I assume you are considering polynomials with integer coefficients.
You've taken the wrong starting point for your investigations; your goal is to find good estimates for the real roots. Looking for an algebraic formula so that you can evaluate it to enough precision is something you can do, but it's not really the right thing to do here. (unless, of course, "the k-th largest real root of a polynomial" is one of your algebraic operations)
A much better starting point is to use Sturm's theorem to isolate the roots of the polynomial. You can then produce better estimates by binary search, but if that's too slow, you can use Newton's method to quickly produce estimates of high precision.
But that's just about finding certificates. There's still the question of what certificates can exist.
First off, I will point out that you can directly compute whether or not two of the roots are exactly $k$ units apart, e.g. by computing $\gcd(p(x), p(x-k))$. You will also have to decide what you want to do about repeated roots and deal with appropriately. I assume you will deal with these case specially.
If we know the two roots are not exactly $k$ units apart, that means that you can produce an estimate of sufficient precision to prove that they are either greater or less than $k$ units apart. e.g. there are two kinds of certificates:
The first kind (proof in the negative) is
- $a$ is not a root of $p$
- $p$ has no roots in $(a-k, a)$
- $p$ has three roots in $(a, \infty)$
The second kind (proof in the positive) is
- $a$ is not a root of $p$
- $p$ has at least two roots in $(a-k,a)$
- $p$ has two roots in $(a, \infty)$
A certificate can be verified by using Sturm's theorem. Now, your question about the size of a certificate boils down to finding how many bits of precision you need to represent $a$.
In other words, what are the bounds on the possible values of $a-b-k$, where $a,b$ are roots of $f$?
I'm not sure of a great approach, but one that should give you something is to observe that all of these values are roots of the polynomial:
$$ g(x) = \mathop{\text{Res}_y}(f(y), f(x + y + k)) $$
Why? Recall that the resultant of two monic polynomials is the product of all differences of their roots, so
$$ g(x) = c^{d^2} \prod_{a,b}(b - (a - x - k))
= \prod_{a,b} (x - (a-b-k))$$
where $c$ is the leading coefficient and $d$ is the degree of $f$. (maybe I've written the formula for $-g(x)$ instead of $g(x)$; I'm never sure on the sign)
So the question is to find estimates for how large the coefficients $g$ can be, and then once you know that, find estimates to how close a root of $g$ can be to zero.
(or, alternatively, find the largest magnitude that a root of the reverse polynomial of $g$ can have; the roots of the reverse polynomial are the inverses of the roots of $g$)
