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**1 - 1**of**1**### On Polynomial System Solving and Multidimensional Realization Theory

"... A giant. A mentor. A friend. We describe how systems of multivariate polynomial equations can be solved by formulating the task as a problem in multidimen-sional realization theory. The Macaulay matrix formulation is used to represent the system of polynomial equations in a linear algebra setting. T ..."

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A giant. A mentor. A friend. We describe how systems of multivariate polynomial equations can be solved by formulating the task as a problem in multidimen-sional realization theory. The Macaulay matrix formulation is used to represent the system of polynomial equations in a linear algebra setting. The null space of the Macaulay matrix, which is spanned by Vandermonde vectors constructed from the roots, is a multidimensional observability matrix. This allows for a natural representation of the problem in realization theory. The for-mulation leads to an eigenvalue-based root-finding algorithm. By considering the problem in the projective space, a description is obtained for both the affine solutions as well as the solutions at infinity. The latter lead to an interpretation as multidimensional descriptor system realizations.