THE ŁOJASIEWICZ EXPONENT FOR WEIGHTED HOMOGENEOUS POLYNOMIAL WITH ISOLATED SINGULARITY

The purpose of this paper is to give an explicit formula of the Łojasiewicz exponent of an isolated weighted homogeneous singularity in terms of its weights.

Topological invariants of germs of real analytic functions

Let f: (ℝn, 0)→ (ℝ,0) be a germ of a real analytic function. Let L and F(f) denote the link of f and the Milnor fibre of fc respectively, i. e., L = {x ∈ Sn−1 | f(x) = 0}, , where 0 ≤ ξ ≪ r ≪ 1, . In [2] Szafraniec introduced the notion of an -germ as a generalization of a germ defined by a weighted homogeneous polynomial satisfying some condition concerning the relation between its degree and weights (definition 1). He also proved that if f is an -germ (presumably with nonisolated singularity) then the number χ(F(f)/d mod 2 is a topological invariant of f, where χ(F(f)) is the Euler characterististic of F(f), and gave the formula for χ(L)/2 mod 2 (it is a well-known fact that F(L) is an even number). As a simple consequence he got the fact that χ(F(f)mod 2 is a topological invariant for any f, which is a generalization of Wall's result [3] (he considered only germs with an isolated singularity).

Topological invariants of weighted homogeneous polynomials

Let ℝn → ℝ be a weighted homogeneous polynomial such that df(0) = 0, L = {x ∈ Sn−1|f(x) = 0}, and let χ(L) be the Euler characteristic of L. The problem is how to calculate χ(L) in terms of f.