| Résume | The Perturbation of Polynomials and Linear Operators is a classical subject which started with Rellich's work in the 1930s. The parameter dependence of the polynomials (resp. operators) ranges from real analytic over C^∞ to differentiable of finite order with often drastically different regularity results for the roots (resp. eigenvalues and eigenvectors). In this talk I will present several recent results such as an optimal estimate of Sobolev regularity of roots monic complex polynomials of one variable with coefficients depending smoothly on one real parameter, multiparameter versions of this result, and the problem of continuity of “coefficients to roots map” with respect to the C^d and the Sobolev norms. Recently, these results were reinterpreted by Antonini, Cavalletti, and Lerario in terms of Wasserstein metric in order to study optimal transport between algebraic hypersurfaces in the complex projective space.
In some cases better regularity of the roots can be obtained under additional assumptions of non-oscilation or finiteness of ordrer of contact between the roots, that is an interesting property if one works with coefficients definable in o-minimal structures.
(based mainly on the joint work with Armin Rainer) |
