Perturbation theory of polynomials and linear operators

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.