Journal of Inequalities in Pure and
Applied Mathematics
ON THE LOCATION OF ZEROS OF COMPLEX POLYNOMIALS
MATTHIAS DEHMER
Technische Universität Darmstadt
Department of Computer Science
Hochschulstraße 10
64289 Darmstadt, Germany.
volume 7, issue 1, article 26,
2006.
Received 05 April, 2004;
accepted 12 December, 2005.
Communicated by: D. Stefanescu
EMail: matthias@dehmer.org
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
073-04
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Abstract
This paper proves bounds for the zeros of complex valued polynomials. The
assertions stated in this work have been specialized in the area of the location
of zeros for complex polynomials in terms of two foci: (i) finding bounds for
complex valued polynomials with special conditions for the coefficients and (ii)
locating zeros of complex valued polynomials without special conditions for the
coefficients − especially we are searching for bounds, which again are positive
roots of concomitant polynomials. As a result we obtain new zero bounds for
univariate polynomials with complex coefficients.
2000 Mathematics Subject Classification: 30C15.
Key words: Complex polynomials, zeros, inequalities.
I thank the referee of the present paper for fruitful comments. Furthermore I thank
Frank Emmert-Streib (Stowers Institute, USA) and Ralf Tandetzky (University of
Jena, Germany) for helpful discussions and comments concerning this work.
Contents
1
2
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Bounds for the Zeros of Complex Polynomials with Special
Conditions for the Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 4
3
Further Bounds for the Zeros of Complex Polynomials . . . . . 11
4
Evaluation of Zero Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
References
On the Location of Zeros of
Complex Polynomials
Matthias Dehmer
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1.
Introduction
The problems in the analytic theory of polynomials concerning locating zeros of
complex polynomials have been frequently investigated. Over many decades,
a large number of research papers, e.g, [1, 2, 3, 5, 6, 8, 9, 12, 13, 14] and
monographs [7, 10, 11] have been published.
The new theorems in this paper provide closed disks in the complex plane
K(z0 , r) := {z ∈ C | |z − z0 | ≤ r},
z0 ∈ C,
r ∈ R+ ,
containing all zeros of a complex valued polynomial. The steps to achieve this
are as follows: Let
n
X
fn : C −→ C; fn (z) =
ai z i , ai ∈ C, n ≥ 1,
On the Location of Zeros of
Complex Polynomials
i=0
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be a complex polynomial. Construct a bound S = S(a0 , a1 , . . . , an ) in such a
way that all zeros of f (z) are situated in the closed disk
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K(z0 , S(a0 , a1 , . . . , an )) := {z ∈ C | |z − z0 | ≤ S(a0 , a1 , . . . , an )} .
Without loss of generality we set z0 = 0. A special case is the existence of
certain conditions, thus resulting in special bounds.
This paper is organized as follows: Section 2 provides such bounds on the
basis of certain conditions, concerning the polynomials’ coefficients. In Section 3 mainly zero bounds are proved which are positive roots of algebraic
equations. The main result of Section 3 is a theorem which is an extension
of a classical result of Cauchy. Section 4 shows applications of the bounds in
such a way that the bounds will be evaluated with certain polynomials. The
paper finishes in Section 5 with conclusions.
Matthias Dehmer
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2.
Bounds for the Zeros of Complex Polynomials
with Special Conditions for the Coefficients
Definition 2.1. Let
fn : C −→ C;
fn (z) =
n
X
ai z i ,
n ≥ 1,
i=0
be a complex polynomial.
fn (z) = an z n + fn−1 (z) ,
(2.1)
On the Location of Zeros of
Complex Polynomials
f0 (z) := a0
Matthias Dehmer
denotes the recursive description of fn (z).
Theorem 2.1. Let P (z) be a complex polynomial, such that P (z) is reducible
in C[z],
P (z) := fn1 (z)gn2 (z) = (bn1 z n1 + fn1 −1 (z))(cn2 z n2 + gn2 −1 (z))
with
|bn1 | > |bi |,
0 ≤ i ≤ n1 − 1,
|cn2 | > |ci |,
0 ≤ i ≤ n2 − 1.
