Let X ⊂ kⁿ be a smooth affine variety of dimension n-r and let 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 (see [2]).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-bc65-0-9,
author = {Zbigniew Jelonek},
title = {On asymptotic critical values and the Rabier Theorem},
journal = {Banach Center Publications},
volume = {65},
year = {2004},
pages = {125-133},
zbl = {1160.58311},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc65-0-9}
}
Zbigniew Jelonek. On asymptotic critical values and the Rabier Theorem. Banach Center Publications, Tome 65 (2004) pp. 125-133. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-bc65-0-9/