617 research outputs found
Towards a Model Theory for Transseries
The differential field of transseries extends the field of real Laurent
series, and occurs in various context: asymptotic expansions, analytic vector
fields, o-minimal structures, to name a few. We give an overview of the
algebraic and model-theoretic aspects of this differential field, and report on
our efforts to understand its first-order theory.Comment: Notre Dame J. Form. Log., to appear; 33 p
Definable transformation to normal crossings over Henselian fields with separated analytic structure
We are concerned with rigid analytic geometry in the general setting of
Henselian fields with separated analytic structure, whose theory was
developed by Cluckers--Lipshitz--Robinson. It unifies earlier work and
approaches of numerous mathematicians. Separated analytic structures admit
reasonable relative quantifier elimination in a suitable analytic language.
However, the rings of global analytic functions with two kinds of variables
seem not to have good algebraic properties such as Noetherianity or excellence.
Therefore the usual global resolution of singularities from rigid analytic
geometry is no longer at our disposal. Our main purpose is to give a definable
version of the canonical desingularization algorithm (the hypersurface case)
due to Bierstone--Milman so that both these powerful tools are available in the
realm of non-Archimedean analytic geometry at the same time. It will be carried
out within a category of definable, strong analytic manifolds and maps, which
is more flexible than that of affinoid varieties and maps. Strong analytic
objects are those definable ones that remain analytic over all fields
elementarily equivalent to . This condition may be regarded as a kind of
symmetry imposed on ordinary analytic objects. The strong analytic category
makes it possible to apply a model-theoretic compactness argument in the
absence of the ordinary topological compactness. On the other hand, our
closedness theorem enables application of resolution of singularities to
topological problems involving the topology induced by valuation. Eventually,
these three results will be applied to such issues as the existence of
definable retractions or extending continuous definable functions.Comment: This is the final version published in the journal Symmetry-Basel,
2019, 11, 93
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
An example of a -minimal structure without definable Skolem functions
We show there are intermediate -minimal structures between the
semi-algebraic and sub-analytic languages which do not have definable Skolem
functions. As a consequence, by a result of Mourgues, this shows there are
-minimal structures which do not admit classical cell decomposition.Comment: 9 pages, (added missing grant acknowledgement
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