F-Diophantine sets over finite fields

Let k ≥ 2, q be an odd prime power, and F ∈ Fq[x₁, …, 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 log q, significantly improving the Θ((log q)1/(k-1)) lower bound in a recent paper by Hammonds-Kim-Miller-Nigam-Onghai-Saikia-Sharma.