A remark on isolated complex hypersurface singularities

TL;DR

This note computes an explicit bound D(n,m) ensuring that sufficiently high-order perturbations of a regular homogeneous polynomial define hypersurface germs analytically equivalent to the original, extending to quasihomogeneous cases.

Contribution

The paper explicitly calculates D(n,m) as n(m-2)+1 and extends the result to quasihomogeneous polynomials, clarifying when hypersurface singularities are analytically stable under perturbations.

Findings

D(n,m) is explicitly computed as n(m-2)+1.

High-order perturbations of a regular homogeneous polynomial yield analytically equivalent hypersurface germs.

Extension of the result to quasihomogeneous polynomials is provided.

Abstract

This is now an expository note about the following classical problem. Let $(X, \bf 0)$ be the germ of a hypersurface in $(\mathbb C^n,\bf 0)$ with an ordinary singularity of multiplicity $m$ at the origin $\bf 0$. A natural question to ask is whether $X$ and its tangent cone at the origin are analytically isomorphic. The answer is negative in general, in view of a theorem of Kioji Saito. However there is an integer $D(n,m)>m$ such that, given a \emph{regular} homogeneous polynomial $f(x_1,\ldots, x_n)$ of degree $m$ (this means that $\{ f=0\}$ is a smooth hypersurface in $\PP^{n-1}$) then, for all $d\geq D(n,m)$, any convergent power series of the form $g=f+ o(d)$ (here, as usual, $o(d)$ stays for a power series of order at least $d$), defines a germ $\{ g=0\}$ which is analytically equivalent to the germ $\{ f=0\}$. In this note we compute $D(n,m)$ explicitly as $n(m-2)+1$. We also give an extension to the case in which $f$ is a quasihomogeneous polynomial. It was pointed out that the value of $D(n,m)$ was already known by \cite[Exercise 7.31]{D}.