Tropical Differential Algebraic Geometry considers difficult or even intractable problems in Differential Equations and tries to extract information on their solutions from a restricted structure of the input. The Fundamental Theorem of Tropical Differential Algebraic Geometry states that the support of solutions of systems of ordinary differential equations with formal power series coefficients over an uncountable algebraically closed field of characteristic zero can be obtained by solving a so-called tropicalized differential system. Tropicalized differential equations work on a completely different algebraic structure which may help in theoretical and computational questions. We show that the Fundamental Theorem can be extended to the case of systems of partial differential equations by introducing vertex sets of Newton polygons.

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热带偏微分代数几何的基本定理

热带微分代数几何考虑微分方程中困难甚至棘手的问题，并试图从输入的受限结构中提取有关其解的信息。热带微分代数几何的基本定理指出，可以通过求解所谓的热带微分系统来获得具有形式幂级数系数的常微分方程组在特征为零的不可数代数闭域上的解的支持。Tropicalized 微分方程适用于完全不同的代数结构，这可能有助于解决理论和计算问题。我们表明，通过引入牛顿多边形的顶点集，基本定理可以扩展到偏微分方程组的情况。