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

TL;DR

This paper explores an analogue of Ruzsa's conjecture within the context of polynomials over finite fields, examining conditions under which certain polynomial functions exhibit specific algebraic properties.

Contribution

The paper formulates and investigates a finite field analogue of Ruzsa's conjecture, extending the understanding of polynomial functions under modular and growth constraints.

Findings

Established conditions under which polynomial functions over finite fields behave similarly to the conjectured properties.

Provided partial results supporting the finite field analogue of Ruzsa's conjecture.

Identified key differences and similarities between integer and finite field cases.

Abstract

In 1971, Ruzsa conjectured that if $f:\ \mathbb{N}\rightarrow\mathbb{Z}$ 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.