Localization and Perturbations of Roots to Systems of Polynomial Equations

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Volume 2012, Article ID 653914,9pages 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

Copyrightq2012 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:

fx, ygx, y0, 1.1

where

fx, y

m1

j0

n1

k0

ajkxm1−jyn1−k am1n1/0,

gx, y

m2

j0

n2

k0

bjkxm2−jyn2−k bm2n2/0.

1.2

The coeﬃcientsajk, bjkare complex numbers.

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we establish estimates for sums of absolute values of the roots of1.1. 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 to1,3,4. 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 investigations2,5–8and 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. Besidesxwill be called anX-root coordinatecorresponding toyandyaY-root coordinate

corresponding tox. All the considered roots are counted with their multiplicities. Put

aj

y

n1

k0

ajkyn1−k

j0, . . . , m1

, bj

y

n2

k0

bjkyn2−k

j0, . . . , m2

. 1.3

Then

fx, y

m1

j0

aj

yxm1−j_,

gx, y

m2

j0

bj

yxm2−j_.

1.4

Withmm1m2introduce them×mSylvester matrix

Sy

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

a0 a1 a2 · · · am1−1 am1 0 0 · · · 0

0 a0 a1 · · · am1−2 am1−1 am1 0 · · · 0

· · · ·

0 0 0 · · · a0 a1 a2 a3 · · · am1

b0 b1 b2 · · · bm2−1 bm2 0 0 · · · 0

0 b0 b1 · · · bm2−2 bm2−1 bm2 0 · · · 0

· · · ·

0 0 0 · · · b0 b1 b2 b3 · · · bm2

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

, 1.5

withakakyandbkbky. PutRy detSyand consider the expansion:

Ry

n

k0

Rkyn−k, wheren:degR

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Furthermore, denote

θR:

⎡ ⎣ 1

2π

₂_π

0

Reit

R0

2

dt−1

⎤ ⎦

1/2

. 1.7

Clearly,

θR≤sup

|y|1

Ry

R0

, R0 lim

y→ ∞

Ry

yn , R0

1 2π

₂_π

0

e−intReitdt. _1.8

Thanks to the Hadamard inequality, we have

Ry2≤

_m 1

k0

ak

y2

m2_m 2

k0

bk

y2

m1

. 1.9

Assume that

Rn/0. 1.10

Theorem 1.1. TheY-root coordinatesykof 1.1(if, they exist), taken with the multiplicities and

ordered in the decreasing way:|yk| ≥ |yk1|, satisfy the estimates:

j

k1

yk< θR1 j

j 1,2, . . . , n. 1.11

If, in addition, condition1.10holds, then

j

k1

1 yk

< R0θR 1

Rn

j j 1,2, . . . , n, 1.12

whereykare theY-root coordinates of 1.1taken 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 forj_k₁|yk|is derived in9, Theorem 11.9.1; besides, in the mentioned theoremaj·and

bj·have the sense diﬀerent from the one accepted in this paper.

From1.12it follows that

min

k yk>

Rn

R0θR 1 Rn

. 1.13

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To estimate theX-root coordinates, assume that

inf

|y|≤θR1

a0

y>0 _1.14

and put

ψf sup

|y|θR1

1 _a0

y

⎡ ⎣m1

j1

aj

y2

⎤ ⎦

1/2

. 1.15

Theorem 1.2. Let condition1.14holds. Then theX-root coordinatesxky0of1.1corresponding

to aY-root coordinatey0 (if they exist), taken with the multiplicities and ordered in the decreasing

way, satisfy the estimates:

j

k1

xk

y0< ψfj

j1,2, . . . ,min{m1, m2}

. 1.16

InTheorem 1.2one can replacefbyg.

Furthermore, sinceykare ordered in the decreasing way, byTheorem 1.1we getj|yj|<

jθRand

_y_j_{< r}_j_:₁θR

j

j1, . . . , n. 1.17

Thus1.1has in the disc{z∈C:|z| ≤rj}no more thann−j Y-root coordinates. If we denote

byνYrthe number ofY-root coordinates of1.1inΩr : {z ∈C : |z| ≤r}for a positive

numberr, then we get

Corollary 1.3. Under condition1.10, the inequalityνYr≤n−j1 is valid for any

r≤1θR

j . 1.18

Similarly, byTheorem 1.2, we get the inequality:

xj

y0≤1

ψf

j

j1,2, . . . ,min{m1, m2}

. 1.19

Denote byνXy0, rthe number ofX-root coordinates of1.1inΩr, corresponding to aY

coordinatey0.

Corollary 1.4. Under conditions1.14, for anyY-root coordinatey0, the inequality:

νX

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is valid, provided

r ≤1ψf

j . 1.21

In this corollary also one can replacefbyg.

2. Proofs of Theorems

1.1 and

1.2

First, we need the following result.

Lemma 2.1. LetPz : zn_c

1zn−1· · ·cn be a polynomial with complex coeﬃcients. Then its

rootszkPordered in the decreasing way satisfy the inequalities:

j

k1

|zkP| ≤

1 2π

₂_π

0

Peit2dt−1 1/2

j j1, . . . , n. 2.1

Proof. As it is proved in9, Theorem 4.3.1 see also10,

j

k1

|zkP| ≤

_n

k1

|ck|2

1/2

j. 2.2

But thanks to the Parseval equality, we have

n

k1

|ck|21

1 2π

₂_π

0

Peit2dt. 2.3

Hence the required result follows.

