Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Temperedness of measures defined by polynomial equations over local
  fields   We investigate the asymptotic growth of the canonical measures on the fibers
of morphisms between vector spaces over local fields of arbitrary
characteristic. For non-archimedean local fields we use a version of the
{\L}ojasiewicz inequality (\cite{lojasiewicz1959},
\cite{hormander1958division}) which follows from Greenberg
\cite{greenberg1966rational}, \cite{bollaerts1990estimate}, together with the
theory of the Brauer group of local fields to construct definite forms of
arbitrarily high degree, and to transfer questions at infinity to questions
near the origin. We then use these to generalize results of H{\"o}rmander
\cite{hormander1958division} on estimating the growth of polynomials at
infinity in terms of the distance to their zero loci. Specifically, when a
fiber corresponds to a non-critical value which is stable, i.e. remains
non-critical under small perturbations, we show that the canonical measure on
the fiber is tempered, which generalizes results of Igusa and Raghavan
\cite{igusa1978lectures}, and Virtanen and Weisbart
\cite{virtanen2014elementary}.

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The categories of this paper are: chem-ph
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The categories of this paper are: supr-con
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The categories of this paper are: nucl-th
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The categories of this paper are: physics
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The categories of this paper are: cond-mat
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The categories of this paper are: q-alg
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-lat
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The categories of this paper are: astro-ph
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The categories of this paper are: stat
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The categories of this paper are: q-bio
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-th
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The categories of this paper are: adap-org
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The categories of this paper are: nlin
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The categories of this paper are: mtrl-th