Exploring tropical differential equations
-
Ethan Cotterill
and
Johana Luviano
Abstract
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) setting. 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. We use this formalism to extend the fundamental theorem of partial differential algebraic geometry to the differential fraction field of the ring of formal power series in arbitrarily (finitely many variables; in doing so we produce new examples of non-Krull valuations that merit further study in their own right.
Funding statement: The second author was supported by CONACYT through Project 299261. The third author was supported by PAPIIT IN108320.
Acknowledgements
The authors thank Lara Bossinger, Alicia Dickenstein, Jeffrey Giansiracusa and Pedro Luis del Ángel for interesting conversations and valuable comments on previous versions of this paper; Yue Ren, from whom we first learned of the potential applications of these methods to p-adic differential equations during a BIRS talk he gave in June 2020; and the anonymous referees, whose constructive criticism has helped to improve the quality of our exposition.
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Articles in the same Issue
- Frontmatter
- Exploring tropical differential equations
- On the bond polytope
- Universal convex covering problems under translations and discrete rotations
- Ehrhart theory of paving and panhandle matroids
- A partial compactification of the Bridgeland stability manifold
- The geometry of discrete L-algebras
- Projective self-dual polygons in higher dimensions
Articles in the same Issue
- Frontmatter
- Exploring tropical differential equations
- On the bond polytope
- Universal convex covering problems under translations and discrete rotations
- Ehrhart theory of paving and panhandle matroids
- A partial compactification of the Bridgeland stability manifold
- The geometry of discrete L-algebras
- Projective self-dual polygons in higher dimensions