K(0, δ),
(2.3)
z
− 4z
n1 +n2 +1
It holds for 1 < δ < 2 +
√
+ 2z
n1 +n2
+z
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where δ > 1 is the positive root of the equation
n1 +n2 +2
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If n1 + n2 > 1, then all zeros of the polynomial P (z) lie in the closed disk
(2.2)
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n2 +1
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+z
n1 +1
− 1 = 0.
J. Ineq. Pure and Appl. Math. 7(1) Art. 26, 2006
2.
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Proof. Expanding fn1 −1 (z), we conclude
fn1 −1 (z) =
nX
1 −1
bi z i ,
i=0
analogously
gn2 −1 (z) =
nX
2 −1
ci z i .
i=0
Assuming
(2.4)
|bn1 | > |bi |,
0 ≤ i ≤ n1 − 1,
|cn2 | > |ci |,
0 ≤ i ≤ n2 − 1,
On the Location of Zeros of
Complex Polynomials
Matthias Dehmer
it follows that
P 1 −1 i
| ni=0
bi z |
|fn1 −1 (z)|
=
|bn1 |
|bn1 |
nX
−1
1
|bi | i
≤
|z|
|b
|
n
1
i=0
<
nX
1 −1
|z|i =
i=0
|z|n1 − 1
.
|z| − 1
With the inequalities (2.4) above, we find that
|P (z)|
= (bn1 z n1 + fn1 −1 (z))(cn2 z n2 + gn2 −1 (z))|
≥ |bn1 ||cn2 ||z|n1 +n2
− {|bn1 ||z n1 ||gn2 −1 (z)| + |cn2 ||z n2 ||fn1 −1 (z)| + |fn1 −1 (z)||gn2 −1 (z)|}
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|gn −1 (z)|
|fn −1 (z)|
= |bn1 ||cn2 | |z|
− | z|n1 2
+ |z|n2 1
|c |
|bn1 |
n2
|fn1 −1 (z)| |gn2 −1 (z)|
+
·
|bn1 |
|cn2 |
|z|n2 − 1
|z|n1 − 1
n1 +n2
> |bn1 ||cn2 | |z|
− |z|n1
+ |z|n2
|z| − 1
|z| − 1
n1
n2
(|z| − 1)(|z| − 1)
+
(|z| − 1)2
|bn1 ||cn2 |
=
{|z|n1 +n2 +2 − 4|z|n1 +n2 +1 + 2|z|n1 +n2 + |z|n2 +1 + |z|n1 +1 − 1},
2
(|z| − 1)
| {z }
n1 +n2
On the Location of Zeros of
Complex Polynomials
Matthias Dehmer
>0
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Let
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H(|z|) := |z|n1 +n2 +2 − 4|z|n1 +n2 +1 + 2|z|n1 +n2 + |z|n2 +1 + |z|n1 +1 − 1,
and we assume n1 + n2 > 1. Hence |P (z)| > 0 if H(|z|) > 0. Applying
Descartes’ Rule of Signs [4] to H(|z|), we can conclude that H(|z|) has either
one or three positive zeros δ1 , δ2 , and δ3 .
Now we examine the positive zeros of H(|z|). At first
H(1) = 0 .
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Dividing H(|z|) by |z| − 1 yields
J. Ineq. Pure and Appl. Math. 7(1) Art. 26, 2006
(|z|n1 +n2 +2 − 4|z|n1 +n2 +1 + 2|z|n1 +n2 + |z|n2 +1 + |z|n1 +1 − 1) : (|z| − 1)
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n1 +n2 +1
= |z|
n1 +n2
− 3|z|
−
n1 +n
X2 −1
j
|z| +
j=n1 +1
n2
X
|z|j .
j=0
Further, let
R(|z|) := |z|n1 +n2 +1 − 3|z|n1 +n2 −
n1 +n
X2 −1
|z|j +
j=n1 +1
n2
X
|z|j .
