A bound for the Milnor number of plane curve singularities

Milnor Number of Weighted-Lê–Yomdin Singularities

Equisingular calculations for plane curve singularities

We present an algorithm which, given a deformation with section of a reduced plane curve singularity, computes equations for the equisingularity stratum (that is, the μ-constant stratum in characteristic 0) in the parameter space of the deformation. The algorithm works for any, not necessarily reduced, parameter space and for algebroid curve singularities C defined over an algebraically closed ...

A bound on the number of points of a plane curve

A conjecture is formulated for an upper bound on the number of points in PG(2, q) of a plane curve without linear components, defined over GF(q). We prove a new bound which is half way from the known bound to the conjectured one. The conjecture is true for curves of low or high degree, or with rational singularity. Let C be a plane curve of degree n, defined over GF(q), without (rational) linea...

The Computational Complexity of the Resolution of Plane Curve Singularities

We present an algorithm which computes the resolution of a plane curve singularity at the origin defined by a power series with coefficients in a (not necessarily algebraically closed) field k of characteristic zero. We estimate the number of £-operations necessary to compute the resolution and the conductor ideal of the singularity. We show that the number of A:-operations is polynomial^ bound...

On Special Values for Pencils of Plane Curve Singularities

Let (Ft : t ∈ P) be a pencil of plane curve singularities and let μ0 be the Milnor number of the fiber Ft. We prove a formula for the jumps μ0 − inf{μ0 : t ∈ P}. As an application, we give a description of the special values of the pencil (Ft : t ∈ P). Introduction. Let (Ft : t ∈ P1), P1 = C ∪ {∞} be a pencil of plane curve singularities defined by two coprime power series f, g ∈ C{X,Y } withou...