On large $F$-Diophantine sets

TL;DR

This paper investigates the existence and construction of polynomials that generate large sets of integers where each pair's polynomial value is a perfect square, revealing infinite families and degree bounds.

Contribution

It demonstrates the existence of infinitely many bivariate polynomials for any finite integer set as a Diophantine set with perfect squares, and establishes degree bounds for such polynomials.

Findings

For any finite set of integers, infinitely many polynomials make it a (F,2)-Diophantine set.

The degree of such polynomials can be as low as 4 times the floor of n/3.

Provides a construction method for these polynomials.

Abstract

Let $F\in\mathbb{Z}[x,y]$ and $m\ge2$ be an integer. A set $A\subset \mathbb{Z}$ is called an $(F,m)$-Diophantine set if $F(a,b)$ is a perfect $m$-power for any $a,b\in A$ where $a\ne b$. If $F$ is a bivariate polynomial for which there exist infinite $(F,m)$-Diophantine sets, then there is a complete qualitative characterization of all such polynomials $F$. Otherwise, various finiteness results are known. We prove that given a finite set of distinct integers $ S$ of size $n$, there are infinitely many bivariate polynomials $F$ such that $ S$ is an $(F,2)$-Diophantine set. In addition, we show that the degree of $F$ can be as small as $\displaystyle 4\lfloor n/3\rfloor$.