The Classification And Recognition Of Boundary Singularities With Three Variables Under R_H~*-Equivalent

It is well known that the classiffication is one of the most important but very difficult problems in singularity.Since the dimension of the spaceε_n consisting of smooth function germs at the origin point is infinite,a basic idea is to reduce the case of infinite dimension to that of finite dimension by using of the Nakayama lemma,substituted the smooth function germs by their Taylor polynomials.So it is natural to guess that a function germ good enough may be equivalent to its some Taylor polynomial through making their jet.Thus the the classiffication of function germs is reduced to that of the finite dimensional vector space of polynomials.For this,there have been many results under low co-dimensions.Thom gives the classiffication of smooth function germs up to codimension 5.By using a sufficient and necessary condition of the contact equivalence,Martin Golubitsky gives the classification of bifurcation problems with one state variable and one bifurcation parameter up to codimension 5 under the contact equivalent group. Arnold,V.I gives the classification of simple boundary singularities under right equivalence. Wang wei studies the recognition condition under right equivalence group. Ren yaoqing gives the classification and recognition of boundary singularities with two variables under right equivalence,up to co-dimension 4.A sufficient and necessary condition of R_H~*-equivalence is got by using Nakayama and the classification and recognition of boundary singularities with three variable under right equivalence group up to co-dimension 3 are given.