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• W2136115389 abstract "Let f and g be holomorphic function germs at 0 in Cn x Cl = {(x, s)}. If dx g A dx f = 0 and if f(x) = f(x, 0) is not a power or a unit, then there exists a germ A at 0 in C x C' such that g(x, s) = A(f(x, s), s). The result has the implication that the notion of an RL-morphism in the unfolding theory of foliation germs generalizes that of a right-left morphism in the function germ case. The notion of an RL-morphism in the unfolding theory of foliation singularities was introduced in [5] to describe the determinacy results and in [6] the versality theorem for these morphisms is proved. This note, which should be considered as an appendix to [5 or 6], contains a factorization theorem implying that an RLmorphism is a generalization of a right-left morphism in the unfolding theory of function germs. It depends on the Mattei-Moussu factorization theorem [1] and is a generalization of a result of Moussu [2]. A codim 1 foliation germ at 0 in C' is a module F = (w) over the ring of holomorphic function germs generated by a germ of an integrable 1-form w (see ?2). An unfolding of F with parameter space Cm = {t} is a codim 1 foliation germ P= () at 0 in C' x Cm with a generator Co whose restriction to Cn x {0} is . We let Ft be the foliation germ generated by the restriction wt of C to Cn x {t}. Let Y' be another unfolding of F with parameter space Cl = {s}. A morphism from Y' to Y is a holomorphic map germ 4I: (Cn x Cl, 0) -* (Cn x Cm, 0) such that (a) ID(x, s) = ( (s, x), ? (s)) for some holomorphic map germs q: (Cn X clC 0) -* (Cn? 0) and V: (Cl,0) -* (Cm, 0), (b) +(x,0) = x and (c) the pull back 4I* of CD by 4I generates Y'. Thus, if we set q$8(x) = X$(x, s), we may think of (q$) as a family of local coordinate changes of (Cn, 0). For an RL-morphism, in place of (c), we only require that q$Uw,(8) generates F' for each s (see (2.1) Definition). Our previous result shows that if F has a generator of the form df for some holomorphic function germ f (strong first integral for F), then every unfolding of F admits a generator of the form df with f an unfolding of f. In the unfolding theory of function germs, there are notions of a right morphism and a right-left morphism. The former involves coordinate changes in the source space (Cn, 0), whereas the latter involves coordinate changes in the target space C as well. It is not difficult to see that our morphism generalizes a right morphism in the sense that when F admits a strong first integral f, then it becomes a (strict) right morphism in the unfolding theory of f. For a foliation without first integrals, it may not seem relevant to talk about Received by the editors May 5, 1987 and, in revised form, July 20, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 32A10, 32G11; Secondary 58C27, 58F14. ?1988 American Mathematical Society 0002-9939/88 $1.00 + $.25 per page" @default.
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• W2136115389 date "1988-09-01" @default.
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• W2136115389 title "A Factorization Theorem for Unfoldings of Analytic Functions" @default.
• W2136115389 doi "https://doi.org/10.2307/2047474" @default.
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