# Localization and Perturbations of  Roots to  Systems of Polynomial Equations

Authors: Michael Gil'
Journal: International Journal of Mathematics and Mathematical Sciences
 (2012)

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## Abstract

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.


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## Body

## 1. Introduction and Statements of the Main Results

Let us consider the system:(1.1)f(x,y)=g(x,y)=0,
where
(1.2)f(x,y)=∑j=0m1‍∑k=0n1ajkxm1-jyn1-k(am1n1≠0),g(x,y)=∑j=0m2‍∑k=0n2‍bjkxm2-jyn2-k(bm2n2≠0).
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 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 f(x~,y~)=g(x~,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(1.3)aj(y)=∑k=0n1ajkyn1-k(j=0,…,m1),bj(y)=∑k=0n2bjkyn2-k(j=0,…,m2).
Then
(1.4)f(x,y)=∑j=0m1aj(y)xm1-j,g(x,y)=∑j=0m2bj(y)xm2-j.
With m=m1+m2 introduce the m×m Sylvester matrix
(1.5)S(y)=(a0a1a2⋯am1-1am100⋯00a0a1⋯am1-2am1-1am10⋯0⋅⋅⋅⋯⋅⋅⋅⋅⋯⋅000⋯a0a1a2a3⋯am1b0b1b2⋯bm2-1bm200⋯00b0b1⋯bm2-2bm2-1bm20⋯0⋅⋅⋅⋅⋅⋅⋅⋅⋯⋅000⋯b0b1b2b3⋯bm2),
with ak=ak(y) and bk=bk(y). Put R(y)=detS(y) and consider the expansion:
(1.6)R(y)=∑k=0nRkyn-k,wheren:=deg⁡R(y).
Furthermore, denote
(1.7)θ(R):=[12π∫02π|R(eit)R0|2dt-1]1/2.
Clearly,
(1.8)θ(R)≤sup⁡|y|=1|R(y)R0|,R0=lim⁡y→∞⁡R(y)yn,R0=12π∫02πe-intR(eit)dt.
Thanks to the Hadamard inequality, we have
(1.9)|R(y)|2≤(∑k=0m1‍|ak(y)|2)m2(∑k=0m2‍|bk(y)|2)m1.
Assume that
(1.10)Rn≠0.Theorem 1.1.
TheY-root coordinates yk of (1.1) (if, they exist), taken with the multiplicities and ordered in the decreasing way: |yk|≥|yk+1|, satisfy the estimates:
(1.11)∑k=1j|yk|<θ(R+1)+j(j=1,2,…,n).
If, in addition, condition (1.10) holds, then
(1.12)∑k=1j1|y~k|<R0(θ(R)+1)Rn+j(j=1,2,…,n),
where y~k are the Y-root coordinates of (1.1) taken with the multiplicities and ordered in the increasing way: |y~k|≤|y~k+1|.This theorem and the next one are proved in the next section. Note that another bound for∑k=1j‍|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
(1.13)min⁡k⁡yk>RnR0(θ(R)+1)+Rn.
So the disc |y|≤Rn/(R0(θ(R)+1)+Rn) is zero free.To estimate theX-root coordinates, assume that
(1.14)inf⁡|y|≤θ(R)+1|a0(y)|>0
and put
(1.15)ψf=sup⁡|y|=θ(R)+11|a0(y)|[∑j=1m1‍|aj(y)|2]1/2.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:
(1.16)∑k=1j‍|xk(y0)|<ψf+j(j=1,2,…,min⁡{m1,m2}).In Theorem1.2 one can replace f by g.Furthermore, sinceyk are ordered in the decreasing way, by Theorem 1.1 we get j|yj|<j+θ(R) and
(1.17)|yj|<rj:=1+θ(R)j(j=1,…,n).
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 getCorollary 1.3.
Under condition (1.10), the inequality νY(r)≤n-j+1 is valid for any
(1.18)r≤1+θ(R)j.Similarly, by Theorem1.2, we get the inequality:
(1.19)|xj(y0)|≤1+ψfj(j=1,2,…,min⁡{m1,m2}).
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:
(1.20)νX(y0,r)≤min⁡{m1,m2}-j+1(j=1,2,…,min⁡{m1,m2})
is valid, provided
(1.21)r≤1+ψfj.In this corollary also one can replacef by g.


