Estimation of a~sum along an~algebraic curve
Matematičeskie zametki, Tome 5 (1969) no. 3, pp. 373-380
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Let $\Gamma$ be an algebraic curve determined over a finite field $k=[q]$; $e$, $\chi$ are subsidiary additive and multiplicative characters of the field $k$; $\varphi$, $\psi$ are functions in $\Gamma$ determined over $k$ and satisfying some natural conditions. If $P$ passes through the points of curve $\Gamma$, rational over $k$, then
$$
\biggl|\sum_{P\in\Gamma}e(\varphi(P))\chi(\psi(P))\biggr|\leqslant C\sqrt q
$$
where constant $C$ depends only on the powers of $\Gamma,\varphi,\psi$.
@article{MZM_1969_5_3_a11,
author = {G. I. Perel'muter},
title = {Estimation of a~sum along an~algebraic curve},
journal = {Matemati\v{c}eskie zametki},
pages = {373--380},
publisher = {mathdoc},
volume = {5},
number = {3},
year = {1969},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/MZM_1969_5_3_a11/}
}
G. I. Perel'muter. Estimation of a~sum along an~algebraic curve. Matematičeskie zametki, Tome 5 (1969) no. 3, pp. 373-380. http://geodesic.mathdoc.fr/item/MZM_1969_5_3_a11/