Singular Zeros of Polynomial Systems

Singular zeros of systems of polynomial equations constitute a bottleneck when it comes to computing, since several methods relying on the regularity of the Jacobian matrix of the system do not apply when the latter has a non-trivial kernel. Therefore they require special treatment. The algebraic information regarding an isolated singularity can be captured by a finite, local basis of differentials expressing the multiplicity structure of the point. In the present article, we review some available algebraic techniques for extracting this information from a polynomial ideal. The algorithms for extracting the, so called, dual basis of the singularity are based on matrix-kernel computations, which can be carried out numerically, starting from an approximation of the zero in question. The next step after obtaining the multiplicity structure is to deflate the root, that is, construct a new system in which the singularity is eliminated. Having a deflated system allows to refine the solution fast and to high accuracy, since the Jacobian matrix is regular and all the usual machinery, e.g. Newton's method or existence and unicity criteria may be applied. Standard verification methods, based e.g. on interval arithmetic and a fixed point theorem, can then be employed to certify that there exists a unique perturbed system with a singular root in the domain.