Squareful points of bounded height

Karl Van Valckenborgh

doi:10.1016/j.crma.2011.05.001

Number Theory/Algebraic Geometry

Squareful points of bounded height

Presented by: the Editorial Board

Karl Van Valckenborgh ¹

¹ K.U. Leuven, Department of Mathematics, Celestijnenlaan 200B, 3001 Leuven, Belgium

Comptes Rendus. Mathématique, Volume 349 (2011) no. 11-12, pp. 603-606.

Abstract (Original language)
Résumé

Let $n ⩾ 5$ . In this Note, we explain how to determine the asymptotic behaviour of the size of the set of rational points $(a_{0} : \dots : a_{n}) \in P^{n} (Q)$ (where $a_{0}, \dots, a_{n} \in Z$ and $\gcd (a_{0}, \dots, a_{n}) = 1$ ) of bounded height $\max_{i = 0, \dots, n} | a_{i} | ⩽ B$ on the hyperplane $\sum_{i = 0}^{n} X_{i} = 0$ such that $a_{i}$ is squareful for each $i \in {0, \dots, n}$ as B goes to infinity. (An integer a is called squareful if the exponent of each prime divisor of a is at least two.) The main tool we will use, is the (classical) Hardy–Littlewood circle method.

Soit $n ⩾ 5$ . Dans cette Note, nous expliquerons comment on peut déterminer le comportement asymptotique du nombre de points rationnels $(a_{0} : \dots : a_{n}) \in P^{n} (Q)$ (avec $a_{0}, \dots, a_{n} \in Z$ et $pgcd (a_{0}, \dots, a_{n}) = 1$ ) de hauteur bornée $\max_{i = 0, \dots, n} | a_{i} | ⩽ B$ sur lʼhyperplan $\sum_{i = 0}^{n} X_{i} = 0$ tels que $a_{i}$ est un entier puissant pour chaque $i \in {0, \dots, n}$ , lorsque B tend vers lʼinfini. (Un entier a est appelé puissant si pour chaque nombre premier p divisant a, on a que $p^{2}$ aussi divise a.) La méthode principale quʼon utilise ici est la méthode du cercle de Hardy–Littlewood (classique).

Received: 2011-03-18
Accepted: 2011-05-02
Published online: 2011-06-01

DOI: 10.1016/j.crma.2011.05.001

Author's affiliations:

Karl Van Valckenborgh ¹

¹ K.U. Leuven, Department of Mathematics, Celestijnenlaan 200B, 3001 Leuven, Belgium

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     author = {Karl Van Valckenborgh},
     title = {Squareful points of bounded height},
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     pages = {603--606},
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     doi = {10.1016/j.crma.2011.05.001},
     language = {en},
}

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Karl Van Valckenborgh. Squareful points of bounded height. Comptes Rendus. Mathématique, Volume 349 (2011) no. 11-12, pp. 603-606. doi : 10.1016/j.crma.2011.05.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2011.05.001/

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Cited by

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