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Let $K,t>0$ be two real constants. Consider the set $P(K,t)$ of monic polynomials with integral coefficients with all roots real, positive, and lying in the interval $(0,K)$, and with absolute trace equal to $t$. By `absolute trace' I mean the sum of all the roots divided by the degree of the polynomial. Is $P(K,t)$ finite?

My understanding is that the answer is `yes' if I additionally bound the degree of the polynomials, or if $t<1.7$.

(This is not precisely the question from the title, but I would expect the answers to be related.)

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    $\begingroup$ Let $n$ be prime, $\zeta = \zeta_{p^2}$ a $p^2$th root of unity. Then the normalized trace of the totally real algebraic integer $$2 + \zeta + \zeta^{-1}$$ is equal to $2$. $\endgroup$
    – user491858
    Commented May 10 at 20:51

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