Localization and Perturbations of Roots to Systems of Polynomial Equations

Michael Gil'

doi:10.1155/2012/653914

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.

Highlights

Introduction and Statements of the MainResultsLet us consider the system: f x, y g x, y 0, f x, y m1 n1ajkxm1−j yn1−k am1n1 / 0, j 0k 0 1.2 g x, y m2 n2bjkxm2−j yn2−k bm2n2 / 0 . j 0k 0The coefficients ajk, bjk are complex numbers
The classical Bezout and Bernstein theorems give us bounds for the total number of solutions of a polynomial system, compared to 1, 2
For many applications, it is very important to know the number of solutions in a given domain

Summary

Introduction

Introduction and Statements of the MainResultsLet us consider the system: f x, y g x, y 0, f x, y m1 n1ajkxm1−j yn1−k am1n1 / 0 , j 0k 0 1.2 g x, y m2 n2bjkxm2−j yn2−k bm2n2 / 0 . j 0k 0The coefficients ajk, bjk are complex numbers. The classical Bezout and Bernstein theorems give us bounds for the total number of solutions of a polynomial system, compared to 1, 2 . The Y -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: j yk < θ R 1 j 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| ≤ |yk 1|.

Results

Conclusion