  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 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 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 latter 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.
