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 $m(P)\geq 0$ 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^[1]^[5] that $m(P)=0$ is an important value in the study of the integer sequences $\Delta _{n}={\text{Res}}(P(x),x^{n}-1)=\prod _{i=1}^{D}(\alpha _{i}^{n}-1)$ 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 $x^{D}P(x^{-1})\neq P(x)$ .

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))\geq {\frac {C}{D\log D}}.

Dobrowolski ^[10] improved this to

\log {\mathcal {M}}(P(x))\geq 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 in 1996 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)\geq {\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)\geq {\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

{\hat {h}}_{E}(Q)\geq {\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

{\hat {h}}_{E}(Q)\geq {\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)\geq 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)\geq M(x^{2}-x-1)\approx 1.618.

For any algebraic number α, let $M(\alpha )$ be the Mahler measure of the minimal polynomial $P_\alpha$ of α. If the field Q(α) is a Galois extension of Q, then Lehmer's conjecture holds for $P_\alpha$ .^[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.
↑ P. Voutier, An effective lower bound for the height of algebraic numbers, Acta Arith. 74 (1996), 81–95.
↑ 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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