Related papers: The {\L}ojasiewicz exponent for weighted homogeneo…

In this article, we consider the singularity of an arbitrary homogeneous polynomial with complex coefficients $f(x_0,\dots,x_n)$ at the origin of $\mathbb C^{n+1}$, via the study of the monodromy characteristic polynomials $\Delta_l(t)$,…

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 wide class of n-tuples of ideals $(I_1,..., I_n)$ in $\O_n$ using the information given by a fixed Newton filtration. In order to obtain this expression we consider a reformulation…

This paper deals with the triangle singularity defined by the \linebreak equation $f=X_1^{p_1}+X_2^{p_2}+X_3^{p_3}$ for weight triple $(p_1,p_2,p_3)$, as well as the category of coherent sheaves over the weighted projective line…

In this note, we study the behaviour of the Lojasiewicz exponent under hyperplane sections and its relation to the order of tangency.

We derive a formula for the weight system of the multivariable Alexander polynomial using determinants, show that it obeys known relations, and satisfies some of the same relations as the single variable polynomial.

The motivation for this paper are computer calculations of complete lists of weight systems of quasihomogeneous polynomials with isolated singularity at 0 up to rather large Milnor numbers. We review combinatorial characterizations of such…

Let $F(x) := (f_{ij}(x))_{i=1,\ldots,p; j=1,\ldots,q},$ be a ($p\times q$)-real polynomial matrix and let $f(x)$ be the smallest singular value function of $F(x).$ In this paper, we first give the following {\em nonsmooth} version of \L…

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 $V$ be a projective hypersurface having only isolated singularities. We show that these singularities are weighted homogeneous if and only if the Koszul syzygies among the partial derivatives of an equation for $V$ are exactly the…

The linearization coefficients for a set of orthogonal polynomials are given explicitly as a weighted sum of combinatorial objects. Positivity theorems of Askey and Szwarc are corollaries of these expansions.

We provide simple criteria and algorithms for expressing homogeneous polynomials as sums of powers of independent linear forms, or equivalently, for decomposing symmetric tensors into sums of rank-1 symmetric tensors of linearly independent…

We provide two new characterizations of bounded orthogonally additive polynomials from a uniformly complete vector lattice into a convex bornological space using harmonic means and completely partitioned weighted geometric means. Our result…

We devise calculus rules for the Kurdyka-\L{}ojasiewicz exponent using the rank theorem and Lie group actions. They apply to a wide class of composite and invariant functions, and are particularly suitable for handling nonisolated local…

The main goal of this paper is to present some explicit formulas for computing the {{\L}}ojasiewicz exponent in the {{\L}}ojasiewicz inequality comparing the rate of growth of two real bivariate analytic function germs.

Polar weighted homogeneous polynomials are the class of special polynomials of real variables $x_i,y_i, i=1,..., n$ with $z_i=x_i+\sqrt{-1} y_i$, which enjoys a "polar action". In many aspects, their behavior looks like that of complex…

Let Y be a weighted homogeneous (singular) subvariety of C^n. The main objective of this paper is to present an explicit formula for solving the d-bar-equation $f=\dbar{g}$ on the regular part of Y, where $f$ is a d-bar-closed $(0,1)$-form…

This note provides a connection between the Hodge spectrum and the Hodge ideals recently developed by Musta\c{t}\v{a}--Popa in the case of weighted homogeneous isolated singularities.

We look for spectral type differential equations satisfied by the generalized Jacobi polynomials, which are orthogonal on the interval [-1,1] with respect to a weight function consisting of the classical Jacobi weight function together with…

This paper concerns an analytic and numerical analysis of a class of weighted singular Cauchy integrals with exponential weights $w:=\exp(-Q)$ with finite moments and with smooth external fields $Q:\mathbb R\to [0,\infty)$, with varying…