On the Łojasiewicz exponent and Newton polyhedron

Abstract

The object of this paper is to give an estimation of the Łojasiewicz exponent of the gradient of a holomorphic function under Kouchnirenko's nondegeneracy condition, using information from the Newton polyhedron.Let f : ðC n ; 0Þ ! ðC; 0Þ be a germ of holomorphic function. The Łojasiewicz exponent of gradient of f , Lð f Þ is by definitionIt is well-known that Lð f Þ < y if and only if f has an isolated singularity at the origin. Chang and Lu [1] proved that for any integer r greater than Lð f Þ, f is a C 0 -su‰cient, r-jet in holomorphic functions, i.e., adding to the function f monomials of order greater than Lð f Þ does note change its topological type. Originally this was proved by Kuo and Kuiper in the real case (see [4, 5]). Teissier [9] showed that C 0 -su‰ciency degree of f (i.e., the minimal integer r such that f is C 0 -su‰cient, r-jet) is equal to ½Lð f Þ þ 1, where ½Lð f Þ denotes the integral part of Lð f Þ. We were motived by the work of Lichtin [7] and Fukui [2] who used the Newton polyhedron of f to give an estimation of Lð f Þ, where f is non-degenerate in the sense of Kouchnirenko. In this note, following this procedure, we estimate the Łojasiewicz exponent of gradient Lð f Þ (Theorem 1 below). However, our estimations are based on other ideas, more precisely, we use the Kouchnirenko's theorem [3] on the Newton number and the geometric characterization of m-constancy in [6, 9].

On combinatorial criteria for non-degenerate singularities

“…where m 0 ð f Þ is a combinatorial number calculated from G þ ð f Þ: Also Abderrahmane [1] gave a similar estimation in terms of nð f Þ: In the case n ¼ 2; 3 there are more exact formula for the Łojasiewicz exponent of non-degenerate holomorphic functions (see [11] and [14]).…”

Łojasiewicz exponents of a certain analytic functions

“…There are two type of inequalities which are shown by S. Lojasiewicz [16,17]. ∂f (z) ≥ c|f (z)| θ , c = 0, 0 ≤ ∃θ < 1, ∀z ∈ U, (1) ∂f (z) ≥ c z η , c = 0, ∃η > 0, ∀z ∈ U (2) where U is a sufficiently small neighborhood of the origin. Here ∂f (z) is the gradient vector ( ∂f ∂z 1 , .…”

Local Łojasiewicz exponents, Milnor numbers and mixed multiplicities of ideals

“…, I r n n ), respectively. By an argument analogous to the previous discussion and considering the definitions of r and r , we have that if p ≥ max{r, r } then σ (I r 1 1 , . .…”