The Newton diagram of an analytic morphism, and applications to differentiable functions

Abstract

(i) ƒ(*)=4*) •»(#*)), where x = (xi , . . . ,xm) , (x) = ( i(x),...,0n(x)) is an analytic mapping, and A(x) is a px q matrix of analytic functions. Given f(x) = {fi{x),..., fp(x))C°°, we seek C°° solutions g(y) = {gi{y),..., gq(y))There is a necessary condition on the Taylor series of ƒ at each point. Special cases are classical: when (x) = x we have the division theorem of Malgrange [7, Chapter VI], and when A(x) = / , the composition problem first studied by Glaeser [5]. We solve the problem in the case that 0(x) and A(x) are algebraic (or Nash), using a Hilbert-Samuel stratification associated to (1). Our methods, however, go far beyond this case. We present algebraic criteria for solving (1), based on a fundamental relationship between two invariants of an analytic morphism and an associated Newton diagram. Hironaka's simple but powerful formal division algorithm [3] is exploited systematically. The only results from differential analysis used are Whitney's extension theorem [7, Chapter I] and Lojasiewicz's inequality [7, Chapter IV]. Let k = R or C. (Some of our assertions hold for other fields.) Let M, N be analytic manifolds (over /c), and 0: M —• AT an analytic mapping. Let A be a p X q matrix of analytic functions on M. For each a G M, let 0a (respectively, da) denote the ring of germs of analytic functions at a (respectively, the completion of 0a in the Krull topology). Let xha be the maximal ideal of O0. In the case k — R, let C°°(M) denote the algebra of C°° functions on M. There is a Taylor series homomorphism ƒ >-» fa from C°°{My onto o P a. The mapping induces ring homomorphisms 0*: C°°(N) -> C°°(M), K' 0 {a) -+ 0a, and fc: o# a) -+ 0aLet $ : C°°{N) C°°{My denote the module homomorphism over 0* defined by $(g) = A(gofy. Let <la : o^ a ) —• 0a denote the analogous module homomorphism over 0*. Let (QC°°(N)*Ydenote the C°°{N)-submodule of C°°{Mf consisting of elements which formally belong to the image $C°°(N) of $; i.e., (^C(AT)«)= {ƒ € C°°{Mf: for all b G 0(M), there exists Gb G Ob such that fa = 9a{Gb) for all a G 0()}Evidently, ( f c C ^ i V ^ i s closed in the C°° topology.