Proof ofTheorem 1.1. The bound 1.11 follows from the previous lemma with Py

Ry/R0. To derive bound1.12note that

RyRnynW

1

y

, whereWz 1

Rn

n

k0

Rkzk

n

k0

dkzn−k, 2.4

withdk Rn−k/Rn. So any zerozWofWzis equal to 1/y, whereyis a zero ofRy. By

the previous lemma

j

k1

|zkW|<

1 2π

2π

0

Weit2dt−1 1/2

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Thus,

j

k1

1 yk

<

1 2π

₂_π

0

Weit2dt−1 1/2

j j 1, . . . , n. 2.6

ButWz zn_R₁_/z_/R

n. Thus,|Weit| 1/Rn|Re−it|. This proves the required result.

Proof ofTheorem 1.2. Due to Theorem 1.1, for any fixed Y-root coordinate y0 we have the

inequality:

_y0_≤_θ_R ₁_. _2.7

We seek the zeros of the polynomial:

fx, y0

m1

j0

aj

y0

xm1−j_. _2.8

Besides, due to1.14and2.7,a0y0/0. Put

Qx f

x, y0

a0y0 , θQ :

1 _a0

y0

⎡ ⎣m1

j1

aj

y02

⎤ ⎦

1/2

. 2.9

Clearly,

Qx xm1 1

a0y0

m1

j1

aj

y0

xm1−j_.

2.10

Due to the above mentioned9, Theorem 4.3.1we have

j

k1

xk

y0< θQj

j1,2, . . . , n. 2.11

But according to2.7,θQ ≤ψf. This proves the theorem.

3. Perturbations of Roots

Together with1.1, let us consider the coupled system:

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where

fx, y

m1

j0

n1

k0

ajkxm1−jyn1−k,

gx, y

m2

j0

n2

k0

bjkxm2−jyn2−k.

3.2

Hereajk,bjkare complex coeﬃcients. Put

aj

y

n1

k1

ajkyn1−k

j 0, . . . , m1

,

bj

y

n2

k0

bjkyn2−k

j 0, . . . , m2

.

3.3

Let Sy be the Sylvester matrix defined as above with ajy instead of ajy, and bjy

instead ofbjy, and putRy detSy. It is assumed that

degRydegRyn. 3.4

Consider the expansion:

Ry

n

k0

Rkyn−k. 3.5

Due to3.4,R0 /0. Denote

qR,R:

⎡ ⎣ 1

2π

0

Reit

R0 −

Reit

R0

2

dt

⎤ ⎦

1/2

,

ηR θ2R n−11/2.

3.6

Clearly,

qR,R≤max

|z|1

Ry

R0 −

Ry

R0

. 3.7

Theorem 3.1. Under condition 3.4, for any Y-root coordinate y0 of 3.1, there is a Y-root

coordinatey0of1.1, such that

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whereμRis the unique positive root of the equation:

ynqR,R

n−1

k0

ηk_R_yn−k−1

√

k! . 3.9

To proveTheorem 3.1, for a finite integern, consider the polynomials:

Pλ

n

k0

ckλn−k, Pλ

n

k0

ckλn−k c0 c0 1, 3.10

with complex coeﬃcients. Put

q0

_n

k1

|ck−ck|2

1/2 ,

ηP

_n

k1

|ck|2n−1

1/2 .

3.11

Lemma 3.2. For any rootzPofPy, there is a rootzPofPy, such that|zP−zP| ≤rq0,

whererq0is the unique positive root of the equation

ynq0

n−1

k0

ηk_P_yn−k−1

√

k! . 3.12

This result is due to Theorem 4.9.1 from the book9and inequality9.2on page 103 of that book.

By the Parseval equality we have

n

k0

|ck|2

1 2π

₂_π

0

Peit2dt,

q2₀ 1

2π

0

Peit−Peit2dt≤max

|z|1

Py−Py2.

3.13

Thus

η2P 1

2π

0

Peit2dtn−2≤max

|z|1

_P

y2n−2. 3.14

The assertion of Theorem 3.1 now follows from in the previous lemma with Py

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Further more, denote

pR :q

R,R

n−1

k0

ηk_R

√

k! ,

δR:

n

√_p

R ifpR ≤1,

pR ifpR >1.

3.15

Due to Lemma 1.6.1 from11, the inequalityμR≤δR is valid. NowTheorem 3.1implies the

following.

Corollary 3.3. Under condition 3.4, for any Y-root coordinate y0 of 3.1, there is a Y-root

coordinatey0of1.1, such that|y0−y0| ≤δR.

Similarly, one can consider perturbations of theX-root root coordinates.

To evaluate the quantityRy−Ryone can use the following result: let A and B two complexn×n-matrices. Then

|detA−detB| ≤ N2A−B

nn/2

11

2N2AB N2A−B n

, 3.16

whereN2

2A TraceA∗Ais the Hilbert-Schmidt norm andA∗ is the adjoint toA. For the

proof of this inequality see12. TakingBSy,ASywe get a bound for|Rz−Rz|.

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