Number Theory
Mordell type exponential sum estimates in fields of prime order
[Estimations de type Mordell pour les sommes exponentielles dans les corps d'ordre premier.]
Comptes Rendus. Mathématique, Tome 339 (2004) no. 5, pp. 321-325.

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*.

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*.

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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, Tome 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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[2] Bourgain, J.; Konyagin, S. Estimates for the number of sums and products and for exponential sums over subgroups in fields of prime order, C. R. Acad. Sci. Paris, Ser. I, Volume 337 (2003) no. 2, pp. 75-80

[3] J. Bourgain, N. Katz, T. Tao, A sum–product theorem in finite fields and applications, GAFA, in press

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[6] Konyagin, S.; Shparlinski, I. Character Sums with Exponential Functions and their Applications, Cambridge University Press, Cambridge, 1999

[7] Mordell, L.J. On a sum analogous to a Gauss' sum, Q. J. Math., Volume 3 (1932), pp. 161-162

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