关于Jacobian猜想和Dru{K}kowski映射的几点注记
Some Remarks on the Jacobian Conjecture and Dru{ż}kowski mappings
摘要：
本文首先证明了非齐次幂线性映射的雅可比猜想在一定条件下成立。其次，我们证明了关于$r\geq 1$维上Jacobian猜想的一个等价命题，并给出了$r\geq 1$的部分结果。最后，对于次数为$d \ge 2$的齐次幂线性Keller映射$F=X+H$，在满足$JH^3=0$的条件下，给出了逆多项式映射。我们将证明，如果$k \le 2$和$JH^{k+1}=0$，则${\operatorname{deg}}(F^{-1})\leq d^k$，但也给出一个$d = 2$和$JH^4=0$的例子，使得${\operatorname{deg}}(F^{-1})> d^3$。
英文摘要：
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 $JH^3=0$. We shall show that ${\operatorname{deg}}(F^{-1})\leq d^k$ if $k \le 2$ and $JH^{k+1}=0$, but also give an example with $d = 2$ and $JH^4=0$ such that ${\operatorname{deg}}(F^{-1})> d^3$.
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