$F$-Diophantine sets over finite fields
Chi Hoi Yip, Semin Yoo
TL;DR
The paper addresses constructing large $F$-Diophantine sets over finite fields ${\mathbb F}_q$ and bounds the maximal size $M(F;{\mathbb F}_q)$. It introduces a monomial-expansion framework, defining type I/II polynomials, and proves a general lower bound $M(F;{\mathbb F}_q)\ge \left\lfloor \frac{1}{d}\bigl(\log_4 q-4\log_4 \log_4 q\bigr)^{1/m}\right\rfloor$ for degree $d$ and $m$ non-constant monomials, via a $V(n)$-construction and Weil bounds on character sums. As a concrete corollary, for $F(x_1,\dots,x_k)=x_1x_2\cdots x_k+1$ one obtains a $k$-Diophantine tuple over ${\mathbb F}_q$ of size at least $(1/k-o(1))\log_4 q$, improving the previous $\Theta((\log q)^{1/(k-1)})$ bound. The method extends to certain sparse homogeneous polynomials and yields strong $F$-Diophantine sets, with discussion of generalizations and non-optimized constants. This work builds bridges between Diophantine-type sets over finite fields and explicit character-sum bounds.
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 $Θ((\log q)^{1/(k-1)})$ lower bound in a recent paper by Hammonds-Kim-Miller-Nigam-Onghai-Saikia-Sharma.