The distribution of polynomials over finite fields, with applications to the Gowers norms

TL;DR

This paper explores the distribution of polynomials over finite fields and their relation to Gowers norms, revealing conditions under which polynomials exhibit low rank and large Gowers norms, with implications for additive combinatorics.

Contribution

It establishes an inverse theorem for Gowers norms of polynomial-exponential functions over finite fields and provides counterexamples in low characteristic, advancing understanding of polynomial distribution and Gowers norm behavior.

Findings

Poorly-distributed polynomials are characterized by low-rank structure.

Large Gowers norms imply correlation with lower-degree polynomials.

Counterexamples show limitations of existing theories in low characteristic fields.

Abstract

In this paper we investigate the uniform distribution properties of polynomials in many variables and bounded degree over a fixed finite field F of prime order. Our main result is that a polynomial P : F^n -> F is poorly-distributed only if P is determined by the values of a few polynomials of lower degree, in which case we say that P has small rank. We give several applications of this result, paying particular attention to consequences for the theory of the so-called Gowers norms. We establish an inverse result for the Gowers U^{d+1}-norm of functions of the form f(x)= e_F(P(x)), where P : F^n -> F is a polynomial of degree less than F, showing that this norm can only be large if f correlates with e_F(Q(x)) for some polynomial Q : F^n -> F of degree at most d. The requirement deg(P) < |F| cannot be dropped entirely. Indeed, we show the above claim fails in characteristic 2 when d = 3 and deg(P)=4, showing that the quartic symmetric polynomial S_4 in F_2^n has large Gowers U^4-norm but does not correlate strongly with any cubic polynomial. This shows that the theory of Gowers norms in low characteristic is not as simple as previously supposed. This counterexample has also been discovered independently by Lovett, Meshulam, and Samorodnitsky. We conclude with sundry other applications of our main result, including a recurrence result and a certain type of nullstellensatz.