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Lehmer's conjecture, also known as the Lehmer's Mahler measure problem, is a problem in
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...
raised by
Derrick Henry Lehmer Derrick Henry "Dick" Lehmer (February 23, 1905 – May 22, 1991), almost always cited as D.H. Lehmer, was an American mathematician significant to the development of computational number theory. Lehmer refined Édouard Lucas' work in the 1930s and ...
. The conjecture asserts that there is an absolute constant \mu>1 such that every
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An ex ...
with integer coefficients P(x)\in\mathbb /math> satisfies one of the following properties: * The Mahler measure \mathcal(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(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 as :P(x)=a_0 (x-\alpha_1)(x-\alpha_2)\cdots(x-\alpha_D), and then set :\mathcal(P(x)) = , a_0, \prod_^ \max(1,, \alpha_i, ). The smallest known Mahler measure (greater than 1) is for "Lehmer's polynomial" :P(x)= x^+x^9-x^7-x^6-x^5-x^4-x^3+x+1 \,, for which the Mahler measure is the Salem number :\mathcal(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.Smyth (2008) p.324


Motivation

Consider Mahler measure for one variable and
Jensen's formula In the mathematical field known as complex analysis, Jensen's formula, introduced by , relates the average magnitude of an analytic function on a circle with the number of its zeros inside the circle. It forms an important statement in the study ...
shows that if P(x)=a_0 (x-\alpha_1)(x-\alpha_2)\cdots(x-\alpha_D) then :\mathcal(P(x)) = , a_0, \prod_^ \max(1,, \alpha_i, ). In this paragraph denote m(P)=\log(\mathcal(P(x)) , which is also called Mahler measure. If P has integer coefficients, this shows that \mathcal(P) is an
algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the p ...
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 polynomial In mathematics, the ''n''th cyclotomic polynomial, for any positive integer ''n'', is the unique irreducible polynomial with integer coefficients that is a divisor of x^n-1 and is not a divisor of x^k-1 for any Its roots are all ''n''th primit ...
s 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 \Delta_n=\text(P(x), x^n-1)=\prod^D_(\alpha_i^n-1) for monic P . If P does not vanish on the circle then \lim, \Delta_n, ^=\mathcal(P). If P 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). As a result, Lehmer 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? 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 /math> 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^DP(x^)\ne P(x). Blanksby and Montgomery and Stewart independently proved that there is an absolute constant C>1 such that either \mathcal(P(x))=1 orSmyth (2008) p.325 :\log\mathcal(P(x))\ge \frac. Dobrowolski improved this to :\log\mathcal(P(x))\ge C\left(\frac\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.


Elliptic analogues

Let E/K be an
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
defined over a number field K, and let \hat_E:E(\bar)\to\mathbb be the canonical height function. The canonical height is the analogue for elliptic curves of the function (\deg P)^\log\mathcal(P(x)). It has the property that \hat_E(Q)=0 if and only if Q is a torsion point in E(\bar). The elliptic Lehmer conjecture asserts that there is a constant C(E/K)>0 such that :\hat_E(Q) \ge \frac for all non-torsion points Q\in E(\bar), where D= (Q):K/math>. If the elliptic curve ''E'' has
complex multiplication In mathematics, complex multiplication (CM) is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visibl ...
, then the analogue of Dobrowolski's result holds: :\hat_E(Q) \ge \frac \left(\frac\right)^3 , due to Laurent.Smyth (2008) p.327 For arbitrary elliptic curves, the best known result is :\hat_E(Q) \ge \frac, due to Masser. For elliptic curves with non-integral
j-invariant In mathematics, Felix Klein's -invariant or function, regarded as a function of a complex variable , is a modular function of weight zero for defined on the upper half-plane of complex numbers. It is the unique such function which is holom ...
, this has been improved to :\hat_E(Q) \ge \frac, by Hindry and Silverman.


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.Smyth (2008) p.328 If further all the coefficients of ''P'' are odd then :M(P) \ge 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 In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base fiel ...
of Q, then Lehmer's conjecture holds for P_\alpha.Smyth (2008) p.329


Relation to structure of compact group automorphisms

The
measure-theoretic entropy In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular. Measure-preserving systems obey the Poincaré recurrence theorem, and are a special c ...
of an ergodic automorphism of a compact metrizable abelian group is known to be given by the logarithmic Mahler measure of a polynomial with integer coefficients if it is finite. As pointed out by Lind, this means that the set of possible values of the entropy of such actions is either all of (0,\infty] or a countable set depending on the solution to Lehmer's problem. Lind also showed that the infinite-dimensional torus either has Ergodic theory, ergodic automorphisms of finite positive entropy or only has automorphisms of infinite entropy depending on the solution to Lehmer's problem. Since an ergodic compact group automorphism is measurably isomorphic to a Bernoulli shift, and the Bernoulli shifts are classified up to measurable isomorphism by their entropy by
Ornstein's theorem In mathematics, the Bernoulli scheme or Bernoulli shift is a generalization of the Bernoulli process to more than two possible outcomes. Bernoulli schemes appear naturally in symbolic dynamics, and are thus important in the study of dynamical sy ...
, 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

*http://wayback.cecm.sfu.ca/~mjm/Lehmer/ is a nice reference about the problem. *{{MathWorld, urlname=LehmersMahlerMeasureProblem, title=Lehmer's Mahler Measure Problem Polynomials Theorems in number theory Conjectures Unsolved problems in number theory