| It is well known that giving types is always the most important and a very basic problem. We have already observed that ε_n is indeed a real vector space of infinite di-mension.In order to give the types of function germs, a basic idea is to reduce infinite dimension to finite dimension.Maybe the work will be easier. So, it is natural to guess the following.If f ∈ ε_n is good enough, it may be R-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 konw the latter is of finite dimension. For this,there have been many wonderful results.Not only this,but also Laurentiu Paunescu,Ould M Abderrahmane and other scholars have studied the characterization and criterion of v-sufficiency from the weighted point of view. There is the finite determination with respect to TZ equivalence from the weighted point of view and its criterion in the following paper.It is critical there is a suitable vector field. At fist we can get a kind of singular Rie-mannian metric by use of a control function.Then we will have the wanted vector field. The classical singularity theory of Mather is used here.We get the critieria of finite determination with respect to R equivalence from the weighted point of view and advance original results. |
