Journal de Théorie des Nombres de Bordeaux

Berend Ringeling

Berend RINGELING
On Lehmer’s question for integer-valued polynomials
Tome 36, no 2 (2024), p. 527-536.
https://doi.org/10.5802/jtnb.1287
© Les auteurs, 2024.
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Journal de Théorie des Nombres
de Bordeaux 36 (2024), 527–536
On Lehmer’s question for integer-valued
polynomials
par Berend RINGELING
Résumé. Nous répondons à une question de type Lehmer sur la mesure de
Mahler des polynômes à valeurs entières.
Abstract. We solve a Lehmer-type question about the Mahler measure of
integer-valued polynomials.
1. Introduction
In the 1930s Lehmer asks, for a monic polynomial P (x) =
∏d
j=1(x−αj) ∈
Z[x], whether the real quantity
∏d
j=1 max{1, |αj |} can be made arbitrarily
close to but larger than 1. This quantity is called the Mahler measure of
P (x) [1]. More generally, for P (x) = c
∏d
j=1(x − αj) ∈ C[x], the Mahler
measure M(P (x)) is defined as |c|
∏d
j=1 max{1, |αj |}. Conjecturally, the
answer to Lehmer’s question is negative and the suspected lower bound
is given by α = 1.176280818 . . . , the unique real zero outside the unit circle
of Lehmer’s polynomial x10+x9−x7−x6−x5−x4−x3+x+1. Here we want to
extend the original question to a bigger class of polynomials, integer-valued
polynomials, that is, polynomials P (x) ∈ Q[x] such that P (k) ∈ Z for all
k ∈ Z. These polynomials often occur in counting problems; basic examples
include binomial coefficients,(
x
n
)
= x(x − 1)(x − 2) . . . (x − n + 1)
n! ∈ Q[x]
for n ∈ N. In fact, any integer-valued polynomial is an integer linear com-
bination of such polynomials (see [4, Part 8, Chapter 2, Section 1]).
Question 1.1. Can M(P (x)) be made arbitrarily close to but larger than 1,
when P (x) is an irreducible integer-valued polynomial?
The irreducibility condition is essential here, since for a reducible integer-
valued polynomial P (x) the bound M(P (x)) ≥ 1 may be violated. This is
Manuscrit reçu le 14 novembre 2022, révisé le 31 août 2023, accepté le 18 octobre 2023.
2020 Mathematics Subject Classification. 11R06, 11R09.
Mots-clefs. Mahler measure; polynomials; asymptotics.
This work is supported by NWO grant OCENW.KLEIN.006.
528 Berend Ringeling
seen from the example
P (x) = x
p − x
p
for primes p. By Fermat’s little theorem P (x) is integer-valued and the
Mahler measure M(P (x)) = 1/p tends to 0 as p increases.
In fact, using the polynomials ap (x
p −x) for p a prime and a an integer, it
is clear that the Mahler measure of a (not necessarily irreducible) integer-
valued polynomial can be arbitrarily close to but larger than 1. Moreover,
it shows that the range of Mahler measures of these polynomials is dense in
the interval [0, ∞). The following statement demonstrates that the bound
M(P (x)) ≥ 1 in Question 1.1 is then best possible.
Lemma 1.2. If P (x) is irreducible and integer-valued, then M(P (x)) ≥ 1.
In fact, one can easily construct infinitely many (non-cyclotomic) irre-
ducible integer-valued polynomials P (x) with M(P (x)) = 1, this is demon-
strated in Example 2.1 below. A goal of this note is to answer Question 1.1
in the affirmative. To accomplish the task, we consider the family of poly-
nomials
fp(x) =
xp − x
p
+ x(p+1)/2 + 1,
for odd integers p. For primes p, the polynomials fp(x) are integer-valued.
We prove the following two statements for them.
Theorem 1.3. For primes p ≡ 3 mod 4, fp(x) is irreducible.
Theorem 1.4. We have the following asymptotics for M(fp(x)):
M(fp(x)) ∼
1 +
√
1 + 4/p2
2 ,
to all orders in p, as p → ∞. In particular, limp→∞ M(fp(x)) = 1. More-
over, the sequence M(fp(x)) is strictly decreasing in p.
