In this paper, valuation theory is used to analyse infinitesimal behaviour of solutions of linear differential equations. For any Picard-Vessiot extension \((F / K, \partial)\) with differential Galois group \(G\), the author looks at the valuations of \(F\) which are left invariant by \(G\). The main reason for this is the following: If a given invariant valuation \(\nu\) measures infinitesimal behaviour of functions belonging to \(F\), then two conjugate elements of \(F\) will share the same infinitesimal behaviour with respect to \(\nu\). This memoir is divided into seven sections.

Table of Contents

• Introduction
• Invariant valuations and solutions of l.d.e.
• Examples and use of invariant valuations
• Continuity of derivations, geometry and examples
• Continuity and field extensions
• Invariant valuations and singularities of l.d.e.
• Existence and geometry of invariant valuations
• Bibliography