Łojasiewicz exponents of non-degenerate holomorohic and mixed functions

Abstract

We consider Lojasiewicz inequalities for a non-degenerate holomorphic function with an isolated singularity at the origin. We give an explicit estimation of the Lojasiewicz exponent in a slightly weaker form than the assertion in Fukui [9]. For a weighted homogeneous polynomial, we give a better estimation in the form which is conjectured by [4] under under some condition (the Lojasiewicz non-degeneracy). We also introduce Lojasiewicz inequality for strongly non-degenerate mixed functions and generalize this estimation for mixed functions.2000 Mathematics Subject Classification. 14P05,32S55.

A note on the Łojasiewicz exponent of non-degenerate isolated hypersurface singularities

“…Lichtin [Lic81], Fukui [Fuk91], Bivià-Ausina [Biv03], Abderrahmane [Abd05], P. Mondal (private communication)). In particular, see the recent paper by Oka [Oka18], who, in the n-dimensional case, obtained estimations from above under additional assumptions.…”

Łojasiewicz exponents of a certain analytic functions

“…For the second inequality, f (z) must have an isolated singularity at the origin. In our previous paper [20], we considered the exponent of the second Lojasiewicz inequality for a nondegenerate holomorphic function f (z) (or a mixed function f (z, z)) with an isolated singularity at the origin. For further information about Lojasiewicz inequality (2), we refer [16,17,1,2,3,5,13,20,21,22].…”

The Łojasiewicz exponent in non-degenerate deformations of surface singularities

“…2.15]. Among other papers concerning the properties of µ-constant non-degenerate families of hypersurface singularities, one can mention [16], [1].…”

“…Hence, f t = f 0 + th t (x, y, z), where ord h t ≥ 2. Applying the procedure from the Splitting Lemma to the variable x, we easily find that there exists a holomorphic change of coordinates of the form x → Φ(t, x, y, z), y → y, z → z, t → t, where ord Φ(0, x, 0, 0) = 1, bringing f t to the form f t = f 0 + t ht (y, z), where ord ht ≥ 2 for small t. Since both µ and L 0 are invariants of stable equivalence (see, e.g., [16,Thm. 21]), we may remove x 2 from f t and infer that (g t ) := (g 0 + t ht (y, z)) is a µ-constant deformation of the isolated plane curve singularity g 0 and L 0 (g t ) = L 0 (f t ).…”