Chapter 3 - The eigenvalue approach to polynomial system solving

Abstract

This chapter discusses the eigenvalue approach to polynomial system solving. Solving a univariate polynomial equation is equivalent to finding the eigenvalues of the related Frobenius matrix. It is proved that for a system of multivariate polynomial equations, one can build a joint eigenvalue problem such that finding the solutions of the system of equations is equivalent to finding all eigenvalues and eigenvectors of a matrix pencil. According to the theorem, to solve an algebraic system, one can construct an eigenvalue problem in the above manner and then solve the resulting joint eigenvalue problem. It is found that writing the equations in matrix form and applying Gaussian elimination, one can get a new system. The algorithm gives either an ordinary eigenvalue problem or a matrix pencil eigenvalue problem in a finite number of steps. In addition, the chapter describes the isolated projection onto some axis of the zeros of the given algebraic system that can be obtained from the eigenvalues of the resulting eigenvalue problems.