Tropical differential equations

Tropical differential equations are introduced and an algorithm is designed which tests solvability of a system of tropical linear differential equations within the complexity polynomial in the size of the system and in its coefficients. Moreover, we show that there exists a minimal solution, and the algorithm constructs it (in case of solvability). This extends a similar complexity bound established for tropical linear systems. In case of tropical linear differential systems in one variable a polynomial complexity algorithm for testing its solvability is designed. We prove also that the problem of solvability of a system of tropical non-linear differential equations in one variable is $NP$-hard, and this problem for arbitrary number of variables belongs to $NP$. Similar to tropical algebraic equations, a tropical differential equation expresses the (necessary) condition on the dominant term in the issue of solvability of a differential equation in power series

INITIAL FORMS AND A NOTION OF BASIS FOR TROPICAL DIFFERENTIAL EQUATIONS

We show that solution sets of systems of tropical differential equations can be characterised in terms of monomial-freeness of an initial ideal. We discuss a candidate definition of tropical differential basis and give a nonexistence result for such bases in an example

Initial forms and a notion of basis for tropical differential equations

We show that solution sets of systems of tropical differential equations can be characterised in terms of monomial-freeness of an initial ideal. We discuss a candidate definition of tropical differential basis and give a nonexistence result for such bases in an example.Comment: 16 page

The fundamental theorem of tropical differential algebra for formal Puiseux series

The fundamental theorem of tropical differential algebra has been established for formal power series solutions of systems of algebraic differential equations. It has been shown that the direct extension to formal Puiseux series solutions fails. In this paper, we overcome this issue by transforming the given differential system and such a generalization is presented. Moreover, we explain why such transformations do not work for generalizing the fundamental theorem to transseries solutions, but show that one inclusion still holds for this case even without using any transformation

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

We prove a fundamental theorem for tropical partial differential equations analogue of the fundamental theorem of tropical geometry in this context. We extend results from Aroca et al., Falkensteiner et al. and from Fink and Toghani, which work only in the case of trivial valuation as introduced by Grigoriev, to differential equations with power series coefficients over any valued field. To do so, a crucial ingredient is the framework for tropical partial differential equations introduced by Giansiracusa and Mereta. Using this framework we also add a fourth statement to the fundamental theorem, seeing the tropicalization as the set of evaluations of points of the differential Berkovich analytification on the generators of a differential algebra for a given presentation. Lastly, as a corollary of the fundamental theorem, we have that the radius of convergence of solutions of an ordinary differential equation over a nontrivially valued field can be computed tropically.Comment: 43 pages, extended the scope from univariate to multivariate case and added another statement to the main theorem, fixed a gap in the proof of Proposition 3.5 of the old versio

Tropical Mathematics and the Lambda Calculus I: Metric and Differential Analysis of Effectful Programs

We study the interpretation of the lambda-calculus in a framework based on tropical mathematics, and we show that it provides a unifying framework for two well-developed quantitative approaches to program semantics: on the one hand program metrics, based on the analysis of program sensitivity via Lipschitz conditions, on the other hand resource analysis, based on linear logic and higher-order program differentiation. To do that we focus on the semantic arising from the relational model weighted over the tropical semiring, and we discuss its application to the study of "best case" program behavior for languages with probabilistic and non-deterministic effects. Finally, we show that a general foundation for this approach is provided by an abstract correspondence between tropical algebra and Lawvere's theory of generalized metric spaces