# Higher derivations of Jacobian type in positive characteristic

## TL;DR

This paper investigates higher derivations of Jacobian type in positive characteristic, providing conditions for polynomial extendability and characterizations of variables and univariate polynomials.

## Contribution

It introduces necessary and sufficient conditions for extendability of polynomial tuples and characterizes variables using higher derivations in positive characteristic.

## Key findings

- Conditions for extendability of polynomial tuples
- Characterizations of variables and univariate polynomials
- Insights into higher derivations in positive characteristic

## Abstract

In this paper, we study higher derivations of Jacobian type in positive characteristic. We give a necessary and sufficient condition for $(n-1)$-tuples of polynomials to be extendable in the polynomial ring in $n$ variables over an integral domain $R$ of positive characteristic. In particular, we give characterizations of variables and univariate polynomials by using the terms of higher derivations of Jacobian type in the polynomial ring in two variables over a field of positive characteristic.