\(C^l-{\mathcal G}_v\)-determinacy of weighted homogeneous function germs on weighted homogeneous analytic varieties (Q934310)

The authors provide estimates on the degree of \( C^l-{\mathcal G}_V \)-determinacy (where \( \mathcal G \) is one of Mather's groups \( \mathcal R \) or \( \mathcal K \)) of weighted homogeneous function germs defined on a weighted homogeneous analytic variety \( V \) and satisfying a suitable Łojasiewicz condition. They give an explicit order such that the \( C^l \) geometric structure of a weighted homogeneous polynomial is preserved under higher order perturbations. This extends results on \( C^l-{\mathcal K} \) determinacy of homogeneous functions germs given by \textit{M. A. Soares Ruas} [Math. Scand. 59, No.~1, 59--70 (1986; Zbl 0632.58013)] and \textit{M. A. Soares Ruas} and \textit{M. J. Saia} [Hokkaido Math. J. 26, No.~1, 89--99 (1997; Zbl 0873.58010)].