Uniform Distribution Theory 9 (2014), no.1, 5–19

ON THE DISTRIBUTION OF POLYNOMIALS WITH
BOUNDED ROOTS II. POLYNOMIALS WITH
INTEGER COEFFICIENTS
Shigeki Akiyama and Attila Pethő∗
ABSTRACT. In the present paper we give a new type of statistical results on
the distribution of integral polynomials of given degree. The main feature of ourformula is that we can see a clear distinction with respect to the signature of polynomials. For example, we see that among certain polynomials in question, totallyreal ones are very rare. Further we show that reducible polynomials are negligiblein every formula. We derive asymptotic results on Pisot, Salem and expanding
polynomials that often appear as dilation constants of dynamical systems.

Communicated by Pierre Liardet
Dedicated to the memory of Gérard Rauzy

1. Introduction
For a real polynomial P (X), its degree d is written as d = r + 2s where r and2s denote the numbers of its real and complex zeroes respectively. The pair (r, s)
is called the signature of P (X). Since d is fixed in the following discussion, wecall s the signature of P (X), for simplicity. In the previous paper [4], we investigated the distribution of polynomials with real coefficients, with zeroes insidethe unit disk and of given signature. The goal of this paper is to derive several
consequences from [4] when we confine ourselves to polynomials with integer
coefficients. An additional tool is a theorem of H. Davenport [8], which gives2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: 11C08, 11K16, 11R09, 11R06.
K e y w o r d s: Distribution of integer polynomials, Pisot numbers, Salem numbers, expanding
polynomials.
The authors are supported by the Japanese Society for the Promotion of Science (JSPS), Grantin aid 21540010 and Invitation Fellowship Program FY2011, L-11514.

5

SHIGEKI AKIYAMA AND ATTILA PETHŐa sharp estimate on the number of lattice points within a given region circumscribed by algebraic surfaces. As we restrict our attention to monic polynomials,
the adjective ”monic” is omitted throughout this paper.
We show a new type of statistical result on the distribution of integral polynomials of given degree and signature. A remarkable feature of our results is that
we can observe explicit dependence on signatures in our asymptotic formulas.
Let us review several known statistical results on polynomials. For a complex
polynomial P (X) let H(P ) denote its height, i.e., the maximum of absolutevalues of the coefficients of P . Let B denote a - typically large - integer or a real
number. Clearly the number of integral polynomials of degree d and height at
most B is (2B)d + O(B d−1 ), where the implied constant depends only on d. B.L.
van der Waerden [20] confirmed a ‘folklore’ belief that reducible polynomials are
negligible within Z[X]. He proved that among such (2B)d +O(B d−1 ) polynomials
the frequency of reducible polynomials with height at most B and of degreed = q + r, which split into factors of degrees q and r, tends to B −q ; if q < r, andto B −r log B; if q = r.
The investigation of the distribution of integral polynomials having a prescribed Galois group goes back to the beginning of the last century and has avast literature. It was proved by K. Dörge [9] in 1925 that the natural densityof integral polynomials of degree d; whose Galois groups are different from thesymmetric group tends to zero. Later P.X. Gallagher [12] proved that the number of the above polynomials is O(B d−1/2 log B). We refer to D. Zywina [21] forrecent developments. Although the statistical theory of integral polynomials has
a rich history, we were not able to find an asymptotic formula for the number ofintegral polynomials P (X) of degree d, signature s and H(P ) ≤ B.
We wish to replace H(P ) by a different quantity, which we call ‘measure’ inthis article. First, let us take |P |, the maximum modulus of zeroes of P (X).
This is called inclusion radius or house of P . This measure is widely used intranscendental number theory [5] and in computer algebra [17]. The following
trivial inequalities|P |≤ H(P ) ≤ (2|P |)ddyield some estimates of the number of integral polynomials of degree d and|P | ≤ B, but it does not lead us to an asymptotic formula in terms of |P |.
As a consequence of [4] and the above-mentioned result of H. Davenport [8], an
asymptotic formula for the number of such polynomials is proved in Theorem 3.1.
Moreover we obtain asymptotic formula for polynomials with a fixed signature.
We also deal with some interesting subsets of Z[X], choosing different but
suitable measures.
6

ON THE DISTRIBUTION OF POLYNOMIALS WITH BOUNDED ROOTS II.

