Topological invariants of plane curve singularities: Polar quotients and Lojasiewicz gradient exponents

摘要

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