On the number of Diophantine $m$-tuples in finite fields

TL;DR

This paper improves the error bounds in the asymptotic count of Diophantine m-tuples over finite fields using a novel argument, refining previous results by Dujella, Kazalicki, Mani, and Rubinstein-Salzedo.

Contribution

It introduces a new method that enhances the accuracy of the asymptotic formula for Diophantine m-tuples in finite fields.

Findings

Sharper error term in the asymptotic formula

Improved understanding of Diophantine m-tuples distribution

Refinement of previous bounds

Abstract

We use a new argument to improve the error term in the asymptotic formula for the number of Diophantine $m$-tuples in finite fields, which is due to A. Dujella and M.Kazalicki (2021) and N. Mani and S. Rubinstein-Salzedo (2021).