PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 49, Number 1, May 1975
SUMSOF POWERS OF CONJUGATES
OF ALGEBRAIC NUMBERS
P. E. BLANKSBY
ABSTRACT.
This
which
have
at least
which
give
lower
of such
lower
to some
for the
work
main
concerned
with
of powers
of S. Chowla,
theorems
with
Theorems
The
used
for sums
conjugates
theorems
of the conjugates.
bounds
numbers
one.
of the
equivalent
and the method
on lower
algebraic
exceeding
of the moduli
are connected
sums
with
modulus
for the maximum
numbers
earlier
P. Turan's
is primarily
conjugate
bounds
algebraic
bounds
note
one
giving
results
depends
of powers
relate
on one of
of complex
numbers.
1. Throughout
of degree
irreducible,
primitive
ing coefficient
and
this
paper
polynomial
d, then
the following
d, there
exists
that
a , a
cl is an algebraic
• .. , a . If ais
over the rational
basic
a positive
numbers
a-
we will call
We have
algebraic
we will assume
n (n > 1) with conjugates
d the
fact.
real
cl of degree
integers,
denominator
For each
number
pair
(ß(n, d),
n and denominator
number
a zero of an
with positive
lead
of a.
of positive
integers
with the property
n
that
for
d,
either fa] < 1,
(1)
where
or
[a] = max,<,<
on the elementary
|aJ«
symmetric
fa] > 1 + l/<f)(n, d),
This
follows
functions
vide bounds
(in terms
of n and
polynomials
for such
a. The problem
with this
property
Received
by the editors
Key words
a , ■■■ , a , which
of finding
here;
on |~a| provide
in turn pro-
of possible
the best
it is proposed
bounds
possible
to pursue
defining
function
this
<7j
else-
March 8, 1974.
classifications
and phrases.
of a
bounds
d) on the coefficients
is not investigated
AMS (MOS) subject
10F10, 12A15.
since
Algebraic
(1970). Primary 12D10, 30A08; Secondary
numbers,
conjugates.
'This
usage is only a matter of convenience
here; clearly
d need not be the
"least
denominator"
of a. in the obvious
sense.
In fact, if D is the least of the
positive
integers
D such that D a is an algebraic
integer,
then D<d<D".
A result
analogous
to our Theorem
3 that follows
then holds,
with 4> replaced
by a corresponding
function
♦ = ♦(«, D).
Copyright © 1475. American Mathematical Society
28
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SUMSOF POWERS OF CONJUGATES OF ALGEBRAIC NUMBERS
where.
It has been shown [l] that <f>(n, 1) = 30k
We define,
for integral
29
log 672 is valid in (I).
k,
S ¿a.) = ±
a*
7= 1
S. Chowla
[2] has proved
Theorem
nomial
(Chowla).
of degree
the following
Let
n with
a n x" + a n
integral
c is a positive
constant
,x"
involving
+...
—i
coefficients,
U \Sk(a-)\ < n for all k satisfying
and
theorem
S,(a).
+ «.\) be a reciprocal
lis
and with zeros
poly-
cl,, a , . . . , a .
0 < k < Hcn f where H = n + Sn=0 \a.\,
independent
of n and
H, then
|a.|
= 1 for
1 <j<n.
In this
results
note
related
Theorem
we will see that
to Chowla's.
1. For every
not a root of unity,
there
results
We prove
algebraic
exists
of the type (1) are connected
in the case
integer- a of degree
an integer
with
d = 1
k satisfying
n (n > l) which
is
1 < k < 2000 n (log 6n)
and such that \S k(a)\ > (30rc3 log 6n)n + 1.
There
could
is no special
be reduced
2. Before
observations
significance
in size
by more detailed
proceeding
about
(that
then the condition
(ii)
\a\
If I a.I = 1 for some
jo] = 1, then
easy
numbers
power
(iii)
a
and so
In the case
when
fa] < 1, then a classical
that
of a),
as in Chowla's
|a, | = 1 for all
a is reciprocal.
|a, | = 1 for all
1 < k < n.
In particular,
1 < k < n. In fact,
1 < k < n, if and only if
is a rational
which
1, we make the following
is a conjugate
< 1 implies
j ex, | = [et] for all
of fa]
2000,
a.
