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Comptes Rendus Mathématique
Volume 339, n° 5
pages 321-325 (septembre 2004)
Doi : 10.1016/j.crma.2004.06.013
Received : 29 May 2004 ;  accepted : 1 June 2004
Mordell type exponential sum estimates in fields of prime order
Estimations de type Mordell pour les sommes exponentielles dans les corps dʼordre premier.
 

Jean Bourgain
IAS, School of Mathematics, Princeton, NJ 08540, USA 

Abstract

We establish a Mordell type exponential sum estimate (see Mordell [Q. J. Math. 3 (1932) 161-162]) for sparse' polynomials   prime, under essentially optimal conditions on the exponents  . The method is based on sum-product estimates in finite fields   and their Cartesian products. We also obtain estimates on incomplete sums of the form   for  , under appropriate conditions on the  . To cite this article: J. Bourgain, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

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Résumé

Nous démontrons une estimée du type Mordell (voir Mordell [Q. J. Math. 3 (1932) 161-162]) pour les sommes exponentielles associées à des polynômes clairsemés  ,  , p premier, sous des hypothèses essentiellement optimales sur les exposants  . La méthode repose sur des estimés « sommes-produits » dans des corps finis   et leurs produits cartésiens. On obtient également des bornes non-triviales sur des sommes incomplètes de la forme   pour  , sous des hypothèses appropriées sur les  . Pour citer cet article : J. Bourgain, C. R. Acad. Sci. Paris, Ser. I 339 (2004).

The full text of this article is available in PDF format.


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