A
RT ¯
URAS
D
UBICKAS
C
HRIS
J. S
MYTH
The Lehmer constants of an annulus
Journal de Théorie des Nombres de Bordeaux, tome
13, n
o2 (2001),
p. 413-420
<http://www.numdam.org/item?id=JTNB_2001__13_2_413_0>
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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques
The
Lehmer
constants
of
anannulus
par DUBICKAS et CHRIS J. SMYTH
RÉSUMÉ. Soit M
(03B1)
la mesure de Mahler d’un nombrealgébrique
03B1, et V un ouvert de C. Alors sa constante de Lehmer L
(V)
estégale à
inf M(03B1)1/
deg(03B1),
l’infimum
étant évalué sur tous les nombresalgébriques
noncyclotomiques
03B1 dont tous lesconjugués
sont à l’extérieur de V. Nous calculons L
(V)
lorsque
V est une couronne centrée en 0. Nous faisons de même pour la constantede Lehmer transfinie L~
(V).
Nous démontrons
également
laréciproque
d’un théorème deLangevin,
qui affirme que L(V)
> 1 si V contient un élément demodule 1, ainsi que le résultat
analogue
avec L~(V).
ABSTRACT. Let M
(03B1)
be the Mahler measure of analgebraic
number 03B1, and V be an open subset of C. Then its Lehmer
con-stant L
(V)
is inf M(03B1)1/
deg(03B1),
the infimumbeing
over all non-zeronon-cyclotomic
03B1lying
with itsconjugates
outside V. Weevaluate L
(V)
when V is any annulus centered at 0. We do thesame for a variant of L
(V),
which we call the transfinite Lehmerconstant L~
(V) .
Also,
we prove the converse toLangevin’s
Theorem,
which states thatL (V)
> 1 if V contains a point of modulus 1. Weprove the
corresponding
result for L~(V) .
1. Introduction and results
Let V be an open subset of C. In
1985,
Langevin
[Lal]
introduced thefollowing
function,
the Lehmerconstantl
of
V(see
[La3]),
defined asL (V)
=where the infimum is taken over every non-zero
non-cyclotomic
algebraic
number a
lying
with itsconjugates
in HereM (a)
is the Mahlermeasure of a:
Manuscrit reçu le 9 novembre 1999.
1 Strictly speaking, we should call L ( V) the best Lehmer constant, according to Langevin’s original definition
where a = al has minimal
polynomial
aozd
+ ... + ad EZ [z],
and theai are
the
conjugates
of a. As M(a) ~
l,
L(V) ~
1.We define also a variant of
L (V),
denotedLoo
(V),
which we call thetransfinite
Lehmer constantof V, given by
where this time the infimum is taken over all a of
degree >
dlying
withtheir
conjugates
inCBV.
Note thathere,
as distinct from in the definitionof L
(V),
a may becyclotomic.
The main aim of the paper is to evaluate both L
(V)
andLoo
(V)
when V is an open annulus centered at 0.So,
for 0 r R letThen our main result is the
following.
Theorem 1. For 0 r R we have
Corollary
1. Let R > 1and,
> 0. ThenLangevin
[Lal]
proved
the remarkable result that L(V)
> 1 if V is anopen set
containing
apoint
of modulus 1.(See
also[M]
and[D]
for otherproofs
ofLangevin’s
Theorem.)
As we show in theproof
of Theorem 2(b),
it follows
immediately
from this that alsoLoo
(V)
> 1. So the newpart
of our second result is that the converses also hold:Theorem 2. Let V be an open subset
of
(C. Then(a)
L(V)
> 1 V contains apoint
of
modulus 1.(b)
Loo
(V)
> 1 V contains apoint
of
modulus 1.Furthermore,
Loo
(V) ~
L(V) .
Note that Theorem 2 is used in the
proof
of Theorem 1.To date L
(V)
has been evaluated for certain sectorsA result of Schinzel
[Sc]
isequivalent
toL (Yo)
= 2
(1
+V5).
Rhin andSmyth
[RS]
evaluated for 0belonging
to nine subintervals of~0, ~r~,
using
acomputational
version of the method outlined in[La2].
Inparticular,
Earlier, Langevin
[La3]
had found a lower bound for L(A)
>1.08,
where A was the interior of the circumcircle of thetriangle
with vertices 0 and(l ±z~/399)
/20.
