On the distribution of polynomials having a given number of irreducible factors over finite fields

TL;DR

This paper derives an asymptotic formula for counting monic polynomials over finite fields with a fixed number of irreducible factors, revealing growth patterns that differ from the integer prime factor case.

Contribution

It provides the first asymptotic count for polynomials with a given number of irreducible factors over finite fields, highlighting differences from integer prime factor distributions.

Findings

Number of such polynomials grows faster than integer analogs when k exceeds log n

Asymptotic formulas are established for fixed q and varying n and k

Growth rate surpasses initial expectations based on integer prime factorization

Abstract

Let $q\geqslant 2$ be a fixed prime power. We prove an asymptotic formula for counting the number of monic polynomials that are of degree $n$ and have exactly $k$ irreducible factors over the finite field $\mathbb{F}_q$. We also compare our results with the analogous existing ones in the integer case, where one studies all the natural numbers up to $x$ with exactly $k$ prime factors. In particular, we show that the number of monic polynomials grows at a surprisingly higher rate when $k$ is a little larger than $\log n$ than what one would speculate from looking at the integer case.