Multidimensional polynomial Szemer\'{e}di 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 Szemer\'{e}di 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.