Lehmer's conjecture

Remarks

The condition $M(f) > 1$ excludes exactly the polynomials $\pm x^k \Phi_{n_1} \cdots \Phi_{n_r}$ built from cyclotomic polynomials, which have $M(f) = 1$. The problem goes back to D. H. Lehmer, who asked it in 1933. The conjecture is open. The smallest known value of $M(f) > 1$ is that of Lehmer's polynomial $x^{10} + x^9 - x^7 - x^6 - x^5 - x^4 - x^3 + x + 1$, about $1.17628$. Smyth proved the conjecture for nonreciprocal polynomials. If $f \in \Z[x]$ is irreducible of degree $d$, $f \ne \pm x$, and $x^d f(1/x) \ne \pm f(x)$, then $M(f) \ge M(x^3 - x - 1) \approx 1.3247$. Dobrowolski proved a lower bound of the form $M(f) \ge 1 + c \left( \frac{\log \log d}{\log d} \right)^3$ for an irreducible noncyclotomic $f$ of degree $d$, with an absolute constant $c > 0$.