In this paper, we study a so-called Condition C1 on square matrices with complex coefficients and a weaker Condition C2. For Druzkowski maps Condition C2 is equivalent to the Jacobian conjecture. We show that these conditions satisfy many good properties and in particular are satisfied by a dense subset of the set of square matrices of a given rank [Formula: see text]. Based on this, we propose a heuristic argument for the truth of the Jacobian conjecture. We propose some new equivalent formulations and some generalizations of the Jacobian conjecture, and some approaches (including computer algebra and numerical methods) toward resolving it. We show that some of these equivalent formulations are automatically satisfied by generic Druzkowski matrices. Applications and experimental results are included.

中文翻译：

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围绕雅可比猜想的一些新的理论和计算结果

在本文中，我们研究了具有复系数和较弱条件 C2 的方阵上的所谓条件 C1。对于 Druzkowski 映射，条件 C2 等价于雅可比猜想。我们证明了这些条件满足许多良好的性质，特别是给定秩的方阵集的密集子集[公式：见文本]。基于此，我们为雅可比猜想的真实性提出了一个启发式论证。我们提出了一些新的等价公式和雅可比猜想的一些推广，以及解决它的一些方法（包括计算机代数和数值方法）。我们表明，通用 Druzkowski 矩阵会自动满足其中一些等价公式。包括应用和实验结果。