Higher-order Image Conjecture for differential operators of polynomial algebras

The interest in images of differential operators of polynomial algebras arose from the research of the Jacobian conjecture. In this paper, we show that the Higher-order Image Conjecture (HIC($n$)) holds for all dimensions $n\geq 1$ in the case of positive characteristic. We also show that HIC($1$) holds for a class of differential operators in the case of characteristic zero.