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 , , p premier, sous des hypothèses essentiellement optimales sur les exposants . La méthode repose sur des estimés « sommes-produits » dans des corps finis et leurs produits cartésiens. On obtient également des bornes non-triviales sur des sommes incomplètes de la forme pour , sous des hypothèses appropriées sur les .
We establish a Mordell type exponential sum estimate (see Mordell [Q. J. Math. 3 (1932) 161–162]) for ‘sparse’ polynomials prime, under essentially optimal conditions on the exponents . The method is based on sum–product estimates in finite fields and their Cartesian products. We also obtain estimates on incomplete sums of the form for , under appropriate conditions on the .
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@article{CRMATH_2004__339_5_321_0, author = {Bourgain, Jean}, title = {Mordell type exponential sum estimates in fields of prime order}, journal = {Comptes Rendus. Math\'ematique}, pages = {321--325}, publisher = {Elsevier}, volume = {339}, number = {5}, year = {2004}, doi = {10.1016/j.crma.2004.06.013}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.crma.2004.06.013/} }
TY - JOUR AU - Bourgain, Jean TI - Mordell type exponential sum estimates in fields of prime order JO - Comptes Rendus. Mathématique PY - 2004 SP - 321 EP - 325 VL - 339 IS - 5 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2004.06.013/ DO - 10.1016/j.crma.2004.06.013 LA - en ID - CRMATH_2004__339_5_321_0 ER -
%0 Journal Article %A Bourgain, Jean %T Mordell type exponential sum estimates in fields of prime order %J Comptes Rendus. Mathématique %D 2004 %P 321-325 %V 339 %N 5 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2004.06.013/ %R 10.1016/j.crma.2004.06.013 %G en %F CRMATH_2004__339_5_321_0
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/
[1] J. Bourgain, Estimates on exponential sums related to the Diffie–Hellman distributions, GAFA, in press
[2] 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] Stepanov's method applied to binomial exponential sums, Q. J. Math., Volume 54 (2003) no. 3, pp. 243-255
[6] Character Sums with Exponential Functions and their Applications, Cambridge University Press, Cambridge, 1999
[7] On a sum analogous to a Gauss' sum, Q. J. Math., Volume 3 (1932), pp. 161-162
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