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A unified approach for real and complex zeros of zero-dimensional ideals.


Ph. Rostalski, J.B. Lasserre, M. Laurent

vol. The IMA Volumes in Mathematics and its Applications, accepted as a book chapter in: "Emerging Applications of Algebraic Geometry","The IMA Volumes in Mathematics and its Applications", Seth Sullivant and Mihai Putinar ed.

Abstract. 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 \in R[x], assuming V_K(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 V_R(I), as shown in the authors’ previous work, but also the complex variety V_C(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 also related.


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