Thus M(fp(x)) > 1. Hence, the affirmative answer to Question 1.1 is
given by the family fp when p ≡ 3 mod 4. Here we tabulate a few values
of M(fp(x)) for small primes p:
p M(fp(x))
3 1.17503 . . .
7 1.02169 . . .
11 1.00821 . . .
19 1.00276 . . .
On Lehmer’s question for integer-valued polynomials 529
2. Properties for the Mahler measure of integer-valued
polynomials
Proof of Lemma 1.2. For polynomials of degree 1, the statement is trivial.
We will assume that the degree is strictly bigger than 1. The Mahler mea-
sure of any polynomial P (x) ∈ Q[x] is bounded from below by the absolute
value of the constant term. Indeed if
P (x) = adxd + · · · + a0 = ad
d∏
j=1
(x − αj),
then
M(P (x)) = |ad|
d∏
j=1
max{1, |αj |} ≥ |ad|
d∏
j=1
|αj | = |ad|
|a0|
|ad|
= |a0|.
Moreover, if P (x) is integer-valued and irreducible, P (0) is guaranteed to
be a non-zero integer. Hence M(P (x)) ≥ 1. □
The following example shows that one can find infinitely many non-
cyclotomic irreducible integer-valued polynomials with Mahler measure ex-
actly 1.
Example 2.1. Consider the integer-valued polynomial
gp(x) =
xp − x
p
+ 1
for primes p > 2. The zeros of gp(x) all lie outside the complex unit circle,
as otherwise for any zeros α of gp inside or on the unit circle, we would
have the contradictary inequality
0 = |gp(α)| =
∣∣∣∣1 + αpp − αp
∣∣∣∣ ≥ 1 − 2p > 0.
As a consequence of this, we find M(gp(x)) = 1.
We want to show that the polynomial pgp(x) is irreducible. If it were
reducible, then at least one of the irreducible factors would have constant
term 1; this is impossible since gp has all the zeros outside the unit cir-
cle. Thus, we have found an infinite family of (non-cyclotomic) irreducible
integer-valued polynomials.
3. Irreducibility
Proof of Theorem 1.3. For a polynomial P (x) of degree d, write P̃ (x) =
xdP (1/x) for its reciprocal. We prove the irreducibility of the polynomials
fp for primes p ≡ 3 mod 4 following the method first used by Ljunggren
in [3], also see the expository notes [2]. The irreducibility of f3 and f7 is
immediate, so we deal with p > 7 from now on.
530 Berend Ringeling
Write
f∗p (x) = pfp(x) = xp + px
p+1
2 − x + p
and
f̃∗p (x) = xpf∗p (1/x) = pxp − xp−1 + px
p−1
2 + 1
for its reciprocal.
Lemma 3.1. The polynomials f∗p and f̃∗p have no zeros in common.
Proof. Suppose α is a zero of both f∗p and f̃∗p , so that
αp − α + pα
p+1
2 + p = 0 and α − αp + pα
p+1
2 + pαp+1 = 0.
The equations imply
(α
p+1
2 + 1)2 = 0,
hence α
p+1
2 = −1. Substituting this in the first equation we find that α =
±1. This is impossible if p ≡ 3 mod 4. □
Suppose f∗p (x) is reducible, i.e. f∗p = gh for g, h ∈ Z[x] of positive degree.
Define an auxiliary polynomial
k = gh̃ = bpxp + · · · + b0;
then kk̃ = f∗p f̃∗p . Note that k ̸= ±f∗p or ±f̃∗p , as otherwise k = gh̃ and
f∗p = gh are equal, up to sign, hence h̃ and h share a common zero, which
is impossible by Lemma 3.1.
We next compute the coefficients of k by comparing the coefficients in
(3.1) kk̃ = f∗p f̃∗p = px2p − x2p−1 + p2x
3p+1
2 − pxp+1
+ 2(p2 + 1)xp − pxp−1 + p2x
p−1
2 − x + p.
Reading off the x2p-coefficient we have b0bp = p. We can assume that
b0 = ±p and bp = ±1, possibly by interchanging k and k̃. Further we may
assume b0 = p and bp = 1 by possibly replacing k with −k (by replacing f∗p
with −f∗p ).
Comparing the xp-coefficient of kk̃ in (3.1), we find
2(p2 + 1) = b20 + · · · + b2p,
therefore
(3.2) p2 + 1 = b21 + · · · + b2p−1.
