Proposed lower bound on the Mahler measure for polynomials with integer coefficients
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 formula_1 such that every polynomial with integer coefficients formula_2 satisfies one of the following properties:
There are a number of definitions of the Mahler measure, one of which is to factor formula_4 over formula_11 as
formula_12
and then set
formula_13
The smallest known Mahler measure (greater than 1) is for "Lehmer's polynomial"
formula_14
for which the Mahler measure is the Salem number
formula_15
It is widely believed that this example represents the true minimal value: that is, formula_16 in Lehmer's conjecture.
Motivation.
Consider Mahler measure for one variable and Jensen's formula shows that if formula_17 then 
formula_13
In this paragraph denote　formula_19 , which is also called Mahler measure.
If formula_20 has integer coefficients, this shows that formula_21 is an algebraic number so formula_22 is the logarithm of an algebraic integer. It also shows that formula_23 and that if formula_24 then formula_20 is a product of cyclotomic polynomials i.e. monic polynomials whose all roots are roots of unity, or a monomial polynomial of formula_7 i.e. a power formula_27 for some formula_28 .
Lehmer noticed that formula_24 is an important value in the study of the integer sequences formula_30 for monic formula_20 . If formula_20 does not vanish on the circle then formula_33. If formula_20 does vanish on the circle but not at any root of unity, then the same convergence holds by Baker's theorem (in fact an earlier result of Gelfond is sufficient for this, as pointed out by Lind in connection with his study of quasihyperbolic toral automorphisms). As a result, Lehmer was led to ask
whether there is a constant formula_35 such that formula_36 provided formula_20 is not cyclotomic?,
or
given formula_35, are there formula_20 with integer coefficients for which formula_40?
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 formula_2 be an irreducible monic polynomial of degree formula_42.
Smyth proved that Lehmer's conjecture is true for all polynomials that are not reciprocal, i.e., all polynomials satisfying formula_43.
Blanksby and Montgomery and Stewart independently proved that there is an absolute constant formula_44 such that either formula_8 or
formula_46
Dobrowolski improved this to
formula_47
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 formula_48 be an elliptic curve defined over a number field formula_49, and let formula_50 be the canonical height function. The canonical height is the analogue for elliptic curves of the function formula_51. It has the property that formula_52 if and only if formula_53 is a torsion point in formula_54. The elliptic Lehmer conjecture asserts that there is a constant formula_55 such that
formula_56 for all non-torsion points formula_57,
where formula_58. If the elliptic curve "E" has complex multiplication, then the analogue of Dobrowolski's result holds:
formula_59
due to Laurent. For arbitrary elliptic curves, the best known result is
formula_60
due to Masser. For elliptic curves with non-integral j-invariant, this has been improved to
formula_61
by Hindry and Silverman.
Restricted results.
Stronger results are known for restricted classes of polynomials or algebraic numbers.
If "P"("x") is not reciprocal then
formula_62
and this is clearly best possible. If further all the coefficients of "P" are odd then
formula_63
For any algebraic number "α", let formula_64 be the Mahler measure of the minimal polynomial formula_65 of "α". If the field Q("α") is a Galois extension of Q, then Lehmer's conjecture holds for formula_65.
Relation to structure of compact group automorphisms.
The measure-theoretic entropy of an ergodic automorphism of a compact metrizable abelian group is known to be given by the logarithmic Mahler measure of a polynomial with integer coefficients if it is finite. As pointed out by Lind, this means that the set of possible values of the entropy of such actions is either all of formula_67 or a countable set depending on the solution to Lehmer's problem. Lind also showed that the infinite-dimensional torus either has ergodic automorphisms of finite positive entropy or only has automorphisms of infinite entropy depending on the solution to Lehmer's problem. Since an ergodic compact group automorphism is measurably isomorphic to a Bernoulli shift, and the Bernoulli shifts are classified up to measurable isomorphism by their entropy by Ornstein's theorem, this means that the moduli space of all ergodic compact group automorphisms up to measurable isomorphism is either countable or uncountable depending on the solution to Lehmer's problem.
References.
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