On the Łojasiewicz exponent, special direction and maximal polar quotient

Details On the Łojasiewicz exponent, special direction and maximal polar quotient On the Łojasiewicz exponent, special direction and maximal polar quotient

2011-12-23

arxiv.org/abs/1112.5578v1

Mathematics

Algebraic Geometry

39 pages, 16 figures

Scientific paper

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.

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