A polynomial in Z[X] is called Pisot if it has one real root greater than one,and the others are less than one in modulus. It is called Salem if it has one realroot greater than one, and others are not greater than one in modulus and at
least one root has modulus one. Pisot polynomials are irreducible and their realroot greater than one is called Pisot number. A Salem polynomial factors into an
irreducible Salem polynomial and cyclotomic polynomials. A Salem number isthe real root greater than one of a Salem polynomial. There is a good overview onthese polynomials and their applications in the book of M.J. Bertin et al. [6]. It is
well known that Pisot and Salem numbers often appear as dilation constants ofself-inducing structures in dynamical systems, and also in numeration systems,
see e.g. [1].
It seems that neither H(P ) nor |P | gives us an asymptotic formula for Pisot
and Salem polynomials. We employ the trace, T (P ) = −pd−1 of a polynomial
P (X) = X d + pd−1 X d−1 + · · · + p0 , as our measure. Clearly for Pisot and Salem
polynomials we have |H(P ) − |T (P )|| ≤ d − 1. In [3] S. Akiyama et al. proved an
asymptotic formula for the number of Pisot polynomials1 of degree d and trace
B. In Corollary 4.2 we obtain a better error term for Pisot polynomials andCorollary 4.1 gives an asymptotic formula for Salem polynomials of degree 2dand trace B. These results shows that Salem polynomials are much less frequent
than Pisot polynomials.
In the last section we are dealing with expanding polynomials, that is, polynomials in Z[X] whose zeroes lie outside the unit disk. The expanding polynomialsalso play an important role in numeration systems, see [1]. Similarly to Pisot
polynomials, H(P ) or |P | does not seem to give an asymptotic formula for thenumber of expanding polynomials. Instead, we choose the norm N (P ) = (−1)d p0
of a polynomial P (X) = X d +pd−1 X d−1 +· · ·+p0 as the measure. For expanding
P , we easily get|N (P )| ≤ H(P ) ≤ 2d |N (P )|.
In [2] S. Akiyama et al. proved an asymptotic formula for the size of the corresponding set. Combining results of [4] with the method of [2] we generalizethat formula for the number of integral expanding polynomials of degree d andwith norm B and with signature s. Finally we prove in a quantitative form that
reducible expanding polynomials are in minority.
Our asymptotic formulae have the common shape cd,s B κ(d) + O(B κ(d)−1 ).
Dependency on signature is far from uniform. Indeed, combining Theorem 3.1
with Theorem 6.1 of Part I [4] we see that the frequency of the totally real
1The proof in [3] indeed shows that the contribution of Salem polynomials is small and falls

into the error term of the formula.

7

SHIGEKI AKIYAMA AND ATTILA PETHŐ
2polynomials among all polynomials in consideration is asymptotically 2−d /2 ,much less than the anticipated uniform frequency 2/d.

2. Preliminary results
Let d be a positive integer. If P (X) ∈ R[X] is of degree d, then its signatures satisfies the inequality 0 ≤ s ≤ bd/2c. Each set of polynomials can be divided
into bd/2c + 1 disjoint classes according to their signatures. Our results andproofs will be true not only for these classes but also for their union, i.e., for theoriginal set as well. To simplify the description, we introduce the ”signature”−1, which means the union of the classes.
Let B > 0, which is typically a big integer or a real number. With the(s)above convention on the signatures denote Ed (B), s = −1, . . . , bd/2c the setof vectors (pd−1 , . . . , p0 ) ∈ Rd such that the corresponding polynomial P (X) =xd + pd−1 xd−1 + · · · + p0 has signature s and satisfies the inequality |P | ≤ B.
(s)
(s)
(s)
We set2 Ed for Ed (1). The d-dimensional Lebesgue measure λd (Ed ) will be(s)denoted by vd . The following theorem was proved by A.T. Fam [10].
Let d ≥ 1 thenmY(j − 1)!4
2

22m
,


(2j − 1)!2
(−1)
j=1
vd
=
m
Y

j!2 (j − 1)!2
2m2 +2m+12,(2j − 1)!(2j + 1)!

