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Affine curves with infinitely many integral points


Author: Dimitrios Poulakis
Journal: Proc. Amer. Math. Soc. 131 (2003), 1357-1359
MSC (2000): Primary 11G30, 14G25, 11D41
DOI: https://doi.org/10.1090/S0002-9939-02-06841-7
Published electronically: October 1, 2002
MathSciNet review: 1949864
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $C \subset {\mathbf{A}}^{n}$ be an irreducible affine curve of (geometric) genus 0 defined by a finite family of polynomials having integer coefficients. In this note we give a necessary and sufficient condition for $C$ to possess infinitely many integer points, correcting a statement of J. H. Silverman (Theoret. Comput. Sci., 2000).


References [Enhancements On Off] (What's this?)

  • 1. L. J. Mordell, Diophantine Equations, Academic Press 1969. MR 40:2600
  • 2. D. Poulakis and E. Voskos, On the Practical Solution of Genus Zero Diophantine Equations, J. Symbolic Computation 30 (2000), 573-582. MR 2001j:11126
  • 3. D. Poulakis and E. Voskos, Solving genus zero diophantine equations with at most two infinite valuations, J. Symbolic Computation 33 (2002), 479-491.
  • 4. A. Schinzel, An improvement of Runge's Theorem on diophantine equations, Pontificia Acad. Sci. 2 (1969), 1-9. MR 43:1922
  • 5. J. H. Silverman, On the distribution of integer points on curves of genus zero, Theoret. Comput. Sci. 235 (2000), no. 1, 163-170. MR 2001h:11080

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Additional Information

Dimitrios Poulakis
Affiliation: Department of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
Email: poulakis@auth.gr

DOI: https://doi.org/10.1090/S0002-9939-02-06841-7
Received by editor(s): March 19, 2001
Received by editor(s) in revised form: January 8, 2002
Published electronically: October 1, 2002
Communicated by: Michael Stillman
Article copyright: © Copyright 2002 American Mathematical Society

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