Back to the Roots: Polynomial System Solving, Linear Algebra, Systems Theory

Abstract

Multivariate polynomial system solving and polynomial optimization problems arise as central problems in many systems theory, identification and control settings. Traditionally, methods for solving polynomial equations have been developed in the area of algebraic geometry. Although a large body of literature is available, it is known as one of the most inaccessible fields of mathematics. In this paper we present a method for solving systems of polynomial equations employing numerical linear algebra and systems theory tools only, such as realization theory, SVD/QR, and eigenvalue computations. The task at hand is translated into the realm of linear algebra by separating coefficients and monomials into a coefficient matrix multiplied with a basis of monomials. Applying realization theory to the structure in the monomial basis allows to find all solutions of the system from eigenvalue computations. Solving a polynomial optimization problem is shown to be equivalent to an extremal eigenvalue problem. Relevant applications are found in identification and control, such as the global optimization of structured total least squares problems.