A characterization of p-bases of rings of constants

TL;DR

This paper provides new criteria involving jacobians and factors to characterize p-bases of rings of constants in polynomial rings over fields of positive characteristic, extending known theorems and reformulating the Jacobian conjecture.

Contribution

It introduces two equivalent conditions for polynomials to form p-bases of rings of constants, extending previous results and offering a new formulation of the Jacobian conjecture in characteristic p.

Findings

Two equivalent conditions for p-bases involving jacobians and factors

Extension of Nousiainen's theorem for the case m=n

A new formulation of the Jacobian conjecture in positive characteristic

Abstract

We obtain two equivalent conditions for m polynomials in n variables to form a p-basis of a ring of constants of some polynomial K-derivation, where K is a UFD of characteristic p>0. One of these conditions involves jacobians, and the second - some properties of factors. In the case of m=n this extends the known theorem of Nousiainen, and we obtain a new formulation of the jacobian conjecture in positive characteristic.