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A general framework for tropical differential equations

Giansiracusa, Jeffrey; Mereta, Stefano

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Authors

Stefano Mereta



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

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Copyright Statement
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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Advance Online Version




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