Subalgebras of the polynomial algebra in positive characteristic and the Jacobian

TL;DR

This paper investigates the structure of subalgebras within polynomial algebras over fields of positive characteristic, focusing on the properties of the Jacobian ideal and its relation to subalgebra generation.

Contribution

It establishes that when the Jacobian ideal is principal, its power forms a subalgebra of a specific polynomial subring, revealing new structural insights.

Findings

If the Jacobian ideal is principal, then its q-th power is a subalgebra of R[x_1^p,...,x_n^p]

The ideal J(R) relates to the algebraic structure of subalgebras in polynomial rings in positive characteristic

Provides conditions under which the Jacobian ideal influences the subalgebra structure in characteristic p

Abstract

Let $k$ be a field of characteristic $p>0$ and $R$ be a subalgebra of $k[X]=k[x_1,...,x_n]$. Let $J(R)$ be the ideal in $k[X]$ defined by $J(R)\Omega_{k[X]/k}^n=k[X]\Omega_{R/k}^n$. It is shown that if it is a principal ideal then $J(R)^q$ is a subalgebra of $R[x_1^p,...,x_n^p]$, where $q=p^n(p-1)/2$.