/Lojasiewicz exponent of the gradient near the fiber

Ha Huy Vui¹ ; Nguyen Hong Duc¹

¹ Institute of Mathematics 18 Hoang Quoc Viet Road Cau Giay District 10307, Hanoi, Vietnam

Annales Polonici Mathematici, Tome 96 (2009) no. 3, pp. 197-207

Résumé

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 $f^{-1}(t_0)$ such that $r$ is the /Lojasiewicz exponent of $\hbox {grad}(f)$ near the fiber $f^{-1}( t_0)$. 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 /Lojasiewicz 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 /Lojasiewicz 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.

Détail

DOI : 10.4064/ap96-3-1

Mots-clés : well known rational number there polynomial complex variables fiber lojasiewicz exponent hbox grad near fiber does remain consider polynomials real variables exactly examples showing rational number lojasiewicz exponent near fiber gradient polynomial real variables second main result paper formula computing lojasiewicz exponent gradient near fiber polynomial real variables particular gives real variables tell whether given value asymptotic critical value