Łojasiewicz exponent of the gradient near the fiber

Ha Huy Vui, and Nguyen Hong Duc. "Łojasiewicz exponent of the gradient near the fiber." Annales Polonici Mathematici 96.3 (2009): 197-207. <http://eudml.org/doc/280842>.

@article{HaHuyVui2009,
abstract = {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 $f^\{-1\}(t₀)$ such that r is the Łojasiewicz exponent of grad(f) near the fiber $f^\{-1\}(t₀)$. 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.},
author = {Ha Huy Vui, Nguyen Hong Duc},
journal = {Annales Polonici Mathematici},
keywords = {Łojasiewicz exponent; Puiseux expansion at infinity; Fedoryuk and Malgrange conditions; asymptotic critical value},
language = {eng},
number = {3},
pages = {197-207},
title = {Łojasiewicz exponent of the gradient near the fiber},
url = {http://eudml.org/doc/280842},
volume = {96},
year = {2009},
}

TY - JOUR
AU - Ha Huy Vui
AU - Nguyen Hong Duc
TI - Łojasiewicz exponent of the gradient near the fiber
JO - Annales Polonici Mathematici
PY - 2009
VL - 96
IS - 3
SP - 197
EP - 207
AB - 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 $f^{-1}(t₀)$ such that r is the Łojasiewicz exponent of grad(f) near the fiber $f^{-1}(t₀)$. 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.
LA - eng
KW - Łojasiewicz exponent; Puiseux expansion at infinity; Fedoryuk and Malgrange conditions; asymptotic critical value
UR - http://eudml.org/doc/280842
ER -