Lehmer's conjecture

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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:

  • 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]

Partial results[edit]

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

Smyth [5] 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[6] and Stewart[7] independently proved that there is an absolute constant C>1 such that either \mathcal{M}(P(x))=1 or[8]

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

Dobrowolski [9] 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.[10]

Elliptic analogues[edit]

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.[11] For arbitrary elliptic curves, the best known result is[11]

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

due to Masser.[12] For elliptic curves with non-integral j-invariant, this has been improved to[11]

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

by Hindry and Silverman.[13]

Restricted results[edit]

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.[14] If further all the coefficients of P are odd then[15]

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.[15]

References[edit]

  1. ^ 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. 
  2. ^ 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. 
  3. ^ Smyth (2008) p.324
  4. ^ 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. 
  5. ^ 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. 
  6. ^ Blanksby, P. E.; Montgomery, H. L. (1971). "Algebraic integers near the unit circle". Acta Arith. 18: 355–369. Zbl 0221.12003. 
  7. ^ Stewart, C. L. (1978). "Algebraic integers whose conjugates lie near the unit circle". Bull. Soc. Math. France 106: 169–176. 
  8. ^ Smyth (2008) p.325
  9. ^ Dobrowolski, E. (1979). "On a question of Lehmer and the number of irreducible factors of a polynomial". Acta Arith. 34: 391–401. 
  10. ^ Smyth (2008) p.326
  11. ^ a b c Smyth (2008) p.327
  12. ^ Masser, D.W. (1989). "Counting points of small height on elliptic curves". Bull. Soc. Math. Fr. 117 (2): 247–265. Zbl 0723.14026. 
  13. ^ 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. 
  14. ^ Smyth (2008) p.328
  15. ^ a b 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. 

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