Check these references
Choosing roots of polynomials smoothly, math.CA/9801026
Choosing roots of polynomials smoothly, II, math.CA/0208228
Dmitri Alekseevsky, Andreas Kriegl, Mark Losik, Peter W. Michor, Choosing roots of polynomials smoothly, Israel J. Math 105 (1998) 203–233, math.CA/9801026,
Andreas Kriegl, Mark Losik, Peter W. Michor, Choosing roots of polynomials smoothly, II, Israel J. Math. 139 (2004) 183–188, math.CA/0208228.
