关于满足雅可比猜想的多项式映射的密度
On the Density of Polynomial Mappings Satisfying the Jacobian Conjecture
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中文总结 AI 辅助
本文针对雅可比猜想这一未解决的代数几何问题，证明对任意维数$n\geq1$，存在非空Zariski稠密开集，其中所有雅可比矩阵可逆的多项式映射均为自同构，推进了该猜想的相关密度研究。
中文摘要 AI 辅助
雅可比猜想是一个著名的未解决问题，也是斯蒂芬·斯梅尔1998年列出的《下世纪的数学问题》清单中的第16号问题。该问题探讨：若多项式映射$F:\mathbb{C}^n\to\mathbb{C}^n$在每一点处的雅可比矩阵都可逆，是否能推出$F$是自同构。$n=1$的情形显然成立，而$n\geq3$的情形近期已被Levent Alpöge提出的反例证明不成立，$n=2$的情形仍是开放问题。本文证明，对所有$n\geq1$，存在一个非空的Zariski稠密开集$U$，使得对所有$F\in U$，若$F$的雅可比矩阵可逆，则$F$是自同构。
英文摘要
The Jacobian Conjecture is a known unsolved problem and it is the problem number 16 of the list ''Mathematical Problems for the Next Century'', made by Stephen Smale, in 1998. The problem asks whether or not the Jacobian matrix of a polynomial mapping $F:\mathbb{C}^n\to\mathbb{C}^n$ at every point being invertible implies that $F$ is an automorphism. The case $n = 1$ is trivially true, while the case $n\geq 3$ has been recently proven to be false by a counter-example provided by Levent Alpöge, and the case $n = 2$ is still an open problem. In this paper, we show that, for all $n \geq 1$, there exists a non-empty Zariski dense open set $U$ such that, for all $F \in U$, if the Jacobian matrix of $F$ is invertible, then $F$ is an automorpshim.
