TY - JOUR
T1 - Bounds of Gauss sums in finite fields
AU - Shparlinski, Igor E.
PY - 2004/10
Y1 - 2004/10
N2 - We consider Gauss sums of the form (Chemical Equation Presented) with a nontrivial additive character χ ≠ χ0 of a finite field Fpm of pm elements of characteristic p. The classical bound |Gn(a)| ≤ (n - 1)pm/2 becomes trivial for n ≥ pm/2 + 1. We show that, combining some recent bounds of Heath-Brown and Konyagin with several bounds due to Deligne, Katz, and Li, one can obtain the bound on |Gn(a)| which is nontrivial for the values of n of order up to pm/2+1/6. We also show that for almost all primes one can obtain a bound which is nontrivial for the values of n of order up to p m/2+1/2.
AB - We consider Gauss sums of the form (Chemical Equation Presented) with a nontrivial additive character χ ≠ χ0 of a finite field Fpm of pm elements of characteristic p. The classical bound |Gn(a)| ≤ (n - 1)pm/2 becomes trivial for n ≥ pm/2 + 1. We show that, combining some recent bounds of Heath-Brown and Konyagin with several bounds due to Deligne, Katz, and Li, one can obtain the bound on |Gn(a)| which is nontrivial for the values of n of order up to pm/2+1/6. We also show that for almost all primes one can obtain a bound which is nontrivial for the values of n of order up to p m/2+1/2.
UR - http://www.scopus.com/inward/record.url?scp=5644237940&partnerID=8YFLogxK
U2 - 10.1090/S0002-9939-04-07133-3
DO - 10.1090/S0002-9939-04-07133-3
M3 - Article
AN - SCOPUS:5644237940
SN - 0002-9939
VL - 132
SP - 2817
EP - 2824
JO - Proceedings of the American Mathematical Society
JF - Proceedings of the American Mathematical Society
IS - 10
ER -