Łojasiewicz exponent of the gradient near the fiber

• Autorzy: Ha Huy Vui , Nguyen Hong Duc
• Języki publikacji: EN

Abstrakty

EN

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.

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Annales Polonici Mathematici
• Rocznik: 2009
• Tom: 96
• Numer: 3
• Strony: 197-207

Daty

wydano

2009

Twórcy

autor

Ha Huy Vui

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

autor

Nguyen Hong Duc

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

Identyfikatory

DOI

10.4064/ap96-3-1