Journal of Inequalities in Pure and
Applied Mathematics
INEQUALITIES ON POLYNOMIAL HEIGHTS
LAURENŢIU PANAITOPOL AND DORU ŞTEFĂNESCU
University of Bucharest
70109 Bucharest 1,
ROMANIA.
EMail: pan@al.math.unibuc.ro
University of Bucharest,
P.O.Box 39–D5, Bucharest 39,
ROMANIA
EMail: stef@irma.u-strasbg.fr
volume 2, issue 1, article 7,
2001.
Received 21 June, 2000;
accepted 26 October 2000.
Communicated by: H. Gauchman
Abstract
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2000
School of Communications and Informatics,Victoria University of Technology
ISSN (electronic): 1443-5756
017-00
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Abstract
We give explicit bounds for the absolute values of the coefficients of the divisors
of a complex polynomial. They are expressed in function of the coefficients and
of upper and lower bounds for the roots. These bounds are compared with
other estimates, in particular with the inequality of Beauzamy [B. Beauzamy,
Products of polynomials and a priori estimates for coefficients in polynomial
decompositions: A sharp result, J. Symbolic Comput., 13 (1992), 463–472].
Through examples it is proved that for some cases our evaluations give better
upper limits.
Inequalities on Polynomial
Heights
Laurenţiu Panaitopol and
Doru Ştefănescu
2000 Mathematics Subject Classification: 12D05, 12D10, 12E05, 26C05.
Key words: Inequalities, Polynomials.
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We are grateful to the referee for the pertinent suggestions concerning Propositions
2.1 and 2.3.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
A Height Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3
A Smallest Divisor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
References
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1.
Introduction
P
If P (X) = di=0 ai X i ∈ C[X], the height of the polynomial P is defined by
H(P
sizes are the
norm ||P || =
qP) = max(|a0 |, |a1 |, . . . , |ad |). Othernpolynomial
o
R1 1
d
2iπθ
logP (e )dθ and Bombieri’s
j=0 |aj |, the measure M(P ) = exp
0 2π
qP
d
2 d
norm [P ]2 =
j=0 |aj | / j .
There exist many estimates for the height of an arbitrary polynomial divisor Q of P . They can be expressed in function of polynomial sizes as the norm,
the measure or Bombieri’s norm. We mention, for example, the estimate
d
H(Q) ≤
M(P ),
bd/2c
where the measure can be easily computed by the method of M. Mignotte [5].
For integer polynomials one of the best height estimates uses the norm of
Bombieri [2] and was obtained by Beauzamy [1]
3/4
H(Q) ≤ ln [P ]2 ,
with
ln =
n/2
3 ·3
.
2(πn)1/2
Inequalities on Polynomial
Heights
Laurenţiu Panaitopol and
Doru Ştefănescu
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2.
A Height Estimation
We present another estimate for the heights of proper divisors of a complex
polynomial. It makes use of an inequality of M. Mignotte [6]. A key step is the
consideration of complex polynomials with roots of moduli greater than 2.
Proposition 2.1. Let P (X) = X n + an−1 X n−1 + · · · + a1 X + a0 ∈ C[X] \ C,
such that P (0) 6= 0 and let µ be a lower bound of the absolute values of the
roots of P . If Q is a monic proper divisor of P in [X], then
H(Q) < H(P )
if µ ≥ 2 ,
2 n−i H(Q) < max ai
if µ < 2 .
0≤i≤n
µ
Proof. We suppose
Laurenţiu Panaitopol and
Doru Ştefănescu
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Q(X) = X k + bk−1 X k−1 + · · · + b1 X + b0 .
Let ξ1 , . . . , ξn ∈ C be the roots of P . Without loss of generality we may suppose
that ξ1 , . . . , ξk are the roots of Q. Note that
(2.1)
Inequalities on Polynomial
Heights
P = (X − ξk+1 ) · · · (X − ξn )Q.
By an inequality of M. Mignotte [6], we have
|ξk+1 | − 1 · · · |ξn | − 1 H(Q) < H(P ).
