ON THE DISTRIBUTION OF THE ROOTS OF A POLYNOMIAL
WITH INTEGRAL COEFFICIENTS
ROBERT BREUSCH
The following question has been raised by D. H. Lehmer1 in connection with prime number problems: "If e is a positive quantity, to
find a polynomial of the form f(z)=zT+aiZr~1+
- ■ ■ +ar where the
a's are integers, such that the absolute value of the product of those
roots of / which lie outside the unit circle, lies between 1 and 1 + e. "
Lehmer calls this absolute value ß(/). Since, for f(z)=z2 —z —1,
0(f) = 1.6 • • • , we may assume e<l, and therefore ar= +1. Then
ß(f) may be written in the form YLn-i I z»| [l"""> where the z„ are the
roots of f(z). It is shown in this paper that no nonreciprocal polynomial f(z) exists with ß(/) < 1.179. Lehmer himself gives an example
of a reciprocal polynomial of degree 10 for which 0(f) is less than
this, namely 1.176 • • • ; no nonreciprocal polynomial seems to be
known for which ß(f) < 1.32. Thus the present result is still far from
being a complete answer to the question.
Let/(z)=zr-r-aizr-1-f■ • • ±1 be a nonreciprocal
polynomial with
integral coefficients, irreducible in the rational field. The proof makes
use of the resultant R(f, g) oif(z) and the polynomial g(z) m ±zrf(í/z)
= zr+ • • • . All the roots of g(z) are of the form zX=zZx where the
z„ are the conjugates of the roots z„ of f(z). Since/(z) and g(z) are
irreducible and different, have integral coefficients and highest coeffi-
cients 1, | R(f, g) | = Xln-i ITm-i I 2»~2m| must be not less than 1.
I R(J, g) I - II
n™l
II \z^-zt\
■\zm-z*n\-f[\zn-
m>n
z*n\=Pi-P2.
n—1
If 0(f) = k, it will be shown in Lemma 2 that Pi < k2r•rT, and in Lemma
3thatPig(4(*-l)/r)'.
Lemma
1. Pi=II«-i
value compatible
Hm>n
|z„-z£|
with the two conditions
Iln-i k| "•»»= * (K*<2)
■[zm-z^j
(I)
[In-i
has
the greatest
[z„|=l,
(II)
for \zi\ =k, |*| = • • • =|zr_i| =1,
\Zr\=l/k.
Proof. For n = \, 2, • • • , r, call bn values of the z„ for which Pi
takes its greatest value when restricted by conditions I and II.
Presented to the Society, April 28, 1951 ; received by the editors January 2, 1951.
1 D. H. Lehmer, Factorization of certain cyclotomic functions, Ann. of Math. vol. 34
(1933) p. 476.
939
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940
ROBERTBREUSCH
[December
Assume that at least two of the b's are absolutely greater than 1, for
example, |¿>i| >1, |i>2| >1. Pi may be rewritten in the form
r
Pi =
(Zl -
Z21)(Z2 — ZV1)' II
*
(Zl — Zn)(Zn — ZTX)(Z2 — Zñ1)(Zn ~ Z2l)
•II
II (Zn - Zm)(Zm— Zn)
n=3
m>n
The expression within the absolute bars is now an analytic function
of Zi and z2 (but not of the remaining z's). Calling this expression
P(zi, z2, z3, • • • , zr), and introducing A =¿>i52 (with 1< | A | ^k), our
assumption implies that F(z, A/z, b3, • • • , br) which is analytic in
1 á | z\ ú\A\ takes its largest value in this region for z = Z»i.But this
is impossible, for bi is by assumption an interior point of this region,
because 1 < | bi | < | A \.
Interchanging
the z„ with the z* does not change Pi; thus not two
of the | ¿>*| can be greater than 1, that is, not two of the | bn\ can be
less than 1.
Therefore (by proper arrangement
of the subscripts) Pi takes its
greatest
value for Zi = £ei(\ z* = (\/k)ei6\
zT=(l/k)eie',
z* = keie',
Zn = eif", z*n=Zn (Kn<r).
LEMMA2.Pi<k2rrr.
Proof.
Using the values found in Lemma 1, we get, for l<«<r'
| zi - zl I • I z„ - z* I = I kem - eie"\-|
=
k
| Zr — Zn | • | Zn — Zr | =
Zr | • | Zr -
keiS»- em \
^-\zi-Zn\2
(since \kei6n —eie,\ = \ei>n —keiSl\ ). Similarly,
| Zl -
k
Zl \ =
| ß"1 -
for l<n<r,
*• | Zr — Zn |2,
Ci9' |2 < | Zl -
Zr |*.
Therefore
r-l
r-l
Pi < | Zl - Zr|2- II I 21- Zn|2- | Zr- Z„|2- IT IT | «. ~ Zm\2
n=2
n*=2 n<,m<r
=n n
n—1
Zn --
*_
Z
«
m>n
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195»]
DISTRIBUTIONOF THE ROOTSOF A POLYNOMIAL
941
The function
««)-^(»-T),n<«-''(T-)"
•n
n
.
iz„-2ro|2
n=2 n<m<r
is analytic for 1 á | z| :£ °°, and has therefore its greatest absolute
value in this region for \z\ =1. But for |z\ = 1, |G(z)| is the absolute
value of the discriminant of the polynomial whose r roots z, ZiZr/z, z2,
■ ■ • , z,—i,all lie on the unit circle. This is known2 to be not greater
than rr. Therefore
\G(z)\ <r* lor |z|>l,
and Pi<|z?|
• | G(zx)|
<k2T-rT.
Lemma 3. // the z„ satisfy conditions I and 11 of Lemma 1, then
P»-]I;-i|*.-*î|3(4(*-l)/r)'.
Proof.
= l+€n>l
If |z„| =l+e„^l
for » = 1, 2, • • • , s, and |z*| —|«»|—*
for n = s+l, ■ ■ ■, r, then |z„-z*| =| |zj -|2»l_1| =1
+€„-l/(l+€n)á2enforn
= l,2, • • ■, r. Also, since Ilñ-i (l+O =*,
and IIfl-«+i(l+«n) =k, 1+ ¿„.i€„á^, 1+ Zn»«+iCn^k,and XXi«n
£2(*-l).
Therefore P2^ ffi-i
Theorem.
(2e.) ^(4(¿-l)/r)'.
If f(z) is a nonreciprocal irreducible polynomial with inte-
gral coefficientand highest coefficient1, then fi(/) = k > 1.179.
Proof.
If r is the degree of f(z), the statement
rect for \ar\ >1, since k^t \ar\.
is obviously cor-
Thus we may assume
by Lemmas 2 and 3, \R(f,g)\ =PiP2<k2rr'(4(k-l)/r)r
This must be not less than 1; therefore ±k2(k-\)
Amherst
\ar\ =1. Then
= (U2(k-l))r.
¿1, *> 1.179.
College
11. Schur, Ueber die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen
mit ganzzahligen Koefiizienten, Math. Zeit. vol. 1 (1918) p. 385.
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