Journal of Symbolic Logic

0-D-valued fields

Nicolas Guzy

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In [12], T. Scanlon proved a quantifier elimination result for valued D-fields in a three-sorted language by using angular component functions. Here we prove an analogous theorem in a different language ℒ₂ which was introduced by F. Delon in her thesis. This language allows us to lift the quantifier elimination result to a one-sorted language by a process described in the Appendix. As a byproduct, we state and prove a “positivstellensatz” theorem for the differential analogue of the theory of real-series closed fields in the valued D-field setting.

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J. Symbolic Logic, Volume 71, Issue 2 (2006), 639-660.

First available in Project Euclid: 2 May 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03C10: Quantifier elimination, model completeness and related topics 12H05: Differential algebra [See also 13Nxx] 12J10: Valued fields

Valued D-fields quantifier elimination real-series closed fields positivstellensatz


Guzy, Nicolas. 0-D-valued fields. J. Symbolic Logic 71 (2006), no. 2, 639--660. doi:10.2178/jsl/1146620164.

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  • F. Delon, Quelques propriétés de corps valués en théorie des modèles, Ph.D. thesis, Université de Paris 7, 1982.
  • R. Farré, A positivstellensatz for chain-closed fields $\mathbb R((t))$ and some related fields, Archives of Mathematics, vol. 57 (1991), pp. 446--455.
  • --------, Model theory for valued and ordered fields and applications, Ph.D. thesis, Universitat de Catalunya, 1993.
  • B. Jacob, A nullstellensatz for $\mathbb R((t))$, Communications in Algebra, vol. 8 (1980), pp. 1083--1094.
  • S. Kochen, Integer valued rational functions over the $p$-adic numbers, a $p$-adic analogue of the theory of real fields, Proceedings of the XII Symposium on Pure Mathematics, 1967, pp. 57--73.
  • --------, The model theory of local fields, Proceedings of the International Summer Institute and Logic Colloquium (ISILC), (Kiel '74), Lecture Notes in Mathematics, vol. 499, 1974, pp. 384--425.
  • A. Macintyre, On definable subsets of $p$-adic fields, Journal of Symbolic Logic, vol. 41 (1976), no. 3, pp. 605--610.
  • D. Marker, M. Messmer, and A. Pillay, Model theory of fields, Lecture Notes in Logic, vol. 5, 1996.
  • C. Michaux, Sur l'élimination des quantificateurs dans les anneaux différentiels, Comptes Rendus de l' Académie des Sciences, Série I, Mathématique, vol. 302 (1986), no. 8, pp. 287--290.
  • --------, Differential fields and machines over the real numbers and automata, Ph.D. thesis, Université de Mons-Hainaut, 1991.
  • P. Ribenboim, Théorie des valuations, Séminaire de Mathématiques Supérieures, No. 9 (Été), vol 1964, Les Presses de l'Université de Montréal, Montréal, Québec, 1968, Deuxième édition multigraphiée.
  • T. Scanlon, Quantifier elimination for the relative Frobenius, Valuation theory and its applications (Saskatoon '99), Vol. II, Fields Institute Communications Series, vol. 33, 1999, pp. 323--352.
  • --------, A model complete theory of valued d-fields, Journal of Symbolic Logic, vol. 65 (2000), no. 4, pp. 1758--1784.
  • M. F. Singer, The model theory of ordered differential fields, Journal of Symbolic Logic, vol. 43 (1978), no. 1, pp. 82--91.