Comptes Rendus
Number Theory/Algebraic Geometry
Squareful points of bounded height
Comptes Rendus. Mathématique, Volume 349 (2011) no. 11-12, pp. 603-606.

Let n5. In this Note, we explain how to determine the asymptotic behaviour of the size of the set of rational points (a0::an)Pn(Q) (where a0,,anZ and gcd(a0,,an)=1) of bounded height maxi=0,,n|ai|B on the hyperplane i=0nXi=0 such that ai is squareful for each i{0,,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 n5. Dans cette Note, nous expliquerons comment on peut déterminer le comportement asymptotique du nombre de points rationnels (a0::an)Pn(Q) (avec a0,,anZ et pgcd(a0,,an)=1) de hauteur bornée maxi=0,,n|ai|B sur lʼhyperplan i=0nXi=0 tels que ai est un entier puissant pour chaque i{0,,n}, lorsque B tend vers lʼinfini. (Un entier a est appelé puissant si pour chaque nombre premier p divisant a, on a que p2 aussi divise a.) La méthode principale quʼon utilise ici est la méthode du cercle de Hardy–Littlewood (classique).

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2011.05.001

Karl Van Valckenborgh 1

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