Tropical differential equations

TL;DR

This paper introduces tropical differential equations, presents algorithms for testing their solvability efficiently in linear cases, and proves NP-hardness for nonlinear cases, advancing the understanding of tropical differential systems.

Contribution

It develops polynomial-time algorithms for solving tropical linear differential equations and establishes complexity bounds for nonlinear cases, extending tropical algebra concepts.

Findings

Polynomial complexity algorithm for tropical linear differential systems in one variable

Existence and construction of minimal solutions for solvable systems

NP-hardness of solving tropical nonlinear differential systems in one variable

Abstract

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.