Lehmer's conjecture

For Lehmer's conjecture about the non-vanishing of τ(n), see Ramanjuan's tau function.

For Lehmer's conjecture about Euler's totient function, see Lehmer's totient problem.

Lehmer's conjecture, also known as the Lehmer's Mahler measure problem, is a problem in number theory raised by Derrick Henry Lehmer.^[1] The conjecture asserts that there is an absolute constant $\mu>1$ such that every polynomial with integer coefficients $P(x)\in\mathbb{Z}[x]$ satisfies one of the following properties:

The Mahler measure $\mathcal{M}(P(x))$ of $P(x)$ is greater than or equal to $\mu$ .

$P(x)$ is an integral multiple of a product of cyclotomic polynomials or the monomial $x$ , in which case $\mathcal{M}(P(x))=1$ . (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 $\mathbb{C}$ as

P(x)=a_0 (x-\alpha_1)(x-\alpha_2)\cdots(x-\alpha_D),

and then set

\mathcal{M}(P(x)) = |a_0| \prod_{i=1}^{D} \max(1,|\alpha_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^[2]

\mathcal{M}(P(x))=1.176280818\dots \ .

It is widely believed that this example represents the true minimal value: that is, $\mu=1.176280818\dots$ in Lehmer's conjecture.^[3]^[4]

Motivation

Consider Mahler measure for one variable and Jensen's formula shows that if $P(x)=a_0 (x-\alpha_1)(x-\alpha_2)\cdots(x-\alpha_D)$ then

\mathcal{M}(P(x)) = |a_0| \prod_{i=1}^{D} \max(1,|\alpha_i|).

In this paragraph denote　 $m(P)=\log(\mathcal{M}(P(x))$ , which is also called Mahler measure.

If $P$ has integer coefficients, this shows that $\mathcal{M}(P)$ is an algebraic number so $m(P)$ is the logarithm of an algebraic integer. It also shows that $m(P)\ge0$ 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^[1]^[5] that $m(P)=0$ is an important value in the study of the integer sequences $\Delta_n=\text{Res}(P(x), x^n-1)=\prod^D_{i=1}(\alpha_i^n-1)$ for monic $P$ . If $P$ does not vanish on the circle then $\lim|\Delta_n|^{1/n}=\mathcal{M}(P)$ 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?,

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.

Partial results

Let $P(x)\in\mathbb{Z}[x]$ be an irreducible monic polynomial of degree $D$ .

Smyth ^[6] proved that Lehmer's conjecture is true for all polynomials that are not reciprocal, i.e., all polynomials satisfying $x^DP(x^{-1})\ne P(x)$ .

Blanksby and Montgomery^[7] and Stewart^[8] independently proved that there is an absolute constant $C>1$ such that either $\mathcal{M}(P(x))=1$ or^[9]

\log\mathcal{M}(P(x))\ge \frac{C}{D\log D}.

Dobrowolski ^[10] improved this to

\log\mathcal{M}(P(x))\ge C\left(\frac{\log\log D}{\log D}\right)^3.

Dobrowolski obtained the value C ≥ 1/1200 and asymptotically C > 1-ε for all sufficiently large D. Voutier obtained C ≥ 1/4 for D ≥ 2.^[11]

Elliptic analogues

Let $E/K$ be an elliptic curve defined over a number field $K$ , and let $\hat{h}_E:E(\bar{K})\to\mathbb{R}$ be the canonical height function. The canonical height is the analogue for elliptic curves of the function $(\deg P)^{-1}\log\mathcal{M}(P(x))$ . It has the property that $\hat{h}_E(Q)=0$ if and only if $Q$ is a torsion point in $E(\bar{K})$ . The elliptic Lehmer conjecture asserts that there is a constant $C(E/K)>0$ such that

\hat{h}_E(Q) \ge \frac{C(E/K)}{D}

for all non-torsion points

Q\in E(\bar{K})

where $D=[K(Q):K]$ . If the elliptic curve E has complex multiplication, then the analogue of Dobrowolski's result holds:

