Memoirs of the American Mathematical Society 2011; 68 pp; softcover Volume: 212 ISBN10: 0821849069 ISBN13: 9780821849064 List Price: US$63 Individual Members: US$37.80 Institutional Members: US$50.40 Order Code: MEMO/212/998
 In this paper, valuation theory is used to analyse infinitesimal behaviour of solutions of linear differential equations. For any PicardVessiot 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
