A Repulsion Motif in Diophantine Equations
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Publication:3174572
DOI10.4169/AMER.MATH.MONTHLY.118.07.584zbMATH Open1270.11058DBLPjournals/tamm/EverestW11arXiv1005.0315OpenAlexW3098259780WikidataQ58150570 ScholiaQ58150570MaRDI QIDQ3174572
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Publication date: 17 October 2011
Published in: The American Mathematical Monthly (Search for Journal in Brave)
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.
Full work available at URL: https://arxiv.org/abs/1005.0315
Rational points (14G05) Elliptic curves over global fields (11G05) Higher degree equations; Fermat's equation (11D41)
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