A Column Space Based Approach to Solve Systems of Multivariate Polynomial Equations

Abstract

We propose a novel approach to solve systems of multivariate polynomial equations, using the column space of the Macaulay matrix that is constructed from the coefficients of these polynomials. A multidimensional realization problem in the null space of the Macaulay matrix results in an eigenvalue problem, the eigenvalues and eigenvectors of which yield the common roots of the system. Since this null space based algorithm uses well-established numerical linear algebra tools, like the singular value and eigenvalue decomposition, it finds the solutions within machine precision. In this paper, on the other hand, we determine a complementary approach to solve systems of multivariate polynomial equations, which considers the column space of the Macaulay matrix instead of its null space. This approach works directly on the data in the Macaulay matrix, which is sparse and structured. We provide a numerical example to illustrate our new approach and to compare it with the existing null space based root-finding algorithm.