1 < / < », then
a is reciprocal
to see that
integral
is
constant
calculations.
to the proof of Theorem
algebraic
(i) If a is reciprocal
result,
in the numerical
if
it is
some positive
number.
d = 1 (that
theorem
is
a is an algebraic
of Kronecker
[3] implies
integer)
that
and
a is a root of
unity.
(iv)
For algebraic
numbers
determined
by the size
is clearly
0, n, or is infinite,
respectively.
Dirichlet's
(l/27r)
For,
theorem
according
satisfying
of |5,(a)|
lim sup,
^^
as
|S,(a)|
k —» °o
equals
\cT\ < 1, [a] = 1, or [ä~| > 1,
0 < e < 1, we can apply
approximations
infinitely
numbers
limit
In fact
as to whether
on simultaneous
complex
the superior
of [a|.
if e is arbitrary,
arg cl. to determine
corresponding
a,
many positive
to the
integers
n real numbers
k such
a^, oz, • • • , ar all lie in the sector
by
< arcrestrictions
cos may
(l apply
- e).
License\d\
or copyright
to redistribution; see http://www.ams.org/journal-terms-of-use
that the
defined
30
P. E. BLANKSBY
For such
k,
\Sk(a)\ > Re S (a) = ¿
Re afe > (1 - e) ¿
7=1
Thus
for these
k we have,
|afe|.
7=1
'
in general,
\Sk(a)\ >(l-f)f^\
and when fa] = 1 (and so |a | = 1 for all 1 < k < n by (ii)),
The above
claims
3. Proof
following
follow
from these
of Theorem
result
1. The theorem
for complex
Theorem
2. For every
the following
two statements
|Sfe(a)| >(l-e)«.
inequalities.
is an immediate
consequence
of the
numbers.
integer
n (n > 1) and every
hold for all lists
real
number
of n complex
<7J (<7j>
> 1),
numbers
z^, z.,
••• , z :
(i) the inequality
max
\zk1 + zk2+ ...
+ zkJ>(n4>)n
+1
T.<k<,n4>\og(n<t>)
implies
the inequality
max
\<k<n
\z\>
1 + .73-1;
(ii) the inequality
max
l<k<n
implies
\z,\
fc
> 1 + r/>-
the inequality
~k +, z*
~h +, •. ..
„k\ ^> í_t.1«
\zk.
• • +, zk\
(n<fj)' + l
max
l<k<20n<p \og(n<i>)
If, as in the hypothesis
not a root of unity,
[l],
fa] > 1 + (30«
z, = a
of Theorem
1, ais
faf > 1, and in fact,
1 follows
Of course,
numbers
tween results
of Chowla.
an algebraic
by the result
integer
already
which
mentioned
directly,
Theorem
since
2 applies
with denominator
30«
equally
the following
the result
log 6n < (6n)
well
to the general
d > 1, and so provides
of the type (1) of the second
For example,
log 6« in Theorem 2(ii),
paragraph
theorem
an equivalence
and results
is also
case
a direct
similar
of algebeto that
application
of Theorem 2.
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Theorem
is
log 672) , where n is the degree of a. Thus on taking
(1 < k < n), and <f>»=(f>(n)= 30n
of Theorem
braic
then
n
3. // <f>- <£>(n, d) is any function
which
is valid
for the basic
SUMS OF POWERS OF CONJUGATES OF ALGEBRAIC NUMBERS
result
(1),
for which
then for every
algebraic
fa] > 1, there
exists
\og(ncf>(n, d)) and such that
4. Proof
(ii)
of Theorem
We follow
[4, Theorem
found
The
an integer
2. (i)
This
// Zj, z , . • • , z
and apply
[5, Corollary
and
a.,
a2, • • ■ , a
satisfying
m+l<k<m
1 /
„_
i
convenience
M=
d
d) •
Turán's
the triangle
second
of this
inequality.
main theorem
inequality
may be
lj:
denote
complex
m> — l there
numbers,
exists
and
an integer
+ n, and
\ "-1
' } - 8 \8e(n + m)J
For notational
1 < k < 20«c/j(«,
using
version
to Theorem
(l — i S. n)> then for any integer
k
after
more recent
\z \\ ^ lzl
;=1
■fi
n and denominator
k satisfying
follows
method
following
in van der Poorten
cl of degree
\Sk(a)\ > (mj>(n, d))n + 1.
