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Given $p\in\mathbb{Z}[x]$, assumed to have a real root, let $r$ be the largest real root of $p$. Now, given $f\in\mathbb{Z}[x]$ (without loss, of lesser degree than $p$), I would like to find out the sign of $f(r)$, using exact arithmetic. Are there known algorithms for that?

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  • $\begingroup$ Mmmmh, I don't understand which role $p$ is playing. Taking an arbitrary $a\in \mathbb{Z}$ and setting $p=(X-a)^{n+1}$, we see that in this case your question becomes: let $f\in \mathbb{Z}[x]$ of degree $\leq n$. Find out the sign of $f(a)$ using exact arithmetic...Do you know algorithms for this particular question ? $\endgroup$
    – GreginGre
    Sep 25, 2019 at 18:37
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    $\begingroup$ @GreginGre: I do not get your comment. Certainly when $r$ (or $a$, why the change of notation?) is an integer, then we can compute $f(a)$ exactly, it is an integer and its sign is no mystery. When $r$ is a general algebraic number seems another matter. $\endgroup$ Sep 25, 2019 at 18:51
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    $\begingroup$ Yes, there are fairly simple algorithms. I suggest you study Lecture VII (Sturm theory) in Chee K. Yap, Fundamental problems in algorithmic algebra. $\endgroup$ Sep 25, 2019 at 19:45
  • $\begingroup$ Thank you, the reference has it! $\endgroup$ Sep 25, 2019 at 23:41

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