Relationship between the zeros of two polynomials
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 polynomials, for example, a polynomial and its polar derivative or Sz.-Nagy’s generalized derivative. We shall introduce some new companion matrices 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 systematic way, a lot of identities relating the sums of powers of zeros of a polynomial to that of the other polynomial.
- [Show abstract] [Hide abstract]
ABSTRACT: For random polynomials with i.i.d. (independent and identically distribu-ted) zeros following any common probability distribution $\mu$ with support contained in the unit circle, the empirical measures of the zeros of their first and higher order derivatives will be proved to converge weakly to $\mu$ a.s. (almost sure(ly)). This, in particular, completes a recent work of Subramanian on the first order derivative case where $\mu$ was assumed to be non-uniform. The same a.s. weak convergence will also be shown for polar and Sz.-Nagy's generalized derivatives, on some mild conditions.09/2014;
Page 1
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.
1
Page 2
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
2
Page 3
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.
3
Page 4
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-
4
Page 5
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.
2Proof 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]:
6
Page 7
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.
7
Page 8
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
8
Page 9
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|.
9
Page 10
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
10
Page 11
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
11
Page 12
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
?
12
Page 13
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.
14
View other sources
Hide other sources
- Available from Wai-Shun Cheung · Oct 20, 2014
- Available from hkumath.hku.hk