The Repulsion Motif in Diophantine Equations

TL;DR

This paper explores the 'repulsion' phenomenon in Diophantine equations, discussing conjectures that suggest solutions tend to be bounded or separated, illustrating deep structural properties of solutions to cubic equations.

Contribution

It introduces the concept of the 'repulsion motif' in Diophantine equations and discusses conjectures related to bounds on solutions near integrality.

Findings

Conjectures suggest solutions exhibit a repulsion effect.

The 'repulsion motif' may explain bounds on solutions.

Connections to Hall's conjecture and integral solutions.

Abstract

Problems related to the existence of integral and rational points on cubic curves date back at least to Diophantus. A significant step in the modern theory of these equations was made by Siegel, who proved that a non-singular plane cubic equation has only finitely many integral solutions. Examples show that simple equations can have inordinately large integral solutions in comparison to the size of their coefficients. A conjecture of Hall attempts to ameliorate this by bounding the size of integral solutions simply in terms of the coefficients of the defining equation. It turns out that a similar phenomenon seems, conjecturally, to be at work for solutions which are close to being integral in another sense. We describe these conjectures as an illustration of an underlying motif - repulsion - in the theory of Diophantine equations.