Valuations and Differential Galois Groups

Book excerpt: 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.