## 2. Proofs of Theorems1.1 and 1.2

First, we need the following result.Lemma 2.1.
LetP(z):=zn+c1zn-1+⋯+cn be a polynomial with complex coefficients. Then its roots zk(P) ordered in the decreasing way satisfy the inequalities:
(2.1)∑k=1j|zk(P)|≤[12π∫02π|P(eit)|2dt-1]1/2+j(j=1,…,n).Proof.
As it is proved in [9, Theorem 4.3.1] (see also [10]),
(2.2)∑k=1j|zk(P)|≤[∑k=1n|ck|2]1/2+j.
But thanks to the Parseval equality, we have
(2.3)∑k=1n‍|ck|2+1=12π∫02π‍|P(eit)|2dt.
Hence the required result follows.Proof of Theorem1.1.
The bound (1.11) follows from the previous lemma with P(y)=R(y)/R0. To derive bound (1.12) note that
(2.4)R(y)=RnynW(1y),whereW(z)=1Rn∑k=0n‍Rkzk=∑k=0n‍dkzn-k,
with dk=Rn-k/Rn. So any zero z(W) of W(z) is equal to 1/y~, where y~ is a zero of R(y). By the previous lemma
(2.5)∑k=1j‍|zk(W)|<[12π∫02π‍|W(eit)|2dt-1]1/2+j(j=1,…,n).
Thus,
(2.6)∑k=1j‍1|y~k|<[12π∫02π‍|W(eit)|2dt-1]1/2+j(j=1,…,n).
But W(z)=znR(1/z)/Rn. Thus, |W(eit)|=(1/Rn)|R(e-it)|. This proves the required result.Proof of Theorem1.2.
Due to Theorem1.1, for any fixed Y-root coordinate y0 we have the inequality:
(2.7)|y0|≤θ(R)+1.
We seek the zeros of the polynomial:
(2.8)f(x,y0)=∑j=0m1‍aj(y0)xm1-j.
Besides, due to (1.14) and (2.7), a0(y0)≠0. Put
(2.9)Q(x)=f(x,y0)a0(y0),θQ:=1|a0(y0)|[∑j=1m1‍|aj(y0)|2]1/2.
Clearly,
(2.10)Q(x)=xm1+1a0(y0)∑j=1m1‍aj(y0)xm1-j.
Due to the above mentioned [9, Theorem 4.3.1] we have
(2.11)∑k=1j‍|xk(y0)|<θQ+j(j=1,2,…,n).
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:
(3.1)f^(x,y)=g^(x,y)=0,
where
(3.2)f^(x,y)=∑j=0m1‍∑k=0n1‍a^jkxm1-jyn1-k,g^(x,y)=∑j=0m2‍∑k=0n2‍b^jkxm2-jyn2-k.
Here a^jk,b^jk are complex coefficients. Put
(3.3)a^j(y)=∑k=1n1‍a^jkyn1-k(j=0,…,m1),b^j(y)=∑k=0n2‍b^jkyn2-k(j=0,…,m2).
Let S^(y) be the Sylvester matrix defined as above with a^j(y) instead of aj(y), and b^j(y) instead of bj(y), and put R^(y)=detS^(y). It is assumed that
(3.4)deg⁡R^(y)=deg⁡R(y)=n.
Consider the expansion:
(3.5)R^(y)=∑k=0n‍R^kyn-k.
Due to (3.4), R^0≠0. Denote
(3.6)q(R,R^)≔[12π∫02π|R(eit)R0-R^(eit)R^0|2dt]1/2,η(R)=[θ2(R)+n-1]1/2.
Clearly,
(3.7)q(R,R^)≤max⁡|z|=1|R(y)R0-R^(y)R^0|.Theorem 3.1.
Under condition (3.4), for any Y-root coordinate y^0 of (3.1), there is a Y-root coordinate y0 of (1.1), such that
(3.8)|y0-y^0|≤μR,
where μR is the unique positive root of the equation:
(3.9)yn=q(R,R^)∑k=0n-1‍ηk(R)yn-k-1k!.To prove Theorem3.1, for a finite integer n, consider the polynomials:
(3.10)P(λ)=∑k=0n‍ckλn-k,P^(λ)=∑k=0n‍c^kλn-k(c0=c^0=1),
with complex coefficients. Put
(3.11)q0=[∑k=1n|ck-c^k|2]1/2,η(P)=[∑k=1n|ck|2+n-1]1/2.Lemma 3.2.
For any rootz(P^) of P^(y), there is a root z(P) of P(y), such that |z(P)-z(P^)|≤r(q0), where r(q0) is the unique positive root of the equation
(3.12)yn=q0∑k=0n-1ηk(P)yn-k-1k!.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(3.13)∑k=0n|ck|2=12π∫02π|P(eit)|2dt,q02=12π∫02π|P(eit)-P~(eit)|2dt≤max⁡|z|=1⁡|P(y)-P~(y)|2.
Thus
(3.14)η2(P)=12π∫02π|P(eit)|2dt+n-2≤max⁡|z|=1⁡|P(y)|2+n-2.The assertion of Theorem3.1 now follows from in the previous lemma with P(y)=R(y)/R0  andP~(y)=R^(y)/R^0.Further more, denote(3.15)pR≔q(R,R^)∑k=0n-1ηk(R)k!,δR≔{pRnifpR≤1,pRifpR>1.
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 y^0 of (3.1), there is a Y-root coordinate y0 of (1.1), such that |y0-y^0|≤δR.Similarly, one can consider perturbations of theX-root root coordinates.To evaluate the quantityR(y)-R^(y) one can use the following result: let A and B two complex n×n-matrices. Then
(3.16)|det⁡(A)-det⁡(B)|≤N2(A-B)nn/2(1+12(N2(A+B)+N2(A-B)))n,
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=S^(y), A=S(y) we get a bound for |R(z)-R^(z)|.



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