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
.
