Square values of several polynomials over a finite field

Abstract

Let $f_1,\dots ,f_m$ be polynomials in n variables with coefficients in a finite field $F_q$. We estimate the number of points $\underset{\_}{x}$ in $F_q^n$ such that each value $f_i\left(\underset{\_}{x}\right)$ is a nonzero square in $F_q$. 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.