IMAGES OF HIGHER-ORDER DIFFERENTIAL OPERATORS OF POLYNOMIAL ALGEBRAS

Abstract

We investigate images of higher-order differential operators of polynomial algebras over a field $k$. We show that, when $\operatorname{char}k>0$, the image of the set of differential operators $\{\unicode[STIX]{x1D709}_{i}-\unicode[STIX]{x1D70F}_{i}\mid i=1,2,\ldots ,n\}$ of the polynomial algebra $k[\unicode[STIX]{x1D709}_{1},\ldots ,\unicode[STIX]{x1D709}_{n},z_{1},\ldots ,z_{n}]$ is a Mathieu subspace, where $\unicode[STIX]{x1D70F}_{i}\in k[\unicode[STIX]{x2202}_{z_{1}},\ldots ,\unicode[STIX]{x2202}_{z_{n}}]$ for $i=1,2,\ldots ,n$. We also show that, when $\operatorname{char}k=0$, the same conclusion holds for $n=1$. The problem concerning images of differential operators arose from the study of the Jacobian conjecture.