j=0
We see that R(1) = 0 and infer
On the Location of Zeros of
Complex Polynomials
H(|z|) = (|z| − 1)R(|z|) = (|z| − 1)2 Q(|z|),
deg(Q(|z|)) = n1 + n2 ,
hence δ1 = δ2 = 1 is a zero with multiplicity two. Altogether, H has exactly
three positive zeros. Now, we note that
sign{H(0)} = −1,
sign{H(∞)} = 1,
and we choose K > 1 such that H(K) > 0. On the other hand, it holds that
H(1) = H 0 (1) = 0 and H 00 (1) < 0. Therefore, there exists a θ > 0 such that
H(z) < 0, 1 ≤ z ≤ 1 + θ. Now, we obtain H(1 + θ) < 0 < H(K) and
therefore δ3 := δ ∈ (1 + θ, K), hence δ > 1. Thus, |P (z)| > 0, if H(|z|) > 0,
|z| > δ. According to this, all zeros of P (z) lie in the closed disk K(0, δ).
In order to examine the value of δ > 1, we replace |z| with δ in equation (2.3)
and examine the equation
R(δ) = δ n1 +n2 +1 − 3δ n1 +n2 −
n1 +n
X2 −1
j=n1 +1
δj +
n2
X
j=0
δ j = 0.
Matthias Dehmer
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The formulas
1 + δ + δ 2 + · · · + δ n1 + δ n1 +1 + · · · + δ n1 +n2 −1 =
⇐⇒
n1 +n
X2 −1
δj =
δ n1 +n2 − 1
− (1 + δ + δ 2 + · · · + δ n1 )
δ−1
δj =
δ n1 +n2 − 1 δ n1 +1 − 1
δ n1 +n2 − δ n1 +1
−
=
δ−1
δ−1
δ−1
j=n1 +1
⇐⇒
n1 +n
X2 −1
δ n1 +n2 − 1
δ−1
j=n1 +1
yield the result
Matthias Dehmer
δ n1 +n2 +1 − 3δ n1 +n2 −
δ n1 +n2 − δ n1 +1 δ n2 +1 − 1
= 0.
+
δ−1
δ
−
1
| {z }
>0
From this equation follows the inequality
δ n1 +n2 +1 − 3δ n1 +n2 −
δ n1 +n2 − δ n1 +1
< 0,
δ−1
and, furthermore,
(2.5)
On the Location of Zeros of
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δ
n1 +1
(δ
n2 +1
− 4δ
n2
+ 2δ
n2 −1
+ 1) < 0 .
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Inequality (2.5) implies
δ
n2
2
δ−4+
δ
< −1
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and we finally conclude
δ 2 − 4δ + 2 < 0 .
(2.6)
To solve inequality (2.6) take a closer look at
ξ 2 − 4ξ + 2 = 0,
and see that
ξ1,2 = 2 ±
√
2.
With the fact that δ > 1, the inequality holds for 1 < δ < 2 +
completes the proof of Theorem 2.1.
√
2. This
On the Location of Zeros of
Complex Polynomials
Matthias Dehmer
Theorem 2.1 can be strengthened by Theorem (2.2).
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Theorem 2.2. Let
P (z) := an z n + an−1 z n−1 + · · · + a0 ,
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an 6= 0 ,
n ≥ 1,
i = 0, 1, . . . , n
be a complex polynomial such that |an | > |ai |, i = 0, 1, . . . , n − 1. Then all
zeros of P (z) lie in the closed disk K(0, 2).
Proof. For |z| ≤ 1 the conclusion of Theorem 2.2 is evident. If we assume that
|z| > 1 we obtain immediately
|P (z)| = |an z n + an−1 z n−1 + · · · + a0 |
≥ |an ||z|n − {|an−1 |z|n−1 + · · · + a0 }
|an−1 | 1
|a0 | 1
n
= |an ||z| 1 −
+ ··· +
.
|an | |z|
|an | |z|n
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Now if |an | > |ai |, i = 0, 1, . . . , n − 1 is assumed, we conclude
|an−1 | 1
|a0 | 1
n
|P (z)| ≥ |an ||z| 1 −
+ ··· +
|an | |z|
|an | |z|n
(
)
n
X
1
> |an ||z|n 1 −
|z|j
j=1
)
(
∞
X
1
> |an ||z|n 1 −
|z|j
j=1
)
(
1
= |an ||z|n 1 −
1 +1
1 − |z|
=
|an ||z|n
{|z| − 2}.
|z| − 1
| {z }
Matthias Dehmer
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>0
Hence, we get
|P (z)| > 0 if
On the Location of Zeros of
Complex Polynomials
|z| > 2.
