Check these references

* Dmitri Alekseevsky, Andreas Kriegl, Mark Losik, Peter W. Michor, _[Choosing roots of polynomials smoothly][1]_, Israel J. Math **105** (1998) 203–233, math.CA/9801026,


* Andreas Kriegl, Mark Losik, Peter W. Michor, _[Choosing roots of polynomials smoothly, II][2]_, Israel J. Math. **139** (2004) 183–188, math.CA/0208228.


  [1]: https://arxiv.org/abs/math/9801026
  [2]: https://arxiv.org/abs/math/0208228