Comparing the x-coefficient in (3.1) we conclude that
−1 = b0bp−1 + b1bp
On Lehmer’s question for integer-valued polynomials 531
implying b1 = −1 − bp−1p. The latter equality is only possible if either
bp−1 = 0 and b1 = −1, or bp−1 = −1 and b1 = p−1, as otherwise (3.2) fails.
Consider the two cases separately.
Case bp−1 = −1 and b1 = p − 1. We obtain from (3.2)
(3.3) 2p − 1 = b22 + · · · + b2p−2.
Compare the x2-coefficient to find that
b0bp−2 + b1bp−1 + b2bp = 0,
so that b2 = −1 − p(bp−2 − 1). According to (3.3), the equality is only
possible if bp−2 = 1 and b2 = −1. Next compare the x3-coefficient to find
that
b0bp−3 + b1bp−2 + b2bp−1 + b3bp = 0,
hence p(bp−3 + 1) + b3 = 0. Again, from (3.3) we conclude that bp−3 = −1
and b3 = 0. We claim that bp−j = (−1)j and bj = 0 for 2 < j < p−12 .
Comparing the xj-coefficient for such j gives
b0bp−j + b1bp−j+1 + b2bp−j+2 + · · · + bjbp = 0.
By induction all the terms bi vanish for 2 < i < j, so that bj = −p(bp−j −
(−1)j). From (3.3) and the fact that p divides bj , we conclude that bj = 0
and bp−j = (−1)j . Finally, compare the coefficient of x
p−1
2 in (3.1):
p2 = b0b p+1
2
+ b1b p+1
2 +1
+ b2b p+1
2 +2
+ · · · + b p−1
2
bp.
This translates into
p2 = p(b p+1
2
− (−1)
p−3
2 ) + b p−1
2
.
Therefore, b p−1
2
is divisible by p, hence b p−1
2
= 0 from (3.3) implying b p+1
2
=
(−1)
p−1
2 + p = p − 1. This calculation contradicts (3.3).
Case bp−1 = 0 and b1 = −1. In this case we have
(3.4) p2 = b22 + · · · + b2p−2.
We claim that bj = 0 for 1 < j < p−12 . Suppose otherwise, let 1 < j
′ < p−12
be the smallest integer such that bj′ ̸= 0. Comparing the xi-coefficient for
1 < i < j′ in (3.1) results in
b0bp−i + b1bp−i+1 · · · + bibp = 0;
it follows by induction that bp−i = 0 for all such i as well.
Comparing the xj′-coefficient in (3.1) we find out that
b0bp−j′ + b1bp−j′+1 + · · · + bj′bp = 0,
532 Berend Ringeling
hence pbp−j′ + bj′ = 0. Since bj′ ̸= 0 by our assumption, we have |bj′ | ≥ p
and |bp−j′ | ≥ 1. Comparing this with (3.4) we find this impossible. The
contradiction implies that bj = 0 and bp−j = 0 for 1 < j < p−12 .
Finally, consider the x
p+1
2 -coefficient in (3.1):
b0bp− p−12
+ b1bp− p−12 +1 + · · · + b p−12 bp = p
2;
this simplifies to
pb p+1
2
+ b p−1
2
= p2.
Comparing with (3.4), the only solution to this equation is b p+1
2
= p and
b p−1
2
= 0. We conclude that k = f∗p , which gives a contradiction.
Thus, f∗p is irreducible. This proves Theorem 1.3. □
4. Asymptotics
For this part, it is more convenient to work with the logarithmic Mahler
measure m(P (x)) = log(M(P (x))). Jensen’s formula allows one to write
it as
(4.1) m(P (x)) = 12πi
∮
|z|=1
log |P (z)| dz
z
.
Denote N = (p − 1)/2 and Qp(x) = (x2 − 1)/p + x and define
mp = m
(
xp − x
p
+ x
p+1
2 + 1
)
= m(xQp(xN ) + 1).
We will show that, for all integers N ,
mp ∼ m(xQp(xN )) = m(Qp(x)) = log
1 +
√
1 + 4/p2
2
to all orders in p, i.e. the difference of mp and m(Qp(x)) is O(pn) for all
n ∈ Z.
