Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Definable completeness of $P$-minimal fields and applications   We show that every definable nested family of closed and bounded subsets of a
$P$-minimal field $K$ has non-empty intersection. As an application we answer a
question of Darni\`ere and Halupczok showing that $P$-minimal fields satisfy
the "extreme value property": for every closed and bounded subset $U\subseteq
K$ and every interpretable continuous function $f\colon U \to \Gamma_K$ (where
$\Gamma_K$ denotes the value group), $f(U)$ admits a maximal value. Two further
corollaries are obtained as a consequence of their work. The first one shows
that every interpretable subset of $K\times\Gamma_K^n$ is already interpretable
in the language of rings, answering a question of Cluckers and Halupczok. This
implies in particular that every $P$-minimal field is polynomially bounded. The
second one characterizes those $P$-minimal fields satisfying a classical cell
preparation theorem as those having definable Skolem functions, generalizing a
result of Mourgues.

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The categories of this paper are: patt-sol
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The categories of this paper are: adap-org
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The categories of this paper are: comp-gas
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The categories of this paper are: mtrl-th
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The categories of this paper are: q-alg
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The categories of this paper are: solv-int
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The categories of this paper are: cs
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The categories of this paper are: nucl-th
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The categories of this paper are: stat
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The categories of this paper are: physics
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The categories of this paper are: ao-sci
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-lat
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The categories of this paper are: hep-ex
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The categories of this paper are: hep-ph
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The categories of this paper are: chem-ph