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Relationship between the zeros of two polynomials

BY W.S. Cheung and T.W. Ng*

Department of Mathematics,

The University of Hong Kong,

Pokfulam, Hong Kong.

E-mail address: wshun@graduate.hku.hk, ntw@maths.hku.hk

Abstract

In this paper, we shall follow a companion matrix approach to study

the relationship between zeros of a wide range of pairs of complex poly-

nomials, for example, a polynomial and its polar derivative or Sz.-Nagy’s

generalized derivative. We shall introduce some new companion matri-

ces and obtain a generalization of the Weinstein-Aronszajn Formula

which will then be used to prove some inequalities similar to Sendov

conjecture and Schoenberg conjecture and to study the distribution of

equilibrium points of logarithmic potentials for finitely many discrete

charges. Our method can also be used to produce, in an easy and sys-

tematic way, a lot of identities relating the sums of powers of zeros of a

polynomial to that of the other polynomial.

1 Introduction and Preliminaries

The concept of differentiators, first introduced by Davis [2], is used to study

the relationship between the zeros of a polynomial and the zeros of its deriva-

tive. In [10] Rajesh Pereira has further developed this idea and applied it

AMS Classification: Primary, 30C10; Secondary, 15A42.

Key words and phrases: Polynomials, zeros, Weinstein-Aronszajn Formula, D-companion

matrices, Schoenberg conjecture, Sendov conjecture.

∗The research was partially supported by a seed funding grant of HKU and RGC grant

HKU 7036/05P.

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successfully to solve several long standing conjectures, including the conjec-

ture of Schoenberg. Similar ideas were also used independently at the same

time by Semen Mark Malamud in [7] and [8] to solve these conjectures. Re-

cently, Cheung and Ng [1] have introduced the D-companion matrix, a matrix

form of the differentiator and applied it to solve de Bruin and Sharma’s con-

jecture. In this paper, we will introduce a generalization of the D-companion

matrix to study the relationship between zeros of two complex polynomials.

Unlike the differentiator, this new tool could be applied to a wide range of

pairs of polynomials, not just a polynomial and its derivative. In particular,

we shall apply our results to study the relationship between zeros of a polyno-

mial and its polar derivative or Sz.-Nagy’s generalized derivative. Our starting

point is to construct a matrix similar to the D-companion matrix when the

two polynomials are related in certain ways. In fact, we have the following

Theorem 1.1 Let A be an n×n matrix with characteristic polynomial p(z) =

Πn

j=1(z − zi) and q(z) be a monic polynomial of degree n given by

q(z)

p(z)= 1 +

n

?

j=1

λj

z − zj.

Then, there exists a rank one matrix H such that the characteristic polynomial

of the matrix A−H is q(z). In particular, if A is the diagonal matrix D formed

by z1,...,zn, then H can be chosen to be the matrix ΛJ =

λ1

...

···

λ1

...

λn ···

λn

,

where Λ is the diagonal matrix formed by λ1,...,λnand J is the n×n all one

matrix.

Theorem 1.2 Let A be an n×n matrix with characteristic polynomial p(z) =

Πn

j=1(z − zi) and q(z) be a monic polynomial of degree n − 1 given by

q(z)

p(z)=

n

?

j=1

λj

z − zj.

There exists a rank one matrix H such that H2= H and the characteristic

polynomial of the matrix A − AH is zq(z). In particular, if A is the diagonal

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matrix D formed by z1,...,zn, then H can be chosen to be the matrix ΛJ =

λn ···

is the n × n all one matrix.

λ1

...

···

λ1

...

λn

, where Λ is the diagonal matrix formed by λ1,...,λnand J

Theorem 1.1 can be considered as a generalization of the famous Weinstein-

Aronszajn Formula. The Weinstein-Aronszajn Formula, a tool to study the

rank one perturbation of Hermitian matrices, is given by

det(zI − A + rww∗)

det(zI − A)

= 1 − r

n

?

j=1

|aj|2

z − zi

where A is a Hermitian matrix with eigenvalues z1,...,znand the correspond-

ing orthonormal basis u1,...,un, r ∈ R, and w = a1u1+ ··· + anunis a unit

vector (see [4, p.134]).

