81 research outputs found
Dimension in the realm of transseries
Let be the differential field of transseries. We establish some
basic properties of the dimension of a definable subset of ,
also in relation to its codimension in the ambient space . The
case of dimension is of special interest, and can be characterized both in
topological terms (discreteness) and in terms of the
Herwig-Hrushovski-Macpherson notion of co-analyzability. The proofs use results
by the authors from "Asymptotic Differential Algebra and Model Theory of
Transseries", the axiomatic framework for "dimension" in [L. van den Dries,
"Dimension of definable sets, algebraic boundedness and Henselian fields", Ann.
Pure Appl. Logic 45 (1989), no. 2, 189-209], and facts about co-analyzability
from [B. Herwig, E. Hrushovski, D. Macpherson, "Interpretable groups, stably
embedded sets, and Vaughtian pairs", J. London Math. Soc. (2003) 68, no. 1,
1-11].Comment: 16 pp; version 2, taking into account comments by the refere
A closedness theorem and applications in geometry of rational points over Henselian valued fields
We develop geometry of algebraic subvarieties of over arbitrary
Henselian valued fields . This is a continuation of our previous article
concerned with algebraic geometry over rank one valued fields. At the center of
our approach is again the closedness theorem that the projections are definably closed maps. It enables application
of resolution of singularities in much the same way as over locally compact
ground fields. As before, the proof of that theorem uses i.a. the local
behavior of definable functions of one variable and fiber shrinking, being a
relaxed version of curve selection. But now, to achieve the former result, we
first examine functions given by algebraic power series. All our previous
results will be established here in the general settings: several versions of
curve selection (via resolution of singularities) and of the \L{}ojasiewicz
inequality (via two instances of quantifier elimination indicated below),
extending continuous hereditarily rational functions as well as the theory of
regulous functions, sets and sheaves, including Nullstellensatz and Cartan's
theorems A and B. Two basic tools applied in this paper are quantifier
elimination for Henselian valued fields due to Pas and relative quantifier
elimination for ordered abelian groups (in a many-sorted language with
imaginary auxiliary sorts) due to Cluckers--Halupczok. Other, new applications
of the closedness theorem are piecewise continuity of definable functions,
H\"{o}lder continuity of definable functions on closed bounded subsets of
, the existence of definable retractions onto closed definable subsets
of , and a definable, non-Archimedean version of the Tietze--Urysohn
extension theorem. In a recent preprint, we established a version of the
closedness theorem over Henselian valued fields with analytic structure along
with some applications.Comment: This paper has been published in Journal of Singularities 21 (2020),
233-254. arXiv admin note: substantial text overlap with arXiv:1704.01093,
arXiv:1703.08203, arXiv:1702.0784
Topological properties of definable sets in tame perfect fields
Let be a field. We continue to study the recently introduced \'etale open
topology on the -points of a -variety . The \'etale open
topology is non-discrete if and only if is large. If is separably,
real, -adically closed then the \'etale open topology agrees with the
Zariski, order, valuation topology, respectively. We show that existentially
definable sets in perfect large fields behave well with respect to this
topology: such sets are finite unions of \'etale open subsets of Zariski closed
sets. This implies that existentially definable sets in arbitrary perfect large
fields enjoy some of the well-known topological properties of definable sets in
algebraically, real, and -adically closed fields. We introduce and study the
class of \'ez fields: is \'ez if is large and every definable set is a
finite union of \'etale open subsets of Zariski closed sets. This should be
seen as a generalized notion of model completeness for large fields.
Algebraically closed, real closed, -adically closed, and bounded
fields are \'ez. (In particular pseudofinite fields and infinite
algebraic extensions of finite fields are \'ez.) We develop the basics of a
theory of definable sets in \'ez fields. This gives a uniform approach to the
theory of definable sets across all characteristic zero local fields and a new
topological theory of definable sets in bounded fields. We also
show that some prominent examples of possibly non-model complete
model-theoretically tame fields (characteristic zero Henselian fields and
Frobenius fields) are \'ez
On finite imaginaries
We study finite imaginaries in certain valued fields, and prove a conjecture
of Cluckers and Denef.Comment: 15p
Expansions which introduce no new open sets
We consider the question of when an expansion of a topological structure has
the property that every open set definable in the expansion is definable in the
original structure. This question is related to and inspired by recent work of
Dolich, Miller and Steinhorn on the property of having o-minimal open core. We
answer the question in a fairly general setting and provide conditions which in
practice are often easy to check. We give a further characterisation in the
special case of an expansion by a generic predicate
Definable equivalence relations and zeta functions of groups
We prove that the theory of the -adics admits elimination
of imaginaries provided we add a sort for for each . We also prove that the elimination of
imaginaries is uniform in . Using -adic and motivic integration, we
deduce the uniform rationality of certain formal zeta functions arising from
definable equivalence relations. This also yields analogous results for
definable equivalence relations over local fields of positive characteristic.
The appendix contains an alternative proof, using cell decomposition, of the
rationality (for fixed ) of these formal zeta functions that extends to the
subanalytic context.
As an application, we prove rationality and uniformity results for zeta
functions obtained by counting twist isomorphism classes of irreducible
representations of finitely generated nilpotent groups; these are analogous to
similar results of Grunewald, Segal and Smith and of du Sautoy and Grunewald
for subgroup zeta functions of finitely generated nilpotent groups.Comment: 89 pages. Various corrections and changes. To appear in J. Eur. Math.
So
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