$C^0$-sufficiency of jets via blowing-up
Satoshi Koike · Kyoto journal of mathematics · 1988
Let erg (n, 1) denote the set of C function germs: (R a , 0)-(R, 0 ) .F or a function f e e [](n , 1), j s f denotes the s-jet of f at 0 e R" (q s).f s (n, 1) is the set of s-jets o f function germs in e[o(n, 1).An s-jet z E r(n, 1) is called 0 -sufficient in e[o(n, 1), if for any functions f, g in Cro(n, 1) such that j s f = j s g=z, there exists a local homeomorphism a: (R e , 0)-(R, , 0) such that f o a = g .Thus 0-sufficiency of jets amounts to saying that all terms of degree >s can be omitted without changing the local topological behavior of the realizations.Concerning a characterization o f C°-sufficiency in e [s](n, 1) o r et 5 + 1 1(n, 1), th e "Kuiper-Kuo theorem" is wellknown (see § 3).In this note, we shall give another characterization of C°-sufficiency of jets by using the "after blowing-up functions".In practice, this criteria is often easier to check than th e above o n e .Here we describe the results about 0-sufficiency in ersi(n, 1) o n ly .Of course, similar results hold also for 0-sufficiency in ers + i i(n, 1).