Comptes Rendus
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
Jean Bourgain 1

1 IAS, School of Mathematics, Princeton, NJ 08540, USA
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Jean Bourgain. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.06.013/

[1] J. Bourgain, Estimates on exponential sums related to the Diffie–Hellman distributions, GAFA, in press

[2] J. Bourgain; S. Konyagin 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

[4] T. Cochrane, C. Pinner, An improved Mordell type bound for exponential sums, Proc. Amer. Math. Soc., submitted for publication

[5] T. Cochrane; C. Pinner Stepanov's method applied to binomial exponential sums, Q. J. Math., Volume 54 (2003) no. 3, pp. 243-255

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

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

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