Cl-contact and Cl-right equivalences of real semi-quasihomogeneous Cl function germs

Abstract

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