Seminar in Real & Complex Geometry

Thursday, 08.11.2012, 16:15-17:30, Schreiber building, room 210

Dmitry Kerner, Ben-Gurion University

On finite determinacy


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.