A zero-test for D-algebraic transseries

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

💡 Research Summary

The paper addresses the problem of deciding whether a given differential polynomial evaluated at D‑algebraic transseries vanishes. Classical zero‑testing algorithms work for D‑algebraic formal power series, but power series lack field properties and cannot represent many functions involving exponentials, logarithms, or infinite sums. To overcome this limitation, the authors work with transseries—formal objects that close the field of power series under exponentiation, logarithms, and infinite summation.

The authors first develop a rigorous algebraic framework for generalized power series. They introduce grid‑based series 𝕂