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A bound for the Milnor number of plane curve singularities

Arkadiusz Płoski — 2014

Open Mathematics

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].

Łojasiewicz exponents and singularities at infinity of polynomials in two complex variables

Janusz GwoździewiczArkadiusz Płoski — 2005

Colloquium Mathematicae

For every polynomial F in two complex variables we define the Łojasiewicz exponents p , t ( F ) 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 p , t ( F ) in terms of the local invariants of singularities of the pencil of projective curves associated with F.

Pinceaux de courbes planes et invariants polaires

Evelia R. García BarrosoArkadiusz Płoski — 2004

Annales Polonici Mathematici

We study pencils of plane curves f t = f - t l N , 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 f t , 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

Evelia García BarrosoTadeusz KrasińskiArkadiusz Płoski — 2005

Annales Polonici Mathematici

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 | g r a d f ( z ) | c | z | θ 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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