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

DOI10.1007/S40993-022-00423-9arXiv2206.12743MaRDI QIDQ6403149

Arghya Datta

Publication date: 25 June 2022

Abstract: Let $qgeqslant2$ 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 $logn$ than what one would speculate from looking at the integer case.

Mathematics Subject Classification ID

Asymptotic results on arithmetic functions (11N37) Polynomials over finite fields (11T06) Arithmetic theory of polynomial rings over finite fields (11T55) Rate of growth of arithmetic functions (11N56)

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