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Let P(x)\in\mathbb{Z}[x] be an irreducible monic polynomial of degree D. Smyth [5] proved that Lehmer's conjecture is true for all polynomials that are not reciprocal, i.e., all polynomials satisfying x^DP(x^{-1})\ne P(x). Blanksby and Montgomery[6] and Stewart[7] independently proved that there is an absolute constant C>1 such that either \mathcal{M}(P(x))=1 or[8] \log\mathcal{M}(P(x))\ge \frac{C}{D\log D}. Dobrowolski [9] improved this to \log\mathcal{M}(P(x))\ge C\left(\frac{\log\log D}{\log D}\right)^3. Dobrowolski obtained the value C ≥ 1/1200 and asymptotically C > 1-ε for all sufficiently large D. Voutier obtained C ≥ 1/4 for D ≥ 2.[10] Elliptic Analogues Let E/K be an elliptic curve defined over a number field K, and let \hat{h}_E:E(\bar{K})\to\mathbb{R} be the canonical height function. The canonical height is the analogue for elliptic curves of the function (\deg P)^{-1}\log\mathcal{M}(P(x)). It has the property that \hat{h}_E(Q)=0 if and only if Q is a torsion point in E(\bar{K}). The elliptic Lehmer conjecture asserts that there is a constant C(E/K)>0 such that \hat{h}_E(Q) \ge \frac{C(E/K)}{D} for all non-torsion points Q\in E(\bar{K}), where D=[K(Q):K]. If the elliptic curve E has complex multiplication, then the analogue of Dobrowolski's result holds: \hat{h}_E(Q) \ge \frac{C(E/K)}{D} \left(\frac{\log\log D}{\log D}\right)^3 , due to Laurent.[11] For arbitrary elliptic curves, the best known result is[11] \hat{h}_E(Q) \ge \frac{C(E/K)}{D^3(\log D)^2}, due to Masser.[12] For elliptic curves with non-integral j-invariant, this has been improved to[11] \hat{h}_E(Q) \ge \frac{C(E/K)}{D^2(\log D)^2}, by Hindry and Silverman.[13] Restricted results Stronger results are known for restricted classes of polynomials or algebraic numbers. If P(x) is not reciprocal then M(P) \ge M(x^3 -x - 1) \approx 1.3247 and this is clearly best possible.[14] If further all the coefficients of P are odd then[15] M(P) \ge M(x^2 -x - 1) \approx 1.618 . If the field Q(α) is a Galois extension of Q then Lehmer's conjecture holds.[15] Lehmer's conjecture, also known as the Lehmer's Mahler measure problem, is a problem in number theory raised by Derrick Henry Lehmer.[1] The conjecture asserts that there is an absolute constant \mu>1 such that every polynomial with integer coefficients P(x)\in\mathbb{Z}[x] satisfies one of the following properties: The Mahler measure \mathcal{M}(P(x)) of P(x) is greater than or equal to \mu. P(x) is an integral multiple of a product of cyclotomic polynomials or the monomial x, in which case \mathcal{M}(P(x))=1. (Equivalently, every complex root of P(x) is a root of unity or zero.) There are a number of definitions of the Mahler measure, one of which is to factor P(x) over \mathbb{C} as P(x)=a_0 (x-\alpha_1)(x-\alpha_2)\cdots(x-\alpha_D), and then set \mathcal{M}(P(x)) = |a_0| \prod_{i=1}^{D} \max(1,|\alpha_i|). The smallest known Mahler measure (greater than 1) is for "Lehmer's polynomial" P(x)= x^{10}+x^9-x^7-x^6-x^5-x^4-x^3+x+1 \,, for which the Mahler measure is the Salem number[2] \mathcal{M}(P(x))=1.176280818\dots \ . It is widely believed that this example represents the true minimal value: that is, \mu=1.176280818\dots in Lehmer's conjecture.[3][4] |
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