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Diophantine geometry over groups V1: Quantifier elimination I

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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.

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References

  • [Gu] V. Guba,Equivalence of infinite systems of equations in free groups and semigroups to finite subsystems, Matematicheskie Zametki40 (1986), 321–324.

    MATH  MathSciNet  Google Scholar 

  • [Me] Yu. I. Merzlyakov,Positive formulae on free groups, Algebra i Logika5 (1966), 257–266.

    Google Scholar 

  • [Sc] P. Scott,Subgroups of surface groups are almost geometric, Journal of the London Mathematical Society17 (1978), 555–565.

    Article  MATH  Google Scholar 

  • [Se1] Z. Sela,Diophantine geometry over groups I: Makanin-Razborov diagrams, Publication Mathématiques de l'IHES93 (2001), 31–105.

    Article  MATH  MathSciNet  Google Scholar 

  • [Se2] Z. Sela,Diophantine geometry over groups II: Completions, closures and formal solutions, Israel Journal of Mathematics134 (2003), 173–254.

    MATH  MathSciNet  Google Scholar 

  • [Se3] Z. Sela,Diophantine geometry over groups III: Rigid and solid solutions, Israel Journal of Mathematics147 (2005), 1–73.

    MATH  MathSciNet  Google Scholar 

  • [Se4] Z. Sela,Diophantine geometry over groups IV: An interative procedure for validation of a sentence, Israel Journal of Mathematics143 (2004), 1–130.

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, The Hebrew University of Jerusalem Givat Ram, 91904, Jerusalem, Israel

    Z. Zela

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  1. Z. Zela
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Partially supported by an Israel Academy of Sciences Fellowship.

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Zela, Z. Diophantine geometry over groups V1: Quantifier elimination I. Isr. J. Math. 150, 1–197 (2005). https://doi.org/10.1007/BF02762378

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  • DOI: https://doi.org/10.1007/BF02762378

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