/Lojasiewicz exponent of the gradient near the fiber
Volume 96 / 2009
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 $f^{-1}(t_0)$ 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.
