A measure of transcendence for singular points on conics
Journal de Théorie des Nombres de Bordeaux, Volume 31 (2019) no. 2, pp. 361-369.

A singular point on a plane conic defined over is a transcendental point of the curve which admits very good rational approximations, uniformly in terms of the height. Extremal numbers and Sturmian continued fractions are abscissa of such points on the parabola y=x 2 . In this paper we provide a measure of transcendence for singular points on conics defined over which, in these two cases, improves on the measure obtained by Adamczewski and Bugeaud. The main tool is a quantitative version of Schmidt subspace theorem due to Evertse.

Un point d’une conique définie sur est dit singulier s’il est transcendant et admet de très bonnes approximations rationnelles, uniformément en termes de la hauteur. Les nombres extrémaux et les fractions continues sturmiennes sont les abscisses de tels points sur la parabole y=x 2 . Nous établissons ici une mesure de transcendance de points singuliers sur les coniques définies sur qui, dans ces deux cas, améliore la mesure obtenue précédemment par Adamczewski et Bugeaud. L’outil principal est une version quantitative du théorème du sous-espace de Schmidt due à Evertse.

Received:
Accepted:
Published online:
DOI: 10.5802/jtnb.1085
Classification: 11J82,  11J13,  11J87
Keywords: Sturmian continued fractions, extremal numbers, transcendental numbers, measure of transcendence, uniform approximation, quantitative subspace theorem, minimal points, conics.
Damien Roy 1

1 Département de Mathématiques Université d’Ottawa 150 Louis Pasteur Ottawa, Ontario K1N 6N5, Canada
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Damien Roy. A measure of transcendence for singular points on conics. Journal de Théorie des Nombres de Bordeaux, Volume 31 (2019) no. 2, pp. 361-369. doi : 10.5802/jtnb.1085. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1085/

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