Abstract
In this paper we propose a unified methodology for computing the set V K (I) of complex (K = ℂ) or real (K = ℝ) roots of an ideal R[x], assuming Vk (I ) is finite. We show how moment matrices, defined in terms of a given set of generators of the ideall, 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 problem. 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 arc outlined and their stopping criteria are related.
AMS(MOS) subject classifications. 12DI0, 12E12, 12Y05, 13A15.
Supported by the french national research agency ANR under grant NT05-3-41612.
Supported by the Netherlands Organization for Scientific Research grant NWO 639.032.203 and by ADONET, Marie Curie Research Training Network MRTN-CT-2003-504438.
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Lasserre, J.B., Laurent, M., Rostalskl, P. (2009). A Unified Approach to Computing Real and Complex Zeros of Zero-Dimensional Ideals. In: Putinar, M., Sullivant, S. (eds) Emerging Applications of Algebraic Geometry. The IMA Volumes in Mathematics and its Applications, vol 149. Springer, New York, NY. https://doi.org/10.1007/978-0-387-09686-5_6
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