The Łojasiewicz exponent of nondegenerate surface singularities

Abstract

In the article we give some estimations of the Łojasiewicz exponent of nondegenerate surface singularities in terms of their Newton diagrams. We also give an exact formula for the Łojasiewicz exponent of such singularities in some special cases. The results are stronger than Fukui inequality [F]. It is also a multidimensional generalization of the Lenarcik theorem [L].

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

“…C O M M E N T S . (i) For n = 3, the case Γ 2 ( f )\E( f ) = ∅ was established by the third-named author in [Ole13,Theorem 1.8]. Namely, in this case, if we denote the variables in C 3 by x, y, z (and permute them if necessary), there is exactly one segment S ∈ Γ 1 ( f ) joining the monomial xy with some monomial z k , where k 2, and then L 0 ( f ) = k − 1.…”

“…The Łojasiewicz exponent, then, turns out to be the greatest coordinate of intersections of the prolongations of the non-exceptional segments with the coordinate axes. Our approach is similar: based on the definition, given by the third-named author in [Ole13], we also distinguish exceptional faces in the Newton polyhedron of a surface singularity. The greatest coordinate of intersections of the prolongations of the non-exceptional 2-dimensional faces with the coordinate axes is exactly the Łojasiewicz exponent of the singularity.…”

A conjecture on the Łojasiewicz exponent

“…, φ k ) = 0. By [O2,Cor. 2.4] there exists S ∈ Γ(g), such that (ord φ i ) k i=1 is a primitive vector of S and 8∇g S (info φ 1 , .…”

“…It is true in the case n = 2 (Lenarcik [L]). For general n only estimations of £ 0 (f ) in terms of Newton diagrams (see [A,B,BE,F,O1,O2]) are known. On the other hand a counter-example to it would disprove the Teissier conjecture that £ 0 (f ) is a topological invariant of f.…”

Łojasiewicz exponent of families of ideals, Rees mixed multiplicities and Newton filtrations

“…The authors applied in [6] the explicit construction of a log-resolution of I to show an effective method to compute L 0 (I). Newton polyhedra have proven to be a powerful tool in the estimation, and determination in some cases, of Lojasiewicz exponents, as can be seen in [2], [10], [18] and [22].…”