Square values of several polynomials over a finite field

Let $f_1,\dots ,f_m$ be polynomials in $n$ variables with coefficients in a finite field ${\mathbb{F}}_q$. We estimate the number of points $\underset{‾}{x}$ in ${\mathbb{F}}_q^n$ such that each value $f_i(\underset{‾}{x})$ is a nonzero square in ${\mathbb{F}}_q$. The error term is especially small when the $f_i$ define smooth projective quadrics with nonsingular intersections. We improve the error term in a recent work by Asgarli–Yip on mutual position of smooth quadrics.