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

Abstract

We provide estimates on the degree of C l − G V -determinacy (G is one of Mather's groups R or 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 − K-determinacy of weighted homogeneous functions germs given by M.A.S. Ruas.