Irreducible Jacobian derivations in positive characteristic

TL;DR

This paper characterizes irreducible Jacobian derivations in positive characteristic fields, linking their structure to p-bases and divergence properties, with specific results for two-variable cases.

Contribution

It provides a complete characterization of irreducible Jacobian derivations in positive characteristic, connecting algebraic properties with geometric and divergence criteria.

Findings

Irreducible polynomial derivations are Jacobian iff they have an (n-1)-element p-basis.

In two variables, these derivations are characterized by divergence and constants.

The paper offers criteria for identifying Jacobian derivations in positive characteristic.

Abstract

We prove that an irreducible polynomial derivation in positive characteristic is a Jacobian derivation if and only if there exists an n-1-element p-basis of its ring of constants. In the case of two variables we characterize these derivations in terms of their divergence and some nontrivial constants.