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Some Remarks on the Jacobian Conjecture and Dru{ż}kowski mappings (1302.5864v3)
Published 24 Feb 2013 in math.AG
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 $r\geq 1$ 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 $JH3=0$. We shall show that ${\operatorname{deg}}(F{-1})\leq dk$ if $k \le 2$ and $JH{k+1}=0$, but also give an example with $d = 2$ and $JH4=0$ such that ${\operatorname{deg}}(F{-1})> d3$.