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3.
Further Bounds for the Zeros of Complex
Polynomials
Now we prove Theorem 3.2, which is similar to Theorem 3.1. Theorem 3.1 is a
classic result for the location of zeros found by Cauchy [7]. The zero bound of
the new Theorem 3.2, as well as Theorem 3.1, depends on an algebraic equation’s positive root.
Theorem 3.1 (Cauchy). Let
n
f (z) = an z + an−1 z
n−1
+ · · · + a0 ,
an 6= 0,
k = 0, 1, . . . , n
On the Location of Zeros of
Complex Polynomials
Matthias Dehmer
be a complex polynomial. All zeros of f (z) lie in the closed disk K(0, ρC ),
where ρC denotes the positive zero of
HC (z) := |a0 | + |a1 |z + · · · + |an−1 |z
n−1
n
− |an |z .
Theorem 3.2. Let
f (z) = an z n + an−1 z n−1 + · · · + a0 ,
an 6= 0,
k = 0, 1, . . . , n
be a complex polynomial.
All zeros of f (z) lie in the closed disk K(0, max(1, δ)),
a where M := max anj and δ 6= 1 denotes the positive root of the equation
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z n+1 − (1 + M )z n + M = 0.
Proof. On the basis of the inequality
(3.1)
|f (z)| ≥ |an | |z|n − M {|z|n−1 + · · · + 1} ,
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we get the modulus of f (z)
|f (z)| ≥ |an | |z|n − M {|z|n−1 + · · · + 1}
|z|n − 1
n
= |an | |z| − M
|z| − 1
n+1
|z|
− |z|n (1 + M ) + M
.
= |an |
|z| − 1
Define
F (z) := z
n+1
n
− z (1 + M ) + M.
Using the Descartes’ Rule of Signs we have that F (z) has exactly two positive
zeros δ1 and δ2 , and F (δ1 = 1) = 0 holds. With
On the Location of Zeros of
Complex Polynomials
Matthias Dehmer
Title Page
sign{F (0)} = 1
Contents
and from the fact that F (z) has exactly two positive zeros, we finally conclude
that
|f (z)| > 0 for |z| > max(1, δ).
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Hence, all zeros of f (z) lie in K(0, max(1, δ)).
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Now, we express a further theorem, the proof of which is similar to that of
Theorem 3.2.
Page 12 of 27
Theorem 3.3. Let
f (z) = an z n + an−1 z n−1 + · · · + a0 ,
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be a complex polynomial. All zeros of f (z) lie in the closed disk K(0, max(1, δ)),
where
an−j − an−j−1 , a−1 := 0
M̃ := max 0≤j≤n
an
and δ 6= 1 denotes the positive root of the equation
z n+2 − (1 + M̃ )z n+1 + M̃ = 0.
Furthermore we can state immediately a new Theorem 3.4 which is very
similar to a well known theorem due to Cauchy [7]. It provides an upper bound
for all zeros of a complex polynomial f (z). In many cases the bound of Theorem 3.4 is sharper than the bound of Cauchy [7] (K(0, 1 + M )).
Theorem 3.4. Let
f (z) = an z n + an−1 z n−1 + · · · + a0 ,
On the Location of Zeros of
Complex Polynomials
Matthias Dehmer
Title Page
an 6= 0,
k = 0, 1, . . . , n
be a complex polynomial. All zeros of f (z) lie in the closed disk K(0, 1 + M̃ ).
Proof. For the zeros with |z| ≤ 1, we have nothing to prove. Assuming |z| > 1
and define
P (z) := (1 − z)f (z) = −an z n+1 + (an − an−1 )z n + · · · + (a1 − a0 )z + a0 .
