$F$-Diophantine sets over finite fields

Abstract

Let $k\ge 2$, $q$ be an odd prime power, and $F\in {\mathbb{F}}_q[x_1,\dots ,x_k]$ be a polynomial. An $F$-Diophantine set over a finite field ${\mathbb{F}}_q$ is a set $A\subset {\mathbb{F}}_q^{\ast }$ such that $F(a_1,a_2,\dots ,a_k)$ is a square in ${\mathbb{F}}_q$ whenever $a_1,a_2,\dots ,a_k$ 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 $F=x_1{x}_2\dots x_k+1$, 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.

7 pages