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arXiv · 2111.03925

A general framework for tropical differential equations

Jeffrey Giansiracusa·Stefano Mereta
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Abstract

We construct a general framework for tropical differential equations based on idempotent semirings and an idempotent version of differential algebra. Over a differential ring equipped with a non-archimedean norm enhanced with additional differential information, we define tropicalization of differential equations and tropicalization of their solution sets. This framework includes rings of interest in the theory of p-adic differential equations: rings of convergent power series over a non-archimedean normed field. The tropicalization records the norms of the coefficients. This gives a significant refinement of Grigoriev's framework for tropical differential equations. We then prove a differential analogue of Payne's inverse limit theorem: the limit of all tropicalizations of a system of differential equations is isomorphic to a differential variant of the Berkovich analytification.

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Jeffrey Giansiracusa, Stefano Mereta. 2021-11-06. A general framework for tropical differential equations. https://arxiv.org/abs/2111.03925

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