On the separation \L ojasiewicz exponents of real analytic sets in the real plane

TL;DR

This paper provides a formula for calculating the separation Łojasiewicz exponents of real analytic sets in the plane using Newton-Puiseux expansions and offers an effective exponent for algebraic sets based on degrees.

Contribution

It introduces a new formula for the separation Łojasiewicz exponents using Newton-Puiseux expansions and derives an effective exponent for algebraic sets in terms of degrees.

Findings

Formula for separation Łojasiewicz exponents via Newton-Puiseux expansions

Effective exponent for algebraic sets based on degrees

Application to real analytic and algebraic sets in the plane

Abstract

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 for the case of two real algebraic sets in terms of their degrees.