Uniform estimates for almost primes over finite fields

TL;DR

This paper derives a new uniform asymptotic formula for counting polynomials with a given number of prime factors over finite fields, extending previous results to larger $k$ and varying $q$ and $n$.

Contribution

It provides a novel asymptotic estimate for almost primes over finite fields with error tending to zero uniformly in both degree and field size, allowing $k$ to grow beyond $ ext{log } n$.

Findings

Error term tends to zero uniformly in $n$ and $q$

Total variation distance between permutation cycles and polynomial prime factors tends to zero at rate $1/(q\sqrt{ ext{log } n})$

Extension of asymptotic formulas to larger $k$ and varying $q$, $n$

Abstract

We establish a new asymptotic formula for the number of polynomials of degree $n$ with $k$ prime factors over a finite field $\mathbb{F}_q$. The error term tends to $0$ uniformly in $n$ and in $q$, and $k$ can grow beyond $\log n$. Previously, asymptotic formulas were known either for fixed $q$, through the works of Warlimont and Hwang, or for small $k$, through the work of Arratia, Barbour and Tavar\'e. As an application, we estimate the total variation distance between the number of cycles in a random permutation on $n$ elements and the number of prime factors of a random polynomial of degree $n$ over $\mathbb{F}_q$. The distance tends to $0$ at rate $1/(q\sqrt{\log n})$. Previously this was only understood when either $q$ is fixed and $n$ tends to $\infty$, or $n$ is fixed and $q$ tends to $\infty$, by results of Arratia, Barbour and Tavar\'{e}.