On the order of determination of a finitely determined germ

Abstract

A map germ f is said to be determined of order k if f is smoothly equivalent to its Taylor polynomial of order k, and any g with the same Taylor polynomial of order k is smoothly equivalent to f A map germ is finitely determined if it is determined of some order k. Given a finitely determined germ f a question immediately arises as to the order of the Taylor polynomial which determines f. The only previous estimate for finitely determined non stable germs is due to Mather and the estimate is astronomical [2]. This paper provides a low order estimate of the order of deter- mination in terms of the power of the maximal ideal contained by tf (O(n)) + o)f (O(p)) and p the dimension of the target space. (1.4) Theorem. Suppose f is in Eo(n,p) and tf(O(n))+mf(O(p)) contains mkO(f), k > O; then f is k(p + 1) determined. (Recall that iff is finitely determined tf(O(n)) + ~of (O(p)) must contain m k O(f) for some k.) Another variant of the generalized Malgrange preparation theorem is used as a tool in the proof. (2.3) Theorem. (Generalized Preparation Theorem). Let f: (IR", 0)-* (~P, 0), g: (~,, 0)--~ (~P', 0) be Coo map germs. Let A be a finitely generated (f, g)* Cp+p, module. If A/g* (rap,) A is finite dimensional as an lR vector space, then A is finitely generated as a g*(Cp,) module. The notation used in this paper is essentially the same as John Mather's papers on stability of C ~ mappings: d is the group of coordinate changes in source and target. Eo(n, p) stands for orign preserving map germs from either ~" to IR p or from I~" to IE p which are either Coo or analytic in the first case or analytic in the second case. C, stands for Coo germs f: (IR", 0)~ IR". Iff is in Eo(n, p), tf(O(n)) is the set of p tuples of germs obtained by applying the differential of f to the set of n tuples of germs in E, ; o)f(O(p)) is the set of p