Proposed lower bound on the Mahler measure for polynomials with integer coefficients
For Lehmer's conjecture about the non-vanishing of τ(
n), see
.
Lehmer's conjecture, also known as the Lehmer's Mahler measure problem, is a problem in
The conjecture asserts that there is an absolute constant
![{\displaystyle \mu >1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/05e590248b5860b9cee369297d0f785d5578827f)
such that every
polynomial with integer coefficients
![{\displaystyle P(x)\in \mathbb {Z} [x]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b681dbdc55e0c6b544392222910a3d01000efb2)
satisfies one of the following properties:
- The Mahler measure[2]
of
is greater than or equal to
.
is an integral multiple of a product of cyclotomic polynomials or the monomial
, in which case
. (Equivalently, every complex root of
is a root of unity or zero.)
There are a number of definitions of the Mahler measure, one of which is to factor
over
as
![{\displaystyle P(x)=a_{0}(x-\alpha _{1})(x-\alpha _{2})\cdots (x-\alpha _{D}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d18df79c9c6531051dec8bbae90e4f8e753b89d5)
and then set
![{\displaystyle {\mathcal {M}}(P(x))=|a_{0}|\prod _{i=1}^{D}\max(1,|\alpha _{i}|).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b6bc1be9cb93926979fef7e407f4a476b445cec)
The smallest known Mahler measure (greater than 1) is for "Lehmer's polynomial"
![{\displaystyle P(x)=x^{10}+x^{9}-x^{7}-x^{6}-x^{5}-x^{4}-x^{3}+x+1\,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/addeb9c17b869a893cb1904f151079ec123dc587)
for which the Mahler measure is the Salem number[3]
![{\displaystyle {\mathcal {M}}(P(x))=1.176280818\dots \ .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9681218f0c17a6fa55e439edbdeed1ad3b416dc)
It is widely believed that this example represents the true minimal value: that is,
in Lehmer's conjecture.[4][5]
Motivation
Consider Mahler measure for one variable and Jensen's formula shows that if
then
![{\displaystyle {\mathcal {M}}(P(x))=|a_{0}|\prod _{i=1}^{D}\max(1,|\alpha _{i}|).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b6bc1be9cb93926979fef7e407f4a476b445cec)
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[1][6] that
is an important value in the study of the integer sequences
for monic
. If
does not vanish on the circle then
. If
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[7]).[8] As a result, Lehmer was led to ask
- whether there is a constant
such that
provided
is not cyclotomic?,
or
- given
, are there
with integer coefficients for which
?
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[9] proved that Lehmer's conjecture is true for all polynomials that are not reciprocal, i.e., all polynomials satisfying
.
Blanksby and
independently proved that there is an absolute constant
![{\displaystyle C>1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/536c544fc840b3b0bc20994827352e7396cf3e5d)
such that either
![{\displaystyle {\mathcal {M}}(P(x))=1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30152583a31d979818b8c0c7a4fd8267e43c321e)
or
[12]
![{\displaystyle \log {\mathcal {M}}(P(x))\geq {\frac {C}{D\log D}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad433e1a6a60c21efa20b29087d080cf1a39572e)
Dobrowolski[13] improved this to
![{\displaystyle \log {\mathcal {M}}(P(x))\geq C\left({\frac {\log \log D}{\log D}}\right)^{3}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a49f67eac29963c274473a4adceaa143d4b5ecd)
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.[14]
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
![{\displaystyle (\deg P)^{-1}\log {\mathcal {M}}(P(x))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff7763a9491daf02c356c55119435abee0894616)
. It has the property that
![{\displaystyle {\hat {h}}_{E}(Q)=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/339def82f4ef2ba4566301bc49c4c139fc825783)
if and only if
![{\displaystyle Q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed)
is a
torsion point
in
![{\displaystyle E({\bar {K}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b9d6893a2d4e4e978fd690d2156f9bb896be76e)
. The
elliptic Lehmer conjecture asserts that there is a constant
![{\displaystyle C(E/K)>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b2eb2c2ca22d7845b5c8e8665f230b4ce6bfeee)
such that
for all non-torsion points
,
where
. If the elliptic curve E has complex multiplication, then the analogue of Dobrowolski's result holds:
![{\displaystyle {\hat {h}}_{E}(Q)\geq {\frac {C(E/K)}{D}}\left({\frac {\log \log D}{\log D}}\right)^{3},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0df38e1c5dbc2d9a01308e85854e7216a6f37ff)
due to Laurent.[15] For arbitrary elliptic curves, the best known result is
![{\displaystyle {\hat {h}}_{E}(Q)\geq {\frac {C(E/K)}{D^{3}(\log D)^{2}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/63fe83eaa65e07b8667bb7a5d1f70dffe001979a)
due to Masser.[16] For elliptic curves with non-integral j-invariant, this has been improved to
![{\displaystyle {\hat {h}}_{E}(Q)\geq {\frac {C(E/K)}{D^{2}(\log D)^{2}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2282fe0322b1ca2a090d3e201522063ce2e7ab91)
by Hindry and Silverman.[17]
Restricted results
Stronger results are known for restricted classes of polynomials or algebraic numbers.
If P(x) is not reciprocal then
![{\displaystyle M(P)\geq M(x^{3}-x-1)\approx 1.3247}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a35de50357527739d69eaaefe746c180f7ae73c6)
and this is clearly best possible.[18] If further all the coefficients of P are odd then[19]
![{\displaystyle M(P)\geq M(x^{2}-x-1)\approx 1.618.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d49fe1a6244e1ba312fd258d34477f2b9ac9f62)
For any algebraic number α, let
be the Mahler measure of the minimal polynomial
of α. If the field Q(α) is a Galois extension of Q, then Lehmer's conjecture holds for
.[19]
Relation to structure of compact group automorphisms
The
As pointed out by Lind, this means that the set of possible values of the entropy of such actions is either all of
![{\displaystyle (0,\infty ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67ad143b1435e7b0039cdd14d280ed40d00c145d)
or a countable set depending on the solution to Lehmer's problem.
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
External links