153 research outputs found

    Non-archimedean tame topology and stably dominated types

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    Let VV be a quasi-projective algebraic variety over a non-archimedean valued field. We introduce topological methods into the model theory of valued fields, define an analogue V^\hat {V} of the Berkovich analytification VanV^{an} of VV, and deduce several new results on Berkovich spaces from it. In particular we show that VanV^{an} retracts to a finite simplicial complex and is locally contractible, without any smoothness assumption on VV. When VV varies in an algebraic family, we show that the homotopy type of VanV^{an} takes only a finite number of values. The space V^\hat {V} is obtained by defining a topology on the pro-definable set of stably dominated types on VV. The key result is the construction of a pro-definable strong retraction of V^\hat {V} to an o-minimal subspace, the skeleton, definably homeomorphic to a space definable over the value group with its piecewise linear structure.Comment: Final versio

    Dimensional groups and fields

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    We shall define a general notion of dimension, and study groups and rings whose interpretable sets carry such a dimensio. In particular, we deduce chain conditions for groups, definability results for fields and domains, and show that pseudofinite groups contain big finite-by-abelian subgroups, and pseudofinite groups of dimension 2 contain big soluble subgroups

    On finite imaginaries

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    We study finite imaginaries in certain valued fields, and prove a conjecture of Cluckers and Denef.Comment: 15p

    Pseudo-finite sets, pseudo-o-minimality

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    We give an example of two ordered structures M, N in the same language L with the same universe, the same order and admitting the same one-variable definable subsets such that M is a model of the common theory of o-minimal L-structures and N admits a definable, closed, bounded, and discrete subset and a definable injective self-mapping of that subset which is not surjective. This answers negatively two questions by Schoutens; the first being whether there is an axiomatization of the common theory of o-minimal structures in a given language by conditions on one-variable definable sets alone. The second being whether definable completeness and type completeness imply the pigeonhole principle. It also partially answers a question by Fornasiero asking whether definable completeness of an expansion of a real closed field implies the pigeonhole principle.Comment: 21 page
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