# Sign of an integer polynomial at a real algebraic number

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?

• 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 ? Sep 25, 2019 at 18:37
• @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. Sep 25, 2019 at 18:51
• Yes, there are fairly simple algorithms. I suggest you study Lecture VII (Sturm theory) in Chee K. Yap, Fundamental problems in algorithmic algebra. Sep 25, 2019 at 19:45
• Thank you, the reference has it! Sep 25, 2019 at 23:41