The level of pairs of polynomials

Given a polynomial $f$ with coefficients in a field of prime characteristic $p$, it is known that there exists a differential operator that raises $1\mathrm{/}f$ 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 $g\mathrm{/}f$ to its $p$th power.