We have
(4.2) mp − m
(
xQp(xN )
)
= m
(
1 + 1
xQp(xN )
)
= 1
N
· m
(
1 + (−1)
N+1
xQp(x)N
)
,
where the last equality follows from the more general observation:
Lemma 4.1. If P (x) is a polynomial and N an integer, then
m
(
1 + (−1)
N+1
xP (x)N
)
= N · m
(
1 + 1
xP (xN )
)
On Lehmer’s question for integer-valued polynomials 533
Proof. Indeed, Jensen’s formula implies that
m
(
1 + (−1)
N+1
xP (x)N
)
= m
(
1 + (−1)
N+1
xN P (xN )N
)
= m
(
1 −
( −1
xP (xN )
)N)
=
∑
ξ : ξN =1
m
(
1 + ξ
xP (xN )
)
,
where the sum is over all roots of unity of degree N . The required identity
follows from noticing that
m
(
1 + ξ
xP (xN )
)
= m
(
1 + 1
xP (xN )
)
,
by substituting ξx for x in the integral (4.1) for the corresponding Mahler
measure. □
Since
1
|Qp(z)|2
=
∣∣∣∣ pz2 + pz − 1
∣∣∣∣2 = p22 + p2 − 2 Re(z2) < 1
for z ∈ C \ {±1}, |z| = 1, we get the convergent expansion
log
(
1 + (−1)
N+1
zQp(z)N
)
=
∞∑
ℓ=1
(−1)ℓN−1
ℓzℓQp(z)ℓN
.
for all such z. From this we find out that
(4.3)
m
(
1 + (−1)
N+1
xQp(x)N
)
= Re 12πi
∮
|z|=1
log
(
1 + (−1)
N+1
zQp(z)N
)
dz
z
= Re
∞∑
ℓ=1
(−1)ℓN−1
ℓ
Fℓ,
where
Fℓ =
1
2πi
∮
|z|=1
dz
zℓ+1Qp(z)ℓN
= p
ℓN
2πi
∮
|z|=1
dz
zℓ+1(z − α1)ℓN (z − α2)ℓN
and α1, α2 are the zeros of Qp(x) ordered by |α2| > 1 > |α1|. We will
examine the asymptotics of Fℓ for ℓ ≥ 1 as p → ∞. We can explicitly
compute these integrals.
Lemma 4.2. For ℓ ≥ 1, we have
Fℓ = (−1)ℓpℓN
ℓN−1∑
j=0
(
2ℓN − 2 − j
ℓN − 1
)(
ℓ + j
j
)
1
(α2 − α1)2ℓN−1−jαℓ+1+j2
.
534 Berend Ringeling
Proof. This follows from Cauchy’s integral theorem. The integrand has pre-
cisely one singularity outside the unit circle. Therefore, the value of the
integral is given by
− Resz=α2
1
zℓ+1Qp(z)ℓN
.
The formula follows by expanding 1/zℓ+1 into a series in z − α2:
(4.4) 1
zℓ+1
=
∞∑
j=0
(−1)j
(
ℓ + j
j
)
1
αℓ+1+j2
(z − α2)j
and extracting the nonpositive powers of z − α2 in the Laurent expansion
of 1/Qp(z)ℓN :
(4.5) 1(z − α1)ℓN (z − α2)ℓN
=
ℓN∑
j=0
(−1)j
(
ℓN + j − 1
j
)
1
(α2 − α1)ℓN+j
(z − α2)j−ℓN + O(z − α2).
Taking the product of (4.4) and (4.5) we conclude with the formula
(−1)ℓ−1pℓN
ℓN−1∑
j=0
(
2ℓN − 2 − j
ℓN − 1
)(
ℓ + j
j
)
1
(α2 − α1)2ℓN−1−jαℓ+1+j2
for the coefficient of 1/(z − α2). □
Using Lemma 4.2, we will estimate |Fℓ| from above.
Lemma 4.3. For ℓ ≥ 1, we have
|Fℓ| ≤
1
pℓ(N+1)
(
2ℓN + ℓ − 1
ℓN
)
.
Proof. The estimates |α2 − α1| ≥ p and |α2| ≥ p imply
|Fℓ| ≤
1
pℓ(N+1)
ℓN−1∑
j=0
(
2ℓN − 2 − j
ℓN − 1
)(
ℓ + j
j
)
= p − 1
p + 1
1
pℓ(N+1)
(
2ℓN + ℓ − 1
ℓN
)
≤ 1
pℓ(N+1)
(
2ℓN + ℓ − 1
ℓN
)
. □
It follows from Lemma 4.3 that Fℓ decays exponentially in ℓN .
