Perturbation theory of polynomials and linear operators

TL;DR

This survey reviews how roots of parameter-dependent polynomials and spectral decompositions of linear operators vary with parameters, highlighting differences between hyperbolic and general complex cases, with detailed proofs of key theorems.

Contribution

It provides a comprehensive overview of perturbation theory for polynomials and operators, including detailed proofs of Rellich's and Bronshtein's theorems, and discusses regularity and spectral dependence.

Findings

Differentiability of roots varies with polynomial regularity.

Hyperbolic polynomials have real roots with specific perturbation properties.

The survey includes full proofs of key theorems.

Abstract

This survey revolves around the question how the roots of a monic polynomial (resp. the spectral decomposition of a linear operator), whose coefficients depend in a smooth way on parameters, depend on those parameters. The parameter dependence of the polynomials (resp. operators) ranges from real analytic over $C^\infty$ to differentiable of finite order with often drastically different regularity results for the roots (resp. eigenvalues and eigenvectors). Another interesting point is the difference between the perturbation theory of hyperbolic polynomials (where, by definition, all roots are real) and that of general complex polynomials. The subject, which started with Rellich's work in the 1930s, enjoyed sustained interest through time that intensified in the last two decades, bringing some definitive optimal results. Throughout we try to explain the main proof ideas; Rellich's theorem and Bronshtein's theorem on hyperbolic polynomials are presented with full proofs. The survey is written for readers interested in singularity theory but also for those who intend to apply the results in other fields.