Back to the Roots – Polynomial System Solving Using Linear Algebra and System Theory

Abstract. We return to the algebraic roots of the problem of finding the solutions of a set of polynomial equations, and review this task from the linear algebra perspective. The system of polynomial equations is represented by a system of homogeneous linear equations by means of a structured Macaulay coefficient matrix multiplied by a vector containing monomials. Two properties are of key importance in the null spaces of Macaulay coefficient matrices, namely the correspondence between linear (in)dependent monomials in the polynomials and the linear (in)dependent rows, and secondly, the occurrence of a monomial multiplication shift structure. Both properties are invariant and occur regardless of the specific numerical basis of the null space of the Macaulay matrix. By exploiting the multiplication structure in the vector of monomials, a (generalized) eigenvalue problem is derived in terms of matrices built up from certain rows of a numerically computed basis for the null space of the Macaulay matrix. The main goal of the paper is to develop a simple solution approach, making the problem accessible to a wide audience of applied mathematicians and engineers.