On Germs of DIfferentiable Functions In Two Variables | Zendy

Masahiro Shiota | Zendy

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On Germs of DIfferentiable Functions In Two Variables

Author(s) -

Masahiro Shiota

Publication year - 1973

Publication title -

publications of the research institute for mathematical sciences

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.786

H-Index - 39

eISSN - 1663-4926

pISSN - 0034-5318

DOI - 10.2977/prims/1195192563

Subject(s) - mathematics , differentiable function , diffeomorphism , analytic function , ring (chemistry) , formal power series , pure mathematics , polynomial ring , taylor series , taylor's theorem , ideal (ethics) , power series , polynomial , maximal ideal , germ , combinatorics , discrete mathematics , mathematical analysis , philosophy , chemistry , organic chemistry , epistemology

In Q4H we have shown some sufficient conditions for a germ (of a differentiable function in two variables) to be transformed into an analytic one or a polynomial through a change of coordinates. Here we shall refine the result above and find a necessary and sufficient condition. We denote respectively by 0, $ the rings of germs at 0 in R of real analytic and C°°-functions, and by J" the ring of formal power series in 2 indeterminates over R. If A is one of the above rings, let m(A) denote the maximal ideal of A. Let Ta denote the Taylor expansion at a. Elements f , g o f & are called equivalent if there exists a local diffeomorphism (of class C°°) r of R around 0 such that f°t=g. If an element / of m(«f) can be factorized into the following form

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