# On asymptotic critical values and the Rabier Theorem

• Volume: 65, Issue: 1, page 125-133
• ISSN: 0137-6934

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## Abstract

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Let X ⊂ kⁿ be a smooth affine variety of dimension n-r and let $f=\left(f₁,...,{f}_{m}\right):X\to {k}^{m}$ be a polynomial dominant mapping. It is well-known that the mapping f is a locally trivial fibration outside a small closed set B(f). It can be proved (using a general Fibration Theorem of Rabier) that the set B(f) is contained in the set K(f) of generalized critical values of f. In this note we study the Rabier function. We give a few equivalent expressions for this function, in particular we compare this function with the Kuo function and with the (generalized) Gaffney function. As a consequence we give a direct short proof of the fact that f is a locally trivial fibration outside the set K(f) (i.e., that B(f) ⊂ K(f)). This generalizes the previous results of the author for $X={k}^{r}$ (see [2]).

## How to cite

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Zbigniew Jelonek. "On asymptotic critical values and the Rabier Theorem." Banach Center Publications 65.1 (2004): 125-133. <http://eudml.org/doc/282534>.

@article{ZbigniewJelonek2004,
abstract = {Let X ⊂ kⁿ be a smooth affine variety of dimension n-r and let $f = (f₁,..., f_m): X → k^m$ be a polynomial dominant mapping. It is well-known that the mapping f is a locally trivial fibration outside a small closed set B(f). It can be proved (using a general Fibration Theorem of Rabier) that the set B(f) is contained in the set K(f) of generalized critical values of f. In this note we study the Rabier function. We give a few equivalent expressions for this function, in particular we compare this function with the Kuo function and with the (generalized) Gaffney function. As a consequence we give a direct short proof of the fact that f is a locally trivial fibration outside the set K(f) (i.e., that B(f) ⊂ K(f)). This generalizes the previous results of the author for $X = k^r$ (see [2]).},
author = {Zbigniew Jelonek},
journal = {Banach Center Publications},
language = {eng},
number = {1},
pages = {125-133},
title = {On asymptotic critical values and the Rabier Theorem},
url = {http://eudml.org/doc/282534},
volume = {65},
year = {2004},
}

TY - JOUR
AU - Zbigniew Jelonek
TI - On asymptotic critical values and the Rabier Theorem
JO - Banach Center Publications
PY - 2004
VL - 65
IS - 1
SP - 125
EP - 133
AB - Let X ⊂ kⁿ be a smooth affine variety of dimension n-r and let $f = (f₁,..., f_m): X → k^m$ be a polynomial dominant mapping. It is well-known that the mapping f is a locally trivial fibration outside a small closed set B(f). It can be proved (using a general Fibration Theorem of Rabier) that the set B(f) is contained in the set K(f) of generalized critical values of f. In this note we study the Rabier function. We give a few equivalent expressions for this function, in particular we compare this function with the Kuo function and with the (generalized) Gaffney function. As a consequence we give a direct short proof of the fact that f is a locally trivial fibration outside the set K(f) (i.e., that B(f) ⊂ K(f)). This generalizes the previous results of the author for $X = k^r$ (see [2]).
LA - eng
UR - http://eudml.org/doc/282534
ER -

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