In this paper, we describe new methods to compute the radical (resp. real radical) of an ideal, assuming it complex (resp. real) variety is nte. The aim is to combine approaches for solving a system of polynomial equations with dual methods which involve moment matrices and semi-de nite programming. While the border basis algorithms of [17] are ecient and numerically stable for computing complex roots, algorithms based on moment matrices [12] allow the incorporation of additional polynomials, e.g., to restrict the computation to real roots or to eliminate multiple solutions. The proposed algorithm can be used to compute a border basis of the input ideal and, as opposed to other approaches, it can also compute the quotient structure of the (real) radical ideal directly, i.e., without prior algebraic techniques such as Grobner bases. It thus combines the strength of existing algorithms and provides a uni ed treatment for the computation of border bases for the ideal, the radical ideal and the real radical ideal.
Additional Metadata
Keywords polynomial equation, real root, border base, semidefinite programming
THEME Other (theme 6)
Publisher Academic Press
Journal Journal of Symbolic Computation
Project Semidefinite programming and combinatorial optimization
Citation
Mourrain, B, Lasserre, J.B, Laurent, M, Rostalski, P, & Trebuchet, P. (2013). Moment matrices, border bases and radical computation. Journal of Symbolic Computation, 51, 63–85.