Definable groups of partial automorphisms

Research output: Contribution to journalArticlepeer-review

Abstract

The motivation for this paper is to extend the known model-the-oretic treatment of differential Galois theory to the case of linear difference equations (where the derivative is replaced by an automorphism). The model-theoretic difficulties in this case arise from the fact that the corresponding theory ACFA does not eliminate quantifiers. We therefore study groups of restricted automorphisms, preserving only part of the structure. We give con-ditions for such a group to be (infinitely) definable, and when these conditions are satisfied we describe the definition of the group and the action explicitly. We then examine the special case when the theory in question is ob-tained by enriching a stable theory with a generic automorphism. Finally, we interpret the results in the case of ACFA, and explain the connection of our construction with the algebraic theory of Picard-Vessiot extensions. The only model-theoretic background assumed is the notion of a defin-able set.

Original languageEnglish
Pages (from-to)295-341
Number of pages47
JournalSelecta Mathematica, New Series
Volume15
Issue number2
DOIs
StatePublished - 1 Aug 2009
Externally publishedYes

Keywords

  • ACFA
  • Definable Galois groups
  • Difference equations
  • Picard-Vessiot theory

ASJC Scopus subject areas

  • Mathematics (all)
  • Physics and Astronomy (all)

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