Complex transseries solutions to algebraic differential equations
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Abstract

In our PhD. we have given an algorithm for the algebraic resolution of algebraic differential equations with real transseries coefficients. Unfortunately, not all equations do admit solutions in this strongly monotonic setting, even though we recently proved an intermediate value theorem.

In this paper we show that the algorithm from our PhD. generalizes to the setting of weakly oscillatory or complex transseries. Modulo a finite number of case separations, we show how to determine the solutions of an arbitrary algebraic differential equation over the complex transseries. We will show that such equations always admit complex transseries solutions. However, the field of complex transseries is not differentially algebraically closed.

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Note: we performed a few corrections with respect to the original version.