# Hokkaido Mathematical Journal

## Hokkaido Mathematical Journal, 37 (2008) pp.309-329

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### 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.

MSC(Primary) 58A35 $C^{l}-\mathcal{R}_{V}$-determinacy, $C^{l}-\mathcal{K}_{V}$-determinacy, weighted homogeneous polynomial function germs, controlled vector field, weighted homogeneous control functions.