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 $I$ 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 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 are outlined and their stopping criteria are related.

Additional Metadata
Keywords Polynomial ideal, zero-dimensional ideal, complex roots, real roots, numerical linear algebra
MSC Polynomials: location of zeros (algebraic theorems) (msc 12D10)
THEME Logistics (theme 3)
Publisher Springer
Editor M. Putinar , S. Sullivant
Series The IMA Volumes in Mathematics and its Applications
Project Semidefinite programming and combinatorial optimization
Citation
Lasserre, J.B, Laurent, M, & Rostalski, P. (2009). A unified approach to computing real and complex zeros of zero-dimensional ideals. In M Putinar & S Sullivant (Eds.), Emerging Applications of Algebraic Geometry (pp. 125–155). Springer.