Now, we obtain the estimation
|P (z)| ≥ |an ||z|n+1 − |an − an−1 ||z|n + |an−1 − an−2 ||z|n−1 + · · ·
+|a1 − a0 ||z| + |a0 |}
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an − an−1 n an−1 − an−2 n−1
n+1
|z|
= |an | |z|
− + ···
|z| + an
an
a1 − a0 a0 − 0 +
|z| + an an (
)
n
X
≥ |an | |z|n+1 − M̃
|z|i
i=0
n+1
|z|
−1
n+1
= |an | |z|
− M̃
|z| − 1
|z|n
n+1
> |an | |z|
− M̃
|z| − 1
n
o
|an |
n+2
n+1
=
|z|
− |z| (1 + M̃ ) .
|z| − 1
| {z }
>0
Finally, we conclude
|z|n+2 − |z|n+1 (1 + M̃ ) = |z|n+1 [|z| − (1 + M̃ )] > 0.
On the Location of Zeros of
Complex Polynomials
Matthias Dehmer
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We infer |P (z)| > 0 if |z| > 1+ M̃ . Hence, all zeros of P (z) lie in K(0, 1+ M̃ ).
Because of the fact that all zeros of f (z) are zeros of P (z), Theorem 3.4 holds
also for f (z).
Now, we prove Theorem 3.5, which provides a zero bound for complex polynomials that are in a more general form than the previous ones.
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Theorem 3.5. Let
f (z) = an z n + an−1 z n−1 + · · · + a0 ,
an 6= 0,
k = 0, 1, . . . , n
be a complex polynomial and H(z) := (f (z))σ , N 3 σ ≥ 1. All zeros of H(z)
lie in the closed disk K(0, ρ), where ρ is the positive zero of
( n−1
!σ
X
|ai |z i
(3.2) H̃(z) := |an |σ z nσ −
i=0
+
σ−1
X
j=1

|an |j z nj
n−1
X
i=0
!σ−j 
σ 
.
|ai |z i

j
Proof. Define H(z) := (f (z))σ , this yields the estimation
!σ !σ n−1
n
X
X
|H(z)| = ai z i = ai z i + an z n i=0
i=0

!σ−j X
n−1
X
σ
σ 
(an z n )j
= ai z i
j
j=0
i=0

!σ−j 
σ−1
n−1
X
X
σ 
(an z n )j
= (an z n )σ +
ai z i
j
j=0
i=0


!σ−j σ−1
n−1
X
X
σ 
(an z n )j
≥ |an z n |σ − ai z i
j
j=0
i=0
On the Location of Zeros of
Complex Polynomials
Matthias Dehmer
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!σ σ−1 
!σ−j 
n−1
n−1
X
X
X
σ 
(an z n )j
= |an z n |σ − ai z i
+
ai z i
j
i=0
j=1
i=0



!
!
σ
σ−j n−1
σ−1
n−1
 X
X
X
σ 
(an z n )j
≥ |an z n |σ − ai z i + ai z i


j
i=0
j=1
i=0
( n−1
!σ
X
≥ |an |σ |z|nσ −
|ai ||z|i
i=0
+
σ−1
X
j=1

|an z n |j
!σ−j 
n−1
X
σ 
|ai ||z|i

j
On the Location of Zeros of
Complex Polynomials
Matthias Dehmer
i=0
Title Page
:= H̃(|z|) .
H̃(z) is a polynomial with only real and positive coefficients. Therefore we
apply the Descartes’ Rule of Signs and obtain that H̃(z) has exactly one positive
zero ρ.
Furthermore, we determine easily H̃(0) = −|a0 |σ < 0. With

!σ−j 
σ−1
n−1
X
X
σ 
(an z n )j
H̃1 (z) := (an z n )σ and H̃2 (z) :=
ai z i
,
j
j=0
i=0
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we infer
deg(H̃1 (z)) > deg(H̃2 (z)).
Finally, it follows that all zeros of H(z) lie in the closed disk K(0, ρ).
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The following corollary states that the bound of Theorem 3.5 is a generalization of Cauchy’s bound concerning Theorem 3.1.
Corollary 3.6. σ = 1 in Equation (3.2) leads to the bound of Cauchy, Theorem 3.1.
