Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A remark on a polynomial mapping $F: \C^n \to \C^{n}$   In \cite{Valette}, Guillaume and Anna Valette associate singular varieties
$V_F$ to a polynomial mapping $F: \C^n \to \C^n$. In the case $F: \C^2 \to
\C^2$, if the set $K_0(F)$ of critical values of $F$ is empty, then $F$ is not
proper if and only if the 2-dimensional homology or intersection homology (with
any perversity) of $V_F$ are not trivial. In \cite{ThuyValette}, the results of
\cite{Valette} are generalized in the case $F: \C^n \to \C^n$ where $n \geq 3$,
with an additional condition. In this paper, we prove that if $F: \C^2 \to
\C^2$ is a non-proper {\it generic dominant} polynomial mapping, then the
2-dimensional homology and intersection homology (with any perversity) of $V_F$
are not trivial. We prove that this result is true also for a non-proper {\it
generic dominant} polynomial mapping $F: \C^n \to \C^n$ ($\, n \geq 3$), with
the same additional condition than in \cite{ThuyValette}. In order to compute
the intersection homology of the variety $V_F$, we provide an explicit
Thom-Mather stratification of the set $K_0(F) \cup S_F$.

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The categories of this paper are: hep-th
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The categories of this paper are: gr-qc
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The categories of this paper are: quant-ph
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The categories of this paper are: chem-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: adap-org
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The categories of this paper are: cmp-lg
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The categories of this paper are: nlin
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The categories of this paper are: econ
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The categories of this paper are: physics
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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: bayes-an
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The categories of this paper are: alg-geom
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The categories of this paper are: eess
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The categories of this paper are: nucl-ex