C-l-contact and C-l-right equivalences of real semi-quasihomogeneous C-l function germs

Abstract

In this paper we investigate the C-l versions of contact and right equivalences of real semi-quasihomogeneous C-l function germs, 1 <= l <= infinity. The C-l-right equivalence implies C-l-contact equivalence for any 1 <= l <= infinity and in this work we show, up to certain conditions, that for semi-quasihomogeneous C-l function germs the converse is also true (Theorem 1). As a consequence, concerning the particular case of quasihomogeneous C-l function germs, we also have a similar result (Corollary 1) which recover a known result of M. Takahashi [14] for l = infinity. We note that we are considering semi-quasihomogeneous function germs with no additional hypothesis of isolated singularity at zero.