54 research outputs found
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
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
Some results of geometry in Hensel minimal structures
We deal with Hensel minimal, non-trivially valued fields of
equicharacteristic zero, whose axiomatic theory was introduced in a recent
paper by Cluckers-Halupczok-Rideau. We additionally require that the classical
algebraic language be induced for the imaginary sort . This condition is
satisfied by the majority of classical tame structures on Henselian fields,
including Henselian fields with analytic structure. The main purpose here is to
carry over many results of our previous papers to the general axiomatic
settings described above, including, among others, the theorem on existence of
the limit, curve selection, the closedness theorem and several non-Archimedean
versions of the Lojasiewicz inequalities. We give examples that curve selection
and the closedness theorem, a key result for numerous applications, may be no
longer true after expanding the language for the leading term structure .
In the case of Henselian fields with analytic structure, we establish a more
precise version of the theorem on existence of the limit (a version of
Puiseux's theorem).Comment: 27 page
Some results of algebraic geometry over Henselian rank one valued fields
We develop geometry of affine algebraic varieties in over Henselian
rank one valued fields of equicharacteristic zero. Several results are
provided including: the projection
and blow-ups of the -rational points of smooth -varieties are definably
closed maps, a descent property for blow-ups, curve selection for definable
sets, a general version of the \L{}ojasiewicz inequality for continuous
definable functions on subsets locally closed in the -topology and extending
continuous hereditarily rational functions, established for the real and
-adic varieties in our joint paper with J. Koll\'ar. The descent property
enables application of resolution of singularities and transformation to a
normal crossing by blowing up in much the same way as over the locally compact
ground field. Our approach relies on quantifier elimination due to Pas and a
concept of fiber shrinking for definable sets, which is a relaxed version of
curve selection. The last three sections are devoted to the theory of regulous
functions and sets over such valued fields. Regulous geometry over the real
ground field was developed by Fichou--Huisman--Mangolte--Monnier.
The main results here are regulous versions of Nullstellensatz and Cartan's
Theorems A and B.Comment: This paper has been published in Selecta Mathematic
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