Theorem 2.1.

if d = 2m,(1)
if d = 2m + 1.

j=1

(s)

For s ≥ 0 we do not have such explicit formula, but we proved in [4] that vd
can be computed by multiple integrals. In the next theorem Resx (P (x), Q(x))
denotes the resultant of the polynomials P (x) and Q(x).
(s)

Let d ≥ 1, 0 ≤ s ≤ bd/2c and r = d − 2s. Then the set Ed
Jordan measurable. Let Rk (x) = x2 − yj x + zj , j = 1, . . . , s and put√√√√
Dr,s = [−1, 1]r × [0, 1] × [−2 z1 , 2 z1 ] × · · · × [0, 1] × [−2 zs , 2 zs ].

Theorem 2.2.

isThen we have
(s)

(s)

vd = λd (Ed ) =

1
r!s!

Z
|∆r |∆s ∆r,s dX,Dr,s2In Part I the sets E (−1) (B) and E (−1) were denoted by E (B) and Edd respectively. We apolddogize for this small difference, but we were not able to find a uniform notation.

8

ON THE DISTRIBUTION OF POLYNOMIALS WITH BOUNDED ROOTS II.

where
∆r

=

Y

(xj − xk ),1≤j<k≤r

∆s

∆r,s

=

=

Y

Resx (Rj (x), Rk (x)),

1≤j<k≤rr YsY

Rk (xj )j=1 k=1

and dX = dx1 . . . dxr dy1 dz1 . . . dys dzs .
The next theorem was proved for s = −1 by I. Schur [19], see also A.T. Famand J.S. Meditsch [11], and for 0 ≤ s ≤ bd/2c by ourselves [4].
(s)Let d ≥ 1 and −1 ≤ s ≤ bd/2c. Then the boundary of Ed
the union of finitely many algebraic surfaces.

Theorem 2.3.

(s)

is

(s)

Now we formulate an easy lemma, which connects Ed and Ed (B). It appeared in a slightly different form as Lemma 4.2 in [3], but the present one ismore appropriate for our purposes.
Lemma 2.1.

Let d ≥ 1 and −1 ≤ s ≤ bd/2c. Then we have
(s)

(s)

Ed (B) = diag(B, . . . , B d )Ed ,(2)

where diag(v1 , . . . , vd ) denotes the d-dimensional diagonal matrix, whose entries
are v1 , . . . , vd .
Moreover(s)(s)λd (Ed (B)) = B d(d+1)/2 λd (Ed ).
(3)
P r o o f. It is clear that the second assertion is an immediate consequence of thefirst one. To prove the first assertion, remark that if the absolute value of theroots of P (X) = X d + pd−1 X d−1 + · · · + p0 are at most one, then the rootsof PB (X) = X d + pd−1 BX d−1 + · · · + p0 B d are of absolute value at most B.
Further, it is obvious that the signature of P and PB is the same. Define the(s)(s)
mapping ψB : Ed 7→ Ed (B) as ψB (z1 , . . . , zd ) = (z1 B, . . . , zd B d ). Thus(s)(s)(pd−1 , . . . , p0 ) ∈ Ed if and only if ψB (pd−1 , . . . , p0 ) ∈ Ed (B).
Later we will estimate the number of elements of bounded subsets of Z[X]. We
will transform such problems into lattice point counting problems in boundedregions. For our purpose the following result of H. Davenport is appropriate.
9

SHIGEKI AKIYAMA AND ATTILA PETHŐLet R be a closed bounded region in the n dimensional space Rn and let N(R) and V(R) denote the number of points with integral
coordinates in R and the volume of R, respectively. Suppose that:
Lemma 2.2 ([8, Theorem]).

