Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Pierce-Birkhoff Conjecture for Smooth Affine Surfaces over Real
  Closed Fields   We will prove that the Pierce-Birkhoff Conjecture holds for non-singular
two-dimensional affine real algebraic varieties over real closed fields, i.e.,
if W is such a variety, then every piecewise polynomial function on W can be
written as suprema of infima of polynomial functions on W. More precisely, we
will give a proof of the so-called Connectedness Conjecture for the coordinate
rings of such varieties, which implies the Pierce-Birkhoff Conjecture.

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The categories of this paper are: cond-mat
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The categories of this paper are: physics
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The categories of this paper are: dg-ga
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The categories of this paper are: cs
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The categories of this paper are: hep-ex
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The categories of this paper are: q-bio
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The categories of this paper are: econ
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The categories of this paper are: hep-th
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The categories of this paper are: eess
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The categories of this paper are: adap-org
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The categories of this paper are: q-alg
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The categories of this paper are: alg-geom
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The categories of this paper are: funct-an
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The categories of this paper are: ao-sci
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The categories of this paper are: plasm-ph
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The categories of this paper are: comp-gas