Some remarks on the Jacobian Conjecture and Drużkowski mappings

Abstract

In this paper, we first show that the Jacobian Conjecture is true for non-homogeneous power linear mappings under some conditions. Secondly, we prove an equivalent statement about the Jacobian Conjecture in dimension r⩾1 and give some partial results for r=2.Finally, for a homogeneous power linear Keller map F=x+H of degree d⩾2, we give the inverse polynomial map under the condition that JH3=0. We shall show that deg(F−1)⩽dk if k⩽2 and JHk+1=0, but also give an example with d=2 and JH4=0 such that deg(F−1)>d3.