(2.2)
We look to µ > 0, which is a lower bound for the absolute values of the roots
of P .
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If µ ≥ 2 then |ξk+1 | − 1, . . . , |ξn | − 1 ≥ 1 and by (2.2) we obtain
H(Q) < H(P ).
(2.3)
If µ < 2 we associate the polynomials
Pµ (X) = (2/µ)n P (µX/2) = X n + an−1 2/µ X n−1 + · · · + a0 (2/µ)n ,
Qµ (X) = (2/µ)k Q(µX/2) = X k + bk−1 2/µ X k−1 + · · · + b0 (2/µ)k .
Inequalities on Polynomial
Heights
Let η1 , . . . , ηn ∈ C be the roots of Pµ . Then
|ηi | = 2ξi /µ ≥ 2
Laurenţiu Panaitopol and
Doru Ştefănescu
and from (2.2) it follows that
n−i H(Q) = max |bj | ≤ max bj (2/µ)k−j = H(Qµ ) < H(Pµ ) = max ai (2/µ) .
j
j
i
Therefore
(2.4)
2 n−i H(Q) < max ai
.
i
µ
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Corollary 2.2. If µ < 2 is a lower bound of the moduli of the roots of the
complex polynomial P and Q is proper divisor of P in [X], then
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H(Q) < (2/µ)n H(P ).
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Proof. Note that 2/µ > 1. Therefore, by Proposition 2.1,
2 n−i H(Q) < max ai
0≤i≤n
µ
<
=
2 n−i
max
0≤i≤n
2 n
µ
µ
· H(P )
H(P ).
Inequalities on Polynomial
Heights
Laurenţiu Panaitopol and
Doru Ştefănescu
Example 2.1. Consider P (X) = X 6 + 2X 5 + 5X 4 + 10X 3 + 21X 2 + 42X + 83.
By the criterion of Eneström–Kakeya (see M. Marden [4], p. 137), the zeros of
a polynomial a0 + a1 X + · · · + an X n with real positive coefficients lie in the
ring
min{ai /ai+1 } ≤ |z| ≤ max{ai /ai+1 } .
Thus, the roots of the polynomial P are in the ring 83/42 ≤ |z| ≤ 5/2. If Q is
a divisor of P we obtain
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H(Q) < 89.183
by Proposition 2.1,
H(Q) < 602.455 by B. Beauzamy [1].
Proposition 2.3. Let P (X) = an X n + an−1 X n−1 + · · · + a1 X + a0 ∈ C[X] \ C
such that P (0) 6= 0 and let ν > 0 be an upper bound for the absolute values of
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the roots of P . If Q is a proper divisor of P in [X], Q(0) = c0 , then
c 1
0
H(Q) < · H(P ),
if ν ≤ ,
a0
2
c 0
H(Q) < · max ai (2ν)i , if ν > 12 .
a0 0≤i≤n
Proof. Let P ∗ and Q∗ be the reciprocal polynomials of P and Q respectively.
Considering
1 ∗
1 n 1
P1 (X) =
P (X) =
X P
a0
a0
X
it is possible to obtain information about the heights with respect to the upper
bounds of the moduli of the roots of P . The polynomial P1 is monic and
1
ν
is a lower bound for the roots of P1 .
P
On the other hand, if P1 (X) = ni=0 bi X i , then bi = an−i /a0 and
2 n−i 1
max bi −1
max ai (2ν)i .
=
0≤i≤n
ν
|a0 | 0≤i≤n
Let Q ∈ C[X] be a proper divisor of P . Therefore
Q1 (X) =
1 ∗
1
Q (X) = X deg(Q) Q(X −1 )
c0
c0
is a proper divisor of P1 . However,
H(Q1 ) =
Inequalities on Polynomial
Heights
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Doru Ştefănescu
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1
H(Q).
|c0 |
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If ν ≤ 1/2, by Proposition 2.1 we have H(Q1 ) < H(P1 ), hence
1
1
· H(Q) <
· H(P ) ,
|c0 |
|a0 |
which gives the first inequality. The second relation in Proposition 2.1 gives the
other inequality.
