Bounds for the smallest integer point of a rational curve
Dimitrios Poulakis (2003)
Acta Arithmetica
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Dimitrios Poulakis (2003)
Acta Arithmetica
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Xavier Xarles (2013)
Journal de Théorie des Nombres de Bordeaux
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In order to study the behavior of the points in a tower of curves, we introduce and study trivial points on towers of curves, and we discuss their finiteness over number fields. We relate the problem of proving that the only rational points are the trivial ones at some level of the tower, to the unboundeness of the gonality of the curves in the tower, which we show under some hypothesis.
Garzón, Alvaro (2003)
Revista Colombiana de Matemáticas
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Fenske, Torsten (1999)
Beiträge zur Algebra und Geometrie
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Alf Bjorn Aure (1984)
Mathematica Scandinavica
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Matt DeLong (2002)
Acta Arithmetica
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Piotr Nayar, Barbara Pilat (2014)
Bulletin of the Polish Academy of Sciences. Mathematics
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In this short note we give an elementary combinatorial argument, showing that the conjecture of J. Fernández de Bobadilla, I. Luengo-Velasco, A. Melle-Hernández and A. Némethi [Proc. London Math. Soc. 92 (2006), 99-138, Conjecture 1] follows from Theorem 5.4 of Brodzik and Livingston [arXiv:1304.1062] in the case of rational cuspidal curves with two critical points.
Stein Arild Stromme (1984)
Mathematica Scandinavica
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Emrah Çakçak, Ferruh Özbudak (2005)
Acta Arithmetica
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Ayla Gafni (2014)
Acta Arithmetica
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We find an asymptotic formula for the number of rational points near planar curves. More precisely, if f:ℝ → ℝ is a sufficiently smooth function defined on the interval [η,ξ], then the number of rational points with denominator no larger than Q that lie within a δ-neighborhood of the graph of f is shown to be asymptotically equivalent to (ξ-η)δQ².
J.-C. Fiorot, P. Jeannin, S. Taleb (1993)
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
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D. Eisenbud, J. Harris (1983)
Inventiones mathematicae
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Lahyane, Mustapha (2002)
International Journal of Mathematics and Mathematical Sciences
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Fumio Sakai, Takashi Matsuoka (1989)
Mathematische Annalen
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J. MacLeod, Allan (2008)
Annales Mathematicae et Informaticae
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