The level of pairs of polynomials

Abstract

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 pth 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 pth power.