Seminar in Real & Complex Geometry

Thursday, 23.10.2014, 11:00-12:30, Schreiber building, room 209

Dmitry Kerner, Ben-Gurion University

Finite determinacy of matrices


Consider the hypersurface singularity {f=0}, where f is a power series with complex coefficients. If the singularity is isolated then f is determined (up to a change of coordinates) by the first N terms of the Taylor expansion (for some large enough N). This phenomenon is called finite determinacy. It has been extensively studied for maps of (germs) of spaces with respect to various subgroups. The finite determinacy implies algebraizability: various analytic/formal objects can be transformed to the algebraic ones. Quite often one needs the (finite) determinacy with respect to some special types of "admissible deformations". In the case of hypersurfaces with non-isolated singularities this study was initiated by de-Jong(s), Pelikaan, Siersma and van Straten. In the case of matrices such admissible deformations are often imposed by the applications. For example, in Commutative Algebra one considers modules over a given ring/singularity, in Representation Theory one wants to preserve the characteristic polynomial of the matrix. We develop the theory of (finite) determinacy for matrices (over a variety of local rings, with respect to a variety of group actions and for various admissible deformations). We give the method of computing the biggest sublocus of the deformation along which the "determinacy holds". The results are ready-to-use, written via some invariants of the matrix (e.g. the Fitting ideals and their singular loci). Joint work with G. Belitski.