Lehmer's conjecture


Lehmer's conjecture, also known as the Lehmer's Mahler measure problem, is a problem in number theory raised by Derrick Henry Lehmer. The conjecture asserts that there is an absolute constant such that every polynomial with integer coefficients satisfies one of the following properties:
There are a number of definitions of the Mahler measure, one of which is to factor over as
and then set
The smallest known Mahler measure is for "Lehmer's polynomial"
for which the Mahler measure is the Salem number
It is widely believed that this example represents the true minimal value: that is, in Lehmer's conjecture.

Motivation

Consider Mahler measure for one variable and Jensen's formula shows that if then
In this paragraph denote　, which is also called Mahler measure.
If has integer coefficients, this shows that is an algebraic number so is the logarithm of an algebraic integer. It also shows that and that if then is a product of cyclotomic polynomials i.e. monic polynomials whose all roots are roots of unity, or a monomial polynomial of i.e. a power for some .
Lehmer noticed that is an important value in the study of the integer sequences for monic . If does not vanish on the circle then and this statement might be true even if does vanish on the circle. By this he was led to ask
or
Some positive answers have been provided as follows, but Lehmer's conjecture is not yet completely proved and is still a question of much interest.

Partial results

Let be an irreducible monic polynomial of degree.
Smyth proved that Lehmer's conjecture is true for all polynomials that are not reciprocal, i.e., all polynomials satisfying.
Blanksby and Montgomery and Stewart independently proved that there is an absolute constant such that either or
Dobrowolski improved this to
Dobrowolski obtained the value C ≥ 1/1200 and asymptotically C > 1-ε for all sufficiently large D. Voutier in 1996 obtained C ≥ 1/4 for D ≥ 2.

Elliptic analogues

Let be an elliptic curve defined over a number field, and let be the canonical height function. The canonical height is the analogue for elliptic curves of the function. It has the property that if and only if is a torsion point in. The elliptic Lehmer conjecture asserts that there is a constant such that
where. If the elliptic curve E has complex multiplication, then the analogue of Dobrowolski's result holds:
due to Laurent. For arbitrary elliptic curves, the best known result is
due to Masser. For elliptic curves with non-integral j-invariant, this has been improved to
by Hindry and Silverman.

Restricted results

Stronger results are known for restricted classes of polynomials or algebraic numbers.
If P is not reciprocal then
and this is clearly best possible. If further all the coefficients of P are odd then
For any algebraic number α, let be the Mahler measure of the minimal polynomial of α. If the field Q is a Galois extension of Q, then Lehmer's conjecture holds for.