Some remarks on the Lehmer conjecture

• Published in: Archiv der Mathematik Vol. 111; no. 1; pp. 33 - 42
• Main Author: de la Peña, José A.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01-07-2018; Springer Nature B.V
• Subjects: Mathematics; Mathematics and Statistics; Polynomials; Coxeter polynomial; 11J97; Primary 16G60; Secondary 11G05; Cyclotomic polynomial; 14H52; Lehmer polynomial; Mahler measure; 11G50
• ISSN: 0003-889X; 1420-8938
• DOI: 10.1007/s00013-018-1165-1

In 1933, Lehmer exhibited the polynomial
L
(
z
)
=
z
10
+
z
9
-
z
7
-
z
6
-
z
5
-
z
4
-
z
3
+
z
+
1
with Mahler measure
μ
0
>
1
. Then he asked if
μ
0
is the smallest Mahler measure, not 1. This question became known as the
Lehmer conjecture
and it was apparently solved in the positive, while this paper was in preparation [
19
]. In this paper we consider those polynomials of the form
χ
A
, that is, Coxeter polynomials of a finite dimensional algebra
A
(for instance
L
(
z
)
=
χ
E
10
). A polynomial in
Z
[
T
]
which is either cyclotomic or with Mahler measure
≥
μ
0
is called a Lehmer polynomial. We give some necessary conditions for a polynomial to be Lehmer. We show that
A
being a tree algebra is a sufficient condition for
χ
A
to be Lehmer.