A Quadratic Vinogradov Mean Value Theorem in Finite Fields

TL;DR

This paper establishes a finite field analogue of the quadratic Vinogradov mean value theorem by deriving bounds on solutions to specific quadratic systems over subsets of finite fields.

Contribution

It introduces new combinatorial geometric techniques to bound solutions of quadratic equations in finite fields, extending classical analytic number theory results.

Findings

Proves an upper bound for the number of solutions to a quadratic system in finite fields.

Develops incidence geometry methods involving modular hyperbolae.

Extends the Vinogradov mean value theorem to quadratic forms over finite fields.

Abstract

Let $p$ be a prime, let $s \geq 3$ be a natural number and let $A \subseteq \mathbb{F}_p$ be a non-empty set satisfying $|A| \ll p^{1/2}$. Denoting $J_s(A)$ to be the number of solutions to the system of equations \[ \sum_{i=1}^{s} (x_i - x_{i+s}) = \sum_{i=1}^{s} (x_i^2 - x_{i+s}^2) = 0, \] with $x_1, \dots, x_{2s} \in A$, our main result implies that \[ J_s(A) \ll |A|^{2s - 2 - 1/9}. \] This can be seen as a finite field analogue of the quadratic Vinogradov mean value theorem. Our techniques involve a variety of combinatorial geometric estimates, including studying incidences between cartesian products $A\times A$ and a special family of modular hyperbolae.