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and this observation permits one to generalize Mahler's measure to polynomials in several variables (see [[#References|[a11]]], [[#References|[a13]]]).
 
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002014.png" /> is an algebraic integer with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002015.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002016.png" /> is either zero or a root of unity. D.H. Lehmer [[#References|[a7]]] asked whether <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002017.png" /> could attain values arbitrarily close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002018.png" />. This subsequently led to the following formulation of Lehmer's conjecture: There exists a positive constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002019.png" /> such that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l130/l130020/l13002020.png" /> is an algebraic integer, not a root of unity, then
+
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
 +
$$
 +
M(\alpha) \ge 1 + \eta \ .
 +
$$
  
<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/l13002021.png" /></td> </tr></table>
+
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 $C$ with the property that if $\alpha$ is a non-zero algebraic integer of degree $N$, not a root of unity, then $\boxed{\alpha}$, the maximal absolute value of a conjugate of $\alpha$ is at least
 +
$$
 +
1+\frac{C}{N}
 +
$$
  
Lehmer's conjecture is equivalent to the existence of ergodic automorphisms of the infinite-dimensional torus having finite [[Entropy|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
+
It is known ([[#References|[a2]]], [[#References|[a14]]]) that Lehmer's conjecture holds for non-reciprocal integers $\alpha$, i.e. algebraic integers whose minimal polynomials do not have $1/\alpha$ as a root. In this case the minimal value for $M(\alpha)$ equals $1.32471\ldots$ and is attained by roots of the polynomial $X^3-X-1$.
 
 
<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>
 
  
 +
In 1971, P.E. Blanksby and H.L. Montgomery [[#References|[a1]]] established, for all algebraic integers $\alpha\ne0$ of degree $N$ that are not roots of unity, the inequality
 +
$$
 +
M(\alpha) \ge 1 + \frac{ 1 }{ 52N\log(6N) }
 +
$$
 
and subsequently E. Dobrowolski [[#References|[a4]]] obtained
 
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>
+
M(\alpha) > 1 + c\left({ \frac{ \log\log N }{ \log N } }\right)^3
 
+
$$
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
+
with $c=1/1200$, whereas for $N \ge N(\epsilon)$ he got $c=1-\epsilon$. Subsequently, several authors increased the value of $c$ to $c=2-\epsilon$ ([[#References|[a3]]], [[#References|[a12]]]) and $c=9/4-\epsilon$ ([[#References|[a9]]]). Since for non-reciprocal integers $\alpha$ one has $M(\alpha) \le \boxed{\alpha}^{N/2}$, 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>
+
\boxed{\alpha} > 1 +  \frac{ 9 }{ 2N } \left({ \frac{ \log\log N }{ \log N } }\right)^3
 
+
$$
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
+
but this has been superseded by A. Dubickas [[#References|[a5]]], who proved for sufficiently large $N$ 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>
+
\boxed{\alpha} > 1 +  \left({ \frac{64}{\pi^2}-\epsilon }\right) \left({ \frac{ \log\log N }{ \log N } }\right)^3
 
+
$$
 
which is the strongest known result toward the Schinzel–Zassenhaus conjecture as of 2000.
 
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]]].
+
The smallest known value of $M(\alpha) > 1$ is$1.17628\ldots$, achieved by the root of $X^{10} + X^9 - X^7 -X^6 - X^5 - X^4 - X^3 + X + 1$ and found in [[#References|[a7]]].
  
 
====References====
 
====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>
+
<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>
  
{{TEX|part}}
+
{{TEX|done}}

Latest revision as of 07:50, 27 March 2018

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 $C$ with the property that if $\alpha$ is a non-zero algebraic integer of degree $N$, not a root of unity, then $\boxed{\alpha}$, the maximal absolute value of a conjugate of $\alpha$ is at least $$ 1+\frac{C}{N} $$

It is known ([a2], [a14]) that Lehmer's conjecture holds for non-reciprocal integers $\alpha$, i.e. algebraic integers whose minimal polynomials do not have $1/\alpha$ as a root. In this case the minimal value for $M(\alpha)$ equals $1.32471\ldots$ and is attained by roots of the polynomial $X^3-X-1$.

In 1971, P.E. Blanksby and H.L. Montgomery [a1] established, for all algebraic integers $\alpha\ne0$ of degree $N$ that are not roots of unity, the inequality $$ M(\alpha) \ge 1 + \frac{ 1 }{ 52N\log(6N) } $$ and subsequently E. Dobrowolski [a4] obtained $$ M(\alpha) > 1 + c\left({ \frac{ \log\log N }{ \log N } }\right)^3 $$ with $c=1/1200$, whereas for $N \ge N(\epsilon)$ he got $c=1-\epsilon$. Subsequently, several authors increased the value of $c$ to $c=2-\epsilon$ ([a3], [a12]) and $c=9/4-\epsilon$ ([a9]). Since for non-reciprocal integers $\alpha$ one has $M(\alpha) \le \boxed{\alpha}^{N/2}$, the last result leads to the inequality $$ \boxed{\alpha} > 1 + \frac{ 9 }{ 2N } \left({ \frac{ \log\log N }{ \log N } }\right)^3 $$ but this has been superseded by A. Dubickas [a5], who proved for sufficiently large $N$ the inequality $$ \boxed{\alpha} > 1 + \left({ \frac{64}{\pi^2}-\epsilon }\right) \left({ \frac{ \log\log N }{ \log N } }\right)^3 $$ which is the strongest known result toward the Schinzel–Zassenhaus conjecture as of 2000.

The smallest known value of $M(\alpha) > 1$ is$1.17628\ldots$, achieved by the root of $X^{10} + X^9 - X^7 -X^6 - X^5 - X^4 - X^3 + X + 1$ 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
How to Cite This Entry:
Lehmer conjecture. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lehmer_conjecture&oldid=43023
This article was adapted from an original article by Władysław Narkiewicz (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article