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

Abstract

Abstract. In this paper we propose a unified methodology for computing the setWe 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.

Border basis relaxation for polynomial optimization

“…It has also been shown that the limit can even be reached in a finite number of steps in some cases, see e.g. (Lasserre et al, 2009;Nie et al, 2006;Marshall, 2009;Ha and Pham, 2010;Nie, 2011;Abril Bucero and Mourrain, 2013). In this case, the relaxation is said to be exact.…”

“…In this section we introduce the notion of optimal linear form for f , involved in the computation of I min (also called generic linear form when f = 0 in (Lasserre et al, 2009(Lasserre et al, , 2012). In order to find this optimal linear form we solve a Semi-Definite Programming (SDP) problem, which involves truncated Hankel matrices associated with the monomial basis and the reduction of their products by the border basis, as described in the previous section.…”

“…If we assume that S = S(g) is finite, then the corresponding border basis relaxation yields the points of S and the generators of g + g 0 . See also (Lasserre et al, 2009(Lasserre et al, , 2012 for zero dimensional real radical computation and (Ma et al, 2013).…”

“…In some cases, the sequence even reaches the infimum in a finite number of steps (Laurent, 2007;Nie et al, 2006;Marshall, 2009;Demmel et al, 2007;Ha and Pham, 2010;Nie, 2011). This approach has proved to be particularly fruitful in many problems (Lasserre, 2009). In contrast with numerical methods such as gradient descent methods, which converge to a local extremum but with no guaranty for the global Email addresses: Marta.Abril_Bucero@inria.fr (Marta Abril Bucero), Bernard.Mourrain@inria.fr (Bernard Mourrain).…”

“…Determining when and how these minimizer points can be computed from the relaxation sequence is a problem that has been addressed, for instance in (Henrion and Lasserre., 2005;Nie, 2012) using full moment matrices. This approach has been used for solving polynomial equations (Laurent, 2007;Lasserre et al, 2008Lasserre et al, , 2009Lasserre, 2009).…”