Key points are not available for this paper at this time.
Let k 2, q be an odd prime power, and F Fqx₁, , xₖ be a polynomial. An F-Diophantine set over a finite field Fq is a set A Fq^* such that F (a₁, a₂, , aₖ) is a square in Fq whenever a₁, a₂, , aₖ 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=x₁x₂ xₖ+1, our construction gives a k-Diophantine tuple over Fq with size ₖ q, significantly improving the ( (q) ^1/ (k-1) ) lower bound in a recent paper by Hammonds-Kim-Miller-Nigam-Onghai-Saikia-Sharma.
Yip et al. (Sat,) studied this question.