Multidimensional Realization Theory and Polynomial System Solving

TL;DR

This paper develops a realization theory for multidimensional systems, linking it to polynomial system solving, and introduces matrix-based methods to analyze and compute solutions, including at infinity and with multiple solutions.

Contribution

It extends realization theory to multidimensional systems and connects it to polynomial equations, providing matrix-based solution techniques and analysis of solution properties.

Findings

Null space of Macaulay matrix as multidimensional observability matrix

Eigenvalue decomposition reduces polynomial system solving

Analysis of solutions at infinity and multiple solutions

Abstract

Multidimensional systems are becoming increasingly important as they provide a promising tool for estimation, simulation and control, while going beyond the traditional setting of one-dimensional systems. The analysis of multidimensional systems is linked to multivariate polynomials, and is therefore more difficult than the well-known analysis of one-dimensional systems, which is linked to univariate polynomials. In the current paper we relate the realization theory for overdetermined autonomous multidimensional systems to the problem of solving a system of polynomial equations. We show that basic notions of linear algebra suffice to analyze and solve the problem. The difference equations are associated with a Macaulay matrix formulation, and it is shown that the null space of the Macaulay matrix is a multidimensional observability matrix. Application of the classical shift trick from realization theory allows for the computation of the corresponding system matrices in a multidimensional state-space setting. This reduces the task of solving a system of polynomial equations to computing an eigenvalue decomposition. We study the occurrence of multiple solutions, as well as the existence and analysis of solutions at infinity, which allow for an interpretation in terms of multidimensional descriptor systems.