Inverse multivariate polynomial root-finding: Numerical implementations of the affine and projective Buchberger-Möller algorithm

We address the inverse problem of multivariate polynomial root-finding: given a finite set Z of points in Cn, find the minimal set of multivariate polynomials that vanish on Z. Two SVD-based algorithms are presented: one algorithm works only for affine roots and as a result almost always returns an overdetermined set of polynomials. This issue is resolved in the second algorithm by introducing projective points and hence adding roots at infinity. In addition, we show how the use of multiplicity structures is required to describe roots with multiplicities. We also derive a suitable tolerance that needs to be used when the roots are not known with infinite precision. A measure of how well the resulting polynomials vanish approximately on Z is shown to be the smallest singular value of a particular matrix. Both affine and projective implementations of our algorithm are applied to the problem of computing continuous-time polynomial dynamical systems from a given set of fixed points, demonstrating the effectiveness and robustness of our proposed methods.