Remark 2.1. If |P (0)| = 1, the inequalities in Proposition 2.3 become
H(Q) < H(P ),
if ν ≤ 12 ,
H(Q) < max ai (2ν)i , if ν > 12 .
Inequalities on Polynomial
Heights
Laurenţiu Panaitopol and
Doru Ştefănescu
0≤i≤n
Indeed, if |P (0)| = 1 we have |a0 | = |c0 | = 1.
The same inequalities are valid if P ∈ [X] and Q is a proper divisor
of P
over Z. In this case a0 and c0 are integers and c0 divides a0 , therefore ac00 ≤ 1.
As in Corollary 2.2 we deduce
Corollary 2.4. If ν > 12 is an upper bound of the moduli of the roots of the
complex polynomial P and Q is proper divisor of P in C[X], then
c 0
H(Q) < (2ν)n H(P ) .
a0
Example 2.2. Let P = 2381X 5 − 597X 4 − 150X 3 − 37X 2 + 9X + 2.
If z ∈ is a root of P , then |z| ≤ 2 max | ai−1
| = 100
, by the criterion of T.
ai
199
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Kojima [3]. If Q is a possible divisor of P , then
H(Q) < 2440.403
By Proposition 2.3,
H(Q) < 10745.533 by B. Beauzamy [1].
Remark 2.2. The estimates from Propositions 2.1 and 2.3 apply to any complex
polynomial, while the estimate of Beauzamy refers only to integer polynomials.
3
Example 2.3. Let P = 381X 5 −95iX 4 +(45−9i)X
+17iX 2 +(2+7i)X +3.
√
If z ∈ is a root of P , then |z| ≤ 2 max | ai−1
| = 2106
, again by the criterion of
ai
95
T. Kojima [3]. If Q is a possible divisor of P , then
H(Q) < 320.703
Laurenţiu Panaitopol and
Doru Ştefănescu
By Proposition 2.3,
H(Q) < 2374.689 by M. Mignotte [5].
7
6
5
4
3
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2
Example 2.4. Let P = 127X + 64X + 32X + 16X + 8X + 4X + 2X + 1.
The roots of P are in the ring 1/2 ≤ |z| ≤ 64/127, by the criterion of Eneström–
Kakeya (M. Marden, loc. cit.). If Q is a possible divisor of P , then
H(Q) < 134.167
Inequalities on Polynomial
Heights
By Proposition 2.3,
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H(Q) < 1472.464 by Beauzamy.
P
Corollary 2.5. Let P [X] = ni=0 ai X i ∈ R[X] \ R be monic and let Q ∈ [X]
be a proper monic divisor of P . If ai > 0 for all i, there exists µ ∈ such that
1
21− n < µ < 2 and ai−1 ≥ µai for i = 1, 2, . . . , n, then
H(Q) < 2 H(P ).
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Proof. From the condition ai−1 ≥ µai for all i and the criterion of Eneström–
Kakeya (M. Marden, loc. cit.) it follows that µ is a lower bound for the absolute
values of the roots of P . Because µ < 2, the inequality (2.4) from Proposition
2.1 is verified. Noting that
1 −1
1
1
< < 2n
2
µ
we obtain
2 n−i
H(Q) < max |ai | · |
|
1≤i≤n
µ
= H(P ) · max
2 n−i
1≤i≤n
< H(P ) · max 2
6
Laurenţiu Panaitopol and
Doru Ştefănescu
µ
n−i
n
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< 2 H(P ).
1≤i≤n
7
Inequalities on Polynomial
Heights
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5
4
3
2
Example 2.5. Let P (X) = X + 2X + 4X + 8X + 15X + 28X +
51X + 92. Then, by the theorem of Eneström–Kakeya (M. Marden loc. cit.),
µ = 51/28 is a lower bound for the absolute values of roots of P . We have
1
21− 7 < 51/28 < 2, so the hypotheses of Corollary 2.5 are fulfilled.
