Estimate of a multidimensional sum with the Legendre symbol for a polynomial of odd degree

G. I. Perel'muter

Saratov State University

Abstract: The estimate
$$ \Bigl|\sum_{x,1\dots,x_n\in F_q}\chi(f(x_1,\dots,x_n))\Bigr|\le(d-1)^nq^{n/2}. $$
is derived for the quadratic character Lambda of a field $F_q$ of $q$ elements and a polynomial $f$ of odd degree $d$ over $F_q$ under certain natural conditions.

UDC: 513.6

Received: 21.03.1975

 English version: Mathematical Notes, 1976, 20:6, 1015–1020

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