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 Tougeron and Fisher 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∞-equations has a formal solution and the derivative satisfies a Lojasiewicz-type condition then the system has a C∞-solution.
- Artin’s approximation theorem
- Equations on groups
- Equations over local rings
- Finite determinacy
- Functional equations
- Tougeron’s Implicit Function Theorem
- Tougeron’s approximation theorem