Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Lojasiewicz exponent of a surface: an intrinsic view

Emel Bilgin, Gülay Kaya, Meral Tosun

Abstract

In this paper we observe that the Łojasiewicz exponent $\mathcal{L}_0(X)$ of an ADE-type singularity $X$ can be computed by means of invariants of certain ideals in the local ring ${\mathcal O}_{X,0}$. After extending the notion of Łojasiewicz exponent to rational singularities of higher multiplicities we make a similar observation for RTP-type singularities.

Lojasiewicz exponent of a surface: an intrinsic view

Abstract

In this paper we observe that the Łojasiewicz exponent of an ADE-type singularity can be computed by means of invariants of certain ideals in the local ring . After extending the notion of Łojasiewicz exponent to rational singularities of higher multiplicities we make a similar observation for RTP-type singularities.

Paper Structure

This paper contains 4 sections, 15 theorems, 30 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Integrally closed ideals in the local ring of a rational singularity
  3. Ł ojasiewicz exponent and length of an ideal
  4. Ł ojasiewicz exponent of rational singularities of higher multiplicity

Key Result

Theorem 1.1

KOP Let $F:\mathbb{C}^N\rightarrow \mathbb{C}$ be a weighted homogeneous polynomial with weight $w=(w_1,\ldots ,w_N)$ and of degree $d$. Assume $d\geq w_i$ for all $i$. Then where the equality holds when $N=3$.

Theorems & Definitions (31)

  • Theorem 1.1
  • Definition 1.2
  • Proposition 1.3
  • Theorem 2.1
  • Theorem 2.2
  • Definition 2.3
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • Corollary 3.3
  • ...and 21 more