Remark 2.3. The same conclusion as in Corollary 2.5 holds if ai > 0 and there
1
exists ν > 0 such that ai−1 ≤ νai and 21 < ν ≤ 2 n −1 .
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3.
A Smallest Divisor
In this section we give a limit for the smallest height of a proper divisor of a
complex polynomial.
Theorem 3.1. Let P be a nonconstant complex polynomial and suppose that its
roots ξ1 , . . . , ξn ∈ C are such that
|ξ1 | ≥ · · · ≥ |ξk | > 1 ≥ |ξk+1 | ≥ · · · ≥ |ξn |.
If P = P1 P2 is a factorization of P over Z, we have
k/2
(n−k)/2
H(P ) > min{H(P1 ), H(P2 )} · |ξk | − 1
· 1 − |ξk+1 |
.
Proof. We observe that
Inequalities on Polynomial
Heights
Laurenţiu Panaitopol and
Doru Ştefănescu
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Y
|ξs | − 1
H(P ) > H(P1 )
s∈J1
and
H(P ) > H(P2 )
Y
|ξt | − 1 ,
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t∈J2
where {J1 , J2 } is a partition of {1, 2, . . . , n}. It follows that
n
Y
H(P ) > H(P1 ) H(P2 ) |ξj | − 1 ,
2
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therefore
v
u n
uY
H(P ) > min H(P1 ), H(P2 ) · t |ξj | − 1 .
j=1
We finally observe that
n
Y
|ξj | − 1 ≥ |ξk | − 1 k/2 · 1 − |ξk+1 | (n−k)/2 ,
j=1
which ends the proof.
Example 3.1. Let P (X) = X 5 − 49/6 X 4 + 59/3 X 3 − 6X 2 − 18X − 9/2. We
have ξ1 = ξ2 = ξ3 = 3, ξ4 = −1/2 and ξ5 = −1/3. Therefore
H(P ) > min{H(P1 ), H(P2 )} · (3 − 1)3/2 · (1 − 1/2)2/2 .
But (3 − 1)3/2 · (1 − 1/2)2/2 > 1.415, hence
1
8
H(P ) <
H(P ).
min{H(P1 ), H(P2 )} <
1.415
11
Remark 3.1. The index k from Theorem 3.1 can be computed by the Schur–
Cohn criterion (see M. Marden [4], p. 198). Since it is usually not possible to
find the roots ξk and ξk+1 , we need to know lower bounds for roots outside the
unit circle, respectively upper bounds for roots outside the unit circle.
Corollary 3.2. If P has no roots on the unit circle, we have
.
k/2
(n−k)/2
min{H(P1 ), H(P2 )} < H(P ) |ξk | − 1
· 1 − |ξk+1 |
.
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Heights
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Doru Ştefănescu
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References
[1] B. BEAUZAMY, Products of polynomials and a priori estimates for coefficients in polynomial decompositions: A sharp result, J. Symbolic Comput.,
13 (1992), 463–472.
[2] B. BEAUZAMY, E. BOMBIERI, P. ENFLO AND H. MONTGOMERY,
Products of polynomials in many variables, J. Number Theory, 36 (1990),
219–245.
[3] T. KOJIMA, On the limits of the roots of an algebraic equation, Tôhoku
Math. J., 11 (1917), 119-127.
[4] M. MARDEN, Geometry of Polynomials, AMS Surveys 3, 4th edition,
Providence, Rhode Island (1989).
[5] M. MIGNOTTE, An inequality about factors of polynomials, Math. Comp.,
28 (1974), 1153 – 1157.
[6] M. MIGNOTTE, An inequality on the greatest roots of a polynomial, Elem.
d. Math., (1991), 86-87.
[7] L. PANAITOPOL AND D. ŞTEFĂNESCU, New bounds for factors of integer polynomials, J. UCS, 1(8) (1995), 599–609.
Inequalities on Polynomial
Heights
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Doru Ştefănescu
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