The Repulsion Motif in Diophantine Equations

Details The Repulsion Motif in Diophantine Equations The Repulsion Motif in Diophantine Equations

2010-05-03

arxiv.org/abs/1005.0315v3

Mathematics

Number Theory

Some minor changes to clarify the flow of the argument

Scientific paper

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.

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