• Any line parallel to one of the n coordinate axes intersects R in a set ofpoints which, if not empty, consists of at most h intervals.
• The same is true (with m in place of n) for any of the m dimensional
regions obtained by projecting R on one of the coordinate spaces defined byequating a selection of n − m of the coordinates to zero; and this conditionis satisfied for all m from 1 to n − 1.
Then
|N(R) − V(R)| ≤n−1
Xhn−m Vm ,m=0where Vm is the sum of the m dimensional volumes of the projections of R on
the various coordinate spaces obtained by equating any n−m coordinates to zero,and V0 = 1 by convention.
The assumptions of Lemma 2.2 are satisfied, if for example the boundary of
R is the union of finitely many algebraic surfaces. We will apply this lemma(s)in case when R = Ed (B) or some transformation of it. By Theorem 2.3 the(s)boundary of Ed is the finite union of algebraic surfaces, then, by Lemma 2.1(s)(s)
the same holds for Ed (B). We obtain the volume of Ed (B) from Theorems
2.1, 2.2 and Lemma 2.1. If we are able to estimate the remaining term precisely
enough, then we obtain the desired result. In the next sections we perform thisprogram.

3. The main distribution results
In this section we study the distribution of polynomials with integer coeffi(s)cients and with bounded roots. For d ≥ 1 and 0 ≤ s ≤ bd/2c let Nd (B) denotethe number of P (X) ∈ Z[X], which are monic, of degree d, with signature s andPbd/2c (s)(−1)
with |P | < B. By our convention Nd (B) = s=0 Nd (B). Our aim is toprove
3.1. Let d ≥ 1, −1 ≤ s ≤ bd/2c and B > 0. Then there exists a
constant c1 depending only on s, d such that
Theorem(s)(s)|Nd (B) − vd B d(d+1)/2 | ≤ c1 B d(d+1)/2−1 .
10ON THE DISTRIBUTION OF POLYNOMIALS WITH BOUNDED ROOTS II.
(s)

P r o o f. In the proof of Lemma 2.1 we introduced the mapping ψB : Ed 7→(s)
Ed (B). It is continuous and bijective and transforms algebraic relations into
algebraic ones. This implies together with Lemma 2.3 that the boundary of(s)
Ed (B) is the union of finitely many algebraic surfaces.
(s)

(s)

Let d, s and B be fixed. By Lemma 2.1 the volume of Ed (B) is vd B d(d+1)/2 .
It is clear that P (X) ∈ Z[X] is monic, of degree d, with signature s and with(s)
|P | < B if and only if the vector of its coefficients belongs to Ed (B). Thus
(s)
(s)
(s)
Nd (B) = |Ed (B) ∩ Zd |, i.e., the number of lattice points in Ed (B).
As for this set the assumptions of Lemma 2.2 are satisfied we obtain(s)(s)|Nd (B) − vd B d(d+1)/2 | ≤d−1
Xhd−m Vm ,m=0where h denotes the maximal number of intervals, which cover the intersection(s)of Ed (B) with any line parallel to one of the d coordinate axis. This number isfinite and is independent from B.
Further Vm is the sum of the m dimensional volumes of the projections of(s)
Ed (B) on the various coordinate spaces obtained by equating any d − m coor(s)(−1)
dinates to zero, and V0 = 1 by convention. Let v = (v1 , . . . , vd ) ∈ Ed ⊆ Ed
and Pv (X) the corresponding polynomial to v. Then, as all roots of Pv (X) belong to the unit disc, we have the trivial bound |vm | < 2d , m = 1, . . . , d. Thus the(s)(s)
above described projections of Ed are bounded. After applying ψB to Ed we see(s)that the length of the projection of Ed (B) to any line parallel to the m-th coordinate axis is covered by an interval of length at most O(B m ), m = 1, . . . , d − 1.
Thus
Vm ≤ O(B d(d+1)/2−(1+···+m) ) ≤ O(B d(d+1)/2−1 ).
The theorem is proved.