Proof. If we set σ = 1 in equation (3.2), we obtain
n
H̃(z) = |an |z −
n−1
X
|ai |z i .
On the Location of Zeros of
Complex Polynomials
i=0
Therefore, we have H̃(z) = −HC (z) and H̃(z) posseses the same positive zero
as HC (z). Thus we have the bound of Cauchy, Theorem 3.1.
The last theorem in this paper expresses another bound for the zeros of H(z).
Here, the belonging concomitant polynomial is easier to solve that the concomitant polynomial of Theorem 3.5.
Theorem 3.7. Let
f (z) = an z n + an−1 z n−1 + · · · + a0 ,
an 6= 0,
k = 0, 1, . . . , n
be a complex polynomial and H(z) := (f (z))σ , N 3 σ ≥ 1. Furthermore let
B := max0≤i≤n−1 |ai | and K := max0≤j≤σ−1 |an |j B σ−j σj . All zeros of H(z)
lie in the closed disk K(0, ρ), where ρ > 1 is the largest positive zero of the
equation
|an |σ (|z| − 1)σ − K|z|σ − B σ |z| + B σ .
Matthias Dehmer
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Proof. Starting from H̃(|z|), obtained in the proof of Theorem 3.5 and the inequality
n−1
n−1
X
X
i
ai |z| ≤ B
|z|i ,
i=0
i=0
we have
σ
nσ
|H(z)| ≥ |an | |z|
σ
n
|z| − 1
σ
− B
|z| − 1
"
n
σ−j #)
σ−1
X
σ
|z|
−
1
+
|an |j |z|nj B σ−j
.
j
|z|
−
1
j=1
On the Location of Zeros of
Complex Polynomials
Matthias Dehmer
If we now assume |z| > 1 we obtain the inequalities
( σ
n
σ−j )
σ−1
n
X
|z|
−
1
|z|
−
1
|H(z)| > |an |σ |z|nσ − B σ
+K
|z|nj
|z| − 1
|z| − 1
j=1
(
)
σ−1
X
|z|nσ
|z|nσ
σ
nσ
σ
+K
> |an | |z| − B
(|z| − 1)σ
(|z| − 1)σ−j
j=1
#)
(
"
σ−1
σ
X
K
B
+
(|z| − 1)j
.
= |z|nσ |an |σ −
(|z| − 1)σ (|z| − 1)σ j=1
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J
On the basis of the relation
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σ−1
X
j=1
(|z| − 1)j <
σ−1
X
j=0
|z|j ,
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we conclude furthermore that
Bσ
K
|z|σ − 1
nσ
σ
|H(z)| > |z|
|an | −
+
·
(|z| − 1)σ (|z| − 1)σ |z| − 1
Bσ
K|z|σ
nσ
σ
> |z|
|an | −
+
(|z| − 1)σ (|z| − 1)σ+1
|z|nσ
=
|an |σ (|z| − 1)σ+1 − K|z|σ − B σ |z| + B σ .
σ+1
(|z| − 1)
|
{z
}
>0
Define P (z) := |an |σ (z − 1)σ+1 − Kz σ − B σ z + B σ . We see immediately that
deg(P (z)) = deg |an |σ (z − 1)σ+1 − Kz σ − B σ z + B σ
!
σ+1
X
σ
+
1
= deg |an |σ
(−1)j
z σ+1−j − Kz σ − B σ z + B σ
j
j=0
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= σ + 1,
hence we infer
lim P (z) = ∞.
(3.3)
z→∞
Now, it holds
(
P (0) =
On the Location of Zeros of
Complex Polynomials
|an |σ + B σ : σ is odd
−|an |σ + B σ : σ is even
Furthermore we obtain P (1) = −K < 0. In the first case P (0) = |an |σ + B σ >
0 it follows that P (z) has at least two positive zeros α1 < α2 and α2 > 1. The
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case P (0) = −|an |σ + B σ > 0 ⇐⇒ B σ > |an |σ leads us to the same situation.