\hat{h}_E(Q) \ge \frac{C(E/K)}{D} \left(\frac{\log\log D}{\log D}\right)^3 ,

due to Laurent.^[12] For arbitrary elliptic curves, the best known result is^[12]

\hat{h}_E(Q) \ge \frac{C(E/K)}{D^3(\log D)^2},

due to Masser.^[13] For elliptic curves with non-integral j-invariant, this has been improved to^[12]

\hat{h}_E(Q) \ge \frac{C(E/K)}{D^2(\log D)^2},

by Hindry and Silverman.^[14]

Restricted results

Stronger results are known for restricted classes of polynomials or algebraic numbers.

If P(x) is not reciprocal then

M(P) \ge M(x^3 -x - 1) \approx 1.3247

and this is clearly best possible.^[15] If further all the coefficients of P are odd then^[16]

M(P) \ge M(x^2 -x - 1) \approx 1.618 .

If the field Q(α) is a Galois extension of Q then Lehmer's conjecture holds.^[16]

References

1 2 Lehmer, D.H. (1933). "Factorization of certain cyclotomic functions". Ann. Math. (2) 34: 461–479. doi:10.2307/1968172. ISSN 0003-486X. Zbl 0007.19904.
↑ Borwein, Peter (2002). Computational Excursions in Analysis and Number Theory. CMS Books in Mathematics. Springer-Verlag. p. 16. ISBN 0-387-95444-9. Zbl 1020.12001.
↑ Smyth (2008) p.324
↑ Everest, Graham; van der Poorten, Alf; Shparlinski, Igor; Ward, Thomas (2003). Recurrence sequences. Mathematical Surveys and Monographs 104. Providence, RI: American Mathematical Society. p. 30. ISBN 0-8218-3387-1. Zbl 1033.11006.
↑ David Boyd (1981). "Speculations concerning the range of Mahler's measure" Canad. Math. Bull. Vol. 24(4)
↑ Smyth, C. J. (1971). "On the product of the conjugates outside the unit circle of an algebraic integer". Bulletin of the London Mathematical Society 3: 169–175. doi:10.1112/blms/3.2.169. Zbl 1139.11002.
↑ Blanksby, P. E.; Montgomery, H. L. (1971). "Algebraic integers near the unit circle". Acta Arith. 18: 355–369. Zbl 0221.12003.
↑ Stewart, C. L. (1978). "Algebraic integers whose conjugates lie near the unit circle". Bull. Soc. Math. France 106: 169–176.
↑ Smyth (2008) p.325
↑ Dobrowolski, E. (1979). "On a question of Lehmer and the number of irreducible factors of a polynomial". Acta Arith. 34: 391–401.
↑ Smyth (2008) p.326
1 2 3 Smyth (2008) p.327
↑ Masser, D.W. (1989). "Counting points of small height on elliptic curves". Bull. Soc. Math. Fr. 117 (2): 247–265. Zbl 0723.14026.
↑ Hindry, Marc; Silverman, Joseph H. (1990). "On Lehmer's conjecture for elliptic curves". In Goldstein, Catherine. Sémin. Théor. Nombres, Paris/Fr. 1988-89. Prog. Math. 91. pp. 103–116. ISBN 0-8176-3493-2. Zbl 0741.14013.
↑ Smyth (2008) p.328
1 2 Smyth (2008) p.329

Smyth, Chris (2008). "The Mahler measure of algebraic numbers: a survey". In McKee, James; Smyth, Chris. Number Theory and Polynomials. London Mathematical Society Lecture Note Series 352. Cambridge University Press. pp. 322–349. ISBN 978-0-521-71467-9.

External links

http://www.cecm.sfu.ca/~mjm/Lehmer/ is a nice reference about the problem.
Weisstein, Eric W., "Lehmer's Mahler Measure Problem", MathWorld.

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