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  1. Hilbert's 17th Problem for Real Closed Rings.Larry Mathews - 1994 - Mathematical Logic Quarterly 40 (4):445-454.
    We recall the characterisation of positive definite polynomial functions over a real closed ring due to Dickmann, and give a new proof of this result, based upon ideas of Abraham Robinson. In addition we isolate the class of convexly ordered valuation rings for which this characterisation holds.
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  • Topological differential fields.Nicolas Guzy & Françoise Point - 2010 - Annals of Pure and Applied Logic 161 (4):570-598.
    We consider first-order theories of topological fields admitting a model-completion and their expansion to differential fields . We give a criterion under which the expansion still admits a model-completion which we axiomatize. It generalizes previous results due to M. Singer for ordered differential fields and of C. Michaux for valued differential fields. As a corollary, we show a transfer result for the NIP property. We also give a geometrical axiomatization of that model-completion. Then, for certain differential valued fields, we extend (...)
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  • Boolean products of real closed valuation rings and fields.Jorge I. Guier - 2001 - Annals of Pure and Applied Logic 112 (2-3):119-150.
    We present some results concerning elimination of quantifiers and elementary equivalence for Boolean products of real closed valuation rings and fields. We also study rings of continuous functions and rings of definable functions over real closed valuation rings under this point of view.
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  • Elimination of quantifiers for ordered valuation rings.M. A. Dickmann - 1987 - Journal of Symbolic Logic 52 (1):116-128.
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  • Ganzstellensätze in theories of valued fields.Deirdre Haskell & Yoav Yaffe - 2008 - Journal of Mathematical Logic 8 (1):1-22.
    The purpose of this paper is to study an analogue of Hilbert's seventeenth problem for functions over a valued field which are integral definite on some definable set; that is, that map the given set into the valuation ring. We use model theory to exhibit a uniform method, on various theories of valued fields, for deriving an algebraic characterization of such functions. As part of this method we refine the concept of a function being integral at a point, and make (...)
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