Now, we assume P (0) = −|an |σ + B σ < 0 ⇐⇒ B σ < |an |σ and it holds that
P (1) < 0. With equation (3.3) we conclude that P (z) has at least one positive
zero α > 1. Let α1 , α2 , . . . , ατ , 1 ≤ τ ≤ deg(P (z)) be the positive zeros of
P (z) and
ρ :=
max
(α1 , α2 , . . . , ατ ).
1≤τ ≤deg(H̃(x))
Finally, we infer that |P (z)| > 0 if |z| > ρ > 1. This inequality completes the
proof of Theorem 3.7.
On the Location of Zeros of
Complex Polynomials
Matthias Dehmer
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4.
Evaluation of Zero Bounds
P
Let f (z) = ni=0 ai z i , an 6= 0, n ≥ 1, be a complex polynomial and z1 , z2 , . . . , zn
be the zeros of f (z). Define ξ := max (|z1 |, |z2 |, . . . , |zn |) and
S1 := max(1, δ), Theorem(2.1)
S2 := δ, Theorem(3.2)
S3 := δ, Theorem(3.3)
S4 := 1 + M̃ , Theorem(3.4)
S5 := ρ, Theorem(3.7)
On the Location of Zeros of
Complex Polynomials
Now, we consider the closed disks |z| ≤ Si , i = 1, 2, . . . , 4 for each complex
polynomial fj (z), j = 1, 2, . . . , 6.
Matthias Dehmer
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1.
f1 (z) := 360 · z 6 − 194z 5 − 79z 4 + 33z 3 + 11z 2 − 12z − 5
= (18z 2 − 7z − 5) · (20z 4 − 3z 3 + z + 1),
therefore it holds
|b2 | > |bi |,
0 ≤ i ≤ 1,
|c4 | > |ci |,
0 ≤ i ≤ 3.
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Zeros of f1 (z):
z1
z2
z3
z4
.
= −0, 3051 − 0, 2775i ,
.
= −0, 3051 + 0, 2775i ,
.
= 0, 3801 − 0, 3864i ,
.
= 0, 3801 + 0, 3864i ,
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.
z5 = −0, 3673 ,
.
z6 = 0, 7562 .
2. f2 (z) := z 5 − iz 4 + iz 2 − z + i.
Zeros of f2 (z):
z1
z2
z3
z4
z5
.
= −1, 1140 − 0, 1223i ,
.
= −0, 8766i ,
.
= 0, 5950i ,
.
= 1, 5262i ,
.
= 1, 1140 − 0, 1223i .
On the Location of Zeros of
Complex Polynomials
Matthias Dehmer
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3. f3 (z) := z 3 − 1000z 2 + 2000z + 1500.
Zeros of f3 (z):
.
z1 = −0, 5810 ,
.
z2 = 2, 5866 ,
.
z3 = 997, 9944 .
4.
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f4 (z) := 10z 4 + z 3 + (3 + 2i)z 2 + z(−2 + i) + i
= (5z 2 − 2z + i) · (2z 2 + z + 1),
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therefore it holds
|b2 | > |bi |, 0 ≤ i ≤ 1, |c2 | > |ci |, 0 ≤ i ≤ 1.
Zeros of f4 (z):
z1
z2
z3
z4
= −0, 25 − 0, 6614i ,
= −0, 25 + 0, 6614i ,
= −0, 1492 + 0, 2863i ,
= −0, 5492 + 0, 2863i .
On the Location of Zeros of
Complex Polynomials
Matthias Dehmer
7
5. f5 (z) := z − z + 1.
Zeros of f5 (z):
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z1
z2
z3
z4
z5
z6
z7
.
= −1, 1127 ,
.
= −0, 6170 − 0, 9008i ,
.
= −0, 6170 + 0, 9008i ,
.
= 0, 3636 − 0, 9525i ,
.
= 0, 3636 + 0, 9525i ,
.
= 0, 8098 − 0, 2628i ,
.
= 0, 8098 + 0, 2628i .
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4
3
2
2
6. f6 (z) := (2z − z + z + 2z − 1) , σ = 2.
