Transformations of Germs of Differentiable Functions through ...
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Transformations of Germs of Differentiable Functions through ...
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Transformations of Germs of Differentiable Functions through Changes of Local Coordinates By Masahiro SHIOTA §1. Introduction We generalize the results in N. Levinson p], £4] and J. Cl. Tougeron Q8] which show that some germs of differentiate (or analytic) functions are transformed through changes of coordinates into polynomials in one (or two) variable with coefficients which are germs in the other variables (§3). H. Whitney [9] has shown that x y( y— x)( y— (3 + *)#)( J-r(0*) (where Y is a transcendental function and f(0) = 4) cannot be transformed into any polynomial through analytic changes of coordinates (locally at the origin), and we prove this function cannot be transformed even through differentiable changes of coordinates (§4). In view of Thorn's Principle, that is, for a germ of an analytic function f having 0 as a topologically isolated singularity, the variety /~1(0) determines the function f, we study particularly whether a germ f$ (such that 0(0) >0) can be transformed into / (§5). As a corollary of the sequence we obtain a sufficient condition for a germ of a differentiable function in two variables to be transformed into a germ of an analytic function (§6). In Appendix we show canonical forms of germs which have un point de naissance or un point critique du type queue d'aronde which is due to Cerf [2]. The method of proofs is to use almost all theorems in Malgrange The author thanks Professor Adachi for his criticism. § 2,
Definitions
According to Malgrange Q6j we denote respectively by On (or 0( Communicated by S. Matsuura, October 4, 1972,
124
MASAHIRO SHIOTA
£n (or #(x)) the rings of germs at 0 in Rw of real analytic and C°°-functions, and by IFn (or ^"(#)) the ring of formal power series in n indeterminates over R. One has a mapping T: gn-+!Fn (Taylor expansion at 0). We regard On^.^n. We denote by w(0 w ) (resp. m(0 and g l 5 ... 9 qn^^n such that
