Abstract: Consider a curve of polynomials of fixed degree n with only real roots and smoothly parameterized by a real parameter t. Can we find n smooth functions which parameterize the roots of P(t) for each t?

Many related results are presented. For example, the roots of P(t) may be parameterized smoothly, if P(t) is smooth and no two different roots meet infinitely flat. Moreover, there always exists a twice differentiable paramterization of the roots, if the curve P(t) is 3n-times continuously differentiable, and this conclusion is best possible.

Moreover, the following generalization of the above problem is investigated: Is it possible to lift a smooth curve in the orbit space of an orthogonal representation of a compact Lie group on a finite dimensional Euclidean vector space to a smooth curve in this vector space. Many results already found for polynomials can be transferred to this general situation. For example: A generic smooth curve in the orbit space admits a smooth lift, and this lift can even be chosen orthogonal to all orbits it meets, if the representation is polar. A sufficiently often differentiable (not necessarily generic) curve in the orbit space can be lifted once differentiably. A smooth (not necessarily generic) curve in the orbit space even allows a once differentiable lift which meets orbits orthogonally.