Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Local Zeta Function for Curves, Non Degeneracy Conditions and Newton
  Polygons   This paper is dedicated to the description of the poles of the Igusa local
zeta functions $Z(s,f,v)$ when $f(x,y)$ satisfies a new non degeneracy
condition, that we have called arithmetic non degeneracy. More precisely, we
attach to each polynomial $f(x,y)$, a collection of convex sets $\Gamma ^{A}$
called the arithmetic Newton polygon of $f(x,y)$, and introduce the notion of
arithmetic non degeneracy with respect to $\Gamma ^{A}(f)$. The set of
degenerate polynomials, with respect to a fixed geometric Newton polygon,
contains an open subset, for the Zariski topology, formed by non degenerate
polynomials with respect to some arithmetic Newton polygon.If $L$ is a number
field, our main result asserts that for almost all non-archimedean valuations
$v$ of $L$, the poles of $Z(s,f,v),$ with $f(x,y)\in L[x,y]$, can be described
explicitly in terms of the equations of the straight segments that conform the
boundaries of the convex sets that belong to $\Gamma ^{A}(f)$. Moreover, our
proof gives an effective procedure to compute $Z(s,f,v)$.

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The categories of this paper are: stat
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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: dg-ga
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The categories of this paper are: atom-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: hep-lat
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The categories of this paper are: plasm-ph
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The categories of this paper are: q-alg
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The categories of this paper are: adap-org
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The categories of this paper are: mtrl-th
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The categories of this paper are: nlin
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The categories of this paper are: math-ph
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The categories of this paper are: quant-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: physics