On the Łojasiewicz exponent, special direction and maximal polar quotient

Abstract

For a local singular plane curve germ f (X, Y ) = 0 we characterize all nonsingular λ ∈ C{X, Y } such that the Lojasiewicz exponent of grad f is not attained on the polar curve J(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. Introduction, main resultsLet C{X, Y } be the ring of convergent power series in two variables.For a nonzero series f = c αβ X α Y β ∈ C{X, Y } we define the order ord f as the minimum of α + β corresponding to nonzero c αβ and the initial form inf = α+β=ordf c αβ X α Y β . We put ord 0 = ∞ by convention. We call f singular if 2 ≤ ord f < ∞, nonsingular if ord f = 1 and a unit if ord f = 0.For f, g ∈ C{X, Y } of positive orders we say that f and g are transverse if the system inf = ing = 0 has no solutions in C 2 \ {0}. Otherwise we call f and g tangent. By t = t(f ) = ord(in f ) red we denote the number of different tangents of f . We callLet f ∈ C{X, Y } be a nonzero series without constant term. The series f defines the curve germ f = 0 at 0 ∈ C 2 . We extend the term: singular (nonsingular, unitangent, multitangent) for germs and the term: tranverse (tangent) for pairs of Mathematics Subject Classification (2000) 32S55; 14H20; Key words and phrases: plane curve singularity, Lojasiewicz exponent, polar curve, polar invariants, polar quotients, Eggers tree, ultrametric space of branches.

Non-degeneracy of the discriminant

“…In [Te1] and [Te3] Teissier introduced the jacobian Newton diagram, which is the Newton diagram of D(u, v). The jacobian Newton diagram depends only on the topological type of ( , f ) (see [Te1] for the case where is generic, Merle [Me] and Ephraim [Eph] for one branch and [Gwo1], [Le2] and [Mi] for general case). Decompositions of the polar curve can be found in the literature (see [Me], [Eph], [Eg], [GB]) .…”