  For a prime $p$ and a polynomial $F(X,Y)$ over a finite field $\mathbb{F}_p$
of $p$ elements, we give upper bounds on the number of solutions $$ F(x,y)=0,
\quad x\in\mathcal{A}, \ y\in \mathcal{B}, $$ where $\mathcal{A}$ and
$\mathcal{B}$ are very small intervals or subgroups. These bounds can be
considered as positive characteristic analogues of the result of Bombieri and
Pila (1989) on sparsity of integral points on curves. As an application we
prove that distinct consecutive elements in sequences generated compositions of
several rational functions are not contained in any short intervals or small
subgroups.