On the other hand, Theorem 1.1 says that for any order n matrix A with

eigenvalues z1,...,znand any λ1,...,λn∈ C,

det(zI − A + H)

det(zI − A)

= 1 +

n

?

j=1

λj

z − zj,

where H is some rank one matrix. If A is Hermitian, then A is unitarily

diagonalizable and all the eigenvalues ziof it are real. Hence, one may assume

that A is the diagonal matrix formed by z1,...,znand H can then be taken as

ΛJ which can be written in the form rww∗easily if λi= −r|aj|2.

In Theorem 1.2, the polynomial

q(z) = p(z)

n

?

j=1

λj

z − zj

is called Sz.-Nagy’s generalized derivative if λ1,...,λnare positive real numbers

such that?n

case, if we take A to be the diagonal matrix D formed by z1,...,zn, then H

j=1λj = n [9]. In particular, if λj =

1

n, then q =

1

np?. In this

can be taken as the matrix

1

nJ and A − AH = D(I −1

nJ) so that the D-

companion matrix of p?(z) introduced in [1] will be the principal submatrix of

U(D(I −1

nJ))U∗for some unitary matrix U.

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When q1is the polar derivative of p (see [11], p.97), then q1(z) = np(z) −

(z − α)p?(z) for some α ∈ C and hence

q1(z) = p(z)

n

?

j=1

α − zj

z − zj.

If?n

i=1zi?= 0, then q(z) =

1

−(n−1)(?n

i=1zi)q1(z) is monic and

q(z)

p(z)=

1

−(n − 1)(

n ?

i=1zi)

n

?

j=1

α − zj

z − zj.

So we can apply Theorem 1.2 and its corollaries to both Sz.-Nagy’s gener-

alized derivatives and polar derivatives of polynomials. In general, a generic

monic polynomial p will have only distinct zeros z1,...,zn. If q is a monic

polynomial with deg(q) < deg(p) = n, then by partial fraction decomposition,

we have

q(z)

p(z)=

n

?

j=1

q(zj)

p?(zj)

1

z − zj. (1)

If deg(q) = deg(p) = n, then deg(q − p) < deg(p) and apply the above

formula, we have

q(z)

p(z)= 1 +

n

?

j=1

λj

z − zj

for some suitable λj. Therefore with the partial fraction decomposition formula

(1), Theorem 1.1 and 1.2 can be applied to a wide range of pairs of polynomials

(when deg(q) < n, one can consider zkq(z) instead where k = n−1−deg(q)).

For example, consider the Dunkl operator Λαon R of index α +1

with the reflection group Z2[3] give by

2associated

Λα(p)(x) = p?(x) + (α +1

2)p(x) − p(−x)

x

,α ≥ −1

2.

Take q to be

1

n+2α+1Λα(p) which is a monic polynomial of degree n−1 and

apply the partial fraction decomposition formula (1) to obtain the λiwhen p

has distinct zeros only.

Theorem 1.1 and 1.2 allow us to apply results in matrix theory to deduce

results concerning the zeros of a pair of polynomials in Section 3 and 4. For ex-

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ample, by applying Theorem 1.1. and 1.2, we prove some results (Corollary 4.2

and Corollary 4.1) in a style similar to the famous Gauss-Lucas Theorem :

Gauss-Lucas Theorem.The zeros of the derivative of a polynomial are

located inside the convex hull of the zeros of the polynomial.

