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73
Germs of Arcs on Singular Algebraic Varieties and Motivic Integration
, 1999
"... Introduction Let k be a field of characteristic zero. We denote by M the Grothendieck ring of algebraic varieties over k (i.e. reduced separated schemes of finite type over k). It is the ring generated by symbols [S], for S an algebraic variety over k, with the relations [S] = [S 0 ] if S is iso ..."
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Cited by 133 (19 self)
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Introduction Let k be a field of characteristic zero. We denote by M the Grothendieck ring of algebraic varieties over k (i.e. reduced separated schemes of finite type over k). It is the ring generated by symbols [S], for S an algebraic variety over k, with the relations [S] = [S 0 ] if S is isomorphic to S 0 , [S] = [S n S 0 ] + [S 0 ] if S 0 is closed in S and [S \Theta S 0 ] = [S] [S 0 ]. Note that, for S an algebraic variety over k, the mapping S 0 7! [S 0 ] from the
A Model Complete Theory Of Valued DFields
 J. Symbolic Logic
, 1999
"... The notion of a Dring, generalizing that of a differential or a difference ring, is introduced. Quantifier elimination and a version of the AxKochenErshov principle is proven for a theory of valued Dfields of residual characteristic zero. The model theory of differential and difference fields has ..."
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Cited by 22 (11 self)
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The notion of a Dring, generalizing that of a differential or a difference ring, is introduced. Quantifier elimination and a version of the AxKochenErshov principle is proven for a theory of valued Dfields of residual characteristic zero. The model theory of differential and difference fields has been extensively studied (see for example [7, 3]) and valued fields have proven to be amenable to model theoretic analysis (see for example [1, 2]). In this paper we subject a theory of valued fields possessing either a derivation or an automorphism interacting strongly with the valuation to such an analysis. Our theory differs from C. Michaux's theory of henselian differential elds [8] on this last point: in his theory, the valuation and derivation have a very weak interaction. In Section 1 we introduce the notion of a Dfield and show that a differential ring may be regarded as a specialization of a difference ring. This formal connection supports the view that differential and difference algebr...
Analytic cell decomposition and analytic motivic integration
 ANN. SCI. ÉCOLE NORM. SUP
, 2006
"... ..."
Decidability of the isomorphism problem for stationary AFalgebras
, 1999
"... The notion of isomorphism of stable AFC ∗algebras is considered in this paper in the case when the corresponding Bratteli diagram is stationary, i.e., is associated with a single square primitive nonsingular incidence matrix. C∗isomorphism induces an equivalence relation on these matrices, call ..."
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Cited by 13 (3 self)
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The notion of isomorphism of stable AFC ∗algebras is considered in this paper in the case when the corresponding Bratteli diagram is stationary, i.e., is associated with a single square primitive nonsingular incidence matrix. C∗isomorphism induces an equivalence relation on these matrices, called C ∗equivalence. We show that the associated isomorphism equivalence problem is decidable, i.e., there is an algorithm that can be used to check in a finite number of steps whether two given primitive nonsingular matrices are C∗equivalent or not.
The model theory of tame valued fields
, 2009
"... A henselian valued field K is called a tame field if its separablealgebraic closure K sep is a tame extension, that is, K sep is equal to the ramification field of the normal extension K sep K. Every algebraically maximal Kaplansky field is a tame field, but not conversely. We prove Ax–Kochen–Ersh ..."
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Cited by 10 (8 self)
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A henselian valued field K is called a tame field if its separablealgebraic closure K sep is a tame extension, that is, K sep is equal to the ramification field of the normal extension K sep K. Every algebraically maximal Kaplansky field is a tame field, but not conversely. We prove Ax–Kochen–Ershov Principles for tame fields. This leads to model completeness and completeness results relative to value group and residue field. As the maximal immediate extensions of tame fields will in general not be unique, the proofs have to use much deeper valuation theoretical results than those for other classes of valued fields which have already been shown to satisfy Ax–Kochen–Ershov Principles. The results of this paper have been applied to gain insight in the Zariski space of places of an algebraic function field, and in the model theory of large fields.
Arithmetic and Geometric Applications of Quantifier Elimination for Valued Fields
, 2000
"... We survey applications of quantifier elimination to number theory and algebraic geometry, focusing on results of the last 15 years. We start with the applications of padic quantifier elimination to padic integration and the rationality of several Poincar series related to congruences f(x) = 0 mo ..."
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Cited by 7 (1 self)
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We survey applications of quantifier elimination to number theory and algebraic geometry, focusing on results of the last 15 years. We start with the applications of padic quantifier elimination to padic integration and the rationality of several Poincar series related to congruences f(x) = 0 modulo a prime power, where f is a polynomial in several variables. We emphasize the importance of padic cell decomposition, not only to avoid resolution of singularities, but especially to obtain much stronger arithmetical results. We survey the theory of padic subanalytic sets, which is needed when f is a power series instead of a polynomial. Next we explain the fundamental results of Lipshitz–Robinson and Gardener–Schoutens on subanalytic sets over algebraically closed complete valued fields, and the connection with rigid analytic geometry. Finally we discuss recent geometric applications of quantifier elimination over C((t)), related to the arc space of an algebraic variety.