The Fundamental theorem of tropical differential algebra over nontrivially valued fields and the radius of convergence of nonarchimedean differential equations

TL;DR

This paper establishes a fundamental theorem for tropical partial differential equations over valued fields, linking tropical geometry with nonarchimedean differential equations and their convergence properties.

Contribution

It extends tropical differential algebra results to nontrivially valued fields and connects the radius of convergence of solutions to tropical computations.

Findings

Proves a fundamental theorem for tropical PDEs over valued fields.

Shows the radius of convergence can be computed tropically.

Extends previous results from trivial valuation to general valued fields.

Abstract

We prove a fundamental theorem for tropical partial differential equations, analogous to the fundamental theorem of tropical geometry in this context. We extend results from Aroca et al., Falkensteiner et al. and from Fink and Toghani for the case of trivial valuation as introduced by Grigoriev to differential equations with power series coefficients over any valued field. Crucial ingredients are the framework for tropical partial differential equations introduced by Giansiracusa and Mereta and a result on infinite intersections of projections of fibers of tropicalizations, which we prove using Hrushovski and Loeser's model-theoretic interpretation of Berkovich analytification. As a corollary of the fundamental theorem, we show that the radius of convergence of solutions of an ordinary differential equation over a nontrivially valued field can be computed tropically.