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 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.

中文翻译：

---

多项式对的水平

摘要 给定一个多项式f，其系数在素数p 的域中，已知存在一个微分算子的p 次方。在超椭圆曲线的情况下，我们首先讨论该微分算子的“级别”与“分层”概念之间的关系。接下来，我们将级别的概念扩展到一对多项式的概念。我们证明了一些基本性质，并在某些特殊情况下计算了这个水平。特别是，我们展示了多项式 g 和 f 的例子，这样就没有微分算子将 g/f 提高到它的 p 次方。