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  1.  19
    Uniformly Defining Valuation Rings in Henselian Valued Fields with Finite or Pseudo-Finite Residue Fields.Raf Cluckers, Jamshid Derakhshan, Eva Leenknegt & Angus Macintyre - 2013 - Annals of Pure and Applied Logic 164 (12):1236-1246.
    We give a definition, in the ring language, of Zp inside Qp and of Fp[[t]] inside Fp), which works uniformly for all p and all finite field extensions of these fields, and in many other Henselian valued fields as well. The formula can be taken existential-universal in the ring language, and in fact existential in a modification of the language of Macintyre. Furthermore, we show the negative result that in the language of rings there does not exist a uniform definition (...)
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  2.  55
    A Version of P-Adic Minimality.Raf Cluckers & Eva Leenknegt - 2012 - Journal of Symbolic Logic 77 (2):621-630.
    We introduce a very weak language L M on p-adic fields K, which is just rich enough to have exactly the same definable subsets of the line K that one has using the ring language. (In our context, definable always means definable with parameters.) We prove that the only definable functions in the language L M are trivial functions. We also give a definitional expansion $L\begin{array}{*{20}{c}} ' \\ M \\ \end{array} $ of L M in which K has quantifier elimination, (...)
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  3.  11
    B-Minimality.Raf Cluckers & François Loeser - 2007 - Journal of Mathematical Logic 7 (2):195-227.
    We introduce a new notion of tame geometry for structures admitting an abstract notion of balls. The notion is named b-minimality and is based on definable families of points and balls. We develop a dimension theory and prove a cell decomposition theorem for b-minimal structures. We show that b-minimality applies to the theory of Henselian valued fields of characteristic zero, generalizing work by Denef–Pas [25, 26]. Structures which are o-minimal, v-minimal, or p-minimal and which satisfy some slight extra conditions are (...)
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  4.  46
    Presburger Sets and P-Minimal Fields.Raf Cluckers - 2003 - Journal of Symbolic Logic 68 (1):153-162.
    We prove a cell decomposition theorem for Presburger sets and introduce a dimension theory for Z-groups with the Presburger structure. Using the cell decomposition theorem we obtain a full classification of Presburger sets up to definable bijection. We also exhibit a tight connection between the definable sets in an arbitrary p-minimal field and Presburger sets in its value group. We give a negative result about expansions of Presburger structures and prove uniform elimination of imaginaries for Presburger structures within the Presburger (...)
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  5.  58
    Grothendieck Rings of ℤ-Valued Fields.Raf Cluckers & Deirdre Haskell - 2001 - Bulletin of Symbolic Logic 7 (2):262-269.
    We prove the triviality of the Grothendieck ring of a Z-valued field K under slight conditions on the logical language and on K. We construct a definable bijection from the plane K 2 to itself minus a point. When we specialized to local fields with finite residue field, we construct a definable bijection from the valuation ring to itself minus a point.
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  6. Stanford University, Stanford, CA March 19–22, 2005.Steve Awodey, Raf Cluckers, Ilijas Farah, Solomon Feferman, Deirdre Haskell, Andrey Morozov, Vladimir Pestov, Andre Scedrov, Andreas Weiermann & Jindrich Zapletal - 2006 - Bulletin of Symbolic Logic 12 (1).
     
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  7.  6
    Grothendieck Rings of $Mathbb{Z}$-Valued Fields.Raf Cluckers & Deirdre Haskell - 2001 - Bulletin of Symbolic Logic 7 (2):262-269.
    We prove the triviality of the Grothendieck ring of a $\mathbb{Z}$-valued field K under slight conditions on the logical language and on K. We construct a definable bijection from the plane K$^2$ to itself minus a point. When we specialized to local fields with finite residue field, we construct a definable bijection from the valuation ring to itself minus a point.
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