Given a univariate polynomial with real coefficients, p(x), with degree n, suppose we know all the zeros x_{j}, and they are all real. Now suppose I perturb each of the coefficients p_{j} (for j ≤ n) by a small real perturbation ε_{j}. What are the conditions on the perturbations (**edit:** for example, how large can they be, by some measure) so that the solutions remain real?

Some thoughts: surely people have thought of this problem in terms of a differential equation valid for small ε that lets you take the known solutions to the new solutions. Before I try to rederive that, does it have a name? This seems like a pretty general solution technique, but perhaps it is so general as to be intractable practically, which might explain why I don't know about it.

If you ignore the smallness of the perturbation, then there is a general question here which seems like it might be related to Horn's problem: given two real polynomials p(x) and q(x) of degree n and their strictly real roots, what can you infer about the roots of p(x)+q(x)? This question is very interesting and I would love to hear what people know about it. But I'm also happy with the perturbed subproblem above, assuming it is indeed simpler.