  Given a polynomial $f$ with coefficients in a field of prime characteristic
$p$, it is known that there exists a differential operator that raises $1/f$ to
its $p$th power. We first discuss a relation between the `level' of this
differential operator and the notion of `stratification' in the case of
hyperelliptic curves. Next we extend the notion of level to that of a pair of
polynomials. We prove some basic properties and we compute this level in
certain special cases. In particular we present examples of polynomials $g$ and
$f$ such that there is no differential operator raising $g/f$ to its $p$th
power.
