Estimate of a multiple sum involving the Legendre symbol

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 $n$.