We shall also prove results (Corollary 3.3 and Corollary 3.7) similar to the

Schoenberg conjecture, now a theorem—after it was proved independently by

Pereira and Malamud in [10]and [8] respectively:

Malamud-Pereira Theorem. Let z1,··· ,znbe the zeros of a polynomial p

of degree n ≥ 2 and w1,··· ,wn−1be the zeros of p?. Then

n−1

?

where equality holds if and only if all zilie on a straight line.

i=1

|wi|2≤

1

n2|

n

?

i=1

zi|2+n − 2

n

n

?

i=1

|zi|2

Theorem 1.1 and 1.2 also allow us to obtain some inequalities about the ze-

ros of polynomials (Corollary 3.4 and Corollary 3.8), and deduce some minmax-

maxmin inequalities (Corollary 3.5 and Corollary 3.9) similar to Sendov con-

jecture:

Sendov conjecture. Let z1,...,zn be the zeros of a polynomial p of degree

n ≥ 2 and w1,...,wn−1be the zeros of p?, the derivative of p. Then,

max

1≤k≤n

min

1≤i≤n−1|wi− zk| ≤ max

1≤k≤n|zk|.

Moreover, we will obtain, in a simple and systematic way, the sum of

powers of the zeros of a polynomial in terms of that of the other polynomial.

Those results (Corollary 3.6 and Corollary 3.10) could be obtained using the

Newton’s formulas, but in a more clumsy way.

Finally, we shall apply Theorem 1.1 to study the distribution of equilibrium

points of logarithmic potentials for finitely many discrete charges.

2 Proof of Theorem 1.1 and 1.2

Proof of Theorem 1.1. For each eigenvalue ziof A, we choose a corresponding

eigenvector vi. Our choice should be that the vector v =?n

5

j=1λjvjis nonzero.

Page 6

Construct a matrix H such that Hvi= v and Hx is a scalar multiple of v for

all x. We claim that the characteristic polynomial of A − H is q(z).

We have

H(A − zI)−1v =

n

?

n

?

?q(z)

j=1

λjH(A − zI)−1vj

=

j=1

λj(zj− z)−1v

?

=

p(z)− 1v.

As H(A−zI)−1is of rank one, q(z)p(z)−1−1 is its unique nonzero eigen-

value. Hence

det(zI − (A − H)) = det(zI − A)det(I + H(zI − A)−1)

= p(z)(1 + (q(z)p(z)−1− 1))

= q(z)

as desired.

?

Proof of Theorem 1.2. As q is monic, we have?n

j=1λj= 1 and therefore

zq(z)

p(z)

= 1 +

n

?

j=1

λjzj

z − zj. (2)

Choose H to be a matrix corresponding to p(z) + q(z) in Theorem 1.1, then

H1= AH is a matrix corresponding to zq(z) in Theorem 1.1. It is straight-

forward to show that H2= H as Hv = (?n

j=1λj)v = v.

?

3 Schoenberg and Sendov type results

The introduction of the two matrices in Theorem 1.1 and 1.2 allows one to

apply results in matrix theory to prove the above mentioned corollaries. For

example, to obtain results similar to Schoenberg’s theorem, we need the fol-

lowing result of Schur[12]:

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Theorem 3.1 . If A is an n×n matrix with eigenvalues w1,...,wn, then we

have

n

?

i=1

|wi|2≤ sum of entries of AA∗= sum of entries of A∗A.

Equality holds if A is normal, i.e. A∗A = AA∗.

To achieve the minmax-maxmin inequalities, we need the following result

in matrix theory:

Theorem 3.2 (Gerschgorin’s Theorem ([6, p.344])) The eigenvalues of

any square matrix A = (aij) of order n ≥ 2, lie in the union G =

Gerschgorin disks

n ?

i=1Giof the

Gi= {z ∈ C : |z − aii| ≤ Ri(A)} ,

where Ri(A) =

n ?

j=1j?=i|aij| , i = 1,...,n.

Finally we shall also make use of the simple fact that the sum of powers

related to the fact that the sum of the n-th power of eigenvalues of a matrix

A is equal to the trace of An.

Now we are ready to prove quite a number of corollaries of Theorem 1.1

and 1.2. We shall divide our results for the zeros of polynomials p and q into

two cases: a) deg(p) = deg(q) and b) deg(p) = deg(q) + 1.

