This article is cited in 3 papers

Estimation of a sum along an algebraic curve

G. I. Perel'muter

Saratov State University named after N. G. Chernyshevsky

Abstract: 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 $\Gamma,\varphi,\psi$.

UDC: 513.6

Received: 17.05.1968

 English version: Mathematical Notes, 1969, 5:3, 223–227

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