The Łojasiewicz Exponent for Weighted Homogeneous Polynomial With Isolated Singularity

Abstract

Abstract. The purpose of this paper is to give an explicit formula of the Lojasiewicz exponent of an isolated weighted homogeneous singularity in terms of its weights.Let f : (C n , 0) → (C, 0) be a holomorphic function with an isolated critical point at 0.It is well known(see [10]) that the Lojasiewicz exponent can be calculated by means of analytic pathswhere ord(φ) := inf i {ord(φ i )} for φ ∈ C{t} n . By definition, we put ord(0) = +∞.Lojasiewicz exponents have important applications in singularity theory, for instance, Teissier [22] showed that C 0 -sufficiency degree of f (i.e., the minimal integer r such that f is topologically equivalent to f + g for all g with ord(g)Despite deep research of experts in singularity theory, it is not proved yet that Lojasiewicz exponent L(f ) is a topological invariant of f (in contrast to the Milnor number). An interesting mathematical problem is to give formulas for L(f ) in terms of another invariants of f or an algorithm to compute it. In the two-dimensional case there are many explicit formulas for L(f ) in various terms (see The aim of this paper is to compute the Lojasiewicz exponent for the classes of weighted homogeneous isolated singularities in terms of the weights. In particular, we generalize a formula for L(f ) of Krasiński, Oleksik and P loski [8] for weighted homogeneous surface singularity. This was already announced by Tan, Yau and Zuo [21], but thier paper seems to have some gaps in the proof of proposition 3.4. We were motived by their papers. However, our considerations are based on other ideas. More precisely, we use the notion of weighted homogenous filtration introduced by Paunescu in [18], the geometric characterization of µ-constancy in [12,22] and the result of Varchenko [23], which described the µ-constant stratum of weighted homogeneous singularities in terms of the mixed Hodge structures.Moreover, we show that the Lojasiewicz exponent is invariant for all µ-constant deformation of weighted homogeneous singularity, which gives an affirmative partial answer to Teissier's conjecture [22].

Łojasiewicz exponents of non-degenerate holomorohic and mixed functions

“…Thus the assertion is obvious. The assertion (2) follows from (1). Of course, (2) is nothing but the existence of a stable radius which is well known by [17] for a general holomorphic function with an islated singularity at the origin.…”

“…Consider an analytic function f (z) with an isolated singularity at the origin. We consider the inequality ∂f (z) ≥ c z θ , ∃c > 0, ∀z ∈ U (1) where U is a sufficiently small neighborhood of the origin and ∂f (z) is the gradient vector ( ∂f ∂z 1 , . .…”

“…, ∂f ∂zn ). The Lojasiewicz exponent ℓ 0 (f ) of f (z) at the origin is the smallest positive number among θ's which satisfy the inequality (1). It is known that there exists such number ℓ 0 (f ) and it is a rational number [16,26].…”

Invariants for bi-Lipschitz equivalence of ideals

“…, n}, and started a systematic study on topology of complex hypersurfaces (see for instance [45,46]). We remark that µ (1) (f ) = ord(f ) − 1. Teissier's works have significant impact, but the question above is still unsolved except for the case n = 2, and is known as the Zariski's multiplicity conjecture (see the survey [20]).…”

“…. , µ (1) (f )), where µ (i) (f ) denotes the Milnor number of the restriction of f to a generic linear i-dimensional subspace of C n , for i ∈ {1, . .…”