An analogue of Ruzsa's conjecture for polynomials over finite fields

In 1971, Ruzsa conjectured that if $f:\mathbb{N}\to \mathbb{Z}$ with $f(n+k)\equiv f(n)$ mod $k$ for every $n,k\in \mathbb{N}$ and $f(n)=O({\theta }^n)$ with $\theta <e$ then $f$ is a polynomial. In this paper, we investigate the analogous problem for the ring of polynomials over a finite field.