A Unified Approach to Computing Real and Complex Zeros of Zero-dimensional Ideals

In this paper we propose a unified methodology for computing the set VK(I) of complex (K = C) or real (K = R) roots of an ideal I ⊆ R[x], assuming VK(I) is finite. We show how moment matrices, defined in terms of a given set of generators of the ideal I, can be used to (numerically) find not only the real variety VR(I), as shown in the authors’ previous work, but also the complex variety VC(I), thus leading to a unified treatment of the algebraic and real algebraic problems. In contrast to the real algebraic version of the algorithm, the complex analogue only uses basic numerical linear algebra because it does not require positive semidefiniteness of the moment matrix and so avoids semidefinite programming techniques. The links between these algorithms and other numerical algebraic methods are outlined and their stopping criteria are related.