Proof of Theorem 1.4. Using Equations (4.2), (4.3) and Lemma 4.3, we
conclude that
|mp − m(Qp(x))| ≤
1
pN+1
(
p − 1
N
)
=: ϵp
On Lehmer’s question for integer-valued polynomials 535
meaning that the difference of the Mahler measures decays exponentially1as
p → ∞. This proves the first part of Theorem 1.4. To show that the sequence
mp for odd p is decreasing, it suffices to prove the inequality
m(Qp(x)) − ϵp > m(Qp+2(x)) + ϵp+2,(4.6)
where ϵp is defined in the proof of Theorem 1.4. We can estimate m(Qp(x))−
m(Qp+2(x)) from below using that log(x) > 1 − 1x for x > 1. Indeed, for
p ≥ 5 we have
m(Qp(x)) − m(Qp+2(x)) = log
1 +
√
1 + 4/p2
1 +
√
1 + 4/(p + 2)2
>
√
1 + 4/p2 −
√
1 + 4/(p + 2)2
1 +
√
1 + 4/p2
≥ 12p2 −
1
2(p + 2)2 ≥
1
p3
.
On the other hand, using
(2n
n
)
≤ 4n for n ≥ 1, we can estimate ϵp + ϵp+2
from above: for p ≥ 7 we obtain
ϵp + ϵp+2 ≤
1
pN+1
4N + 1(p + 2)N+2 4
N+1 ≤
(4
p
)N+1
≤
(3
4
)p
≤ 1
p3
.
This implies inequality (4.6) for p ≥ 7. Together with
m3 = 0.16129 . . . , m5 = 0.04920 . . . , m7 = 0.02145 . . . ,
it concludes our proof of Theorem 1.4. □
5. Discussion
The choice for the family of polynomials fp(x) is far from optimal: among
integer-valued polynomials of prime degree p ≡ 3 mod 4, it is not the one
with smallest Mahler measure larger than 1. This can already be seen when
p = 3: an integer-valued polynomial with the smallest Mahler measure is
Q3(x) =
2
3x
3 − 12x
2 − 16x − 1,
with the Mahler measure 1.02833 . . . much smaller than M(f3(x)) =
1.17503 . . . .
For d = 2, 3, . . . , define Qd(x) to be an irreducible integer-valued poly-
nomial of degree d with smallest Mahler measure larger than 1. Then the
following questions arise.
1In fact, the decay is much faster than exponentially as the leading term of the asymptotics
in ϵp is given by √
2
π
·
2p−1
p
p+1
2
√
p − 1
as p → ∞.
536 Berend Ringeling
Question 5.1. How to (efficiently) compute these polynomials Qd(x)?
Question 5.2. What can be said about the asymptotics of M(Qd(x)) for
d → ∞?
Acknowledgments. The author would like to thank François Brunault,
David Hokken and Wadim Zudilin for interesting discussions and helpful
comments. The author would also like to thank Matilde Lalín, Don Zagier
and the referee for carefully reading the manuscript and for giving useful
suggestions.
References
[1] F. Brunault & W. Zudilin, Many variations of Mahler measures: a lasting symphony,
Australian Mathematical Society Lecture Series, vol. 28, Cambridge University Press, 2020,
xv+167 pages.
[2] K. Conrad, “Irreducibility of xn − x − 1”, unpublished note, available at https://kconrad.
math.uconn.edu/blurbs/ringtheory/irredselmerpoly.pdf.
[3] W. Ljunggren, “On the irreducibility of certain trinomials and quadrinomials”, Math. Scand.
8 (1960), p. 65-70.
[4] G. Pólya & G. Szegő, Problems and theorems in analysis. II. Theory of functions, ze-
ros, polynomials, determinants, number theory, geometry, Classics in Mathematics, Springer,
1998, translated from German by C. E. Billigheimer, reprint of the 1976 English translation,
xii+392 pages.
Berend Ringeling
Department of Mathematics, IMAPP
Radboud University, PO Box 9010
6500 GL Nijmegen, Netherlands
E-mail: bjringeling@gmail.com
URL: https://sites.google.com/view/berendringeling/


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