Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2012, Article ID 653914, 9 pages
doi:10.1155/2012/653914
Research Article
Localization and Perturbations of Roots to Systems
of Polynomial Equations
Michael Gil’
Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel
Correspondence should be addressed to Michael Gil’, gilmi@bezeqint.net
Received 20 March 2012; Accepted 28 May 2012
Academic Editor: Andrei Volodin
Copyright q 2012 Michael Gil’. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We establish estimates for the sums of absolute values of solutions of a zero-dimensional
polynomial system. By these estimates, inequalities for the counting function of the roots are
derived. In addition, bounds for the roots of perturbed systems are suggested.
1. Introduction and Statements of the Main Results
Let us consider the system:
f x, y g x, y 0,
1.1
where
n1
m1 f x, y ajk xm1 −j yn1 −k
0,
am1 n1 /
j0 k0
g x, y n2
m2 1.2
bjk x
m2 −j
y
n2 −k
bm2 n2 / 0.
j0 k0
The coefficients ajk , bjk are complex numbers.
The classical Beźout and Bernstein theorems give us bounds for the total number
of solutions of a polynomial system, compared to 1, 2. But for many applications, it is
very important to know the number of solutions in a given domain. In the present paper
2
International Journal of Mathematics and Mathematical Sciences
we establish estimates for sums of absolute values of the roots of 1.1. By these estimates,
bounds for the number of solutions in a given disk are suggested. In addition, we discuss
perturbations of system 1.1. Besides, bounds for the roots of a perturbed system are
suggested.
We use the approach based on the resultant formulations, which has a long history;
the literature devoted to this approach is very rich, compared to 1, 3, 4. We combine it with
the recent estimates for the eigenvalues of matrices and zeros of polynomials. The problem of
solving polynomial systems and systems of transcedental equations continues to attract the
attention of many specialists despite its long history. It is still one of the burning problems
of algebra, because of the absence of its complete solution, compared to the very interesting
recent investigations 2, 5–8 and references therein. Of course we could not survey the whole
subject here.
A pair of complex numbers x,
y
is a solution of 1.1 if fx,
y
gx,
y
0.
Besides x will be called an X-root coordinate corresponding to y
and y a Y -root coordinate
corresponding to x.
All the considered roots are counted with their multiplicities.
Put
n1
ajk yn1 −k
aj y n2
bjk yn2 −k
bj y j 0, . . . , m1 ,
k0
j 0, . . . , m2 .
1.3
k0
Then
m1
f x, y aj y xm1 −j ,
j0
1.4
m2
g x, y bj y xm2 −j .
j0
With m m1 m2 introduce the m × m Sylvester matrix
⎛
a0
⎜0
⎜
⎜·
⎜
⎜
⎜0
S y ⎜
⎜ b0
⎜
⎜0
⎜
⎝·
0
a1
a0
·
0
b1
b0
·
0
a2
a1
·
0
b2
b1
·
0
···
···
···
···
···
···
·
···
am1 −1 am1
am1 −2 am1 −1
·
·
a1
a0
bm2 −1 bm2
bm2 −2 bm2 −1
·
·
b1
b0
0
am1
·
a2
0
bm2
·
b2
0
0
·
a3
0
0
·
b3
⎞
··· 0
··· 0 ⎟
⎟
··· · ⎟
⎟
· · · am1 ⎟
⎟
⎟,
··· 0 ⎟
⎟
··· 0 ⎟
⎟
··· · ⎠
· · · bm 2
1.5
with ak ak y and bk bk y. Put Ry det Sy and consider the expansion:
n
Rk yn−k ,
R y k0
where n : deg R y .
1.6
International Journal of Mathematics and Mathematical Sciences
3
Furthermore, denote
⎡
1
θR : ⎣
2π
⎤1/2
2
2π it Re
dt − 1⎦ .
0 R0 1.7
Clearly,
R y θR ≤ sup ,
R
0
|y|1
R y
1 2π −int it R0 lim
,
R
e R e dt.
0
y → ∞ yn
2π 0
1.8
Thanks to the Hadamard inequality, we have
2
R y ≤
m
m1
m
2 1 2 m2 2
ak y bk y .
k0
1.9
k0
Assume that
Rn /
0.
1.10
Theorem 1.1. The Y -root coordinates yk of 1.1 (if, they exist), taken with the multiplicities and
ordered in the decreasing way: |yk | ≥ |yk1 |, satisfy the estimates:
j yk < θR 1 j
j 1, 2, . . . , n .
1.11
k1
If, in addition, condition 1.10 holds, then
j
1
R θR 1
< 0
j
k Rn
k1 y
j 1, 2, . . . , n ,
1.12
where yk are the Y -root coordinates of 1.1 taken with the multiplicities and ordered in the increasing
way: |yk | ≤ |yk1 |.