Thisimplied
that L(V1r/2)
> 1.08. The results of[RS]
alsogive
lower bounds > 1 for L for0 ~ () ~
3 .
To our
knowledge
Theorem 1 contains the first exact evaluation of any(V)
> 1.However, Mignotte
[M]
showedthat,
for 9 > 0 there isan effective
positive
constant c such thatLoo
(V~_a) ~
1 +Also,
itis natural to
conjecture
thatLoo
(VO)
=22,
where 1 = 1.31427... is theconstant
given
in[Sm],
Theorem 1.Langevin’s
choice of the name ’Lehmer constant’ for the functions wehave called L
(V)
is because of Lehmer’s 1933question
[Lel]
as to whetherthere is an absolute constant c > 1 such that M
(a) >
c for anyalgebraic
number a with
M (a)
> 1.Langevin’s
resultquoted
in Theorem 2(a)
shows that if a and its
conjugates
areslightly
restricted tokeep
away froma
neighbourhood
of asingle
point
on the unitcircle,
then for some c(V)
> 1an
exponentially
stronger
result M(a) ~
c(V)deg(a)
holds. In his reviewof
[La3],
Lehmer[Le2]
has somedry
observations on his still-unansweredquestion.
2. Proof of Theorem 1
For the
proof,
we need thecorollary
to thefollowing
lemma.Lemma 1. Let C > 2 and
a, b E N‘ ,
the setof
positive integers.
Then thepolynomial
+ + 1 has b zeros in the annulusand a zeros in the annulus
Proof.
Sincehas, by
Rouché’sTheorem,
a zeros inI
1,
and so b zeros inIzl I
> 1. From z°+C+z-a = 0we have for
Izl I
> 1and for
lzl
1whence the results. The condition C > 2 ensures that the annuli do not
We
apply
the lemma to apolynomial
which was obtainedby slightly
perturbing
the coefficients of thepolynomial
(Z mb
+(Z ma
+ Iwhich
clearly
has rrzb zeros onlzl
= Aa and rrza zeros onIzl
=A-b.
Corollary
2. Let > 1 and e > 0 begiven.
Thenf or
msufficiently
large
thepolynomial
has ma zeros in the annulus
and mb zeros in the annulus
The
corollary
followseasily by applying
the lemma to P(21/(m(a+b» z)
/2.
We can now prove Theorem 1. We first do the case r 1 R. It is convenient to consider this case in the
Corollary
1form,
i.e. when r = R-7for 7
= - log r/ log R
> 0. It isenough
to prove that L(A (R-’R,
R) )
and thatLoo
(A (R-7, R)) ~
R7~~1+’~~.
For then the full result inthe case r 1 R follows from the
inequality
L of Theorem 2.So,
we first prove thatL(.4(R"~R)) ~
R’~~(1+7>,
Let a, ofdegree
d,
lie with its
conjugates
inCBA (R-’Y, R),
with say ..., R and ..., R--t. Then on the one handwhile on the other hand
Hence
so that
giving
therequired
inequality.
Note that for theproof
of thisinequality
we have not made use of the arithmetical nature of the coefficients of the minimal
polynomial
of a.Also,
it is easy to see from thesteps
of theproof
that the
inequality
is anequality
precisely
when a is a unitlying
withits
conjugates
on the two circlesIzl
= Rand Izi
= R--t. See[DS]
for aFor the other
inequality
Loo
(A (R-’~,
R))
R’Y/(l+’Y),
we useCorollary
2to prove the existence of monic irreducible
polynomials
withinteger
coef-ficientshaving
all their zeros close to one or other of the circlesbounding
the annulus. To do
this,
we must choose values ofA,
a and b toapply
Corollary
2.Essentially,
b/a
will be a rationalapproximation
to ~y, and Aa close
approximation
toRi/a.
To beprecise,
we want a, b and A so thatand
It is easy to check that all these
inequalities
will be satisfiedprovided
thatb/a
is in the intervaland,
having
chosen such ab/a,
thatlog A
is in the intervalNote that both of these intervals are
non-empty!
Having
chosen a, b and A in this way, we see fromCorollary
2 that P(z)
has all its zeros outside A
(R-~’, R) .
Note too that P has rrLb zeros in theannulus
Izl [
e Awith,
for esufficiently
small,
the other ma zerosinside the unit circle.
