特征值空间中具有断开曲线的非奇异平面映射的注入性

Pub Date : 2023-09-23 DOI:10.12775/tmna.2022.073

Marco Sabatini

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引用次数: 0

摘要

Fessler和Gutierrez \cite{Fe}, \cite{Gu}证明了如果一个非奇异平面映射在$(0,+\infty)$中具有没有特征值的雅可比矩阵，则该映射是内射的。我们证明了用与上(下)复半平面分离的任意无界曲线代替$(0,+\infty)$成立。另外，我们证明了如果$\partial P/\partial x + \partial Q/\partial y$不是满射函数，则雅可比映射$(P,Q)$是内射。

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Injectivity of non-singular planar maps with disconnecting curves in the eigenvalues space

Fessler and Gutierrez \cite{Fe}, \cite{Gu} proved that if a non-singular planar map has Jacobian matrix without eigenvalues in $(0,+\infty)$, then it is injective. We prove that the same holds replacing $(0,+\infty)$ with any unbounded curve disconnecting the upper (lower) complex half-plane. Additionally we prove that a Jacobian map $(P,Q)$ is injective if $\partial P/\partial x + \partial Q/\partial y$ is not a surjective function.

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