Chowla's
IX].
number
31
mm
1<j<n
|flj
+ • • •+ a.\ |z,|
7
k
we write
max \zk\
and
S, = |z* + z\ + • •. + z*|.
\<k<n
By Turán's
result,
there
exists
an integer
S >
Mk Í
k~
Assuming
S
n > 2, m > 1, and
~
8
the integer
(l + <f>~ )
m + 1 < k < m + n and
\"-1
+ m)
M > 1 we get
> ¿ Mrn+ l(2Aem)l-n
8
the inequalities
n- 1
\8e(n
> ±-Mm + 1(8e(m + 2))1""
If [x] denotes
k with
8
part
> 2 and
of x, define
> Mm + 1(24^)"n.
m = [l8«c¿
log («có)].
e log x < x, we derive
s\/n > (1 + c^1)18* lo«(^)(432ew<7J logUc/J))"1
> 2181og("*) (432 («á)2)-1
_ _J_
432
Thus
we have
Z„^181og
Z„A\3/2 % Z„^(«
(«cÄ)181°«2-2
2"2 x>(«0)3/2>(«c¿)("
S, > (n<p)n
for some
18 «c/j log (mf>)<k<l8ncß
which
concludes
the proof of (ii).
+
+ l)/n
1)
k satisfying
log («0) + « < 20 ncf>log («ci),
License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
Using
k
32
P. E. BLANKSBY
REFERENCES
1. P. E. Blanksby
and H. L. Montgomery,
Acta Arith. 18 (1971), 355-369.
2. S. Chowla,
On polynomials
Zwei
integers
roots
lie on the unit
all of whose
Angew. Math. 222 ( 1966), 69-70.
3. L. Kronecker,
Algebraic
near
the unit
circle,
MR 45 #5082.
circle,
J. Reine
MR 32 #5604.
Satze
über
Gleichungen
mit ganzzahligen
Coefficienten,
J. Reine Angew. Math. 53 (1857), 173-175.
4. Paul
Turan,
Eine
neue
Methode
in der Analysis
und deren
Anwendungen,
Akad. Kiadó, Budapest, 1953. MR 15, 688.
5. A. J. van der Poorten,
Generalisations
bounds for sums of powers, Bull. Austral.
of Turan
s main
theorems
Math. Soc. 2(1970),
on lower
15-37.
MR 42 #217.
DEPARTMENT OF PURE MATHEMATICS, UNIVERSITY OF ADELAIDE, ADELAIDE,
SOUTH AUSTRALIA, AUSTRALIA 5001
PROCEEDINGS OF THE
AMERICAN MATHEMATICALSOCIETY
Volume 49, Number 1, May 1975
THE DIMENSIONOF THE RING OF COEFFICIENTS
IN A POLYNOMIALRING
JIMMYT. ARNOLD
ABSTRACT.
A and
such
A and B are commutative
B are stably
that
equivalent
the polynomial
isomorphic.
If A and
rings
B are
rings with
provided
there
/ILA7., •••
stably
identity.
exists
, X ] and
equivalent,
We say
a positive
ß[y,,
•••
they
have
then
that
integer
n
, Y i are
equal
Krull
dimension.
The question
concerning
equivalent
denotes
rings
by Eakin
rings,
the Krull
Received
has given
Eakin
dimension
by the editors
AMS (MOS) subject
arises
from recent
an example
need not be isomorphic.
and Heinzer
then
paper
of the ring of coefficients
In [6], Höchster
equivalent
are posed
in this
the uniqueness
(cf. [l]—[6]).
stably
answered
in [5].
and Heinzer
of the ring
February
classifications
ask whether
R).
in a polynomial
which
Several
In particular,
We shall
investigations
illustrates
related
if A and
ring
that
questions
B are stably
dim A = dim B (dim R
presently
show that this
1, 1974.
( 1970). Primary 13B25; Secondary
13A15,
13C15.
Key words
and phrases.
Polynomial
ring,
Krull
dimension,
coefficient
ring.
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Copyright © 1975, American Mathematical Society