Expanding polynomials over finite fields of large characteristic, and a regularity lemma for definable sets

Let $P: \F \times \F \to \F$ be a polynomial of bounded degree over a finite field $\F$ of large characteristic. In this paper we establish the following dichotomy: either $P$ is a \emph{moderate asymmetric expander} in the sense that $|P(A,B)| \gg |\F|$ whenever $A, B \subset \F$ are such that $|A| |B| \geq C |\F|^{2-1/8}$ for a sufficiently large $C$, or else $P$ takes the form $P(x,y) = Q(F(x)+G(y))$ or $P(x,y) = Q(F(x) G(y))$ for some polynomials $Q,F,G$. This is a reasonably satisfactory classification of polynomials of two variables that moderately expand (either symmetrically or asymmetrically). We obtain a similar classification for weak expansion (in which one has $|P(A,A)| \gg |A|^{1/2} |\F|^{1/2}$ whenever $|A| \geq C |\F|^{1-1/16}$), and a partially satisfactory classification for almost strong asymmetric expansion (in which $|P(A,B)| = (1-O(|\F|^{-c})) |\F|$ when $|A|, |B| \geq |\F|^{1-c}$ for some small absolute constant $c>0$). The main new tool used to establish these results is an \emph{algebraic regularity lemma} that describes the structure of dense graphs generated by definable subsets over finite fields of large characteristic. This lemma strengthens the Sz\'emeredi regularity lemma in the algebraic case, in that while the latter lemma decomposes a graph into a bounded number of components, most of which are $\eps$-regular for some small but fixed $\epsilon$, the former lemma ensures that all of the components are $O(|\F|^{-1/4})$-regular. This lemma, which may be of independent interest, relies on some basic facts about the \'etale fundamental group of an algebraic variety.

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