On the null spaces of the Macaulay matrix

In this article both the left and right null space of the Macaulay matrix are described. The left null space is shown to be linked with the occurrence of syzygies in its row space. It is also demonstrated how the dimension of the left null space is described by a piecewise function of polynomials. We present two algorithms that determine these polynomials. Furthermore we show how the finiteness of the number of basis syzygies results in the notion of the degree of regularity. This concept plays a crucial role in describing a basis for the right null space of the Macaulay matrix in terms of differential functionals. We define a canonical null space for the Macaulay matrix in terms of the projective roots of a polynomial system and extend the multiplication property of this canonical basis to the projective case. This results in an algorithm to determine the upper triangular commuting multiplication matrices. Finally, we discuss how Stetter's eigenvalue problem to determine the roots of a multivariate polynomial system can be extended to the case where a multivariate polynomial system has both affine roots and roots at infinity.