Square values of several polynomials over a finite field

TL;DR

This paper estimates the count of solutions over finite fields where multiple polynomial values are nonzero squares, improving error bounds especially for smooth quadrics with nonsingular intersections.

Contribution

It advances previous results by providing sharper error estimates for the distribution of polynomial values as squares over finite fields, focusing on smooth quadrics.

Findings

Improved error bounds for counting solutions where polynomial values are nonzero squares.

Enhanced understanding of the distribution of polynomial values on smooth quadrics.

Refined estimates in the context of intersections of smooth projective quadrics.

Abstract

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.