F-Diophantine sets over finite fields

Let [Formula: see text], q be an odd prime power, and [Formula: see text] be a polynomial. An F-Diophantine set over a finite field [Formula: see text] is a set [Formula: see text] such that [Formula: see text] is a square in [Formula: see text] whenever [Formula: see text] 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 [Formula: see text], our construction gives a k-Diophantine tuple over [Formula: see text] with size [Formula: see text], significantly improving the [Formula: see text] lower bound in a recent paper by Hammonds–Kim–Miller–Nigam–Onghai–Saikia–Sharma.