$F$-Diophantine sets over finite fields

TL;DR

This paper introduces a method to construct large $F$-Diophantine sets over finite fields for certain polynomials, notably improving known bounds for the case $F=x_1x_2...x_k+1$, with implications for Diophantine tuples.

Contribution

It provides a new construction strategy for large $F$-Diophantine sets over finite fields, especially for polynomials with specific monomial properties, improving previous bounds.

Findings

Constructs large $F$-Diophantine sets when $F$ has a monomial expansion property.

Achieves size $ ext{gg}_k ext{log} q$ for $k$-Diophantine tuples with $F=x_1x_2...x_k+1$.

Significantly improves lower bounds over previous results.

Abstract

Let $k \geq 2$, $q$ be an odd prime power, and $F \in \mathbb{F}_q[x_1, \ldots, x_k]$ be a polynomial. An $F$-Diophantine set over a finite field $\mathbb{F}_q$ is a set $A \subset \mathbb{F}_q^*$ such that $F(a_1, a_2, \ldots, a_k)$ is a square in $\mathbb{F}_q$ whenever $a_1, a_2, \ldots, a_k$ are distinct elements in $A$. In this paper, we provide a strategy to construct a large $F$-Diophantine set, provided that $F$ has a nice property in terms of its monomial expansion. In particular, when $F=x_1x_2\ldots x_k+1$, our construction gives a $k$-Diophantine tuple over $\mathbb{F}_q$ with size $\gg_k \log q$, significantly improving the $\Theta((\log q)^{1/(k-1)})$ lower bound in a recent paper by Hammonds-Kim-Miller-Nigam-Onghai-Saikia-Sharma.