On the Łojasiewicz exponent, special direction and maximal polar quotient

For a local singular plane curve germ $f(X,Y)=0$ we characterize all nonsingular $λ\in\bbC\{X,Y\}$ such that the Łojasiewicz exponent of $\grad\,f$ is not attained on the polar curve $\bJ(λ,f)=0$. When $f$ is not Morse we prove that for the same $λ$'s the maximal polar quotient $q_0(f,λ)$ 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.

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