A zero-test for D-algebraic transseries

2026-02-01

AI Summary

This study addresses the problem of determining whether a D-algebraic transseries—defined by a system of polynomial differential equations together with initial conditions—satisfies a given algebraic polynomial identity \(P(f_1,\dots,f_k)=0\). To this end, the authors extend, for the first time, zero-testing algorithms from formal power series to the broader setting of transseries. By integrating techniques from differential algebra, transseries theory, and symbolic computation, they develop an algorithm capable of deciding algebraic relations among D-algebraic transseries. This work not only advances the boundary of computability in differential algebra but also provides an effective tool for verifying algebraic dependencies among solutions of nonlinear differential equations.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Deep Learning Algorithms
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

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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.

Problem

• zero-test
• D-algebraic
• transseries
• differential equations
• polynomial

Innovation

• transseries
• zero-test
• D-algebraic
• polynomial differential equations
• formal power series