In this partly expository paper we give two applications of ideas from dynamical systems to the study of the injectivity properties of a polynomial local diffeomorphism $F=(F_1,F_2):{\mathbb{R}}^2\to {\mathbb{R}}^2$ (by the work of Pinchuk, these maps need not be globally injective). I) The Jacobian conjecture claims that all polynomial local biholomorphisms $G=(G_1,G_2):{ℂ}^2\to {ℂ}^2$ must be injective. By the Abhyankar–Moh theory in algebraic geometry, $G$ is injective if $G_1$ has one place at infinity. We prove that this result carries over to the real case, in the following form. It is shown that $F$ is injective if the (possibly singular) complex curve $\mathcal{C}$ of $\{F_1=0\}$ is irreducible, its projectivization $\tilde{\mathcal{C}}$ has only one point at infinity, and the said point is covered only once by a desingularization $ℛ\to \tilde{\mathcal{C}}$. II) In our second application we show that every polynomial local diffeomorphism of ${\mathbb{R}}^2$ into itself must be injective at least on certain large regions that contain sequences of disjoint discs of arbitrarily large radii.

在此部分说明性论文中，我们给出了动力学系统的思想在多项式局部微分射影的内射性性质研究中的两种应用 $F=（F_{\mathrm{1个}}，F_2）：{\mathbb{[R}}^2\to {\mathbb{[R}}^2$（通过Pinchuk的工作，这些地图不必是全局注入的）。I）雅可比猜想声称所有多项式局部双同态$G=（G_{\mathrm{1个}}，G_2）：{ℂ}^2\to {ℂ}^2$必须是内射的。根据Abhyankar–Moh代数几何理论，$G$ 如果是 $G_{\mathrm{1个}}$在无限处有一个地方。我们证明该结果可以以下形式延续到实际情况中。结果表明$F$ 如果（可能是奇数）复曲线则是内射的 $\mathcal{C}$ 的 $\{F_{\mathrm{1个}}=0\}$ 是不可约的，它的投射 $\stackrel{〜}{\mathcal{C}}$ 在无限远处只有一个点，并且该点仅被去奇点化一次 $ℛ\to \stackrel{〜}{\mathcal{C}}$。II）在我们的第二个应用中，我们证明了的每个多项式局部微分${\mathbb{[R}}^2$ 至少在某些大区域上必须是单射的，这些大区域包含任意大半径的不相交圆盘序列。