Case a : deg(p) = deg(q)

For this case, we take A − H to be D − ΛJ.

Corollary 3.3 Let p be a monic polynomial of degree n with zeros z1,...,zn.

Suppose q is a monic polynomial given by

q(z)

p(z)= 1 +

n

?

j=1

λj

z − zj.

If w1,...,wnare zeros of q, then

n

?

i=1

|wi|2≤

n

?

i=1

|zi|2− 2

n

?

i=1

Re(zi¯λi) + (

n

?

i=1

|λi|)2.

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Proof. We apply Theorem 3.1 to D − Λ1/2JΛ1/2which is similar to D − ΛJ.

?

Corollary 3.4 Let p be a monic polynomial of degree n with zeros z1,...,zn.

Suppose q is a monic polynomial given by

q(z)

p(z)= 1 +

n

?

j=1

λj

z − zj.

Then for any zero w of q, there exists zkand zlsuch that

|w − zk− λk| ≤

?

j?=k

|λj|

and

|w − zl− λl| ≤ (n − 1)|λl|.

Proof. Apply Gerschgorin’s theorem to D − ΛJ.

A direct consequence of Corollary 3.4 minmax-maxmin inequality similar

?

to Sendov conjecture is the following:

Corollary 3.5 Let p be a monic polynomial of degree n with zeros z1,...,zn.

Suppose q is a monic polynomial given by

q(z)

p(z)= 1 +

n

?

j=1

λj

z − zj.

If w1,...,wnare zeros of q, then

max

1≤i≤nmin

1≤k≤n|wi− zk| ≤

n

?

j=1

|λj| ≤ n max

1≤j≤n|λj|.

Corollary 3.5 can be proved directly: Let wibe a root of q which is not a

root of p, then

1 =

?????

n

?

j=1

λj

wi− zj

?????≤

n

?

j=1

|λj|

|wi− zj|≤ n

max

1≤j≤n|λj|

min

1≤k≤n|wi− zk|.

The trace of a matrix of A, denoted by tr(A), is the sum of eigenvalues

of A. By the fact that tr(AB) = tr(BA) and that JEJ = tr(E)J for any

diagonal matrix E, we can obtain relations among the sum of powers of zeros

of p and the zeros of q. For instance, we have

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Corollary 3.6 Let p be a monic polynomial of degree n with zeros z1,...,zn.

Suppose q is a monic polynomial given by

q(z)

p(z)= 1 +

n

?

j=1

λj

z − zj.

If w1,...,wnare zeros of q, then

n

?

?

i=1

wi=

n

?

i=1

zi−

n

?

i=1

λi,

n

?

i=1

w2

i=

n

i=1

z2

i− 2

n

?

?

i=1

λizi+

?

n

?

i=1

λi

?2

,

and

n

?

i=1

w3

i=

n

?

i=1

z3

i− 3

n

?

i=1

λiz2

i+ 3

n

?

i=1

λi

?

n

?

i= tr(D − ΛJ)r, r = 1,2,3.

j=1

λjzj−

?

n

?

i=1

λi

?3

.

Proof. The three equalities follows from?n

?

i=1wr

Case b : deg(p) = deg(q) + 1

Corollary 3.7 Let p be a monic polynomial of degree n with zeros z1,...,zn.

Suppose q is a monic polynomial of degree n − 1 given by

q(z)

p(z)=

n

?

j=1

λj

z − zj.

If w1,...,wn−1are zeros of q, then

n

?

Proof. It follows from equation (2) and Corollary 3.3.

i=1

|wi|2≤

n

?

i=1

(1 − 2Reλi)|zi|2+ (

n

?

i=1

|λizi|)2.

?

Corollary 3.8 Let p be a monic polynomial of degree n with zeros z1,...,zn.

Suppose q is a monic polynomial of degree n − 1 given by

q(z)

p(z)=

n

?

j=1

λj

z − zj.

