A zero-test for D-algebraic transseries

TL;DR

This paper introduces an algorithm to determine whether a polynomial relation holds among D-algebraic transseries solutions of differential equations, extending existing power series methods to a broader class of formal series.

Contribution

The paper develops a novel zero-test algorithm specifically for D-algebraic transseries, expanding the applicability of algebraic relation testing beyond power series.

Findings

Algorithm successfully tests polynomial relations among D-algebraic transseries.

Extends zero-test methods from power series to transseries.

Provides a computational tool for formal series solutions of differential equations.

Abstract

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.