Mordell type exponential sum estimates in fields of prime order

Jean Bourgain

doi:10.1016/j.crma.2004.06.013

Number Theory

Mordell type exponential sum estimates in fields of prime order

Presented by: Jean Bourgain

Jean Bourgain ¹

¹ IAS, School of Mathematics, Princeton, NJ 08540, USA

Comptes Rendus. Mathématique, Volume 339 (2004) no. 5, pp. 321-325.

Abstract
Résumé

We establish a Mordell type exponential sum estimate (see Mordell [Q. J. Math. 3 (1932) 161–162]) for ‘sparse’ polynomials $f (x) = \sum_{i = 1}^{r} a_{i} x^{k_{i}}, (a_{i}, p) = 1, p$ prime, under essentially optimal conditions on the exponents $1 ⩽ k_{i} < p - 1$ . The method is based on sum–product estimates in finite fields $F_{p}$ and their Cartesian products. We also obtain estimates on incomplete sums of the form $\sum_{s = 1}^{t} e_{p} (\sum_{i = 1}^{r} a_{i} θ_{i}^{s})$ for $t > p^{ɛ}$ , under appropriate conditions on the $θ_{i} \in F_{p}^{*}$ .

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 $f (x) = \sum_{i = 1}^{r} a_{i} x^{k_{i}}$ , $(a_{i}, p) = 1$ , p premier, sous des hypothèses essentiellement optimales sur les exposants $1 ⩽ k_{i} < p - 1$ . La méthode repose sur des estimés « sommes-produits » dans des corps finis $F_{p}$ et leurs produits cartésiens. On obtient également des bornes non-triviales sur des sommes incomplètes de la forme $\sum_{s = 1}^{t} e_{p} (\sum_{i = 1}^{r} a_{i} θ_{i}^{s})$ pour $t > p^{ɛ}$ , sous des hypothèses appropriées sur les $θ_{i} \in F_{p}^{*}$ .

Received: 2004-05-29
Accepted: 2004-06-01
Published online: 2004-08-06

DOI: 10.1016/j.crma.2004.06.013

Author's affiliations:

Jean Bourgain ¹

¹ IAS, School of Mathematics, Princeton, NJ 08540, USA

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     title = {Mordell type exponential sum estimates in fields of prime order},
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     pages = {321--325},
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     doi = {10.1016/j.crma.2004.06.013},
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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/

References
Cited by

[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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