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].
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].
Abstract. We show that every µ-constant family of isolated hypersurface singularities satisfying a nondegeneracy condition in the sense of Kouchnirenko, is topologically trivial, also is equimultiple.Let f : (C n , 0) → (C, 0) be the germ of a holomorphic function with an isolated singularity. The Milnor number of a germ f , denoted by µ(f ), is algebraically defined as the dim O n /J(f ), where O n is the ring of complex analytic function germs : (C n , 0) → (C, 0) and J(f ) is the Jacobian ideal in O n generated by the partial derivatives {We recall that the multiplicity m(f ) is defined as the lowest degree in the power se-are germs of holomorphic functions. We use the notation F t (z) = F (z, t) when t is fixed. Let m t denote the multiplicity and µ t denote the Milnor number of F t at the origin. The deformation F is equimultiple (resp. µ-constant) if m 0 = m t (resp. µ 0 = µ t ) for small t. It is well-known by the result of Lê-Ramanujam [9]. that for n = 3, the topological type of the family F t is constant under µ-constant deformations. The question is still open for n = 3. However, under some additional assumption, positive answers have been given. For example, if F t is non-degenerate in the sense of Kouchnirenko [7] and the Newton boundary Γ(F t ) of F t is independent of t, i.e., Γ(F t ) = Γ(f ), it follows that µ * (F t ) is constant, and hence F t is topologically trivial (see [12,16] for details). Motivated by the Briançon-Speder µ-constant family F t (z) = z 5 1 + z 2 z 7 3 + z 15 2 + tz 1 z 6 3 , which is topologically trivial but not µ * -constant, M. Oka [13] shows that any non-degenerate family of type F (z, t) = f (z) + tz A for A = (A 1 , . . . , A n ) ∈ N n , where N is the set of nonnegative integers and z A = zn as usual, is topologically trivial, under the assumption of µ-constancy. Our purpose of this paper is to generalize this result, more precisely, we show that every µ-constant non-degenerate family F t with not necessarily Newton boundary Γ(F t ) independent of t, is topologically trivial. Moreover, we show that F is equimultiple, which gives a positive answer to a question of Zariski [19,5,14] for a nondegenerate family. To prove the main result (Theorem 1.1 below), we shall use the notion of (c)-regularity in stratification theory, introduced by K. Bekka in [3], which is weaker than Whitney regularity, nevertheless (c)-regularity implies topological triviality. First, we give a characterization of (c)-regularity (Theorem 2.1 below). By using it, we can show that the µ-constancy condition for a non-degenerate family implies Bekkas (c)-regularity condition and then obtain the topological triviality as a corollary.
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