Universality for low degree factors of random polynomials over finite fields

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 $p\le \exp (n^{1/13})$ where $n$ is the degree of the polynomial. Our proofs use Fourier analysis, and rely on tools recently applied by Breuillard and Varjú 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.

arXiv commentv2: updated introduction, 34 pages, 4 figures. Comments are welcome!

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