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A strong version of implicit function theorem. (English) Zbl 1348.47049

Summary: We suggest the necessary/sufficient criteria for existence of a (order-by-order) solution \(y(x)\) of a functional equation \(F(x,y)=0\) over a ring. In full generality, the criteria hold in the category of filtered groups, this includes the wide class of modules over (commutative, associative) rings. The classical Implicit Function Theorem and its strengthening obtained by J.-C. Tougeron [Ann. Inst. Fourier 18, No. 1, 177–240 (1968; Zbl 0188.45102)] and B. Fisher [Proc. Am. Math. Soc. 125, No. 11, 3185–3189 (1997; Zbl 0883.13021)] appear to be (weaker) particular forms of the general criterion. We obtain a special criterion for solvability of equations arising from group actions \(g(w)=w+u\), here \(u\) is “small”. As an immediate application we re-derive the classical criteria of determinacy, in terms of the tangent space to the orbit. Finally, we prove the Artin-Tougeron-type approximation theorem: if a system of \(C^\infty\)-equations has a formal solution and the derivative satisfies a Łojasiewicz-type condition then the system has a \(C^\infty\)-solution.

MSC:

47J07 Abstract inverse mapping and implicit function theorems involving nonlinear operators
26B10 Implicit function theorems, Jacobians, transformations with several variables
30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
39B05 General theory of functional equations and inequalities
65Q20 Numerical methods for functional equations
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