Estimate of a~multiple sum involving the Legendre symbol

G. I. Perel'muter

Matematičeskie zametki, Tome 18 (1975) no. 3, pp. 421-427.

Résumé

Suppose $f(x_1,\dots,x_n)$ is a polynomial of even degree $d$ having coefficients in the finite field $k=[q]$ and satisfying certain natural conditions, and let $\chi$ be the quadratic character of $k$. Then $$ \Bigl|\sum x_1,\dots,x_n\in kx(f(x_1,\dots,x_n))\Bigr|\le Cq^{n/2}, $$ where the constant $C$ depends only on $d$ and $n$.