A zero-test for D-algebraic transseries

Abstract

Consider formal power series $f_1,\dots ,f_k\in \mathbb{Q}[[z]]$ that are defined as the solutions of a system of polynomial differential equations together with a sufficient number of initial conditions. Given $P\in \mathbb{Q}[F_1,\dots ,F_k]$, several algorithms have been proposed in order to test whether $P(f_1,\dots ,f_k)=0$. In this paper, we present such an algorithm for the case where $f_1,\dots ,f_k$ are so-called transseries instead of power series.