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Valuations and differential Galois groups


About this Title

Guillaume Duval, 1 Chemin du Chateau, 76430 Les Trois Pierres, France

Publication: Memoirs of the American Mathematical Society
Publication Year: 2011; Volume 212, Number 998
ISBNs: 978-0-8218-4906-4 (print); 978-1-4704-0615-8 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-2010-00606-9
Published electronically: December 3, 2010
MathSciNet review: 2839103
Keywords:Differential Galois theory, valuations, Hardy fields
MSC: Primary 34M15; Secondary 12J20

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Table of Contents


Chapters

  • Chapter 1. Introduction
  • Chapter 2. Invariant valuations and solutions of l.d.e.
  • Chapter 3. Examples and use of invariant valuations
  • Chapter 4. Continuity of derivations, geometry and examples
  • Chapter 5. Continuity and field extensions
  • Chapter 6. Invariant valuations and singularities of l.d.e.
  • Chapter 7. Existence and geometry of invariant valuations

Abstract


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 $G$, we look 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 $\nu$. The article is divided into seven sections as follows. In section

1, we give a brief account on Picard-Vessiot theory and valuation theory. In section 2, we explore the links between invariant valuations and solutions of linear differential equations. To this respect, Corollary 2 is a kind of Riemann-Roch property which states that some solutions of linear differential equations must have poles at invariant valuations. In section 3, we give examples of invariant valuations. We also use the above theory to give a new proof of a result due to Drach and Kolchin about elliptic functions (Theorem 26). In section

4, we analyse the properties of valuations which are describing the analytic shape of functions. The notion of continuity of a derivation w.r.t. a valuation, plays a central role. This is justified geometrically thanks to the theory of vector fields in Corollary

38. In section 5, the permanence of the continuity by field extensions is proved for algebraic extensions (Theorem 3), and for invariant valuations of Liouvillian extensions (Theorem 4). In Theorem

5 of section 6, we show that in general, invariant analytic valuations are related to singularities of linear differential equations. In section 7, we prove the existence of invariant valuations (Theorem 6) for Picard-Vessiot extensions with connected differential Galois group.

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