Romanoff's theorem for polynomials over finite fields revisited
Volume 200 / 2021
Acta Arithmetica 200 (2021), 1-15
MSC: 11T06, 11B05, 11N36.
DOI: 10.4064/aa190618-1-4
Published online: 9 September 2021
Abstract
Let $g$ be a given polynomial of positive degree over a finite field. Shparlinski and Weingartner proved that the proportion of monic polynomials of degree $n$ which can be represented as $h+g^k$ has the order of magnitude $1/\mathrm {deg} g$, where $h$ is chosen from the set of irreducible monic polynomials of degree $n$ and $k\in \mathbb {N}$. In this paper, we show that the proportion of monic polynomials of degree $n$ which can be written as $l+g^p$, where $l$ is the product of two monic irreducible polynomials with $\mathrm {deg} l=n$ and $p$ is a prime number, still has the order of magnitude $1/\mathrm {deg} g$.
