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.
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