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

2021/03/23 by Kuca, Borys · 1 citation
#11B30 (Primary) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2103.12606

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 \mathbbF_p^D, we show that each subset of \mathbbF_p^D lacking a nontrivial configuration of the form x, x + v₁ P₁(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.

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