Topological invariants of plane curve singularities: Polar quotients and \L ojasiewicz gradient exponents

TL;DR

This paper investigates topological invariants of plane curve singularities, demonstrating that polar quotients and ojasiewicz gradient exponents are topologically invariant and can be computed from polar quotients, with applications to polynomial inequalities.

Contribution

It establishes the topological invariance of polar quotients and ojasiewicz exponents, and provides methods to compute these invariants from polar quotients.

Findings

Polar quotients are topological invariants.

ojasiewicz gradient exponents can be derived from polar quotients.

Effective estimates for ojasiewicz exponents in polynomial inequalities.

Abstract

In this paper, we study polar quotients and \L ojasiewicz exponents of plane curve singularities, which are {\em not necessarily reduced}. We first show that the polar quotients is a topological invariant. We next prove that the \L ojasiewicz gradient exponent can be computed in terms of the polar quotients, and so it is also a topological invariant. As an application, we give effective estimates of the \L ojasiewicz exponents in the gradient and classical inequalities of polynomials in two (real or complex) variables.