The Łojasiewicz exponent of a set of weighted homogeneous ideals

Abstract

a b s t r a c tWe give an expression for the Łojasiewicz exponent of a set of ideals which are pieces of a weighted homogeneous filtration. We also study the application of this formula to the computation of the Łojasiewicz exponent of the gradient of a semi-weighted homogeneous function (C n , 0) → (C, 0) with an isolated singularity at the origin.

The Łojasiewicz exponent of nondegenerate surface singularities

“…In 2010 the paper by Tan, Yau, Zuo ([TYZ]) appeared, in which Theorem 1.9 was given in analogous form for n-variables, n > 3, but their proof is false (the proof of their Proposition 3.4 is false). Some results for quasihomogeneous singularities in n-dimensional case were also given by Bivia-Ausina and Encinas ( [BE1], [BE2]).…”

The sequence of mixed Łojasiewicz exponents associated to pairs of monomial ideals

“…Since we assume that I ⊆ J, Corollary 5.8 implies the equality L In the following example we see that, in general, inequality (59) can be strict (we will see that this is not the case when J is diagonal). , provided that (f, g) is a sufficiently general element of I ⊕ J (see [7,Theorem 3.6]). Let H = f, g .…”

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

“…. , A rn ) is non-degenerate on Γ + (see [7,Proposition 4.2] for details). For the sake of completeness, we show in Proposition 3.5 a reformulation of [8, Theorem 3.3] considering the notion of Rees mixed multiplicity.…”

“…, w 1 ···wn wn e n } and by Γ w the Newton boundary of Γ w + . It is straightforward to see that Γ w has only one compact facet, which is supported by w, and that the weighted homogeneous filtration induced by w (see [7,Section 4]) coincides with the Newton filtration of O n induced by Γ w + .…”

“…Example 4.14. Let us consider the function f : (C 4 , 0) → (C, 0) of [7,Example 4.12], that is f (x, y, z, t) = z 9 −y 11 t+yt 5 +x 27 . This function is weighted homogeneous of degree 27 with respect to w, where w = (1, 2, 3, 5).…”