It is well-known that if r is a rational number from [-1,0), then there is no polynomial f in two complex variables and a fiber such that r is the Łojasiewicz exponent of grad(f) near the fiber . We show that this does not remain true if we consider polynomials in real variables. More exactly, we give examples showing that any rational number can be the Łojasiewicz exponent near the fiber of the gradient of some polynomial in real variables. The second main result of the paper is the formula computing the Łojasiewicz exponent of the gradient near a fiber of a polynomial in two real variables. In particular, this gives, in the case of two real variables, a way to tell whether a given value is an asymptotic critical value or not.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ap96-3-1,
author = {Ha Huy Vui and Nguyen Hong Duc},
title = {\L ojasiewicz exponent of the gradient near the fiber},
journal = {Annales Polonici Mathematici},
volume = {95},
year = {2009},
pages = {197-207},
zbl = {1182.14062},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap96-3-1}
}
Ha Huy Vui; Nguyen Hong Duc. Łojasiewicz exponent of the gradient near the fiber. Annales Polonici Mathematici, Tome 95 (2009) pp. 197-207. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ap96-3-1/