Abstract
Mather gave the necessary and sufficient conditions for the finite determinacy smooth function germs with no more than codimension 4. The theorem is very effective on determining low codimension smooth function germs. In this paper, the concept of right equivalent for smooth function germs ring generated by two ideals finitely is defined. The containment relationships of function germs still satisfy finite k-determinacy under sufficiently small disturbance which are discussed in orbit tangent spaces. Furthermore, the methods in judging the right equivalency of Arnold function family with codimension 5 are presented.
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