The purpose of this paper is fourfold. The first is to develop the theory of tropical differential algebraic geometry from scratch; the second is to present the tropical fundamental theorem for differential algebraic geometry, and show how it may be used to extract combinatorial information about the set of power series solutions to a given system of differential equations, both in the archimedean (complex analytic) and in the non-archimedean (e.g., $p$-adic) settings. A third and subsidiary aim is to show how tropical differential algebraic geometry is a natural application of semiring theory, and in so doing, contribute to the valuative study of differential algebraic geometry. Finally, the methods we have used in formulating and proving the fundamental theorem reveal new examples of non-classical valuations that merit further study in their own right.

翻译：本文的目的有四个方面:第一个方面是从零开始发展热带差异代数几何理论;第二个方面是提出差异代数几何的热带基本理论,并展示如何利用它来提取关于特定差异方程系统的一系列动力序列解决方案的组合信息,无论是在考古(复合分析)和非考古(例如,$p$-adic)设置中。 第三个方面和次要方面的目的是说明热带差异代数几何如何是半数理论的自然应用,从而对差异代数几何几何法的数值研究作出贡献。 最后,我们在制订和证明基本理论时所使用的方法揭示了非古典估价的新例子,这些例子本身值得进一步研究。