Square values of several polynomials over a finite field

Let f₁, , fₘ be polynomials in n variables with coefficients in a finite field Fq. We estimate the number of points x in Fqⁿ such that each value fᵢ (x) is a nonzero square in Fq. The error term is especially small when the fᵢ 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.