Research Article Localization and Perturbations of Roots to Systems of Polynomial Equations

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1 International Mathematics and Mathematical Sciences Volume 2012, Article ID , 9 pages doi: /2012/ 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 f x, y ) m 1 n 1 a jk x m1 j y n 1 k j 0 g x, y ) m 2 n 2 b jk x m2 j y n 2 k j 0 a m1 n 1 / 0, b m2 n 2 / The coefficients a jk,b jk 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 2 International 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, ỹ is a solution of 1.1 if f x, ỹ g x, ỹ 0. Besides x will be called an X-root coordinate corresponding to ỹ and ỹ a Y-root coordinate corresponding to x. All the considered roots are counted with their multiplicities. Put ) n 1 a j y a jk y n ) ) n 2 1 k j 0,...,m1, bj y b jk y n ) 2 k j 0,...,m Then f x, y ) m 1 ) a j y x m 1 j, j 0 g x, y ) m 2 ) b j y x m 2 j. j With m m 1 m 2 introduce the m m Sylvester matrix a 0 a 1 a 2 a m1 1 a m a 0 a 1 a m1 2 a m1 1 a m1 0 0 S y ) a 0 a 1 a 2 a 3 a m1, 1.5 b 0 b 1 b 2 b m2 1 b m b 0 b 1 b m2 2 b m2 1 b m b 0 b 1 b 2 b 3 b m2 with a k a k y and b k b k y. PutR y det S y and consider the expansion: R y ) n R k y n k, where n : deg R y ). 1.6

3 International Mathematics and Mathematical Sciences 3 Furthermore, denote θ R : 1 2π 0 R e it 2 dt 1 R 0 1/ Clearly, R y ) θ R sup, y 1 R 0 ) R R y 0 lim y y n, R 0 1 2π 0 e int R e it) dt. 1.8 Thanks to the Hadamard inequality, we have ) m1 ) m2 R y 2 ) m1 m2 ak y 2 ) bk y 2). 1.9 Assume that R n / Theorem 1.1. The Y-root coordinates y k of 1.1 if, they exist), taken with the multiplicities and ordered in the decreasing way: y k y, satisfy the estimates: j ) yk <θ R 1 j j 1, 2,...,n If, in addition, condition 1.10 holds, then j 1 < R 0 θ R 1 j ỹk R n j 1, 2,...,n ), 1.12 where ỹ k are the Y-root coordinates of 1.1 taken with the multiplicities and ordered in the increasing way: ỹ k ỹ. This theorem and the next one are proved in the next section. Note that another bound for j y k is derived in 9, Theorem ; besides, in the mentioned theorem a j and b j have the sense different from the one accepted in this paper. From 1.12 it follows that R n miny k >. k R 0 θ R 1 R n 1.13 So the disc y R n / R 0 θ R 1 R n is zero free.

4 4 International Mathematics and Mathematical Sciences To estimate the X-root coordinates, assume that ) inf a0 y > 0 y θ R and put ψ f sup y θ R 1 1 ) m 1 ) aj y 2 a0 y j 1 1/ Theorem 1.2. Let condition 1.14 holds. Then the X-root coordinates x k y 0 of 1.1 corresponding to a Y-root coordinate y 0 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,m 2 } ) In Theorem 1.2 one can replace f by g. Furthermore, since y k are ordered in the decreasing way, by Theorem 1.1 we get j y j < j θ R and y j <r j : 1 θ R j j 1,...,n ) Thus 1.1 has in the disc {z C : z r j } no more than n j Y-root coordinates. If we denote by ν Y r the number of Y-root coordinates of 1.1 in Ω r : {z C : z r} for a positive number r, then we get Corollary 1.3. Under condition 1.10, the inequality ν Y r n j 1 is valid for any r 1 θ R j Similarly, by Theorem 1.2, we get the inequality: xj y0 ) 1 ψ f j j 1, 2,...,min{m1,m 2 } ) Denote by ν X y 0,r the number of X-root coordinates of 1.1 in Ω r, corresponding to a Y coordinate y 0. Corollary 1.4. Under conditions 1.14, for any Y-root coordinate y 0, the inequality: ν X y0,r ) min{m 1,m 2 } j 1 j 1, 2,...,min{m1,m 2 } ) 1.20

5 International Mathematics and Mathematical Sciences 5 is valid, provided r 1 ψ f j 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 : z n c 1 z n 1 c n be a polynomial with complex coefficients. Then its roots z k P ordered in the decreasing way satisfy the inequalities: j z k P [ 1 ] 1/2 P e it) 2 dt 1 j j 1,...,n ). 2.1 Proof. As it is proved in 9, Theorem see also 10, [ ] j n 1/2 z k P c k 2 j. 2.2 But thanks to the Parseval equality, we have n c k π 0 P e it) 2 dt. 2.3 Hence the required result follows. Proof of Theorem 1.1. The bound 1.11 follows from the previous lemma with P y R y /R 0. To derive bound 1.12 note that R y ) R n y n W ) 1, where W z 1 n R k z k y R n n d k z n k, 2.4 with d k R n k /R n.soanyzeroz W of W z is equal to 1/ỹ, where ỹ is a zero of R y. By the previous lemma j z k W < [ 1 ] 1/2 W e it) 2 dt 1 j j 1,...,n ). 2.5

