The Closedness Subspace Method for Computing the Multiplicity Structure of a Polynomial System

The multiplicity structure of a polynomial system at an isolated zero is identified with the dual space consisting of differential functionals vanishing on the entire polynomial ideal. Algorithms have been developed for computing dual spaces as the kernels of Macaulay matrices. These previous algorithms face a formidable obstacle in handling Macaulay matrices whose dimensions grow rapidly when the problem size and the order of the differential functionals increase. This paper presents a new algorithm based on the closedness subspace strategy that substantially reduces the matrix sizes in computing the dual spaces, enabling the computation of multiplicity structures for large problems. Comparisons of timings and memory requirements demonstrate a substantial improvement in computational efficiency.