This theorem and the next one are proved in the next section. Note that another bound
for k1 |yk | is derived in 9, Theorem 11.9.1; besides, in the mentioned theorem aj · and
bj · have the sense different from the one accepted in this paper.
From 1.12 it follows that
j
minyk >
k
Rn
.
R0 θR 1 Rn
So the disc |y| ≤ Rn /R0 θR 1 Rn is zero free.
1.13
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International Journal of Mathematics and Mathematical Sciences
To estimate the X-root coordinates, assume that
a0 y > 0
inf
|y|≤θR1
1.14
and put
⎡
⎤1/2
m1 1
2
ψf sup ⎣ aj y ⎦ .
a0 y j1
|y|θR1
1.15
Theorem 1.2. Let condition 1.14 holds. Then the X-root coordinates xk y0 of 1.1 corresponding
to a Y -root coordinate y0 (if they exist), taken with the multiplicities and ordered in the decreasing
way, satisfy the estimates:
j
xk y0 < ψf j
j 1, 2, . . . , min{m1 , m2 } .
1.16
k1
In Theorem 1.2 one can replace f by g.
Furthermore, since yk are ordered in the decreasing way, by Theorem 1.1 we get j|yj | <
j θR and
yj < rj : 1 θR
j
j 1, . . . , n .
1.17
Thus 1.1 has in the disc {z ∈ C : |z| ≤ rj } no more than n − j Y -root coordinates. If we denote
by νY r the number of Y -root coordinates of 1.1 in Ωr : {z ∈ C : |z| ≤ r} for a positive
number r, then we get
Corollary 1.3. Under condition 1.10, the inequality νY r ≤ n − j 1 is valid for any
r ≤1
θR
.
j
1.18
Similarly, by Theorem 1.2, we get the inequality:
ψ
xj y0 ≤ 1 f
j
j 1, 2, . . . , min{m1 , m2 } .
1.19
Denote by νX y0 , r the number of X-root coordinates of 1.1 in Ωr , corresponding to a Y
coordinate y0 .
Corollary 1.4. Under conditions 1.14, for any Y -root coordinate y0 , the inequality:
νX y0 , r ≤ min{m1 , m2 } − j 1
j 1, 2, . . . , min{m1 , m2 }
1.20
International Journal of Mathematics and Mathematical Sciences
5
is valid, provided
r ≤1
ψf
.
j
1.21
In this corollary also one can replace f by g.
2. Proofs of Theorems 1.1 and 1.2
First, we need the following result.
Lemma 2.1. Let P z : zn c1 zn−1 · · · cn be a polynomial with complex coefficients. Then its
roots zk P ordered in the decreasing way satisfy the inequalities:
j
1
|zk P | ≤
2π
k1
1/2
2π 2
it j
P e dt − 1
j 1, . . . , n .
2.1
0
Proof. As it is proved in 9, Theorem 4.3.1 see also 10,
1/2
j
n
2
j.
|zk P | ≤
|ck |
k1
2.2
k1
But thanks to the Parseval equality, we have
n
|ck |2 1 k1
1
2π
2π 2
P eit dt.
2.3
0
Hence the required result follows.
Proof of Theorem 1.1. The bound 1.11 follows from the previous lemma with P y Ry/R0 . To derive bound 1.12 note that
R y Rn yn W
1
,
y
where Wz n
n
1 Rk zk dk zn−k ,
Rn k0
k0
2.4
with dk Rn−k /Rn . So any zero zW of Wz is equal to 1/y,
where y is a zero of Ry. By
the previous lemma
j
|zk W| <
k1
1
2π
1/2
2π 2
it j
W e dt − 1
0
j 1, . . . , n .
2.5
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International Journal of Mathematics and Mathematical Sciences
Thus,
1/2
j
1
1 2π it 2
<
j
W e dt − 1
k 2π 0
k1 y
j 1, . . . , n .
2.6
But Wz zn R1/z/Rn . Thus, |Weit | 1/Rn |Re−it |. This proves the required result.
Proof of Theorem 1.2. Due to Theorem 1.1, for any fixed Y -root coordinate y0 we have the
inequality:
y0 ≤ θR 1.
2.7
m1
f x, y0 aj y0 xm1 −j .
2.8
We seek the zeros of the polynomial:
j0
Besides, due to 1.14 and 2.7, a0 y0 / 0. Put
f x, y0
Qx ,
a0 y0
⎡
⎤1/2
m1 2
1
θQ : ⎣ aj y0 ⎦ .
a0 y0 j1
2.9
Clearly,
Qx xm1 m1
1 aj y0 xm1 −j .
a0 y0 j1
2.10
Due to the above mentioned 9, Theorem 4.3.1 we have
j
xk y0 < θQ j
j 1, 2, . . . , n .
