Lehmer conjecture
A conjecture about the minimal Mahler measure of a non-zero algebraic integer which is not a root of unity. The Mahler measure $M(\alpha)$ of an algebraic number $\alpha$ is defined by $$ M(\alpha) = a_0 \prod_{i=1}^N \max(1,|\alpha_i|) \ , $$ where $a_0$ denotes the leading coefficient and $N$ the degree of the minimal polynomial $f$ (with integral coefficients) of $\alpha$ (cf. also Algebraic number) and $\alpha=\alpha_1,\alpha_2,\ldots,\alpha_N$ are its conjugates. Since $M(\alpha)$ depends only on $f$, it is also denoted by $M(f)$ and called the Mahler measure of $f$. Jensen's formula (cf. also Jensen formula) implies the equality $$ M(f) = \exp\left({ \int_0^1 \log\left\vert{ f\left({ e^{2\pi i t} }\right) }\right\vert d t }\right) $$ and this observation permits one to generalize Mahler's measure to polynomials in several variables (see [a11], [a13]).
A theorem of L. Kronecker implies that if $\alpha$ is an algebraic integer with $M(\alpha\le1$, then $\alpha$ is either zero or a root of unity. D.H. Lehmer [a7] asked whether $M(\alpha$ could attain values arbitrarily close to $1$. This subsequently led to the following formulation of Lehmer's conjecture: There exists a positive constant $\eta$ such that if $\alpha\ne0$ is an algebraic integer, not a root of unity, then $$ M(\alpha) \ge 1 + \eta \ . $$
Lehmer's conjecture is equivalent to the existence of ergodic automorphisms of the infinite-dimensional torus having finite entropy [a8] and its truth would imply the following conjecture stated by A. Schinzel and H. Zassenhaus [a16]: There exists a positive constant with the property that if
is a non-zero algebraic integer of degree
, not a root of unity, then
, the maximal absolute value of a conjugate of
is at least
![]() |
It is known ([a2], [a14]) that Lehmer's conjecture holds for non-reciprocal integers , i.e. algebraic integers whose minimal polynomials do not have
as a root. In this case the minimal value for
equals
and is attained by roots of the polynomial
.
In 1971, P.E. Blanksby and H.L. Montgomery [a1] established, for all algebraic integers of degree
that are not roots of unity, the inequality
![]() |
and subsequently E. Dobrowolski [a4] obtained
![]() |
with , whereas for
he got
. Subsequently, several authors increased the value of
to
([a3], [a12]) and
([a9]). Since for non-reciprocal integers
one has
, the last result leads to the inequality
![]() |
but this has been superseded by A. Dubickas [a5], who proved for sufficiently large the inequality
![]() |
which is the strongest known result toward the Schinzel–Zassenhaus conjecture as of 2000.
The smallest known value of is
, realized by the root of
and found in [a7].
References
< TR>[a1] | P.E. Blanksby, H.L. Montgomery, "Algebraic integers near the unit circle" Acta Arith. , 18 (1971) pp. 355–369 |
[a2] | K. Breusch, "On the distribution of the roots of a polynomial with integral coefficients" Proc. Amer. Math. Soc. , 3 (1951) pp. 939–941 |
[a3] | D.G. Cantor, E.G. Straus, "On a conjecture of D.H. Lehmer" Acta Arith. , 42 (1982) pp. 97–100; 325 |
[a4] | E. Dobrowolski, "On a question of Lehmer and the number of irreducible factors of a polynomial" Acta Arith. , 34 (1979) pp. 391–401 |
[a5] | A. Dubickas, "On algebraic numbers of small measure" Liet. Mat. Rink. , 35 (1995) pp. 421–431 |
[a6] | A. Dubickas, "Algebraic conjugates outside the unit circle" , New Trends in Probability and Statistics , 4 (1997) pp. 11–21 |
[a7] | D.H. Lehmer, "Factorization of certain cyclotomic functions" Ann. Math. , 34 : 2 (1933) pp. 461–479 |
[a8] | D.A. Lind, K. Schmidt, T. Ward, "Mahler measure and entropy for commuting automorphisms of compact groups" Invent. Math. , 101 (1990) pp. 503–629 |
[a9] | R. Louboutin, "Sur la mesure de Mahler d'un nombre algébrique" C.R. Acad. Sci. Paris , 296 (1983) pp. 707–708 |
[a10] | K. Mahler, "An application of Jensen's formula to polynomials" Mathematika , 7 (1960) pp. 98–100 |
[a11] | K. Mahler, "On some inequalities for polynomials in several variables" J. London Math. Soc. , 37 (1962) pp. 341–344 |
[a12] | U. Rausch, "On a theorem of Dobrowolski about the product of conjugate numbers" Colloq. Math. , 50 (1985) pp. 137–142 |
[a13] | A. Schinzel, "The Mahler measure of polynomials" , Number Theory and its Applications (Ankara, 1996) , M. Dekker (1999) pp. 171–183 |
[a14] | C.J. Smyth, "On the product of the conjugates outside the unit circle of an algebraic integer" Bull. London Math. Soc. , 3 (1971) pp. 169–175 |
[a15] | C.L. Stewart, "Algebraic integers whose conjugates lie near the unit circle" Bull. Soc. Math. France , 196 (1978) pp. 169–176 |
[a16] | A. Schinzel, H. Zassenhaus, "A refinement of two theorems of Kronecker" Michigan J. Math. , 12 (1965) pp. 81–85 |
Lehmer conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lehmer_conjecture&oldid=43027