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 μ > 1 such that every polynomial with integer coefficients P ( x ) ∈ Z [ x ] satisfies one of the following properties:
The Mahler measure M ( P ( x ) ) of P ( x ) is greater than or equal to μ . P ( x ) is an integral multiple of a product of cyclotomic polynomials or the monomial x , in which case 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 C as
P ( x ) = a 0 ( x − α 1 ) ( x − α 2 ) ⋯ ( x − α D ) , and then set
M ( P ( x ) ) = | a 0 | ∏ i = 1 D max ( 1 , | α 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
M ( P ( x ) ) = 1.176280818 … . It is widely believed that this example represents the true minimal value: that is, μ = 1.176280818 … in Lehmer's conjecture.
Consider Mahler measure for one variable and Jensen's formula shows that if P ( x ) = a 0 ( x − α 1 ) ( x − α 2 ) ⋯ ( x − α D ) then
M ( P ( x ) ) = | a 0 | ∏ i = 1 D max ( 1 , | α i | ) . In this paragraph denote m ( P ) = log ( M ( P ( x ) ) , which is also called Mahler measure.
If P has integer coefficients, this shows that M ( P ) is an algebraic number so m ( P ) is the logarithm of an algebraic integer. It also shows that m ( P ) ≥ 0 and that if m ( P ) = 0 then P is a product of cyclotomic polynomials i.e. monic polynomials whose all roots are roots of unity, or a monomial polynomial of x i.e. a power x n for some n .
Lehmer noticed that m ( P ) = 0 is an important value in the study of the integer sequences Δ n = Res ( P ( x ) , x n − 1 ) = ∏ i = 1 D ( α i n − 1 ) for monic P . If P does not vanish on the circle then lim | Δ n | 1 / n = M ( P ) and this statement might be true even if P does vanish on the circle. By this he was led to ask
whether there is a constant
c > 0 such that
m ( P ) > c provided
P is not cyclotomic?,
or
given
c > 0 , are there
P with integer coefficients for which
0 < m ( P ) < c ?
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.
Let P ( x ) ∈ Z [ x ] be an irreducible monic polynomial of degree D .
Smyth proved that Lehmer's conjecture is true for all polynomials that are not reciprocal, i.e., all polynomials satisfying x D P ( x − 1 ) ≠ P ( x ) .
Blanksby and Montgomery and Stewart independently proved that there is an absolute constant C > 1 such that either M ( P ( x ) ) = 1 or
log M ( P ( x ) ) ≥ C D log D . Dobrowolski improved this to
log M ( P ( x ) ) ≥ C ( log log D log D ) 3 . 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.
Let E / K be an elliptic curve defined over a number field K , and let h ^ E : E ( K ¯ ) → R be the canonical height function. The canonical height is the analogue for elliptic curves of the function ( deg P ) − 1 log M ( P ( x ) ) . It has the property that h ^ E ( Q ) = 0 if and only if Q is a torsion point in E ( K ¯ ) . The elliptic Lehmer conjecture asserts that there is a constant C ( E / K ) > 0 such that
h ^ E ( Q ) ≥ C ( E / K ) D for all non-torsion points
Q ∈ E ( K ¯ ) ,
where D = [ K ( Q ) : K ] . If the elliptic curve E has complex multiplication, then the analogue of Dobrowolski's result holds:
h ^ E ( Q ) ≥ C ( E / K ) D ( log log D log D ) 3 , due to Laurent. For arbitrary elliptic curves, the best known result is
h ^ E ( Q ) ≥ C ( E / K ) D 3 ( log D ) 2 , due to Masser. For elliptic curves with non-integral j-invariant, this has been improved to
h ^ E ( Q ) ≥ C ( E / K ) D 2 ( log D ) 2 , by Hindry and Silverman.
Stronger results are known for restricted classes of polynomials or algebraic numbers.
If P(x) is not reciprocal then
M ( P ) ≥ M ( x 3 − x − 1 ) ≈ 1.3247 and this is clearly best possible. If further all the coefficients of P are odd then
M ( P ) ≥ M ( x 2 − x − 1 ) ≈ 1.618.
For any algebraic number α, let M ( α ) be the Mahler measure of the minimal polynomial P α of α. If the field Q(α) is a Galois extension of Q, then Lehmer's conjecture holds for P α .