The next theorem gives a similar asymptotic formula for the number of irreducible polynomials P (X) ∈ Z[X] of degree d, signature s and with |P | ≤ B.
(s)
This number is denoted by Id (B). The theorem is a quantitative version ofCorollary of [18] on p. 47.
3.2. Let d ≥ 1, −1 ≤ s ≤ bd/2c and B > 0. Then there exists a
constant c2 depending only on s, d such that
Theorem(s)(s)|Id (B) − vd B d(d+1)/2 | ≤ c2 B d(d+1)/2−1 .
P r o o f. It is clear that we obtain the set of irreducible polynomials with therequired properties if we remove from all polynomials the reducible ones. If a
11

SHIGEKI AKIYAMA AND ATTILA PETHŐpolynomial of degree d is reducible then it has a factor of degree in the interval[dd/2e, d − 1]. Notice that the signature of the divisors may differ from thedividend, which we have to take into account. Thus


d−1
X
(s)
(s)
Id (B) ≥ Nd (B) − 
Nj (B)Nd−j (B) .
j=dd/2e

Using Theorem 3.1 we obtain

(s)

(s)

Id (B) ≥ vd B d(d+1)/2 − 

d−1
X


(−1)

vj

(−1)

B j(j+1)/2 vd−j B (d−j)(d−j+1)/2 

j=dd/2e+ O(Bd(d+1)/2−1).

Now
B j(j+1)/2 B (d−j)(d−j+1)/2 = B j(j+1)/2+(d−j)(d−j+1)/2
and we have the inequalityd(d + 1)(d − j)(d − j + 1) j(j + 1)
+
≤
−1
2
2
2
for the exponents. Thus
(s)

(s)

Id (B) ≥ vd B d(d+1)/2 − dO(B d(d+1)/2−1 ) + O(B d(d+1)/2−1 )(s)= vd B d(d+1)/2 − O(B d(d+1)/2−1 ).
The lower bound
(s)

(s)

Id (B) ≥ vd B d(d+1)/2 + O(B d(d+1)/2−1 )is an immediate consequence of Theorem 3.1. Thus the assertion is completely
proved.
The following corollary is an immediate consequence of Theorems 3.1 and 3.2.
3.1. Let d ≥ 1, −1 ≤ s ≤ bd/2c and B > 0. Then the numberof reducible polynomials P (X) ∈ Z[X] of degree d, signature s and such that
|P | ≤ B is O(B d(d+1)/2−1 ).

CorollaryThis means that there are much more irreducible polynomials than reducible(s)ones in each signature classes. Theorems 3.1 and 3.2 show that Nd (B) and(s)
Id (B) have for each s the same growth rate in B. Hence the frequency of the(s)appearance of a signature s depends on the volume vd . We quantify this by thenext corollary.
12

ON THE DISTRIBUTION OF POLYNOMIALS WITH BOUNDED ROOTS II.
Corollary 3.2.

Let d ≥ 1, 0 ≤ s ≤ bd/2c and B > 0. Then(s)

Nd (B)
(−1)

Nd

(s)

=

(B)

vd

(−1)

+ O(B −1 ).

vd

(s)

In Part I we studied the quotients

vd

(−1)

vd

. We proved among others that they

are rational numbers, Theorem 5.1. In the case s = 0 we were able to show
2
that the size of this quotient is 2−d /2 , Theorem 6.1. This means that totallyreal polynomials are extremely rare. On the other hand for even d, in the same(d/2)

theorem, we obtained the conditional boundvd∼ cd−3/8 , i.e. totally com-

(−1)

vd

plex polynomials has much bigger frequency as the average. It is an interesting(s)

problem to describe the asymptotic behavior

vd

(−1)

vd

for other indices s.

4. Distribution of polynomials with a dominating root(s)

Let d, s be as earlier, a ≥ 1 be fixed and B ∈ Z. Denote by Bd,a (B) the set ofpolynomials P (X) ∈ Z[X] with trace B, signature s and such that the absolutevalue of all but one of its zeroes is at most a. The set containing the irreducible(s)(s),irr
elements of Bd,a (B) will be denoted by Bd,a (B). From the correspondence
(s)

(s)

P (X) with P (−X), we easily have |Bd,a (B)| = |Bd,a (−B)|, i.e., we may assume
B > 0 without loss of generality. Taking a = 1 and s = −1 we obtain the eminent
example of this concept, the set of Pisot and Salem polynomials.
In [3], S. Akiyama et al. proved
(−1)

