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Multiplicative valued difference fields

Published online by Cambridge University Press:  12 March 2014

Koushik Pal
Koushik Pal*
Affiliation:
Department of Mathematics, University of California,Berkeley, Berkeley, CA 94720-3840, USA, E-mail: koushik@math.berkeley.edu
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Abstract

The theory of valued difference fields (K, σ, υ,) depends on how the valuation υ interacts with the automorphism σ. Two special cases have already been worked out - the isometric case, where υ(σ(x)) = υ(x) for all x Є K, has been worked out by Luc Belair, Angus Macintyre and Thomas Scanlon; and the contractive case, where υ(σ(x)) > nυ(x) for all x Є K× with υ(x) > 0 and n Є ℕ, has been worked out by Salih Azgin. In this paper we deal with a more general version, the multiplicative case, where υ(σ(x)) = ρ · υ(x), where ρ (> 0) is interpreted as an element of a real-closed field. We give an axiomatization and prove a relative quantifier elimination theorem for this theory.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 2 , June 2012 , pp. 545 - 579
Copyright
Copyright © Association for Symbolic Logic 2012

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References

REFERENCES

[1] Azgin, S., Valued fields with contractive automorphism and Kaplansky fields, Journal of Algebra, vol. 324 (2010), no. 10, pp. 2757–2785.Google Scholar
[2] Azgin, S. and van den Dries, L., Elementary theory of valued fields with a valuation-preserving automorphism. Journal of the Institute of Mathematics of Jussieu, vol. 10 (2011), no. 1, pp. 1–35.Google Scholar
[3] Basarab, S. and Kuhlmann, F. V., An isomorphism theorem for Henselian algebraic extensions of valued fields, Manuscripta Mathematica, vol. 77 (1992), no. 1, pp. 113–126.Google Scholar
[4] Bélair, L., Macintyre, A., and Scanlon, T., Model theory of the Frobenius on the Witt vectors, American Journal of Mathematics, vol. 129 (2007), no. 3, pp. 665–721.CrossRefGoogle Scholar
[5] Bourbaki, N., Elements of mathematics—commutative algebra, Addison-Wesley Publishing Company, Reading, Massachusetts, 1972.Google Scholar
[6] Cherlin, G., Model theoretic algebra—selected topics, Lecture Notes in Mathematics, vol. 521, Springer-Verlag, Berlin, 1976.Google Scholar
[7] Cohn, R. M., Difference algebra, Interscience Publishers, John Wiley & Sons, New York, London, Sydney, 1965.Google Scholar
[8] Flenner, J., The relative structure of Henselian valued fields, Ph.D. thesis, University of California, Berkeley, 2008, available at http://www.nd.edu/~jflenner/papers/dissertation.pdf.Google Scholar
[9] Kaplansky, I., Maximal fields with valuations, Duke Mathematical Journal, vol. 9 (1942), pp. 303–321.Google Scholar
[10] Kikyo, H. and Shelah, S., The strict order property and generic automorphisms, this Journal, vol. 67 (2002), no. 1, pp. 214–216.Google Scholar
[11] Marker, D., Model theory: An introduction, Springer-Verlag, Berlin, 2002.Google Scholar
[12] Neumann, B. H., On ordered division rings, Transactions of the American Mathematical Society, vol. 66 (1949), no. 1, pp. 202–252.Google Scholar
[13] Point, F., Existentially closed ordered difference fields and rings, Mathematical Logic Quarterly, vol. 56 (2010), no. 3, pp. 239–256.Google Scholar
[14] Scanlon, T., A model complete theory of valued D-fields, this Journal, vol. 65 (2000), no. 4, pp. 1758–1784.Google Scholar
[15] Scanlon, T., Quantifier elimination for the relative Frobenius, Valuation theory audits applications, Fields Institute Communication Series, vol. II, American Mathematical Society, Providence, 2003, pp. 323–352.Google Scholar
[16] van den Dries, L., Quantifier elimination for linear formulas over ordered and valued fields, Bulletin de la Société Mathématique de Belgique Série B, vol. 33 (1981), no. 1, pp. 19–31.Google Scholar