with $p\leq \exp({n^{1/13}})$ where $n$ is the degree of the polynomial. Our proofs use Fourier analysis, and rely on tools recently applied by Breuillard and Varj\'u to study the $ax+b$ process, which show equidistribution for $f(\alpha)$ at a single point. We extend this to handle multiple roots and the Hasse derivatives of $f$, which allow us to study the irreducible factors with multiplicity.