Related papers: Lojasiewicz exponent of a surface: an intrinsic vi…

We consider \L ojasiewicz inequalities for a non-degenerate holomorphic function with an isolated singularity at the origin. We give an explicit estimation of the \L ojasiewicz exponent in a slightly weaker form than the assertion in…

Let $(X,o)$ be a complex analytic normal surface singularity and let ${\mathcal O}_{X,o}$ be its local ring. We investigate the normal reduction number of ${\mathcal O}_{X,o}$ and related numerical analytical invariants via resolutions…

The main aim of the paper is to give a formula for computing the separation \L ojasiewicz exponents for two real analytic set germs via the Newton--Puiseux expansions of their defining functions. Moreover, we present an effective exponent…

In this note I give a description of Lyubeznik's local cohomology invariants for a certain natural class of local rings, namely the ones which have the same local cohomology vanishing as one expects from an isolated singularity. This…

Let $X$ be a projective scheme over a field $k$ and let $A$ be the local ring at the vertex of the affine cone of $X$ under some embedding $X\hookrightarrow\mathbb{P}^n_k$. We prove that, when $\ch(k)>0$, the Lyubeznik numbers…

We define a Hodge-theoretical refinement of the Lyubeznik numbers for local rings of complex algebraic varieties. We prove that these numbers are independent of the choices made in their definition and that, for the local ring of an…

Let $R=\oplus_{m\geq 0}R_m$ be a standard graded equidimensional ring over a field $R_0$, and $I\subseteq J$ be two non-nilpotent graded ideals in $R$. Then we give a set of numerical characterizations of the integral dependence of $I$ and…

Let H be the supremum of finitely many real polynomials of degree d and assume that H has a strict local minimum at 0. We prove a \L ojasiewicz-type inequality $H(x_1,...,x_n) > ||(x_1,...,x_n)||^s$ where s depends only on d and n. This…

The Lojasiewicz exponent at infinity of an entire function measures of the infimal rate of growth of its gradient. The authors compute the Lojasiewicz exponents at infinity of the 3-variable complex polynomials x - 3 x^{2n+1} y^{2q} + 2…

We prove that any surjective self-morphism with $\delta_f > 1$ on a potentially dense smooth projective surface defined over a number field $K$ has densely many $L$-rational points for a finite extension $L/K$.

We describe the set of all locally nilpotent derivations of the quotient ring $\mathbb{K}[X,Y,Z]/(f(X)Y - \varphi(X,Z))$ constructed from the defining equation $f(X)Y = \varphi(X,Z)$ of a generalized Danielewski surface in $\mathbb K^3$ for…

Let $S = K[x_1, ..., x_n ]$ be a polynomial ring over a field $K$, and $E = K < y_1, ..., y_n >$ an exterior algebra. The "linearity defect" $ld_E(N)$ of a finitely generated graded $E$-module $N$ measures how far $N$ departs from…

We introduced an $\tilde{\mathcal{A}}$-invariant for quasi-ordinary parameterizations and we consider it to describe quasi-ordinary surfaces with one generalized characteristic exponent admitting a countable moduli.

In this paper, we study the Kurdyka-{\L}ojasiewicz (KL) exponent, an important quantity for analyzing the convergence rate of first-order methods. Specifically, we develop various calculus rules to deduce the KL exponent of new (possibly…

We exhibit a global bound for the Lyubeznik numbers of a ring of prime characteristic. In addition, we show that for a monomial ideal, the Lyubeznik numbers of the quotient rings of its radical and its polarization are the same.…

Using the semicontinuity of the Milnor number and the {\L}ojasiewicz exponent, we give a simple proof of Yau's characterization of isolated homogenous hypersurface singularities.

A maximal surface $\sb$ with isolated singularities in a complete flat Lorentzian 3-manifold $\N$ is said to be entire if it lifts to a (periodic) entire multigraph $\tilde{\sb}$ in $\l^3.$ In addition, $\sb$ is called of finite type if it…

Let $K$ be a totally real number field of degree $r=[K:\mathbb{Q}]$ and let $p$ be an odd rational prime. Let $K_{\infty}$ denote the cyclotomic $\mathbb{Z}_{p}$-extension of $K$ and let $L_{\infty}$ be a finite extension of $K_{\infty}$,…

$(L_p, L_q)$ estimates are obtained for oscillatory potentials $(K^\alphaf)(x)=\int\limits_{R^n}\frac{\exp(i|y|)}{|y|^{n-\alpha}}f(x-y)dy$, $0<\alpha<n$, $n\geq 2$, whose symbol has a singularity on the unit sphere. These potentials are…

Let $X$ be a genus zero compact polyhedral surface (the Riemann sphere equipped with a flat conical metric $m$). We derive the variational formulas for the determinant of the Laplacian, ${\rm det}\,\Delta^m$, on $X$ under infinitesimal…