153 research outputs found
Non-archimedean tame topology and stably dominated types
Let 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 of the Berkovich analytification of ,
and deduce several new results on Berkovich spaces from it. In particular we
show that retracts to a finite simplicial complex and is locally
contractible, without any smoothness assumption on . When varies in an
algebraic family, we show that the homotopy type of takes only a
finite number of values. The space is obtained by defining a
topology on the pro-definable set of stably dominated types on . The key
result is the construction of a pro-definable strong retraction of
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
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
We study finite imaginaries in certain valued fields, and prove a conjecture
of Cluckers and Denef.Comment: 15p
Pseudo-finite sets, pseudo-o-minimality
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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