Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the separation Łojasiewicz exponents of real analytic sets in the real plane

Phi Dung Hoang, Hong Duc Nguyen

Abstract

The main aim of the paper is to give a formula for computing the separation Łojasiewicz exponents for two real analytic set germs via the Newton--Puiseux expansions of their defining functions. Moreover, we present an effective exponent for the case of two real algebraic sets in terms of their degrees.

On the separation Łojasiewicz exponents of real analytic sets in the real plane

Abstract

The main aim of the paper is to give a formula for computing the separation Łojasiewicz exponents for two real analytic set germs via the Newton--Puiseux expansions of their defining functions. Moreover, we present an effective exponent for the case of two real algebraic sets in terms of their degrees.
Paper Structure (6 sections, 6 theorems, 54 equations, 1 algorithm)

This paper contains 6 sections, 6 theorems, 54 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Newton polygon relative to an arc
  4. Approximations of Puiseux series
  5. Separation Ł ojasewicz exponent of two real analytic subsets
  6. Effective estimation for separation Ł ojasiewicz exponent

Key Result

Lemma 2.1

Suppose that $\phi$ is not a root of $f = 0$. Consider a series where $c \in \mathbb{K},0<\rho \in \mathbb{Q}$ and $o(\rho)$ denotes a Puiseux series of order greater than $\rho$. Then the following statements hold:

Theorems & Definitions (20)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Definition 2.4: Nguyen2019
  • Remark 2.5
  • Definition 2.6
  • Example 2.7
  • ...and 10 more