6 6 International Mathematics and Mathematical Sciences Thus, j 1 < ỹk [ 1 ] 1/2 W e it) 2 dt 1 j j 1,...,n ). 2.6 But W z z n R 1/z /R n.thus, W e it 1/R n R e it. This proves the required result. Proof of Theorem 1.2. Due to Theorem 1.1, for any fixed Y-root coordinate y 0 we have the inequality: y0 θ R We seek the zeros of the polynomial: f ) m 1 ) x, y 0 a j y0 x m 1 j. j Besides, due to 1.14 and 2.7, a 0 y 0 / 0. Put Q x f x, y 0 ) a 0 y0 ), 1 θ Q : ) m 1 ) aj y0 2 a0 y0 j 1 1/ Clearly, Q x x m 1 1 m 1 ) ) a j y0 x m 1 j. a 0 y0 j Due to the above mentioned 9, Theorem we have j ) xk y0 <θq j j 1, 2,...,n ) But according to 2.7, θ Q ψ f. This proves the theorem. 3. Perturbations of Roots Together with 1.1, let us consider the coupled system: f x, y ) ĝ x, y ) 0, 3.1

7 International Mathematics and Mathematical Sciences 7 where f x, y ) m 1 n 1 â jk x m1 j y n1 k, j 0 ĝ x, y ) m 2 n 2 b jk x m2 j y n2 k. j Here â jk, b jk are complex coefficients. Put ) n 1 â j y â jk y n ) 1 k j 0,...,m1, 3.3 ) n 2 b j y b jk y n ) 2 k j 0,...,m2. Let Ŝ y be the Sylvester matrix defined as above with â j y instead of a j y, and b j y instead of b j y, and put R y det Ŝ y. It is assumed that deg R y ) deg R y ) n. 3.4 Consider the expansion: R y ) n R k y n k. 3.5 Due to 3.4, R 0 / 0. Denote ) q R, R : 1 R e it R ) e it 2 1/2 dt, R 0 R [ ] 1/2. η R θ 2 R n 1 Clearly, ) q R, R R y ) max R ) y z 1 R 0 R Theorem 3.1. Under condition 3.4, for any Y-root coordinate ŷ 0 coordinate y 0 of 1.1, such that y0 ŷ 0 μr, of 3.1, thereisay-root 3.8

8 8 International Mathematics and Mathematical Sciences where μ R is the unique positive root of the equation: y n q R, R )n 1 η k R y n k 1 k!. 3.9 To prove Theorem 3.1, for a finite integer n, consider the polynomials: n P λ c k λ n k, n P λ ĉ k λ n k c 0 ĉ 0 1, 3.10 with complex coefficients. Put [ ] n 1/2, q 0 c k ĉ k 2 [ n 1/2 η P c k 2 n 1] Lemma 3.2. For any root z P of P y, there is a root z P of P y, such that z P z P r q 0, where r q 0 is the unique positive root of the equation n 1 y n η k P y n k 1 q k! This result is due to Theorem from the book 9 and inequality 9.2 on page 103 of that book. By the Parseval equality we have n c k 2 1 2π 0 P e it) 2 dt, q P e it) P e it) 2 dt max P y ) P y ) 2. z Thus η 2 P 1 P e it) 2 ) dt n 2 max P y 2 n 2. z The assertion of Theorem 3.1 now follows from in the previous lemma with P y R y /R 0 and P y R y / R 0.

9 International Mathematics and Mathematical Sciences 9 Further more, denote p R : q R, R )n 1 η k R k!, { n pr if p R 1, δ R : p R if p R > Due to Lemma from 11, the inequality μ R δ R is valid. Now Theorem 3.1 implies the following. Corollary 3.3. Under condition 3.4, for any Y-root coordinate ŷ 0 of 3.1, thereisay-root coordinate y 0 of 1.1, such that y 0 ŷ 0 δ R. Similarly, one can consider perturbations of the X-root root coordinates. To evaluate the quantity R y R y one can use the following result: let A and B two complex n n-matrices. Then det A det B N 2 A B 1 1 n n n/2 2 N 2 A B N 2 A B ), 3.16 where N 2 2 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 Ŝ y, A S y we get a bound for R z R z. References 1 I. M. Gel fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, Mathematics: Theory & Applications, Birkhäuser, Boston, Mass, USA, B. Sturmfels, Solving algebraic equations in terms of A-hypergeometric series, Discrete Mathematics, vol. 210, no. 1 3, pp , B. Mourrain, Computing the isolated roots by matrix methods, Symbolic Computation,vol. 26, no. 6, pp , 1998, Symbolic numeric algebra for polynomials. 4 B. Sturmfels, Solving Systems of Polynomial Equations, vol. 97 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, W. Forster, Homotopies for systems of polynomial equations, Acta Mathematica Vietnamica, vol. 22, no. 1, pp , J. McDonald, Fiber polytopes and fractional power series, Pure and Applied Algebra, vol. 104, no. 2, pp , J. McDonald, Fractional power series solutions for systems of equations, Discrete & Computational Geometry, vol. 27, no. 4, pp , Z.K.Wang,S.L.Xu,andT.A.Gao,Algebraic Systems of Equations and Computational Complexity Theory, vol. 269 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 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, M. I. Gil, Bounds for counting functions of finite order entire functions, Complex Variables and Elliptic Equations, vol. 52, no. 4, pp , M. I. Gil, Operator Functions and Localization of Spectra, vol of Lecture Notes in Mathematics, Springer, Berlin, Germany, M. I. Gil, A bound for functions defined on a normed space, Physics A, vol. 40, no. 15, pp , 2007.

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