To reiterate Serre's question (Open Problem 2.5 in Odlyzko's survey): *Must there be only finitely many polynomials having root discriminant below a given bound?*

With this answer I just want to note that the much stronger statement formulated in the last paragraph of user631's answer, which concerned the existence of a $c > 1$ such that all irreducible polynomials of discriminant $\leq (cd)^d$ are essentially cyclotomic, is false without additional assumptions. Since we may take $c = M(f)^2$ as Mahler proved, this is a natural statement to consider: its truth would have strengthened Lehmer's conjecture.

A counterexample is given by $x^d-x-1$, whose discriminant has absolute value $d^d + (-1)^{d}(d-1)^{d-1}$. We may take more generally any sequence of irreducible trinomials with $\pm 1$ coefficients and degree going to infinity. This follows by the explicit calculation of the discriminant of the general trinomial; the statement and simple derivation of the formula is given as Theorem 2 in Swan's 1962 paper *Factorization of polynomials over finite fields* in the Pacific Journal of Mathematics. (Another reference is Prasolov's book *Polynomials*, which reproduces the same calculation).

A counterexample of a rather different kind (in particular, having unbounded Mahler measure under all integer translations) is provided by the minimum polynomial of the generator $\zeta_n+\zeta_n^{-1} = 2\cos(2\pi/n)$ of the integer ring of the maximal totally real subfield of $\mathbb{Q}(\zeta_n)$. Further counterexamples would include the minimum polynomials of the generators over $\mathbb{Z}$ of the integer rings of other bounded index monogenic subfields, if such exist, of either $\mathbb{Q}(\zeta_n)$ or the splitting fields of the $\pm 1$ trinomials considered in the previous paragraph.

However, one may wish to restrict to reciprocal polynomials; for Lehmer's problem this would be sufficient. For these the statement seems very interesting, and could well be true; by the formula in Swan's article, it certainly holds for all trinomials. The above examples are of no use here: the only reciprocal $\pm 1$ trinomials are $x^{2n} \pm x^n +1$, and those have only cyclotomic factors.

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