2.11
k1
But according to 2.7, θQ ≤ ψf . This proves the theorem.
3. Perturbations of Roots
Together with 1.1, let us consider the coupled system:
f x, y g x, y 0,
3.1
International Journal of Mathematics and Mathematical Sciences
7
where
n1
m1 ajk xm1 −j yn1 −k ,
f x, y j0 k0
g x, y n2
m2 3.2
bjk x
m2 −j
y
n2 −k
.
j0 k0
Here ajk , bjk are complex coefficients. Put
n1
ajk yn1 −k
aj y j 0, . . . , m1 ,
k1
n2
bj y bjk yn2 −k
3.3
j 0, . . . , m2 .
k0
Let Sy
be the Sylvester matrix defined as above with aj y instead of aj y, and bj y
instead of bj y, and put Ry
det Sy.
It is assumed that
y deg R y n.
deg R
3.4
n
y k yn−k .
R
R
3.5
Consider the expansion:
k0
0 Due to 3.4, R
/ 0. Denote
⎡
it 2 ⎤1/2
2π it
R
e Re
1
: ⎣
q R, R
−
dt⎦ ,
2π 0 R0
0 R
3.6
1/2
.
ηR θ2 R n − 1
Clearly,
R y
y R
−
q R, R ≤ max
.
0 |z|1 R0
R
3.7
Theorem 3.1. Under condition 3.4, for any Y -root coordinate y0 of 3.1, there is a Y -root
coordinate y0 of 1.1, such that
y0 − y0 ≤ μR ,
3.8
8
International Journal of Mathematics and Mathematical Sciences
where μR is the unique positive root of the equation:
n−1 k
η Ryn−k−1
.
yn q R, R
√
k!
k0
3.9
To prove Theorem 3.1, for a finite integer n, consider the polynomials:
P λ n
ck λn−k ,
P λ k0
n
ck λn−k
c0 c0 1,
3.10
k0
with complex coefficients. Put
q0 n
1/2
|ck − ck |
2
,
k1
1/2
n
2
ηP .
|ck | n − 1
3.11
k1
Lemma 3.2. For any root zP of P y, there is a root zP of P y, such that |zP − zP | ≤ rq0 ,
where rq0 is the unique positive root of the equation
yn q0
n−1 k
η P yn−k−1
.
√
k!
k0
3.12
This result is due to Theorem 4.9.1 from the book 9 and inequality 9.2 on page 103
of that book.
By the Parseval equality we have
n
k0
|ck |2 1
2π
2π 2
P eit dt,
0
2
1 2π it it 2
q02 P e − P e dt ≤ maxP y − P y .
2π 0
|z|1
3.13
Thus
η2 P 1
2π
2π 2
2
P eit dt n − 2 ≤ maxP y n − 2.
0
|z|1
3.14
The assertion of Theorem 3.1 now follows from in the previous lemma with P y 0.
R
Ry/R0 andP y Ry/
International Journal of Mathematics and Mathematical Sciences
9
Further more, denote
n−1 k
η R
pR : q R, R
√ ,
k!
k0
√
n p
R if pR ≤ 1,
δR :
pR
if pR > 1.
3.15
Due to Lemma 1.6.1 from 11, the inequality μR ≤ δR is valid. Now Theorem 3.1 implies the
following.
Corollary 3.3. Under condition 3.4, for any Y -root coordinate y0 of 3.1, there is a Y -root
coordinate y0 of 1.1, such that |y0 − y0 | ≤ δR .
Similarly, one can consider perturbations of the X-root root coordinates.
To evaluate the quantity Ry − Ry
one can use the following result: let A and B two
complex n × n-matrices. Then
|detA − detB| ≤
n
1
N2 A − B
,
1
B
N
−
B
A
A
N
2
2
2
nn/2
3.16
where N22 A TraceA∗ A is the Hilbert-Schmidt norm and A∗ is the adjoint to A. For the
proof of this inequality see 12. Taking B Sy,
A Sy we get a bound for |Rz − Rz|.
References
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3 B. Mourrain, “Computing the isolated roots by matrix methods,” Journal of Symbolic Computation, vol.
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5 W. Forster, “Homotopies for systems of polynomial equations,” Acta Mathematica Vietnamica, vol. 22,
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9 M. Gil’, Localization and Perturbation of Zeros of Entire Functions, vol. 258 of Lecture Notes in Pure and
Applied Mathematics, CRC Press, Boca Raton, Fla, USA, 2010.
10 M. I. Gil’, “Bounds for counting functions of finite order entire functions,” Complex Variables and
Elliptic Equations, vol. 52, no. 4, pp. 273–286, 2007.
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