Computing the Multiplicity Structure in Solving Polynomial Systems

B. H. Dayton, Z. Zeng



This paper presents a duality analysis and an algorithm for computing the multiplicity structure of a zero to a polynomial system, while the zero can be exact or approximate with the system being intrinsic or empirical. The main algorithm is developed through formulating multiplicity matrices and identifying the multiplicity as nullity while a null space is isomorphic to the dual space of the polynomial ideal at the multiple zero. Consequently, the multiplicity structure can be computed using either symbolic or numerical rank-revealing where inexact data and/or zeros are permitted. A preliminary implementation of the algorithm is carried out. Computing results show the algorithm is robust and accurate for problems of moderate sizes. As an application, the dual space theory and methodology are utilized to analyze deflation methods in solving polynomial systems, to establish tighter deflation bound, and to derive special case algorithms.