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
leftright 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.
