Abstract
In [1], Guillaume and Anna Valette associate singular varieties VF to a polynomial mapping . In the case , if the set K0(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 VF is not trivial. In [2], the results of [1] are generalized in the case where n≥3, with an additional condition. In this paper, we prove that for a class of non-proper generic dominant polynomial mappings, the results in [1] and [2] hold also for the case that the set K0(F) is not empty.
Highlights
In [1], Guillaume and Anna Valette provide a criteria for properness of a polynomial mapping. They construct a real algebraic singular variety satisfying the following property: if the set of critical values of F is empty F is not proper if and only if the 2-dimensional homology or intersection homology of VF is not trivial ([1], Theorem 3.2). This result provides a new approach for the study of the well-known Jacobian Conjecture, which is still open until today, even in the two-dimensional case
Is a non-proper generic dominant polynomial mapping, the 2-dimensional homology and intersection homology of VF are not trivial. We prove that this result is true for a non-proper generic dominant polynomial mapping
We briefly recall the definition of intersection homology
Summary
In [1], Guillaume and Anna Valette provide a criteria for properness of a polynomial mapping. ( K0 ( F ) ∪ SF ) ×{0 }q , where K0 ( F ) is the set of critical values and SF is the asymptotic set of F. is a non-proper generic dominant polynomial mapping, the 2-dimensional homology and intersection homology (with any perversity) of VF are not trivial. For instance: in general, the set K0 ( F ) is not closed; the set K0 ( F ) ∪ SF is not smooth; K0 ( F ) ∪ SF is not pure dimensional if F is not dominant Via these examples, we make clear the well-known Thom-Mather partition of K0 ( F ) defined by Thom in [6]
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