Elementary solutions of ordinary tropical differential equations, and vanishing orders of solutions of algebraic differential equations

Abstract

Our aim is to use tropical differential algebra to systematically build a combinatorial basis for the study of the set of (formal) power series solutions to nonlinear algebraic ordinary differential equations (over $\mathbb{C}$) expanded around the point $t_0=0\in\mathbb{C}$, which may also be effectively computed using standard tropical algebra. This paper is divided into two parts. First, given an ordinary tropical differential equation in one differential variable $P=P(y)$ of (differential) order $k$, we study the sets $Sol_{\mathbb{B}[\![t^Γ]\!],k}(P)\supset μ(Sol_{\mathbb{B}[\![t^Γ]\!]}(P)\!)$ of tropical elementary $k$-solutions and minimal tropical solutions, respectively; we show that these two sets bear many similarities. We do this for tropical solutions $y=φ(t)\in \mathbb{B}[\![t^Γ]\!]$ (with coefficients in the boolean semifield $\mathbb{B}$) of $P$ having support in different relevant submonoids $Γ$ of $(\mathbb{R},+,0)$. Then, given an ordinary algebraic differential equation $\mathfrak{P}$ (with meromorphic coefficients in one differential variable $\mathfrak{P}=\mathfrak{P}(y)$ and of differential order $k$), we consider the set $S(\mathfrak{P},0):=ord_t(Sol_{\mathbb{C}[\![t^{\mathbb{R}}]\!]}(\mathfrak{P})\!)\subset\mathbb{R}$ of $t$-adic orders of formal Hahn solutions of $\mathfrak{P}$, which is an algebraic object that gives information about the nature of the germs of solutions of $\mathfrak{P}$ at the point $t_0=0\in\mathbb{C}$. We show that this set is contained in the set of $t$-adic orders of Hahn elementary $k$-solutions of its tropicalization $P=trop(\mathfrak{P})$, this is $S(\mathfrak{P},0)\subset ord_t(Sol_{\mathbb{B}[\![t^{\mathbb{R}}]\!],k}(P))$. In most cases, the set $Sol_{\mathbb{B}[\![t^{\mathbb{R}}]\!],k}(P)$ is a finite family of univariate tropical polynomials.