Israel Journal of Mathematics

, Volume 150, Issue 1, pp 1–197

Diophantine geometry over groups V1: Quantifier elimination I

Article

Abstract

This paper is the first part (out of two) of the fifth paper in a sequence on the structure of sets of solutions to systems of equations in a free group, projections of such sets, and the structure of elementary sets defined over a free group. In the two papers on quantifier elimination we use the iterative procedure that validates the correctness of anAE sentence defined over a free group, presented in the fourth paper, to show that the Boolean algebra ofAE sets defined over a free group is invariant under projections, and hence show that every elementary set defined over a free group is in the Boolean algebra ofAE sets. The procedures we use for quantifier elimination, presented in this paper and its successor, enable us to answer affirmatively some of Tarski's questions on the elementary theory of a free group in the sixth paper of this sequence.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [Gu] V. Guba,Equivalence of infinite systems of equations in free groups and semigroups to finite subsystems, Matematicheskie Zametki40 (1986), 321–324.MATHMathSciNetGoogle Scholar
  2. [Me] Yu. I. Merzlyakov,Positive formulae on free groups, Algebra i Logika5 (1966), 257–266.Google Scholar
  3. [Sc] P. Scott,Subgroups of surface groups are almost geometric, Journal of the London Mathematical Society17 (1978), 555–565.MATHCrossRefGoogle Scholar
  4. [Se1] Z. Sela,Diophantine geometry over groups I: Makanin-Razborov diagrams, Publication Mathématiques de l'IHES93 (2001), 31–105.MATHCrossRefMathSciNetGoogle Scholar
  5. [Se2] Z. Sela,Diophantine geometry over groups II: Completions, closures and formal solutions, Israel Journal of Mathematics134 (2003), 173–254.MATHMathSciNetGoogle Scholar
  6. [Se3] Z. Sela,Diophantine geometry over groups III: Rigid and solid solutions, Israel Journal of Mathematics147 (2005), 1–73.MATHMathSciNetGoogle Scholar
  7. [Se4] Z. Sela,Diophantine geometry over groups IV: An interative procedure for validation of a sentence, Israel Journal of Mathematics143 (2004), 1–130.MATHMathSciNetGoogle Scholar

Copyright information

© Hebrew University 2005

Authors and Affiliations

  • Z. Zela
    • 1
  1. 1.Institute of MathematicsThe Hebrew University of Jerusalem Givat RamJerusalemIsrael

Personalised recommendations