Milnor numbers of deformations of semi-quasi-homogeneous plane curve singularities

TL;DR

This paper characterizes the range of Milnor numbers achievable through deformations of semi-quasi-homogeneous plane curve singularities, providing explicit bounds based on the Newton diagram.

Contribution

It establishes a precise description of all possible Milnor numbers obtainable from deformations of irreducible, nondegenerate semi-quasi-homogeneous singularities, using Newton diagram data.

Findings

All Milnor numbers between μ(f) and μ(f)-r(p-r) are attainable.

The bounds depend explicitly on the Newton diagram of the singularity.

The results apply to irreducible, nondegenerate cases.

Abstract

The aim of this paper is to show the possible Milnor numbers of deformations of semi-quasi-homogeneous isolated plane curve singularities. Main result states that if $f$ is irreducible and nondegenerate, by deforming $f$ one can attain all Milnor numbers ranging from $\mu(f)$ to $\mu(f)-r(p-r)$, where $r$ and $p$ are easily computed from the Newton diagram of $f$.