Multidimensional polynomial Szemer\'{e}di theorem in finite fields for polynomials of distinct degrees

TL;DR

This paper proves a finite-field version of the multidimensional polynomial Szemerédi theorem for polynomials of distinct degrees, providing polynomial bounds on the size of subsets lacking certain polynomial configurations.

Contribution

It introduces a polynomial upper bound for the finite-field multidimensional polynomial Szemerédi theorem involving distinct-degree polynomials, extending Gowers norms with a vector-adapted approach.

Findings

Sets lacking the polynomial configurations are of size at most O(p^{D-c})

Extension of Gowers norms along vectors from ergodic theory

Application to finite fields with polynomial bounds

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.