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.