Romanoff type theorems for polynomials over finite fields

Source 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 $ρ_g=\varphi_q(g)/|g|$, 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.