A Quadratic Vinogradov Mean Value Theorem in Finite Fields

Let p be a prime, let s ≥ 3 be a natural number and let A ⊆ Fp be a non-emptyset satisfying |A| ≪ p1/2. Denoting Js(A) to be the number of solutions to the system ofequationsXsi=1(xi − xi+s) = Xsi=1(x2i − x2i+s) = 0,with x1, . . . , x2s ∈ A, our main result implies thatJs(A) ≪ |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 studyingincidences between Cartesian products A × A and a special family of modular hyperbolae.