617 research outputs found

    Towards a Model Theory for Transseries

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    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

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    We are concerned with rigid analytic geometry in the general setting of Henselian fields KK 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 KK. 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

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    We develop geometry of algebraic subvarieties of KnK^{n} over arbitrary Henselian valued fields KK. 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 Kn×Pm(K)→KnK^{n} \times \mathbb{P}^{m}(K) \to K^{n} 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 KnK^{n}, the existence of definable retractions onto closed definable subsets of KnK^{n}, 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 PP-minimal structure without definable Skolem functions

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    We show there are intermediate PP-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 PP-minimal structures which do not admit classical cell decomposition.Comment: 9 pages, (added missing grant acknowledgement
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