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

Abstract

In this paper we investigate the C ` versions of contact and right equivalences of real semi-quasihomogeneous C ` function germs, 1 ≤ ` ≤ ∞. The C ` -right equivalence implies C ` -contact equivalence for any 1 ≤ ` ≤ ∞ and in this work we show, up to certain conditions, that for semi-quasihomogeneous C ` function germs the converse is also true. As a consequence, we recover some known results about C∞-right and C∞-contact equivalences of C∞ function germs. We note that we are considering semi-quasihomogeneous function germs with no additional hypothesis of isolated singularity at zero.