J. Ineq. Pure and Appl. Math. 7(1) Art. 26, 2006
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fi
f1
f2
f3
f4
f5
f6
S1
3,2918
3,1746
-
S2
1,4895
1,9659
1501,0
1,7897
1,9919
2,7494
S3
2,5366
2,4069
3001,0
1,8595
1,2301
4,2499
S4
2,5388
2,4142
3001,0
1,9
3,0
4,25
S5
ξ
2,1757 (σ = 1)
0,756
1,5262
997,9914
1.6683 (σ = 1)
1,25
1,1127
3,850 (σ = 2)
0,4184
Figure 1: Comparison of zero bounds
On the Location of Zeros of
Complex Polynomials
Matthias Dehmer
Zeros of f6 (z):
z1
z2
z3
z4
z5
z6
z7
z8
.
= −0, 8935 ,
.
= 0, 4184 ,
.
= 0, 4875 − 1, 0485i ,
.
= 0, 4875 + 1, 04857i ,
.
= −0, 8935 ,
.
= 0, 4184 ,
.
= 0, 4875 − 1, 04857i ,
.
= 0, 4875 + 1, 04857i .
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5.
Conclusions
Table 1 shows the overall results of the comparison of the new zero bounds. In
general, the comparison between zero bounds is very difficult. Hence, it is not
possible to obtain general assertions for describing the quality of zero bounds.
Table 1 shows that the quality of the proven bounds depends on the polynomial
under consideration. We observe that the bounds S3 , S4 are very large for f3 (z).
a This is due to the fact that the number M := max anj is very large. In the
0≤j≤n−1
case of a polynomial H(z) := (f (z))σ , the bound of Theorem 3.5 is directly
applicable (without expanding H(z)) in order to determine the value of the
bound. The new bounds of the present paper can be used in many applications.
The characteristic property of our new bounds is that we can compute these zero
bounds more effectively than the classic bound of Theorem 3.1.
On the Location of Zeros of
Complex Polynomials
Matthias Dehmer
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References
[1] H. ALZER, Bounds for the zeros of polynomials, Complex Variables, Theory Appl., 8(2) (2001), 143–150.
[2] M. DEHMER, Schranken für die Nullstellen komplexwertiger Polynome,
Die Wurzel, 8 (2000), 182–185.
[3] F. HEIGL, Neuere Entwicklungen in der Geometrie der Polynome, Jber.
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[4] P. HENRICI, Applied and computational complex analysis. Volume I, Wiley Classics Library Edition, (1988).
[5] F. KITTANEH, Bounds for the zeros of polynomials from matrix inequalities, Arch. Math., 81(5) (2003), 601–608.
[6] S. LIPKA, Über die Abgrenzung der Wurzeln von algebraischen Gleichungen, Acta. Litt. Sci. (Szeged), 3 (1927), 73–80.
[7] M. MARDEN, Geometry of polynomials, Mathematical Surveys. Amer.
Math. Soc., Rhode Island, 3 (1966)
On the Location of Zeros of
Complex Polynomials
Matthias Dehmer
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[8] Q.G. MOHAMMAD, Location of the zeros of polynomials, Amer. Math.
Monthly, 74 (1962), 290–292.
Close
[9] Q.G. MOHAMMAD, On the zeros of polynomials, Amer. Math.
Monthly, 72 (1966), 631–633.
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[10] N. OBRESCHKOFF, Verteilung und Berechnung der Nullstellen reeller
Polynome, Hochschulbücher für Mathematik. 55. Berlin, VEB Deutscher
Verlag der Wissenschaften, (1963)
[11] Q.I. RAHMAN AND G. SCHMEISSER, Analytic Theory of Polynomials.
Critical Points, Zeros and Extremal Properties, Clarendon Press, (2002)
[12] D.M. SIMEUNOVIĆ, On the location of the zeros of polynomials, Math.
Morav., 2 (1998), 91–96.
[13] W. SPECHT, Die Lage der Nullstellen eines Polynoms, Mathematische
Nachrichten, 15 (1956), 353–374.
[14] E.C. WESTERFIELD, A new bound for the zeros of polynomials, Amer.
Math. Monthly, 40 (1933), 18–23.
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Complex Polynomials
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