Number Theory/Algebraic Geometry
On uniform boundedness of a rational distance set in the plane
[Sur une borne uniforme dʼun ensemble de distances rationelles sur un plan]

Présenté par : Claire Voisin

Makhul, Mehdi1 ; Shaffaf, Jafar1
1 Department of Mathematical Sciences, Shahid Beheshti University, G.C., P.O. Box 1983963113, Tehran, Iran
Comptes Rendus. Mathématique, Tome 350 (2012) no. 3-4, pp. 121-124.

Un ensemble sur un plan est appelé rationnel si la distance entre tous ses points est rationnelle. Une question posée par Ulam en 1945 demande sʼil existe un ensemble rationnel et partout dense sur un plan. Solymosi et de Zeeuw ont démontré que lʼintersection de toute courbe algébrique irréductible définie sur R avec tout ensemble rationnel sur un plan est un ensemble fini sauf si la courbe est une ligne droite ou un cercle. Comme application de la conjecture dʼuniformité en géométrie arithmétique, une conséquence de la conjecture de Lang, nous démontrons que si S est un ensemble rationnel et infini sur un plan dont lʼintersection avec toute ligne droite est un ensemble fini alors il existe une borne uniforme sur le cardinal de ces ensembles dʼintersection.

A rational set in the plane is a point set with all its pairwise distances rational. Ulam asked in 1945 if there is an everywhere dense rational set. Solymosi and de Zeeuw proved that every rational distance subset of the plane has only finitely many points in common with an irreducible algebraic curve defined over R unless the curve is a line or circle. As an application of uniformity conjecture in arithmetic algebraic geometry which is a consequence of Lang conjecture we prove that if S is an infinite rational distance subset of the plane that has only finitely many points on any line then there is a uniform bound (independent of S) on the number of these points.

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Accepté le :
Publié le :
DOI : 10.1016/j.crma.2012.01.010
Makhul, Mehdi 1 ; Shaffaf, Jafar 1

1 Department of Mathematical Sciences, Shahid Beheshti University, G.C., P.O. Box 1983963113, Tehran, Iran 
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Makhul, Mehdi; Shaffaf, Jafar. On uniform boundedness of a rational distance set in the plane. Comptes Rendus. Mathématique, Tome 350 (2012) no. 3-4, pp. 121-124. doi : 10.1016/j.crma.2012.01.010. https://www.numdam.org/articles/10.1016/j.crma.2012.01.010/

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