Implicit function theorem says: if at least one derivative of the function
at a point is not zero, then after a local change of variables the function
is linear. This holds in the smooth, formal and analytic categories.
If the function of one variable has a critical point, then (after a change
of coordinates and scaling) it becomes a monomial f(x)=x^p.
In general, the functions are often determined (locally) by just a few first
terms of their Taylor expansion. This phenomenon is called 'finite
determinacy'.
Functions that are not finitely determined are rare in a very strong sense.
In the case of maps of smooth germs one would expect such a finite
determinacy to hold generically too (for reasonably rich group of
equivalence). This indeed holds for the contact equivalence. But for the
left-right equivalence this happens only for some "nice pairs of dimensions"
(of the domain and the target), [Mather 1970].
Jointly with G.Belitski we study the finite determinacy of matrices over
complete rings (aka modules over formal germs). In this case the finite
determinacy implies the algebraizability (when a module is the stalk of an
algebraic sheaf) In this case even for the contact equivalence the notion
of nice/bad dimensions appears, we prove a simple description of these.
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