A note on an expansion formula with application to nonlinear DAE's

TL;DR

This paper extends an expansion formula for derivatives to nonlinear differential algebraic systems, providing new insights into power series solutions and an extension of Tougeron's implicit function theorem.

Contribution

It generalizes existing expansion formulas to systems of nonlinear differential algebraic equations and introduces a simple extension of Tougeron's implicit function theorem.

Findings

Extended expansion formula for nonlinear DAE systems

Conditions for power series solutions of algebraic differential equations

A new extension of Tougeron's implicit function theorem

Abstract

In [DL] systems of differential polynomials are investigated with respect to properties of Artin approximation type. The key tool in [DL] is an extended version of a formula by Hurwitz [Hu] expressing high order derivatives of an expansion by lower ones. The formula is further refined in [VFZ] to deliver sufficient conditions concerning the existence of power series solutions of scalar algebraic differential equations of order n. In the paper at hand, the main results from [VFZ] are first reproduced and further extended to systems of nonlinear differential algebraic equations. In addition, a simple extension of Tougeron's implicit function theorem is given in a specific constellation. The results follow from [S1], [S2] where Artin approximation is treated within a Banach space setting, thereby constructing an expansion formula that expresses accurately the required dependency of low and high order derivatives within the system of undetermined coefficients.