We prove that, for a tropical rational map if for any point the convex hull of Jacobian matrices at smooth points in a neighborhood of the point does not contain singular matrices then the map is an isomorphism. We also show that a tropical polynomial map on the plane is an isomorphism if all the Jacobians have the same sign (positive or negative). In addition, for a tropical rational map we prove that if the Jacobians have the same sign and if its preimage is a singleton at least at one regular point then the map is an isomorphism.

中文翻译：

---

雅可比猜想的热带版本

我们证明，对于热带有理映射，如果对于任何点，该点邻域中平滑点的雅可比矩阵的凸包不包含奇异矩阵，则该映射是同构的。我们还表明，如果所有雅可比矩阵具有相同的符号（正或负），则平面上的热带多项式映射是同构的。此外，对于热带有理映射，我们证明，如果雅可比矩阵具有相同的符号，并且其原像至少在一个规则点上是单例，那么该映射是同构的。