Perturbations of Polynomials, Generalizations, and Applications

In the late nineteen-thirties F. Rellich developed the one parameter analytic perturbation theory of linear operators which culminated with the celebrated monograph of T. Kato. Rellich proved that the roots of a real analytic curve of monic univariate hyperbolic (all roots real) polynomials allow a real analytic parameterization. Using this he showed that the eigenvalues and the eigenvectors of a real analytic curve of symmetric matrices (or even unbounded selfadjoint operators in a Hilbert space with common domain of definition and compact resolvent) can be chosen real analytically. Smooth perturbations of polynomials have been studied intensively ever since, predominantly, one parameter perturbations of hyperbolic polynomials. In the last decade several new contributions to this subject appeared. Some of them are based on a recent deeper understanding of resolution of singularities, which opens new ways to study multiparameter perturbation of polynomials. In this research project perturbations of polynomials will be studied with emphasis on the smooth multiparameter complex (not necessarily hyperbolic) case. Resolution of singularities of quasianalytic function classes will constitute an integral part. The results will have applications to the perturbation theory of unbounded normal operators. This requires a differential calculus for quasianalytic function classes beyond Banach spaces. So the development of the convenient setting for quasianalytic Denjoy-Carleman differentiable mappings is a further main aim of this research project. In a second line of research the following natural generalization of the perturbation problem for polynomials will be investigated: Consider a rational complex finite dimensional representation of a reductive linear algebraic group. The algebra of invariant polynomials is finitely generated, and its embedding in the algebra of all polynomials on the representation space induces a projection to the categorical quotient. This projection can be identified with the mapping built by a system of generators. Given a smooth mapping into the categorical quotient, considered as subset of the affine complex space, we can ask whether there exists a smooth lift into the representation space. This problem of lifting mappings over invariants fits into the larger project of studying the analytic and geometric properties of orbit spaces. A closely related lifting problem is the question to what extend normalizations are pseudoimmersions, i.e., have the universal property of smooth immersions. Another goal of this research project is to find a natural smooth lifting condition for normalizations. Apart from the perturbation theory of linear operators and the study of the structure of orbit spaces, one may expect applications to the Cauchy problem in PDEs.

The question how regular the roots of a polynomial depending on parameters may be chosen is very fundamental with important applications to several mathematical disciplines such as partial differential equations, perturbation theory of linear operators, or singularity theory. Being absolutely elementary, this question could have been already posed with the invention of differential calculus 300 years ago. The conjecture of Spagnolo from 2000, as to whether the roots of a smooth curve of polynomials may be parameterized by absolutely continuous functions, was considered to be the main open problem in this field. We proved this conjecture. As a direct application one gets local solvability of certain systems of partial differential equation. The problem of choosing regular roots is naturally related to perturbation theory for linear operators which is ubiquitous in physics (e.g. in quantum mechanics) and engineering science. However, in the presence of some structural regularity of the operators, like selfadjointness or normality that usually are satisfied in applications, the perturbation results for linear operators are essentially stronger than those for polynomials. A goal of the project was to rigorously generalize to the infinite dimensional setting results on smooth perturbation of the spectral decomposition that we had previously obtained for finite dimensional matrices. The implementation of this goal led to the surprising discovery that the perturbation theory for normal operators, although normality being a weaker condition, works just as well as that for selfadjoint operators. The treatment of perturbation theory in infinite dimensions required a differential calculus for (in particular, quasianalytic) classes of mappings between general infinitely dimensional linear spaces. One possible approach, that is especially suited for many questions of global analysis, goes by the name convenient setting". We developed the convenient setting for Denjoy-Carleman classes; these are classes of functions given by growth conditions on the iterated derivatives with wide-ranging applications (e.g. in partial differential equations). The class is called quasianalytic if the functions in the class are uniquely determined by the sequence of all derivatives in any single point. Quasianalytic and non-quasianalytic classes have very different qualitative behaviour. Nevertheless, we found a uniform proof of the convenient setting for all Denjoy-Carleman classes no matter whether quasianalytic or not. We applied our convenient calculus to manifolds of mappings and diffeomorphism groups. The problem of choosing regular roots is a special case of a general abstract lifting problem. In fact, the roots of a polynomial may be permuted without changing the polynomial. Thus the space of polynomials of a fixed degree is identical to the space of orbits with respect to the group of permutations acting on the space of roots. In this picture a polynomial depending on parameters is a mapping in the orbit space and a choice of the roots is a lifting of this mapping over the orbit projection. We considered this problem for abstract group actions and partly generalized our results for polynomials.