Sarkozy's theorem in function fields

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 $k\ge 2$ be an integer and let $q$ be a prime power. Set $c(k,q):=(2k^2{D}_q(k{)}^2\log q{)}^{-1}$, 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 $\mathrm{\mid }A\mathrm{\mid }>2q^{(1-c(k,q))n}$, then $A$ contains distinct polynomials $p(T),p^{\mathrm{\prime }}(T)$ such that $p(T)-p^{\mathrm{\prime }}(T)=b(T{)}^k$ for some $b \in $\mathbb{F}$_q[T]$.