On a class of rational cuspidal plane curves.
Hubert Flenner, Mikhail Zaidenberg (1996)
Manuscripta mathematica
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Hubert Flenner, Mikhail Zaidenberg (1996)
Manuscripta mathematica
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Stein Arild Stromme (1984)
Mathematica Scandinavica
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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.
Nguyen Van Chau (2011)
Annales Polonici Mathematici
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In certain cases the invertibility of a polynomial map F = (P,Q): ℂ²→ ℂ² can be characterized by the irreducibility and the rationality of the curves aP+bQ = 0, (a:b) ∈ ℙ¹.
D. Eisenbud, J. Harris (1983)
Inventiones mathematicae
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Fumio Sakai, Takashi Matsuoka (1989)
Mathematische Annalen
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Dimitrios Poulakis (2003)
Acta Arithmetica
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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.
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.
Arnaldo Garcia, Luciane Quoos (2001)
Acta Arithmetica
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Xian Wu (1994)
Manuscripta mathematica
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E. Artal Bartolo (1997)
Manuscripta mathematica
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Ragni Piene (1981)
Mathematische Annalen
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Ziv Ran (2004)
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze
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We compute the number of irreducible rational curves of given degree with 1 tacnode in or 1 node in meeting an appropriate generic collection of points and lines. As a byproduct, we also compute the number of rational plane curves of degree passing through given points and tangent to a given line. The method is ‘classical’, free of Quantum Cohomology.