Lehmer's conjecture for polynomials satisfying a congruence divisibility condition and an analogue for elliptic curves
Let \(h(\alpha)\) be the absolute logarithmic height, so that Lehmer's conjecture asserts that \newline \(\sum_{f(\alpha)=0} h(\alpha) \geq C>0\) for each monic polynomial \(f \in \mathbb{Z}[X]\) without roots of unity. The author proves that for all \(\varepsilon>0\) there is a constant \(C(\varepsilon)>0\) such that for each monic polynomial \(f \in \mathbb{Z}[X]\) without roots of unity, \(\deg f=D\), such that \(f(X)\) is divisible by \(X^{n-1}+\dots+X+1\) in the ring \((\mathbb{Z}/m \mathbb{Z})[X]\) for some integers \(m\geq 2\) and \(n\geq \max(\varepsilon D, 2)\) then \[ \sum_{f(\alpha)=0} h(\alpha) \geq C(\varepsilon) \log m. \] The proof uses some Fejér kernel estimate which in the context of Mahler measure was first used by \textit{P. E. Blanksby} and \textit{H. L. Montgomery} in 1971 [Acta Arith. 18, 355--369 (1971; Zbl 0221.12003)]. Some more precise conditions on a polynomial under which Lehmer's conjecture holds are also given. Finally, the author gives an elliptic analogue of this result.
scientific article; zbMATH DE number 6142433
scientific article