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. 



Finding the new zeros of a "perturbed" polynomial
Assume that the roots of $p(x)$ are real and distinct. Then we may let $\mu > 0$ denote the minimum distance between any two roots. Let $q(x)$ be any polynomial of degree $< n$ such that $|q(\alpha)| < (\mu/2)^n$ for any root $\alpha$ of $p(x)$. Then I claim that $g(x) = p(x) + q(x)$ has real roots. 


Note that for a root $\alpha$ of $p(x)$, we have $|g(\alpha)| = |q(\alpha)| < (\mu/2)^n$. Yet


$$|g(\alpha)| = \prod |\alpha - \beta\_i|$$


where the $\beta\_i$ are the roots of $g$. It follows that $g(x)$ has a root $\beta$ such that
$|\alpha - \beta| < \mu/2$. By the triangle inequality, $\alpha$ is uniquely determined by $\beta$ and this inequality.
In particular, $g(x)$ has exactly one root within $\mu/2$ of
each root of $p(x)$. Since $g(x)$ and $p(x)$ have the same degree, this exhausts all the roots of $g(x)$. Yet if $\beta$ was complex, then $|\alpha - \beta| = |\alpha -
\overline{\beta}| < \mu/2$, a contradiction.


If the roots of $p(x)$ are not distinct, then one is in trouble, as the example 
$p(x) = x^2$ shows.


This can be thought of as an application of the $\mathbf{R}$-version of Krasner's Lemma, and is, in particular, an (exact) analog of the argument that the splitting field of a separable polynomial $f(x)$ over the $p$-adics is locally constant. 




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EDIT:
I am confused about your second question. Let $f(x)$ be any polynomial of degree $n$ with real coefficients. Let $A$ and $B$ denote the maximum and minimum values of $f(x)$ on the interval $[0,n]$. Choose any $M > \max(|A|,|B|)$.
A polynomial of degree $n$ may be defined by specifying $n+1$ of its values.
Let $p(x)$ be the polynomial of degree $n$ such that
the values $p(k)$ for $k = 0,\ldots,n$ are alternatively $-M$ and $+M$.
By the intermediate value theorem, $p(x)$ has $n$ real roots.


Let $q(x) = f(x) - p(x)$. The signs of $q(k)$ also alternate for $k = 0,
\ldots,n$ by construction. It follows that $q(x)$ also has $n$ real roots.
Yet $p(x) + q(x) = f(x)$, and thus the sum of two polynomials with real roots does not satisfy any restrictions.


