The Newton diagram of an analytic morphism, and applications to differentiable functions
Edward Bierstone, Pierre D. Milman · Bulletin of the American Mathematical Society · 1983
Consider a system of equations of the form (i) ƒ(*)=4*) •»(#*)), where x = (xi,...,x m ), (x) = (i(x),...,0 n (x)) is an analytic mapping, and A(x) is a px q matrix of analytic functions.Given f(x) = {fi{x),..., f p (x))C°°, we seek C°° solutions g(y) = {gi{y),..., g q (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 /ç), 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 0 a (respectively, d a ) denote the ring of germs of analytic functions at a (respectively, the completion of 0 a in the Krull topology).Let xh a be the maximal ideal of Ô 0 .In the case k -R, let C°°(M) denote the algebra of C°° functions on M.There is a Taylor series homomorphism ƒ >-» f a from C°°{My onto ô P a .The mapping induces ring homomorphisms 0*: C°°(N) -> C°°(M), K' 0 {a) -+ 0a, and fc: ô# a ) -+ 0 a -Let $: C°°{N) C°°{My denote the module homomorphism over 0* defined by $(g) = A-(gofy.Let