Let $f$ be a monic univariate polynomial with real coefficients:

$$f_A(x) = x^n + a_{n-1}x^{n-1} + ... + a_{0}$$

The values of $A=(a_{n-1},...,a_0)$ are unknown, but are estimated as $B=(b_{n-1},...,b_0)$ with error $\epsilon$. Therefore, $b_i - \epsilon \leq a_i \leq b_i + \epsilon$. Furthermore, the value of $\epsilon$ can be reduced arbitrarily, but cannot be set to 0. (Changing $\epsilon$ will alter the $B$ estimates). $lim_{\epsilon\rightarrow 0} B = A$

**Question 1) Is there any method of determining if $f_A$ has a repeated root using $B$ and the ability to reduce $\epsilon$?**

I've worked out how to isolate the roots of $f_A$ if $f_A$ does not have repeated roots as follows:

$D(A)$, the discriminant of $f_A$, must be non-zero. As computing $D(B)$ from the Sylvester resultant of $f_B$ and $f_B'$ can be accomplished with only $+$ and $*$ operators, $g(\epsilon,B)$, the error propagated while calculating the resultant, must strictly increase. We can decrease the value of $\epsilon$ until $D(B) + g(\epsilon,B) < 0$ or $D(B) - g(\epsilon,B) > 0$. This gives us the bound $|D(A)| \geq \min(|D(B) + g(\epsilon,B)|,|D(B) - g(\epsilon,B)|)$. This lower bound for $|D(A)|$ can be plugged into Rump's inequality to generate a root separation bound. Then Sturm's theorem can be applied until the roots are sufficiently isolated (including the error generated from $\epsilon$).

**Question 2) If $f_A$ has multiple roots, is there any way of isolating these roots?**

In this case, Sturm's theorem can isolate the roots to an arbitrary degree of precision, but without a root separation bound, there is no way to disambiguate between repeated roots and close roots.

If we try to calculate the number of distinct zeros as $n - \gcd(f_B,f_B')$, there will always be a remainder term $r$. By reducing $\epsilon$, we might be able to make $r$ arbitrarily close to 0, but we run into the same problem as with Sturm's theorem in that we don't have an bound on $r$ which differentiates between $r=0$ and $r$ is just small.

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