Neighborhood radius estimation for Arnold's miniversal deformations of complex and p-adic matrices

Abstract

V.I. Arnold (1971) constructed a simple normal form to which all complex matrices B in a neighborhood U of a given square matrix A can be reduced by similarity transformations that smoothly depend on the entries of B. We calculate the radius of the neighborhood U. A.A. Mailybaev (1999, 2001) constructed a reducing similarity transformation in the form of Taylor series; we construct this transformation by another method. We extend Arnold's normal form to matrices over the field Qp of p-adic numbers and the field F((T)) of Laurent series over a field F.