View source for Lehmer conjecture

A conjecture about the minimal [[Mahler measure|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 [[#References|[a11]]], [[#References|[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 [[#References|[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]] [[#References|[a8]]] and its truth would imply the following conjecture stated by A. Schinzel and H. Zassenhaus [[#References|[a16]]]: There exists a positive constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002022.png" /> with the property that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002023.png" /> is a non-zero algebraic integer of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002024.png" />, not a root of unity, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002025.png" />, the maximal absolute value of a conjugate of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002026.png" /> is at least

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002027.png" /></td> </tr></table>

It is known ([[#References|[a2]]], [[#References|[a14]]]) that Lehmer's conjecture holds for non-reciprocal integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002028.png" />, i.e. algebraic integers whose minimal polynomials do not have <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002029.png" /> as a root. In this case the minimal value for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002030.png" /> equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002031.png" /> and is attained by roots of the polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002032.png" />.

In 1971, P.E. Blanksby and H.L. Montgomery [[#References|[a1]]] established, for all algebraic integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002033.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002034.png" /> that are not roots of unity, the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002035.png" /></td> </tr></table>

and subsequently E. Dobrowolski [[#References|[a4]]] obtained

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002036.png" /></td> </tr></table>

with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002037.png" />, whereas for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002038.png" /> he got <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002039.png" />. Subsequently, several authors increased the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002040.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002041.png" /> ([[#References|[a3]]], [[#References|[a12]]]) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002042.png" /> ([[#References|[a9]]]). Since for non-reciprocal integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002043.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002044.png" />, the last result leads to the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002045.png" /></td> </tr></table>

but this has been superseded by A. Dubickas [[#References|[a5]]], who proved for sufficiently large <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002046.png" /> the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002047.png" /></td> </tr></table>

which is the strongest known result toward the Schinzel–Zassenhaus conjecture as of 2000.

The smallest known value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002048.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002049.png" />, realized by the root of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002050.png" /> and found in [[#References|[a7]]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  P.E. Blanksby,   H.L. Montgomery,   "Algebraic integers near the unit circle"  ''Acta Arith.'' , '''18'''  (1971)  pp. 355–369</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  K. Breusch,   "On the distribution of the roots of a polynomial with integral coefficients"  ''Proc. Amer. Math. Soc.'' , '''3'''  (1951)  pp. 939–941</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  D.G. Cantor,   E.G. Straus,   "On a conjecture of D.H. Lehmer"  ''Acta Arith.'' , '''42'''  (1982)  pp. 97–100; 325</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  E. Dobrowolski,   "On a question of Lehmer and the number of irreducible factors of a polynomial"  ''Acta Arith.'' , '''34'''  (1979)  pp. 391–401</TD></TR>
<TR><TD valign="top">[a5]</TD> <TD valign="top">  A. Dubickas,   "On algebraic numbers of small measure"  ''Liet. Mat. Rink.'' , '''35'''  (1995)  pp. 421–431</TD></TR><
TR><TD valign="top">[a6]</TD> <TD valign="top">  A. Dubickas,   "Algebraic conjugates outside the unit circle" , ''New Trends in Probability and Statistics'' , '''4'''  (1997)  pp. 11–21</TD></TR>
<TR><TD valign="top">[a7]</TD> <TD valign="top">  D.H. Lehmer,   "Factorization of certain cyclotomic functions"  ''Ann. Math.'' , '''34''' :  2  (1933)  pp. 461–479</TD></TR>
<TR><TD valign="top">[a8]</TD> <TD valign="top">  D.A. Lind,   K. Schmidt,   T. Ward,   "Mahler measure and entropy for commuting automorphisms of compact groups"  ''Invent. Math.'' , '''101'''  (1990)  pp. 503–629</TD></TR>
<TR><TD valign="top">[a9]</TD> <TD valign="top">  R. Louboutin,   "Sur la mesure de Mahler d'un nombre algébrique"  ''C.R. Acad. Sci. Paris'' , '''296'''  (1983)  pp. 707–708</TD></TR>
<TR><TD valign="top">[a10]</TD> <TD valign="top">  K. Mahler,   "An application of Jensen's formula to polynomials"  ''Mathematika'' , '''7'''  (1960)  pp. 98–100</TD></TR>
<TR><TD valign="top">[a11]</TD> <TD valign="top">  K. Mahler,   "On some inequalities for polynomials in several variables"  ''J. London Math. Soc.'' , '''37'''  (1962)  pp. 341–344</TD></TR>
<TR><TD valign="top">[a12]</TD> <TD valign="top">  U. Rausch,   "On a theorem of Dobrowolski about the product of conjugate numbers"  ''Colloq. Math.'' , '''50'''  (1985)  pp. 137–142</TD></TR>
<TR><TD valign="top">[a13]</TD> <TD valign="top">  A. Schinzel,   "The Mahler measure of polynomials" , ''Number Theory and its Applications (Ankara, 1996)'' , M. Dekker  (1999)  pp. 171–183</TD></TR>
<TR><TD valign="top">[a14]</TD> <TD valign="top">  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</TD></TR>
<TR><TD valign="top">[a15]</TD> <TD valign="top">  C.L. Stewart,   "Algebraic integers whose conjugates lie near the unit circle"  ''Bull. Soc. Math. France'' , '''196'''  (1978)  pp. 169–176</TD></TR>
<TR><TD valign="top">[a16]</TD> <TD valign="top">  A. Schinzel,   H. Zassenhaus,   "A refinement of two theorems of Kronecker"  ''Michigan J. Math.'' , '''12'''  (1965)  pp. 81–85</TD></TR>
</table>

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