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 .
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 .
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Jean Bourgain 1
@article{CRMATH_2004__339_5_321_0, author = {Jean Bourgain}, 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}, }
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/
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