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  1. Exponential Constructible Functions in P-Minimal Structures.Saskia Chambille, Pablo Cubides Kovacsics & Eva Leenknegt - forthcoming - Journal of Mathematical Logic:2050005.
    Exponential-constructible functions are an extension of the class of constructible functions. This extension was formulated by Cluckers and Loeser in the context of semi-algebraic and sub-analytic structures, when they studied stability under integration. In this paper, we will present a natural refinement of their definition that allows for stability results to hold within the wider class of [Formula: see text]-minimal structures. One of the main technical improvements is that we remove the requirement of definable Skolem functions from the proofs. As (...)
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  • Integration and Cell Decomposition in P-Minimal Structures.Pablo Cubides Kovacsics & Eva Leenknegt - 2016 - Journal of Symbolic Logic 81 (3):1124-1141.
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  • Cell Decomposition and Definable Functions for Weak P‐Adic Structures.Eva Leenknegt - 2012 - Mathematical Logic Quarterly 58 (6):482-497.
    We develop a notion of cell decomposition suitable for studying weak p-adic structures definable). As an example, we consider a structure with restricted addition.
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  • On P ‐Adic Semi‐Algebraic Continuous Selections.Athipat Thamrongthanyalak - 2020 - Mathematical Logic Quarterly 66 (1):73-81.
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  • Cell Decomposition for Semibounded P-Adic Sets.Eva Leenknegt - 2013 - Archive for Mathematical Logic 52 (5-6):667-688.
    We study a reduct ${\mathcal{L}_*}$ of the ring language where multiplication is restricted to a neighbourhood of zero. The language is chosen such that for p-adically closed fields K, the ${\mathcal{L}_*}$ -definable subsets of K coincide with the semi-algebraic subsets of K. Hence structures (K, ${\mathcal{L}_*}$ ) can be seen as the p-adic counterpart of the o-minimal structure of semibounded sets. We show that in this language, p-adically closed fields admit cell decomposition, using cells similar to p-adic semi-algebraic cells. From (...)
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  • Reducts of P-Adically Closed Fields.Eva Leenknegt - 2014 - Archive for Mathematical Logic 53 (3-4):285-306.
    In this paper, we consider reducts of p-adically closed fields. We introduce a notion of shadows: sets Mf={∈K2∣|y|=|f|}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${M_f = \{ \in K^2 \mid |y| = |f|\}}$$\end{document}, where f is a semi-algebraic function. Adding symbols for such sets to a reduct of the ring language, we obtain expansions of the semi-affine language where multiplication is nowhere definable, thus giving a negative answer to a question posed by Marker, Peterzil and Pillay. The second (...)
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  • Computable Valued Fields.Matthew Harrison-Trainor - 2018 - Archive for Mathematical Logic 57 (5-6):473-495.
    We investigate the computability-theoretic properties of valued fields, and in particular algebraically closed valued fields and p-adically closed valued fields. We give an effectiveness condition, related to Hensel’s lemma, on a valued field which is necessary and sufficient to extend the valuation to any algebraic extension. We show that there is a computable formally p-adic field which does not embed into any computable p-adic closure, but we give an effectiveness condition on the divisibility relation in the value group which is (...)
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  • Cell Decomposition and Classification of Definable Sets in P-Optimal Fields.Luck Darnière & Immanuel Halpuczok - 2017 - Journal of Symbolic Logic 82 (1):120-136.
    We prove that forp-optimal fields a cell decomposition theorem follows from methods going back to Denef’s paper [7]. We derive from it the existence of definable Skolem functions and strongp-minimality. Then we turn to stronglyp-minimal fields satisfying the Extreme Value Property—a property which in particular holds in fields which are elementarily equivalent to ap-adic one. For such fieldsK, we prove that every definable subset ofK×Kdwhose fibers overKare inverse images by the valuation of subsets of the value group is semialgebraic. Combining (...)
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  • A Version of P-Adic Minimality.Raf Cluckers & Eva Leenknegt - 2012 - Journal of Symbolic Logic 77 (2):621-630.
    We introduce a very weak language L M on p-adic fields K, which is just rich enough to have exactly the same definable subsets of the line K that one has using the ring language. (In our context, definable always means definable with parameters.) We prove that the only definable functions in the language L M are trivial functions. We also give a definitional expansion $L\begin{array}{*{20}{c}} ' \\ M \\ \end{array} $ of L M in which K has quantifier elimination, (...)
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  • Linear Extension Operators for Continuous Functions on Definable Sets in the P -Adic Context.Athipat Thamrongthanyalak - 2017 - Mathematical Logic Quarterly 63 (1-2):104-108.
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  • Clustered Cell Decomposition in P -Minimal Structures.Saskia Chambille, Pablo Cubides Kovacsics & Eva Leenknegt - 2017 - Annals of Pure and Applied Logic 168 (11):2050-2086.
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  • A P-Minimal Structure Without Definable Skolem Functions.Pablo Cubides Kovacsics & Kien Huu Nguyen - 2017 - Journal of Symbolic Logic 82 (2):778-786.
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  • Topological Cell Decomposition and Dimension Theory in P-Minimal Fields.Pablo Cubides Kovacsics, Luck Darnière & Eva Leenknegt - 2017 - Journal of Symbolic Logic 82 (1):347-358.
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