An explicit polynomial analogue of Romanoff's theorem

TL;DR

This paper investigates the distribution of polynomials over finite fields that can be expressed as the sum of an irreducible polynomial and a polynomial power, establishing an asymptotic proportion related to the degree of the polynomial g.

Contribution

It introduces an explicit polynomial analogue of Romanoff's theorem, quantifying the density of such polynomials over finite fields.

Findings

Proportion of polynomials of degree n of the form h + g^k is approximately 1/deg g.

The result provides an explicit asymptotic estimate for the distribution.

Connects polynomial analogues to classical number theory results.

Abstract

Given a polynomial $g$ of positive degree over a finite field, we show that the proportion of polynomials of degree $n$, which can be written as $h+g^k$, where $h$ is an irreducible polynomial of degree $n$ and $k$ is a nonnegative integer, has order of magnitude $1/\deg g$.