A general framework for tropical differential equations

Giansiracusa, Jeffrey; Mereta, Stefano

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.

Citation

Giansiracusa, J., & Mereta, S. (2024). A general framework for tropical differential equations. manuscripta mathematica, 173(3-4), 1273-1304. https://doi.org/10.1007/s00229-023-01492-5

• Journal Article Type: Article
• Acceptance Date: May 30, 2023
• Online Publication Date: Jul 27, 2023
• Publication Date: Mar 1, 2024
• Deposit Date: Jul 6, 2023
• Publicly Available Date: Aug 22, 2023
• Journal: manuscripta mathematica
• Print ISSN: 0025-2611
• Electronic ISSN: 1432-1785
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 173
• Issue: 3-4
• Pages: 1273-1304
• DOI: https://doi.org/10.1007/s00229-023-01492-5
• Public URL: https://durham-repository.worktribe.com/output/1168767
• Publisher URL: https://www.springer.com/journal/229