Moment matrices, border bases and real radical computation

Abstract

In this paper, we describe new methods to compute the radical (resp. real radical) of an ideal, assuming it complex (resp. real) variety is finte. The aim is to combine approaches for solving a system of polynomial equations with dual methods which involve moment matrices and semi-definite programming. While the border basis algorithms of [17] are efficient 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 Gröbner bases. It thus combines the strength of existing algorithms and provides a unified treatment for the computation of border bases for the ideal, the radical ideal and the real radical ideal.J.B. Lasserre, LAAS,

Border basis relaxation for polynomial optimization

“…It can be seen as a type of border basis algorithm, in which in the main loop we compute the optimal linear form (section 4), we then check when the minimun is reached (section 5) and finally we compute the minimizer points (section 6). This algorithm is closely connected to the real radical border basis algorithm presented in (Lasserre et al, 2012).…”

“…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.…”

“…By Theorem 3.14 of (Lasserre et al, 2012),Λ has a decomposition of the form Λ = r i=1 ω i 1 ξi with ω i > 0 and ξ i ∈ R n . By Lemma 3.5 of (Lasserre et al, 2012), V is isomorphic to R[x]/I(ξ 1 , .…”

“…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).…”

“…(Riener et al, 2013)) or polynomial reduction (see e.g. (Lasserre et al, 2012)). In this paper, we extend this latter approach.…”