(−1)

|Bd,1 (B)| − vd−1 B d−1 = O(B d−1−1/(d−1) ).
(s)

A new embedding of Bd,a (B) into a d − 1-dimensional lattice together with
Theorem 2.2, especially the case s = 0, allow us to estimate the number of Salem
polynomials, hence the Pisot polynomials as well. The main result of this sectionisTheorem 4.1. Let d, s, B be integers and a ∈ R such that d, B ≥ 1, a > 0 and−1 ≤ s ≤ bd/2c. Then

(s)

(s)

|Bd,a (B)| − vd−1 ad(d−1)/2 B d−1 = O(B d−2 ),where the constant in the O depends only on d, s, a.
We obtain similar result for irreducible polynomials.
13

SHIGEKI AKIYAMA AND ATTILA PETHŐTheorem 4.2.

Let d, s, a, B as in Theorem 4.1. Then(s),irr

|Bd,a

(s)

(B)| − vd−1 ad(d−1)/2 B d−1 = O(B d−2 ),where the constant involved in the O depends only on d, s, a.
Before proving these theorems we formulate their consequences for Pisot andSalem polynomials. You find a good overview on these polynomials and theirapplications in the book of M.J. Bertin et al. [6]. It is well known that a Salem
polynomial has to be reciprocal and its degree is even. Let d ≥ 1, B be integers.
Denote S2d (B) the number of Salem polynomials P of degree 2d and with T (P ) =B. By the explanation of the beginning of this section we may restrict ourselvesto the case B > 0. Finally the number of irreducible polynomials among theirr
Salem polynomials will be denoted by S2d(B).
Corollary 4.1.

With the above notations(0)

S2d (B) − vd−1 2d(d−1)/2 B d−1 = O(B d−2 )
and
(0)

irr
S2d
(B) − vd−1 2d(d−1)/2 B d−1 = O(B d−2 ),where the constants in O depend only on the degree 2d.
(s)Finally, let Pd (B) denote the number of Pisot polynomials of degree d,with signature s and with trace B. We may assume B > 0 again. As a Pisot
polynomial always has a real zero we have to modify the range of signatures asfollows: if d is odd, then −1 ≤ s ≤ (d−1)/2 and if d is even then −1 ≤ s ≤ d/2−1.
Notice that Pisot polynomials are always irreducible, thus we do not need to
introduce extra counting functions for them.
Corollary 4.2.

With the above notations
(s)

(s)

Pd (B) − vd−1 B d−1 = O(B d−2 ),where the constant involved in the O depends only on d.
Now we turn to the proof of the statements.
(s)

Proof of Theorem 4.1. Let M be a positive integer and Ad,a (M ) be the setof vectors (b0 , b1 , . . . , bd−2 ) ∈ Rd−1 such that all but one root of xd − M xd−1 −bd−2 xd−2 − · · · − b0 has modulus less than a. From the formula(xd−1 + rd−2 xd−2 + · · · + r0 )(x − M − rd−2 ) = xd − M xd−1 − bd−2 xd−2 − · · · − b0 .
14

ON THE DISTRIBUTION OF POLYNOMIALS WITH BOUNDED ROOTS II.

we define the map (r0 , . . . , rd−2 ) 7→ (b0 , . . . , bd−2 ). More explicitly, for fixed(s)(s)
integers d, s, M and real number a we define χM : Ed−1 (a) 7→ Ad,a (M ) byχM (r0 , . . . , rd−2 ) = (rd−2 (M +rd−2 )−rd−3 , . . . , r1 (M +rd−2 )−r0 , r0 (M +rd−2 )).
This map is continuous and surjective. By uniqueness of the polynomial factorization, χM is even injective provided M > da. This is true because the modulusof the additional root, M + rd−2 , is larger than a. Thus for M > da we have
(s)

(s)

|Bd,a (M )| = |χM (Ed−1 (a)) ∩ Zd−1 |.
(s)

As χM is a linear mapping and the boundary of Ed−1 (a) is the union of finitely(s)

many algebraic surfaces, the same is true for χM (Ed−1 (a)).
(s)

To apply Lemma 2.2 we have to compute the volume of χM (Ed−1 (a)). Computation of the Jacobian leads to the formula:Z(s)λd−1 (χM (Ed−1 (a))) =| det(J1 )| dr0 . . . drd−2(s)Ed−1 (a)

with




J1 = 



0
0

...
0

−1
M + rd−2

M + rd−2
0

...
..
.
0
0

0
−1

−1
M + rd−2..
.

