# The level of pairs of polynomials

## TL;DR

This paper explores the concept of the 'level' of pairs of polynomials over fields of prime characteristic, relating it to stratification in hyperelliptic curves and providing computations and examples of its properties.

## Contribution

It extends the notion of polynomial 'level' to pairs of polynomials, establishing basic properties and computing this level in specific cases, including counterexamples.

## Key findings

- Relation between level and stratification in hyperelliptic curves
- Extension of level concept to pairs of polynomials
- Existence of polynomial pairs without differential operators raising g/f to pth power

## 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 $p$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.