The Classification And Recognition Of Boundary Singularities With Two Variable Under R_H~*-Equivalent

It is well known that giving types is the most important and a very basic problem.We have already observed thatε_n is a real vector space of infinite dimension. In order to give the classification of function germs,a basic idea is to Using Nakayama lemma,reduce infinite dimension to finite dimension.Therefore,infinite dimension problem can be transformed into finite dimension problem.Maybe the work will be easier.So,it is natural to guess the following.If f∈ε_n is good enough,it maybe right equivalent to a some Taylor polynomial.So,the problem of giving the types of function germs is reduced the question of sorting the vector space of polynomials. We know the latter is finite dimension.For this,there have been many results about classification of low orders.Thom gives the derivable function germs classification of codimension no more than 5.Using the Nakayama lemma and a sufficient and necessary condition,Martin Golubitsky gives the classification of bifurcation problem with a state variable and a parameter under the K- equivalent.Arnold,V.I gives the classification of simple boundary singularities under the R_H~* - equivalent.Wang wei gives a sufficient and necessary condition of R_H~*-equivalent.Using Nakayama lemma and a sufficient and necessary condition of R_H~*- equivalent, Chapter two of this paper gives the classification and recognition of boundary singularities with two variable under R_H~*-equivalent,up to co-dimension 4.Chapter three gives several lemmas and the classification with trivial solution of two state variables and a parameter under the action of the K-equivalent.