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Affine curves with infinitely many integral points

Author(s): Dimitrios Poulakis
Journal: Proc. Amer. Math. Soc. 131 (2003), 1357-1359.
MSC (2000): Primary 11G30, 14G25, 11D41
Posted: October 1, 2002
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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 to possess infinitely many integer points, correcting a statement of J. H. Silverman (Theoret. Comput. Sci., 2000).

References:

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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