[duetosymmetry]

I was just pondering about this. I imagine that moving a coefficient a small amount (and so long as the roots are not too close) that Newton-Raphson, initialized with a previously-known root, would converge very quickly to the new root while preserving the order.

If there truly are roots of multiplicity >= 2, then preserving the order of the multiple roots doesn't matter. Another issue is that Newton-Raphson might fail if the roots are too close.