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

TL;DR

This paper calculates the neighborhood radius for Arnold's miniversal deformations of matrices over complex, p-adic, and Laurent series fields, extending Arnold's normal form to these settings and providing a new construction method.

Contribution

It computes the neighborhood radius for Arnold's normal form and extends the deformation theory to p-adic and Laurent series matrices with a novel construction approach.

Findings

Calculated the neighborhood radius for Arnold's normal form.

Extended Arnold's normal form to p-adic and Laurent series matrices.

Provided a new method for constructing the similarity transformation.

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 $\mathbb Q_p$ of $p$-adic numbers and the field $\mathbb F((T))$ of Laurent series over a field $\mathbb F$.