Analytic sets which are biholomorphically equivalent to quasihomogeneous sets (Q1065202)

A formal series \(f\in Q_ n={\mathbb C}[[x_ 1,...,x_ n]]\) is called quasihomogeneous of degree \(k\) with weights \(\alpha_ i\), if \(\sum_{i}\alpha_ i\frac{\partial f}{\partial x_ i}=kf\). An ideal \(i\in Q_ n\) is quasihomogeneous, if it possesses a system of quasihomogeneous generators. The authors asks, when an ideal \(i\) can be made quasihomogeneous by an automorphism of \(Q_ n\), supposing that \(\dim Q_ n/i<\infty\). The sufficient condition is the existence of a derivation \(d: Q_ n\to Q_ n\) such that \(d(i)=i\) and that the induced map \(i/iM\to i/iM\) is not degenerate, where \(M\) is the maximal ideal of \(Q_ n\). If \(n=2\) or if \(i\) admits \(n\) generators, this condition is also necessary.