Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Studies on the discrete integrable equations over finite fields   Discrete dynamical systems over finite fields are investigated and their
integrability is discussed. In particular, the discrete Painlev\'{e} equations
and the discrete KdV equation are defined over finite fields and their special
solutions are obtained. Their investigation over the finite fields has not been
done thoroughly, partly because of the indeterminacies that appear in defining
the equations. In this paper we introduce two methods to well-define the
equations over the finite fields and apply the methods to several classes of
discrete integrable equations. One method is to extend the space of initial
conditions through blowing-up at the singular points. In case of discrete
Painlev\'{e} equations, we prove that an finite field analog of the Sakai
theory can be applied to construct the space of initial conditions. The other
method is to define the equations over the field of $p$-adic numbers and then
reduce them to the finite fields. The mapping whose time evolutions commute
with the reduction is said to have a `good reduction'. We generalize good
reduction in order to be applied to integrable mappings. The generalized good
reduction is formulated as `almost good reduction' and is proved to be a
$p$-adic analog of the singularity confinement test. (*Note that this paper is
intended for the author's thesis and it draws from several of our already
published papers. Also note that several minor modifications have been made
from the original thesis.)

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The categories of this paper are: physics
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The categories of this paper are: q-alg
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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: cond-mat
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The categories of this paper are: econ
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The categories of this paper are: plasm-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: atom-ph
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The categories of this paper are: q-fin
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The categories of this paper are: nucl-th
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The categories of this paper are: solv-int
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The categories of this paper are: adap-org
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The categories of this paper are: quant-ph
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The categories of this paper are: hep-ph
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The categories of this paper are: chao-dyn