M + 2rd−2
rd−3

...
...

0
0

r1
r0





.



Readily det(J1 ) is a monic polynomial in M of degree d − 1 and its other coefficients are polynomials in r0 , . . . , rd−2 , i.e. they are bounded in absolute value
by some polynomial in a. ThusZ(s)λd−1 (χM (Ed−1 (a))) = M d−1
dr0 . . . drd−2(s)Ed−1 (a)


Z
d−2
X
+ O
Mj
j=0


(s)

Ed−1 (a)pj (r0 , . . . , rd−2 ) dr0 . . . drd−2 (s)= λd−1 (Ed−1 (a))M d−1 + O(M d−2 )(s)= vd−1 ad(d−1)/2 M d−1 + O(M d−2 ).
For the last step we used Lemma 2.1. From now on we may repeat the proof(s)of Theorem 3.1 because the assumptions of Lemma 2.2 hold for χB (Ed−1 (a)).
15

SHIGEKI AKIYAMA AND ATTILA PETHŐ

Finally we obtain
(s)

(s)

|Bd,a (B)| = vd−1 ad(d−1)/2 B d−1 + O(B d−2 ).
2
Proof of Theorem 4.2. Like in the proof of Theorem 3.2 we count the(s)number of reducible polynomials in Bd,a (B). We assume that B > ad. Let(s)

P (X) ∈ Bd,a (B) be reducible and denote by β its dominating root, which exists
by the proof of the last theorem. There exist monic polynomials Q(X), R(X) in
Z[X] such that d − 1 ≥ deg Q ≥ deg R ≥ 1 and P (X) = Q(X)R(X). It is clearthat β can be a zero only one of Q and R, the zeroes of the other factor arebounded in absolute value by a. Using the estimates of Theorems 3.1 and 4.1(s)
the number of reducible elements in Bd,a (B) is bounded by
d−1
X

(−1)

(−1)

vm−1 am(m−1)/2 B m−1 vd−m a(d−m)(d−m−1)/2 + O(B d−3 )m=bd/2c= O(B d−2 ),where the constant in O depends only on d and a. Combining this estimate withthe result of Theorem 4.1 we complete the proof. 2
Now we turn to the proof of the Corollaries.
Proof of Corollary 4.1 It is well known, see e.g. [6, 14], that the degree of a
Salem polynomial is even, it has two real roots one of which is larger, the otheris less then one and all others are non-real complex numbers, lying on the unit
circle. Moreover they are reciprocal polynomials, i.e., P (X) = X d P (1/X). Let
B be an integer and assume that P (X) is a Salem polynomial of degree 2d andtrace B. Let β denote the dominating root of P (X).
Dividing P (X) by X d leads to a polynomial Q(y) in y = X +1/X with integer
coefficients and of degree d. This polynomial has only real roots and its trace
is B. If γ denotes a zero of P (X) then γ + 1/γ is a zero of Q(y). Moreover if
γ 6= β, 1/β then |γ + 1/γ| ≤ 2. Thus(0)(0),irrirr
S2d (B) = |Bd,2 (B)| and S2d(B) = |Bd,2(B)|

and the statements follow immediately from Theorems 4.1 and 4.2. 2
Proof of Corollary 4.2 Let B be a fixed integer. It is clear that if P (X) =xd − Bxd−1 + pd−2 xd−2 + · · · + p0 ∈ Z[X] is such that all but one of its roots lie
in the unit disk then it is a Pisot or Salem polynomial. Since the contributionof Salem polynomials is by Corollary 4.1 much smaller, we obtain the result.
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ON THE DISTRIBUTION OF POLYNOMIALS WITH BOUNDED ROOTS II.

