Perturbations of polynomials and applications

TL;DR

This paper investigates how small changes in polynomial coefficients affect their roots and common divisors, applying these insights to matrix deformations and eigenvalue stability in finite-dimensional linear operators.

Contribution

It reinterprets the continuity of polynomial roots through deformation theory and analyzes the stability of polynomial common divisors under perturbations, with applications to matrix eigenvalues.

Findings

Stability of polynomial roots under coefficient perturbations

Continuity of roots in deformation framework

Applications to eigenvalue analysis of matrices

Abstract

After reconsidering the theorem of continuity of the roots of a polynomial in terms of its coefficients in the deformation framework, we study the stability of the greater common divisor of two polynomials compared to perturbations on their roots. We apply this results to the study of deformations of a linear operator in finite dimension and in particular to the roots study of deformed matrices.