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 $P_1,...,P_t$ are nonconstant integer polynomials of distinct degrees and $v_1,...,v_t$ 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 $$x,x+v_1{P}_1(y),...,x+v_t{P}_t(y)$$ 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.