$C^{l}-\mathcal{G}_{V}-$ determinacy of weighted homogeneous function germs on weighted homogeneous analytic varieties

Abstract

We provide estimates on the degree of $C^{l}-\mathcal{G}_{V}$-determinacy ($\mathcal{G}$ is one of Mather's groups $\mathcal{R} $ or $\mathcal{K}$) of weighted homogeneous function germs which are defined on weighted homogeneous analytic variety $V$ and satisfies a convenient Lojasiewicz condition. The result gives an explicit order such that the $C^{l}$-geometrical structure of a weighted homogeneous polynomial function germ is preserved after higher order perturbations, which generalize the result on $C^{l}-\mathcal{K}$-determinacy of weighted homogeneous functions germs given by M. A. S. Ruas.