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 (Formula presented.) 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.