A Jacobian identity in positive characteristic

Abstract

In this note, we present several new results on derivations in characteristic $p\neq0$, together with a Jacobian identity that we recently discovered through a miscalculation. Our main identity states that, if $k$ is a field of characteristic $p$ and $f_{1},\ldots,f_{n}$ belong to the polynomial ring $k\left[x_{1},\ldots,x_{n}\right]$ and $J(f)$ equals the determinant of the $n\times n$ Jacobian matrix, $[\partial f_{i}/\partial x_{j}]$, then {\footnotesize \[ \sum^{p-1}_{i_{1}=1}\!\!\cdots\!\!\sum^{p-1}_{i_{n}=1}\!\!f_{1}^{i_{1}}\!\!\cdots \! f_{n}^{i_{n}}\nabla\!\left(\!f_{1}^{p-1-i_{1}}\!\cdots f_{n}^{p-1-i_{n}}\!\right)\!=\!\left(-1\right)^{n}\left(J\left(f\right)\right)^{p-1}\!, \]} where $\nabla=\partial^{n(p-1)}/\partial x_{1}^{p-1}\cdots\partial x_{n}^{p-1}$. We conclude with a brief discussion of nilpotent derivations in characteristic $p$ in connection with the degree less than $p$ version of the Jacobian conjecture.