We establish bounds on exponential sums where , p prime, and ψ an additive character on . They extend the earlier work of Bourgain, Glibichuk, and Konyagin to fields that are not of prime order . More precisely, a non-trivial estimate is obtained provided n satisfies for all , , where is arbitrary.
On etabli des bornes sur les sommes d'exponentielles où , p est premier et ψ est un caractère additif de . Il s'agit d'une extension des résultats de Bourgain, Glibichuk, et Konyagin pour un corps qui n'est pas d'ordre premier, c'est-à-dire . On obtient une estimée non-triviale pour tout n satisfaisant la condition pour tout et où est arbitraire.
Accepted:
Published online:
Jean Bourgain 1; Mei-Chu Chang 2
@article{CRMATH_2006__342_9_643_0, author = {Jean Bourgain and Mei-Chu Chang}, title = {A {Gauss} sum estimate in arbitrary finite fields}, journal = {Comptes Rendus. Math\'ematique}, pages = {643--646}, publisher = {Elsevier}, volume = {342}, number = {9}, year = {2006}, doi = {10.1016/j.crma.2006.01.022}, language = {en}, }
Jean Bourgain; Mei-Chu Chang. A Gauss sum estimate in arbitrary finite fields. Comptes Rendus. Mathématique, Volume 342 (2006) no. 9, pp. 643-646. doi : 10.1016/j.crma.2006.01.022. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.01.022/
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[2] J. Bourgain, A. Glibichuk, S. Konyagin, Estimates for the number of sums and products and for exponential sums in fields of prime order, J. London Math. Soc., in press
[3] A sum-product estimate in finite fields and their applications, GAFA, Volume 14 (2004) no. 1, pp. 27-57
[4] Character Sums with Exponential Functions and their Applications, Cambridge Univ. Press, Cambridge, 1999
[5] Bounds on Gauss sums in finite fields, Proc. Amer. Math. Soc., Volume 132 (2006) no. 10, pp. 2817-2824
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