5. Distribution of expanding polynomials
A polynomial is called expanding, if its zeroes lie outside the unit disk. There
are only finitely many expanding polynomials with integer coefficients of degreed and with fixed constant term B. By the argument of the beginning of the lastsection we may assume B > 0. Denoting this set by Cd (B) it was proved by S.
Akiyama et al. [2] that
|Cd (B)|(−1)
lim= vd−1 .
B→∞ B d−1
Later M. Madritsch and A. Pethő3 [16] proved a formula with error term:(−1)|Cd (B)| − vd−1 B d−1 = O(B d−1−1/d ).
Of course Cd (B) can also be split in disjoint union of subsets according thesignature of the occurring polynomials. In accordance of the earlier definitions(s)these subsets will be denoted by Cd (B), −1 ≤ s ≤ bd/2c. Combining the methodof [16] with Theorem 3.1 it is easy to prove
Theorem 5.1.

With the above notations
(s)

(s)

|Cd (B)| − vd−1 B d−1 = O(B d−1−1/d ).
As we being not able to improve the error term, we omit the details.
Through this paper we show that the number of irreducible polynomials is atleast one magnitude larger than the reducible ones in the investigated sets. This
was neither done in [2] nor in [16]. At the end of this paper we fill this gap. Let(s),irr(s)Cd(B) denote the subset of Cd (B), which contains its irreducible elements.
Theorem 5.2.

With the above notations(s),irr|Cd(s)(B)| − vd−1 B d−1 = O(B d−1−1/d ).

P r o o f. As in the above proofs we estimate the number of reducible elements(s)in Cd (B). Assume that P (X) is such an element and P (X) = Q(X)R(X) with
R, Q ∈ Z[X], deg R and deg Q ≥ 1. Of course both are expansive and one ofthem is of degree at least bd/2c. Moreover, if the constant term of Q is q, thenq is a divisor of B and the constant term of R is B/q. Thus the number of
reducible polynomials is at most
X

d−1
X

(−1)

(−1)
|Cm
(q)||Cd−m (B/q)|.

q|B m=bd/2c
3In both cited papers slightly different notation was used.

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SHIGEKI AKIYAMA AND ATTILA PETHŐEach term of the inner sum is estimated in Theorem 5.2 by O(B d−2 ), which
implies the same estimate for the whole inner sum. Hence the number of reducible(s)polynomials in Cd (B) is at mostd(B)O(B d−2 ),where d(B) denotes the number of divisors of B, which is o(B), see e.g. [15].


6. Acknowledgments
The first author moved from Niigata University to the current address in August 2012.
The paper was written, when the second author was visiting Niigata University as along term research fellow of JSPS. Both of us wish to express our deep gratitude to
all the staffs in Department of Mathematics, Niigata University and the support from
JSPS, Grant in aid 21540010 and Invitation Fellowship Program FY2011, L-11514.
The second author was partially supported by the OTKA grant K104208 and by theTÁMOP-4.2.2.C-11/1/KONV-2012-0001 project. The project is implemented throughthe New Hungary Development Plan, co-financed by the European Social Fund andthe European Regional Development Fund.
The authors thank Lajos Rónyai for his valuable hints to the literature. We also
thank the anonymous referee for detailed comments, which considerably improved thepresentation of this paper.

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reine angew. Math., 148 (1918), 122–145.4[20] B. L. van der Waerden, Die Seltenheit der reduziblen Gleichungen und der Gleichungenmit Affekt, Monatsh. Math. Phys. 43 (1936), 133-147.
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Received July 20, 2012
Accepted June 16, 2013

Institute of Mathematics
University of Tsukuba
Tennodai 1-1-1, Tsukuba, Ibakari, JAPANE-mail: akiyama@math.tsukuba.ac.jp
Department of Computer Science
University of DebrecenH-4010 Debrecen
P.O. Box 12, HUNGARY
E-mail: Petho.Attila@inf.unideb.hu

4Reprinted in Schur’s collected papers: I. Schur, Gesammelte Abhandlungen. Band I.-III. (Ger-man) Herausgegeben von Alfred Brauer und Hans Rohrbach. Springer-Verlag, Berlin-New
York, 1973.

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