Then for any zeros w of q, there exists zkand zlsuch that

|w − zk− λkzk| ≤

?

j?=k

|λjzj|

and

|w − zl− λlzl| ≤ (n − 1)|λlzl|.

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Proof. It follows from equation (2) and Corollary 3.4.

?

By applying Corollary 3.8 for polynomials p(z + a) and q(z + a) where a

is any complex number, we obtain the next minmax-maxmin inequality.

Corollary 3.9 Let p be a monic polynomial of degree n with zeros z1,...,zn.

Suppose q is a monic polynomial of degree n − 1 given by

q(z)

p(z)=

n

?

j=1

λj

z − zj.

If w1,...,wn−1are zeros of q, then

max

1≤i≤n−1min

1≤k≤n|wi− zk| ≤ min

a∈C

n

?

j=1

|λj||zj− a| ≤ n max

1≤j≤n|λj|min

a∈Cmax

1≤k≤n|zj− a|.

Again, we have the relations involving the sum of powers of q and the zeros

of p and we list the first three below.

Corollary 3.10 Let p be a monic polynomial of degree n with zeros z1,...,zn.

Suppose q is a monic polynomial given by

q(z)

p(z)=

n

?

j=1

λj

z − zj.

If w1,...,wn−1are zeros of q, then

n−1

?

i=1

wi=

n

?

i=1

(1 − λi)zi,

n−1

?

i=1

w2

i=

n

?

i=1

(1 − 2λi)z2

i+

?

n

?

i=1

λizi

?2

,

and

n−1

?

i=1

w3

i=

n

?

i=1

(1 − 3λi)z3

i+ 3

?

n

?

i=1

λizi

?

n

?

j=1

λjz2

j−

?

n

?

i=1

λizi

?3

.

4Distribution of equilibrium points

So far we make no restrictions on λi. In this section, we will mainly consider

real λiin order to study the distribution of equilibrium points of logarithmic

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potentials for finitely many discrete charges. In fact, when λiare positive real

numbers, the zeros of functions of the form

f(z) =

n

?

j=1

ak

z − zj,aj> 0,zj∈ C

are called equilibrium points of logarithmic potential U generated by the

charged particles with charges aj> 0 at zjwhere

U(z) =

n

?

j=1

ajlog|1 −z

zj|.

The critical points of U (which are the zeros of f) coincide with equilibrium

points of electrostatic field. It would therefore be interesting to locate the zeros

of f in terms of the poles zj. The following results are well-known (see [11,

p.76] and [4, p.134]) but we shall give a matrix theoretical proof here.

Corollary 4.1 Let p be a monic polynomial of degree n with zeros z1,...,zn.

Suppose q is a monic polynomial given by

q(z)

p(z)=

n

?

j=1

λj

z − zj.

If λ1,...,λn ≥ 0, then the zeros of q are located inside the convex hull of

the zeros of p. If furthermore, z1 ≥ z2 ≥ ··· ≥ zn and the zeros of q are

w1≥ w2≥ ··· ≥ wn−1, then z1≥ w1≥ z2≥ w2≥ ··· ≥ wn−1≥ zn.

Proof. We first recall that the eigenvalues of a matrix lie inside the numerical

range of a matrix, and the numerical range of a normal matrix is exactly the

convex hull of its eigenvalues. Now suppose that zi= 0 for some i. Let e be

the all one vector and

V =

I − Λ1/2JΛ1/2

Λ1/2e

eTΛ1/2

0

.

We have V2= I and that V is Hermitian and hence unitary. Since zq(z) is the

characteristic polynomial of D(I−Λ) which is similar to (I−Λ1/2JΛ1/2)D(I−

Λ1/2JΛ1/2), a principal submatrix of the normal matrix V (D ⊕ 0)V , we have

the zeros of zq(z) lying inside the numerical range of D⊕0 which is the convex

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hull of the zeros of p. Furthermore if the zeros of p are real, then V (D ⊕ 0)V

is Hermitian and the interlacing property holds (see [6, p.185]).

