Sidon basis in polynomial rings over finite fields

TL;DR

This paper establishes the existence of Sidon bases of order 3 in polynomial rings over finite fields, extending classical additive number theory results to algebraic structures over finite fields.

Contribution

It proves the existence of Sidon bases of order 3 in polynomial rings over finite fields, an analogue of Erdős's conjecture for integers.

Findings

Existence of a $B_2[2]$ sequence of non-zero polynomials forming an asymptotic basis of order 3.

Existence of a Sidon basis of order $3 + ext{epsilon}$ for any epsilon > 0.

Any large degree polynomial can be expressed as a sum of four elements from the sequence, with one having small degree.

Abstract

Let $\mathbb{F}_q[t]$ denote the ring of polynomials over $\mathbb{F}_q$, the finite field of $q$ elements. Suppose the characteristic of $\mathbb{F}_q$ is not $2$ or $3$. In this paper, we prove an $\mathbb{F}_q[t]$-analogue of results related to the conjecture of Erd\H{o}s on the existence of infinite Sidon sequence of positive integers which is an asymptotic basis of order 3. We prove that there exists a $B_2[2]$ sequence of non-zero polynomials in $\mathbb{F}_q[t]$, which is an asymptotic basis of order $3$. We also prove that for any $\varepsilon> 0$, there exists a sequence of non-zero polynomials in $\mathbb{F}_q[t]$, which is a Sidon basis of order $3 + \varepsilon$. In other words, there exists a sequence of non-zero polynomials in $\mathbb{F}_q[t]$ such that any $n \in \mathbb{F}_q[t]$ of sufficiently large degree can be expressed as a sum of four elements of the sequence, where one of them has a degree less than or equal to $\varepsilon \text{deg } n.$