Some Remarks on the Jacobian Conjecture and Dru{ż}kowski 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 $r\ge 1$ and give some partial results for $r=2$. Finally, for a homogeneous power linear Keller map $F=X+H$ of degree $d\ge 2$, we give the inverse polynomial map under the condition that $J{H}^3=0$. We shall show that $\mathrm{deg}(F^{-1})\le d^k$ if $k\le 2$ and $J{H}^{k+1}=0$, but also give an example with $d=2$ and $J{H}^4=0$ such that $\mathrm{deg}(F^{-1})>d^3$.

15 pages

References in corpus (4)

• Some Remarks on the Jacobian Conjecture and Dru{ż}kowski mappings
• The strong nilpotency index of a matrix
• A note on the Jacobian Conjecture
• Some remarks on the Jacobian conjecture and polynomial endomorphisms

Cited by in corpus (4)

• Some Remarks on the Jacobian Conjecture and Dru{ż}kowski mappings
• Some approaches toward the Jacobian conjecture
• Jacobian conjecture in ${\mathbb{R}}^2$
• Some observations on the properness of identity plus linear powers: part 2