Abstract
We generalize and unify various tools from the study of valued fields equipped with an automorphism to obtain a relative quantifier elimination result for such fields. Along the way we point how the techniques we employ relate to a classical result from tropical geometry. The quantifier elimination result we provide is then applied to the transseries field equipped with the automorphism which sends \(f(x)\) to \(f(x+1)\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
The conjecture is indeed false.
That is, valued difference fields where \(\sigma (\gamma )>n\gamma \) for all \(n\) whenever \(\gamma >0\).
With \({\varvec{i}}=(0,1,0 \ldots ,0)\), the multi-index \((0,0,1,0,\ldots ,0)\) can not be written as \({\varvec{i}}+{\varvec{j}}\), and hence we do not claim \(\sigma (\gamma _1)<\sigma ^2(\gamma _1)\), which is not true in general.
See [2], section 2, for the definition of complexity.
Generic elements from [12] correspond to regular elements.
The first author would like to thank Koushik Pal for bringing up this point, which has been neglected in [1].
It is also possible to obtain these maps with additional assumptions. For example if the value difference group is flat as a \(\mathbb Z [\sigma ]\)-module then the field admits a cross section in an elementary extension.
References
Azgin, S.: Valued fields with contractive automorphism and Kaplansky fields. J. Algebra 324(10), 2757–2785 (2010)
Azgin, S., van den Dries, L.: Equivalence of valued fields with value preserving automorphisms. J. Inst. Math. Jussieu 10(1), 1–35 (2011)
Basarab, S., Kuhlmann, F.-V.: An isomorphism theorem for henselian algebraic extensions of valued fields. Manuscr. Math. 77(1), 113–126 (1992)
Bélair, L., Macintyre, A., Scanlon, T.: Model theory of frobenius on witt vectors. Am. J. Math. 129, 665–721 (2007)
Chernikov, A., Hils, M.: Valued difference fields and ntp2. arXiv:1208.1341
Cohn, R.M.: Difference Algebra. Wiley Interscience Publishers, New York (1965)
Einsiedler, M., Kapranov, M., Lind, D.: Non-archimedean amoebas and tropical varieties. J. reine und angew. Math. 601, 139–158 (2006)
Hrushovski, E.: The elementary theory of the frobenius automorphism. arXiv:math/0406514
Kaplansky, I.: Maximal fields with valuations. Duke Math. J. 9, 303–321 (1942)
Kochen, S.: The model theory of local fields. In: Proceedings of International Summer Institute and Logic Colloquium Kiel, 1974, pp. 384–425. Springer, Berlin (1975)
Onay, G.: Modules Valus: en vue d’applications la thorie des corps valus de caractristique positive. Ph.D. thesis, Universit Paris Diderot-Paris VII (2011)
Pal, K.: Multiplicative valued difference fields. J. Symb. Log. 77(2), 545–579 (2012)
van der Hoeven, J.: Transseries and Real Differential Algebra. Lecture Notes in Mathematics, vol. 1888. Springer, Berlin (2006)
van der Hoeven, J.: Meta-expansion of transseries. J. Symb. Comput. 46(4), 339–359 (2011)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Durhan, S., Onay, G. Quantifier elimination for valued fields equipped with an automorphism. Sel. Math. New Ser. 21, 1177–1201 (2015). https://doi.org/10.1007/s00029-015-0183-0
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00029-015-0183-0