# Wydawnictwa / Czasopisma IMPAN / Annales Polonici Mathematici / Wszystkie zeszyty

## /Lojasiewicz exponent of the gradient near the fiber

### Tom 96 / 2009

Annales Polonici Mathematici 96 (2009), 197-207 MSC: Primary 14R25; Secondary 32A20, 32S05, 14R25. DOI: 10.4064/ap96-3-1

#### Streszczenie

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.

#### Autorzy

• Ha Huy VuiInstitute of Mathematics
Cau Giay District
10307, Hanoi, Vietnam
e-mail
• Nguyen Hong DucInstitute of Mathematics