Perturbation of polynomials and lifting over invariants

In the late nineteen-thirties F. Rellich developed the one parameter analytic perturbation theory of linear operators which culminates with the celebrated monograph of T. Kato. To study the behaviour of the eigenvalues of symmetric matrices depending analytically on a parameter, Rellich proved that the roots of a real analytic curve of monic univariate hyperbolic (all roots real) polynomials with fixed degree allow a real analytic parameterization. Smooth perturbations of polynomials have been studied intensively ever since, predominantly, one parameter perturbations of hyperbolic polynomials. Only recently, multiparameter real analytic perturbation theory for hyperbolic polynomials was tackled by K. Kurdyka and L. Paunescu as well as one parameter smooth perturbation theory of complex polynomials without restrictions by the applicant. In this research project the study of perturbations of polynomials shall be continued with emphasis on the smooth multiparameter complex case. I expect that (under suitable conditions) a smooth family of polynomials allows parameterizations of its roots with locally integrable first order partial derivatives. The problem of choosing the roots of a family of polynomials in a regular way has a natural generalization: Consider a complex finite dimensional representation V of a reductive algebraic group G. The algebra of G- invariant polynomials on V is finitely generated, say by p_1,...,p_n, and its embedding in the algebra of polynomials on V defines a projection of V to the categorical quotient V//G. This projection can be identified with the mapping p=(p_1,...,p_n). Given a smooth mapping F into V//G, considered as subset of the affine complex space, we can ask whether there exists a smooth lift of F into V. The investigation of that lifting problem constitutes the second part of the research project. It is reasonable to expect that the results for the perturbation problem for polynomials generalize to this setting. This has already be demonstrated for the real counterpart (which corresponds to the hyperbolic case), where V is a real finite dimensional representation of a compact Lie group G. One can expect applications to the perturbation theory for linear operators, to the theory of PDE, and to the study of the structure of orbit spaces.