  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 $\underline{x}$
in $\mathbb{F}_q^n$ such that each value $f_i(\underline{x})$ 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.
