On special values for pencils of plane curve singularities.
Discriminant and the Lojasiewicz exponent.
Formulae for the singularities at infinity of plane algebraic curves.
A bound for the Milnor number of plane curve singularities
Let f = 0 be a plane algebraic curve of degree d > 1 with an isolated singular point at 0 ∈ ℂ2. We show that the Milnor number μ0(f) is less than or equal to (d−1)2 − [d/2], unless f = 0 is a set of d concurrent lines passing through 0, and characterize the curves f = 0 for which μ0(f) = (d−1)2 − [d/2].
Multiplicity and the Lojasiewicz exponent
Newton polygons and the Łojasiewicz exponent of a holomorphic mapping of
Polar quotients and singularities at infinity of polynomials in two complex variables
Using the notion of the maximal polar quotient we characterize the critical values at infinity of polynomials in two complex variables. As an application we give a necessary and sufficient condition for a family of affine plane curves to be equisingular at infinity.
Characterization of non-degenerate plane curve singularities.
On the approximate roots of polynomials
We give a simplified approach to the Abhyankar-Moh theory of approximate roots. Our considerations are based on properties of the intersection multiplicity of local curves.
Łojasiewicz exponents and singularities at infinity of polynomials in two complex variables
For every polynomial F in two complex variables we define the Łojasiewicz exponents measuring the growth of the gradient ∇F on the branches centered at points p at infinity such that F approaches t along γ. We calculate the exponents in terms of the local invariants of singularities of the pencil of projective curves associated with F.
Pinceaux de courbes planes et invariants polaires
We study pencils of plane curves , t ∈ ℂ, using the notion of polar invariant of the plane curve f = 0 with respect to a smooth curve l = 0. More precisely we compute the jacobian Newton polygon of the generic fiber , t ∈ ℂ. The main result gives the description of pencils which have an irreducible fiber. Furthermore we prove some applications of the local properties of pencils to singularities at infinity of polynomials in two complex variables.
The Łojasiewicz numbers and plane curve singularities
For every holomorphic function in two complex variables with an isolated critical point at the origin we consider the Łojasiewicz exponent ₀(f) defined to be the smallest θ > 0 such that near 0 ∈ ℂ² for some c > 0. We investigate the set of all numbers ₀(f) where f runs over all holomorphic functions with an isolated critical point at 0 ∈ ℂ².
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