Romanoff type theorems for polynomials over finite fields

Abstract

Given a polynomial \(g\) of degree \(δ>0\) over a finite field, we study monic polynomials \(f\) of degree \(n\) that can be written as \(f=h+g^k\), where \(h\) is a monic irreducible polynomial of degree \(n\) and \(k\in\mathbb{N}\) with \(δk<n\). Following Erdős, we show that some \(f\) admit at least \(c\log n\) such representations. If \(δρ_g<1\), where \(ρ_g=φ_q(g)/|g|\), then a positive proportion of degree-\(n\) polynomials admits at least two such representations. More generally, for sums of powers \(g^{\lfloor k_i^{r_i}\rfloor}\), we prove that a positive proportion of polynomials of degree \(n\) is representable whenever \(\sum_{i=1}^{t}r_i^{-1}\ge 1\), while certain residue classes modulo some irreducible polynomial contain no representable polynomial.