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Quantifier elimination in valued Ore modules

  • Luc Bélair (a1) and Françoise Point (a2)


We consider valued fields with a distinguished isometry or contractive derivation as valued modules over the Ore ring of difference operators. Under certain assumptions on the residue field, we prove quantifier elimination first in the pure module language, then in that language augmented with a chain of additive subgroups, and finally in a two-sorted language with a valuation map. We apply quantifier elimination to prove that these structures do not have the independence property.



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[1]Adler, H., An introduction to theories without the independence property, to appear,
[2]Ax, J., The elementary theory offinite fields, Annals of Mathematics, vol. 88 (1968), pp. 239271.
[3]Azgin, S., Model theory of valued difference fields, Ph.D. thesis, University of Illinois at Urbana-Champaign, 2007.
[4]Bélair, L., Types dans les corps values munis d'applications coefficients, Illinois Journal of Mathematics, vol. 43 (1999), no. 2, pp. 410425.
[5]Bélair, L. and Macintyre, A., L'automorphisme de Frobenius des vecteurs de Witt, Comptes Rendus de l'Académie des Sciences de Paris. Série I, vol. 331 (2000), no. 1, pp. 14.
[6]Bélair, L., Macintyre, A., and Scanlon, T., Model theory of the Frobenius on the Witt vectors, American Journal of Mathematics, vol. 129 (2007), pp. 665721.
[7]Bélair, L. and Point, F., Élimination des quantificateurs dans les équations aux différences linéaires sur les vecteurs de Witt, Comptes Rendus de l'Académie des Sciences de Paris. Série I, vol. 346 (2008), pp. 703706.
[8]Blum, L., Cucker, F., Shub, M., and Smale, S., Complexity and real computation, Springer-Verlag, New York, 1998.
[9]Cohn, P. M., Skew fields, Encyclopedia of Mathematics and its Applications, vol. 57, Cambridge University Press, Cambridge, 1995.
[10]Dellunde, P., Delon, F., and Point, F., The theory of modules of separably closedfields. II, Annals of Pure and Applied Logic, vol. 129 (2004), pp. 181210.
[11]van den Dries, L., Quantifier elimination for linear formulas over ordered and valued fields, Proceedings of the Model Theory Meeting (Univ. Brussels, Brussels/Univ. Mons, Mons, 1980), vol. 33, 1981, pp. 1931.
[12]Goodearl, K. R. and Warfield, R. B. Jr., An introduction to noncommutative rings, London Mathematical Society Student Texts, vol. 16, Cambridge University Press, Cambridge, 1989.
[13]Gurevich, Y. and Schmitt, P. H., The theory of ordered abelian groups does not have the independence property, Transactions of the American Mathematical Society, vol. 284 (1984), no. 1, pp. 171182.
[14]Hellegouarch, Y., Modules de Drinfeld généralisés, Approximations Diophantiennes et Nombres Transcendants (Luminy, 1990) (Philippon, P., editor), de Gruyter, Berlin, 1992, pp. 123164.
[15]Hellegouarch, Y. and Recher, F., Generalized t-modules, Journal of Algebra, vol. 187 (1997), pp. 323372.
[16]Hodges, W., Model theory, Cambridge University Press, Cambridge, 1993.
[17]Hrushovski, E. and Point, F., On von Neumann regular rings with an automorphism, Journal of Algebra, vol. 315 (2007), pp. 76120.
[18]Jacobson, N., Basic algebra. I, second ed., W. H. Freeman and Company, New York, 1985.
[19]Jacobson, N., Basic algebra. II, second ed., W. H. Freeman and Company, New York, 1989.
[20]Kaplansky, I., Selected papers and other writings, Springer-Verlag, New York, 1995.
[21]Karpinski, M. and Macintyre, A., Approximating volumes and integrals in o-minimal and p-minimal theories, Connections between model theory and algebraic and analytic geometry, vol. 6, Dipartimento di Matematica, Seconda Universita di Napoli, Caserta, 2000, pp. 149177.
[22]Magid, A. R., Lectures on differential Galois theory, University Lecture Series, vol. 7, American Mathematical Society, Providence, RI, 1994.
[23]Michaux, C. and Rivière, C., Quelques remarques concernant la théorie des corps ordonnés différentiellement clos, Bulletin of the Belgian Mathematical Society. Simon Stevin, vol. 12 (2005), pp. 341348.
[24]Poizat, B., Cours de théorie des modèles, Nur Al-Mantiq Wal-Ma'rifah, 1985.
[25]Prest, M., Model theory and modules, vol. 130, Cambridge University Press, Cambridge, 1988.
[26]Priess-Crampe, S. and Ribenboim, P., Differential equations over valued fields (and more), Journal für die Reine und Angewandte Mathematik, vol. 576 (2004), pp. 123147.
[27]Rohwer, T., Valued difference fields as modules over twisted polynomial rings, Ph.D. thesis, University of Illinois at Urbana-Champaign, 2003.
[28]Scanlon, T., Quantifier elimination for the relative Frobenius, Valuation theory and its applications, Vol. II (proceedings of the first international conference on valuation theory, saskatoon 1999) (Kuhlmann, al., editors), Fields Institute Communications, vol. 33, American Mathematical Society, Providence, RI, 2003, pp. 323352.
[29]Weispfenning, V., Quantifier elimination and decision procedures for valued fields, Models and sets (Aachen, 1983), Lecture Notes in Mathematics, vol. 1103, Springer, Berlin, 1984, pp. 419472.

Quantifier elimination in valued Ore modules

  • Luc Bélair (a1) and Françoise Point (a2)


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