On the {\L}ojasiewicz exponent, special direction and maximal polar quotient

Abstract

For a local singular plane curve germ $f(X,Y)=0$ we characterize all nonsingular \lambda\in\bbC\{X,Y\} such that the {\L}ojasiewicz exponent of \grad\,f is not attained on the polar curve \bJ(\lambda,f)=0. When $f$ is not Morse we prove that for the same $\lambda$'s the maximal polar quotient $q_0(f,\lambda)$ is strictly less than its generic value $q_0(f)$. Our main tool is the Eggers tree of singularity constructed as a decorated graph of relations between balls in the space of branches defined by using a logarithmic distance.Comment: 39 pages, 16 figure