For general p and q, we have the zeros of q(z + z1) lying inside the convex

hull of the zeros of p(z + z1), thus the conclusion follows.

?

Finally, we have

Corollary 4.2 Let A be an n×n matrix with characteristic polynomial p(z) =

Πn

j=1(z − zi) and q(z) be a monic polynomial of degree n given by

q(z)

p(z)= 1 +

n

?

j=1

λj

z − zj.

We have

(i) If there exists a zero v of q such that

λj

zi−v≥ 0 for all j, then the other

zeros of q are located inside the convex hull of the zeros of p.

(ii) If z1≥ ··· ≥ znand λ1,...,λnare all nonnegative or all nonpositive,

then the zeros of q are also real. Furthermore, suppose that w1≥ ··· ≥

wn, if λj’s are nonnegative then w1≥ z1≥ w2≥ z2≥ ··· ≥ wn≥ zn

and if λj’s are nonpositive then z1≥ w1≥ z2≥ w2≥ ··· ≥ zn≥ wn.

Proof.

(i) Without loss of generality, we may assume that v = 0.Then q(z) = zq1(z)

n ?

q1(z)

p(z)=

j=1

and

j=1

λj

zj= 1. Hence we have

n

?

λj

zj(z − zj).

By Corollary 4.1, the zeros of q1are located inside the convex hull of the

zeros of p.

(ii) Suppose λj’s are nonnegative. For z from zndown to −∞, we have

increasing from −∞ to 1, and hence there exists a zero v < znof q. Thus

λj

zj−v≥ 0 for all j. Apply part (i) and Corollary 4.1 again. The case that

all λj’s are nonpositive is similar.

q

p

?

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Acknowledgement

We would like to thank the referee for telling us that Theorem 3.1 comes from

[12] and that there is a simple direct proof of Corollary 3.5.

References

[1] W. S. Cheung and T. W. Ng, A companion matrix approach to the study of

zeros and critical points of a polynomial, J. Math. Anal. Appl., 319 (2006),

690–707.

[2] C. Davis, Eigenvalues of compressions, Bull. Math. Soc. Math. Phys. RPR

51 (1959) 3-5.

[3] C. F. Dunkl, Differential-difference operators associated to reflexion

groups, Trans. Amer. Math. Soc., 311, (1989), 167-183.

[4] H. Flaschka and J. Millson, Bending Flows for Sums of Rank One Matrices,

Canad. J. Math. 57, (2005), 114-158.

[5] W.K. Hayman, Research problems in function theory. The Athlone Press

University of London, London 1967.

[6] R.A. Horn and C.R. Johnson, Matrix analysis. Cambridge University Press,

Cambridge, 1990.

[7] S.M. Malamud, An Analog of the Poincar´ e Separation Theorem for Nor-

mal Matrices and the Gauss-Lucas Theorem, Functional Analysis and Its

Applications 37 (2003), no.3, 232-235.

[8] S.M. Malamud, Inverse spectral problem for normal matrices and the

Gauss-Lucas theorem. Trans. Amer. Math. Soc. 357 (2005), no. 10, 4043–

4064.

[9] Sz.-Nagy, Gyula Verallgemeinerung der Derivierten in der Geometrie der

Polynome. Acta Univ. Szeged. Sect. Sci. Math. 13, (1950). 169–178.

13

Page 14

[10] R. Pereira, Differentiators and the geometry of polynomials. J. Math.

Anal. Appl. 285 (2003), no. 1, 336–348.

[11] Q.I. Rahman, G. Schmeisser, Analytic theory of polynomials. London

Mathematical Society Monographs. New Series, 26. The Clarendon Press,

Oxford University Press, Oxford, 2002.

[12] I. Schur,¨Uber die charakteristischen Wurzeln einer linearen Substitution

mit einer Anwendung auf die Theorie der Integralgleichungen, Mathema-

tische Annalen 66 (1909), no.4, 488-510.

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