Number Theory
Mordell type exponential sum estimates in fields of prime order
Comptes Rendus. Mathématique, Volume 339 (2004) no. 5, pp. 321-325.

We establish a Mordell type exponential sum estimate (see Mordell [Q. J. Math. 3 (1932) 161–162]) for ‘sparse’ polynomials f(x)=i=1raixki,(ai,p)=1,p prime, under essentially optimal conditions on the exponents 1ki<p1. The method is based on sum–product estimates in finite fields Fp and their Cartesian products. We also obtain estimates on incomplete sums of the form s=1tep(i=1raiθis) for t>pɛ, under appropriate conditions on the θiFp*.

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 f(x)=i=1raixki, (ai,p)=1, p premier, sous des hypothèses essentiellement optimales sur les exposants 1ki<p1. La méthode repose sur des estimés « sommes-produits » dans des corps finis Fp et leurs produits cartésiens. On obtient également des bornes non-triviales sur des sommes incomplètes de la forme s=1tep(i=1raiθis) pour t>pɛ, sous des hypothèses appropriées sur les θiFp*.

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DOI: 10.1016/j.crma.2004.06.013
Bourgain, Jean 1

1 IAS, School of Mathematics, Princeton, NJ 08540, USA
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Bourgain, Jean. Mordell type exponential sum estimates in fields of prime order. Comptes Rendus. Mathématique, Volume 339 (2004) no. 5, pp. 321-325. doi : 10.1016/j.crma.2004.06.013. http://www.numdam.org/articles/10.1016/j.crma.2004.06.013/

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