  S\'ark\"ozy proved that dense sets of integers contain two elements differing
by a $k$th power. The bounds in quantitative versions of this theorem are
rather weak compared to what is expected. We prove a version of S\'ark\"ozy's
theorem for polynomials over $\mathbb{F}_q$ with polynomial dependencies in the
parameters.
  More precisely, let $P_{q,n}$ be the space of polynomials over $\mathbb{F}_q$
of degree $< n$ in an indeterminate $T$. Let $k \geq 2$ be an integer and let
$q$ be a prime power. Set $c(k,q) := (2 k^2 D_q(k)^2\log q)^{-1}$, where
$D_q(k)$ is the sum of the digits of $k$ in base $q$. If $A \subset P_{q,n}$ is
a set with $|A| > 2q^{(1 - c(k,q))n}$, then $A$ contains distinct polynomials
$p(T), p'(T)$ such that $p(T) - p'(T) = b(T)^k$ for some $b \in
\mathbb{F}_q[T]$.
