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 of an algebraic number is defined by

(a1)

where denotes the leading coefficient and is the degree of the minimal polynomial (with integral coefficients) of (cf. also Algebraic number) and are its conjugates. Since depends only on , it is also denoted by and called the Mahler measure of . Jensen's formula (cf. also Jensen formula) implies the equality

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 is an algebraic integer with , then is either zero or a root of unity. D.H. Lehmer [a7] asked whether could attain values arbitrarily close to . This subsequently led to the following formulation of Lehmer's conjecture: There exists a positive constant such that if is an algebraic integer, not a root of unity, then

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but this has been superseded by A. Dubickas [a5], who proved for sufficiently large the inequality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

[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