Also,
P isirreducible, by
Eisenstein’sCriterion,
sofor a a zero of P we have
using
(1).
Since
f
(e, R) ~
as e -~0,
we haveL~
(A
(R--t, R)) ~
asrequired.
Thiscompletes
theproof
of the theorem when r 1 R.The
proof
of the casesr ~
1 orR ~
1 followsimmediately
from3. Proof of Theorem 2
For this
proof,
we need to recall the nthChebyshev
polynomial
of the secondkind,
U~
( x ) ,
definedby
It is well-known
(and
easily
shown)
that(x)
is a monicpolynomial
withinteger coefficients,
with all its zeroslying
in the interval~-2, 2J.
SoU~
(x)
is of
degree
n - 1 withleading
coefficients n, and all zeros in(-2, 2).
Lemma 2. When n is even, the
only cyclotomic factor of
is z2
+ 1. When n isodd,
thepolynomial
has nocyclotorrzic factors.
This is an
example
in[BS],
where analgorithm
isgiven
forfinding
allpoints
on a curve whose coordinates are roots ofunity.
Thealgorithm
isbased on the observation that any root of
unity w
isconjugate
to one of-w,
w2,
or-w2.
Using
thisobservation,
it can be shown that any suchpoint
on the curve also lies on another curve, sothat, by intersecting
thetwo curves, all these
points
can be found. In the case of thislemma,
weapply
thealgorithm
to a curve obtained from(z
+1 /z)
essentially
by making zn
a new variable.Curiously,
the result of Lemma 2 contrasts with thecorresponding
fac-torisation withUn
replaced by
Tn
the nthChebyshev polynomial
of the firstkind,
definedby
Tn
(z
+ + z-n. SinceTn
(x) =
(x),
all of the factors of( z
+1 /z)
arecyclotomic!
To prove
part
(a)
of Theorem2,
recallthat,
from[Lal],
L (V)
> 1 if V contains apoint
of modulus 1.Conversely,
suppose that V contains nosuch
points.
Putting
we know that all zeros of P are
on lzl
= 1 andthat,
by
Lemma2,
none ofthe zeros are roots of
unity
for n odd.Suppose
that,
for a fixed odd n >1,
P factorises over Z as P =floj ,
j
where
Qj
hasdegree
dj,
leading
coeflicient fj )
1,
and say{3j
as one ofits zeros. As we have
just
seen, no{3j
is a root ofunity.
Thenflfj
= n,j
~~
= 2n - 2and,
since all the(3j
and theirconjugates
have modulus1,
i
Hence
Now as the
weights
(2n -
2)
sum to1,
we must havefor at least
one j .
Hence,w
w--To prove
(b),
suppose first that V contains nopoint
of modulus 1. Then all the nth roots ofunity
wnlie,
with theirconjugates,
in But M1,
so thatLoo
(V)
= 1.Now suppose, on the
contrary,
that V contains apoint
of modulus 1.Then contains
only finitely
manyconjugate
sets of roots ofunity.
Such a can therefore beignored
in the definition ofLoo
(V),
showing
thatLoo
(V) ~
L(V)
>1,
using
(a).
This proves(b).
For the final
remark,
we havejust
seen thatLoo
(V ) ~
L(V)
if V contains apoint
ofIzl
=1,
while from(a)
and(b)
Loo
(V)
= L(V)
if it does notcontain such a
point.
Acknowledgements.
Wegratefully
acknowledge
thesupport
of the Lon-don MathematicalSociety,
enabling
the first-named author to visitEdin-burgh,
where much of this work was done. He was alsosupported by
theLithuanian State Studies and Science Foundation.
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[Le2] D. H. LEHMER, Review of [La3]. Math. Rev. 89j:11025.
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[RS] G. RHIN, C. J. SMYTH, On the absolute Mahler measure of polynomials having all zeros in a sector. Math. Comp. 65 (1995), 295-304.
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Arturas DUBICKAS
Dept. of Mathematics and Informatics Vilnius University
Naugarduko 24
Vilnius 2006 Lithuania
arturas. dubickasQmaf. vu. It
Chris J. SMYTH
Dept. of Mathematics and Statistics
Edinburgh University J.C.M.B, King’s Buildings Mayfield road Edinburgh EH9 3JZ Scotland UK E-mail : chris0maths.ed.ac.uk