On the estimates of the order of finitelya k -determined map-germs

In this paper we deal with the estimates of the order of determinacy of finitely determined map-germs. This problem has been noted by mathematicians for more than ten years. But the results given by some authors for the estimates of the order of determinacy are stated with respect toa, while those given in this paper are stated with respect toa _k , wherek may be any non-negative integer, and we havea ₀=a in particular. This paper was completed during the author's visit to the Institute of Mathematics, Now I'd like to express my heart-felt thanks to the Institute for its financial support and Prof. Li Peixin for his instruction.