TITLE: Finding the new zeros of a "perturbed" polynomial
QUESTION [14 upvotes]: Given a univariate polynomial with real coefficients, p(x), with degree n, suppose we know all the zeros xj, and they are all real.  Now suppose I perturb each of the coefficients pj (for j ≤ n) by a small real perturbation εj.  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.

REPLY [4 votes]: The following criterion might help (see for example these lecture notes): 

all roots of a polynomial are real if
  and only if the $n \times n$ Hankel
  matrix defined by $H_{ij}=s_k$ for all
  $i+j=k-2$ is positive definite,

where $s_k$ is the sum of the $k$-th powers of the $n$ roots, i.e. $s_k = \sum_{i=1}^n x_i^k$ with your notation. These sums can be computed directly from the polynomial coefficients using Newton's identities.