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

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 $So{l}_{\mathbb{B}[\text{\hspace{0.17em}⁣}[t^{\mathrm{\Gamma }}]\text{\hspace{0.17em}⁣}],k}(P)\supset \mu (So{l}_{\mathbb{B}[\text{\hspace{0.17em}⁣}[t^{\mathrm{\Gamma }}]\text{\hspace{0.17em}⁣}]}(P)\text{\hspace{0.17em}⁣})$ 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 $y=\phi (t)\in \mathbb{B}[\text{\hspace{0.17em}⁣}[t^{\mathrm{\Gamma }}]\text{\hspace{0.17em}⁣}]$ (with coefficients in the boolean semifield $\mathbb{B}$) of $P$ having support in different relevant submonoids $\mathrm{\Gamma }$ 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 $S(\mathfrak{P},0):=or{d}_t(So{l}_{\mathbb{C}[\text{\hspace{0.17em}⁣}[t^{\mathbb{R}}]\text{\hspace{0.17em}⁣}]}(\mathfrak{P})\text{\hspace{0.17em}⁣})\subset \mathbb{R}$ 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 $S(\mathfrak{P},0)\subset or{d}_t(So{l}_{\mathbb{B}[\text{\hspace{0.17em}⁣}[t^{\mathbb{R}}]\text{\hspace{0.17em}⁣}],k}(P))$. In most cases, the set $So{l}_{\mathbb{B}[\text{\hspace{0.17em}⁣}[t^{\mathbb{R}}]\text{\hspace{0.17em}⁣}],k}(P)$ is a finite family of univariate tropical polynomials.