Multidimensional polynomial Szemerédi theorem in finite fields for polynomials of distinct degrees

Abstract

We obtain a polynomial upper bound in the finite-field version of the multidimensional polynomial Szemerédi theorem for distinct-degree polynomials. That is, if P₁, …, P_t are nonconstant integer polynomials of distinct degrees and v₁, …, v_t are nonzero vectors in $\mathbb{F}_p^D$, we show that each subset of $\mathbb{F}_p^D$ lacking a nontrivial configuration of the form

$${\bf{x}},{\bf{x}} + {{\bf{v}}_1}{P_1}(y), \ldots,{\bf{x}} + {{\bf{v}}_t}{P_t}(y)$$

has at most O(p^D−c) elements. In doing so, we apply the notion of Gowers norms along a vector adapted from ergodic theory, which extends the classical concept of Gowers norms on finite abelian groups.