Related papers: {\L}ojasiewicz exponents of non-degenerate holomor…

In the article we give some estimations of the {\L}ojasiewicz exponent of nondegenerate surface singularities in terms of their Newton diagrams. We also give an exact formula for the {\L}ojasiewicz exponent of such singularities in some…

The purpose of this paper is to give an explicit formula of the {\L}ojasiewicz exponent of an isolated weighted homogeneous singularity in terms of its weights.

We consider the exponent of \L ojasiewicz inequality $\|\partial\,f(\mathbf z)\| \ge c |f(\mathbf z|^\theta$ for two classes of analytic functions and we will give an explicit estimation for $\theta$. First we consider certain…

Let $f: (\mathbb{C}^n,0) \rightarrow (\mathbb{C},0)$ be a semiquasihomogeneous function. We give a formula for the local {\L}ojasiewicz exponent $\mathcal{L}_{0}(f)$ of $f$, in terms of weights of $f$. In particular, in the case of a…

We present a global version of the {\L}ojasiewicz inequality on comparing the rate of growth of two polynomial functions in the case the mapping defined by these functions is (Newton) non-degenerate at infinity. In addition, we show that…

The purpose of this paper is to give the exact value of the {\L}ojasiewicz exponent for an isolated weighted homogeneous polynomials of two real variaibles in terms of its weights.

We prove the constancy of the {\L}ojasiewicz exponent in non-degenerate $\mu$-constant deformations of surface singularities. This is a positive answer to a question posed by B. Teissier.

We give an expression for the {\L}ojasiewicz 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 {\L}ojasiewicz exponent of the gradient…

Given a hypersurface singularity $(X,0) \subset (\mathbb{C}^{n+1},0)$ defined by a holomorphic function $f:(\mathbb{C}^{n+1},0) \to (\mathbb{C},0)$, we introduce an alternating version of Teissier's Jacobian Newton polygon, associated with…

Mixed polynomials $f:\mathbb{C}^2\to\mathbb{C}$ are polynomial maps in complex variables $u$ and $v$ as well as their complex conjugates $\bar{u}$ and $\bar{v}$. They are therefore identical to the set of real polynomial maps from…

We give a criterion for the existence of a non-degenerate quasihomogeneous polynomial in a configuration, i.e. in the space of polynomials with a fixed set of weights, and clarify the relation of this criterion to the necessary condition…

Let $f$ be an isolated singularity at the origin of $\mathbb{C}^n$. One of many invariants that can be associated with $f$ is its {\L}ojasiewicz exponent $\mathcal{L}_0 (f)$, which measures, to some extent, the topology of $f$. We give, for…

Let $f$ be a germ of a smooth function at the orirgin in $\RR^n.$ We show that if $f$ is Kouchnirenko's nondegenerate and satisfies the so called Kamimoto--Nose condition then it admits the \L ojasiewicz inequalities. We compute the \L…

Let $f$ be a (possibly Newton degenerate) weighted homogeneous polynomial defining an isolated surface singularity at the origin of $\mathbb{C}^3$, and let $\{f_s\}$ be a generic deformation of its coefficients such that $f_s$ is Newton…

We study pluricomplex Green functions on algebraic sets. Let $f$ be a proper holomorphic mapping between two algebraic sets. Given a compact set $K$ in the range of $f$, we show how to estimate the pluricomplex Green functions of $K$ and of…

Mixed functions are analytic functions in variables $z_1,..., z_n$ and their conjugates $\bar z_1,..., \bar z_n$. We introduce the notion of Newton non-degeneracy for mixed functions and develop a basic tool for the study of mixed…

We present an extension theorem for a separately holomorphic function which is polynomial/rational in some variables.

We establish the unique solvability in weighted mixed-norm Sobolev spaces for a class of degenerate parabolic and elliptic equations in the upper half space. The operators are in nondivergence form, with the leading coefficients given by…

This note presents three resonances in commutative algebra and analytic geometry of the concept of Lojasiewicz inequality. The first is the interpretation in complex analytic geometry of the best possible exponent for a function g with…

We strengthen some estimations of the local and global {\L}ojasiewicz exponent for polynomial mappings on closed semialgebraic sets obtained by K.Kurdyka, S.Spodzieja and A.Szlachci\'nska.