Semidefinite Characterization and Computation of Zero-Dimensional Real Radical Ideals

Abstract

For an ideal I ⊆ R [x] given by a set of generators, a new semidefinite characterization of its real radical I(V R (I)) is presented, provided it is zero-dimensional (even if I is not). Moreover we propose an algorithm using numerical linear algebra and semidefinite optimization techniques, to compute all (finitely many) points of the real variety V R (I) as well as a set of generators of the real radical ideal. The latter is obtained in the form of a border or Gröbner basis. The algorithm is based on moment relaxations and, in contrast to other existing methods, it exploits the real algebraic nature of the problem right from the beginning and avoids the computation of complex components.

Sums of Squares, Moment Matrices and Optimization Over Polynomials

“…(Recall Corollary 6.9 for the radical zerodimensional case.) Theorem 6.15 below extends a result of Laurent [81] and uses ideas from Lasserre et al [75].…”

“…The question we now address is how to find the v i 's from the moment matrix M s (y). We present the method proposed by Henrion and Lasserre [54], although our description differs in some steps and follows the implementation proposed by Jibetean and Laurent [60] and presented in detail in Lasserre et al [75].…”

“…As in (6.9), K min p denotes the set of global minimizers of p over the set K. Thus K min p = ∅, e.g., when K is compact. The next result is based on [54] combined with ideas from [75].…”

“…The second part of Theorem 6.18, asserting that all global minimizers are found when having a maximum rank solution, relies on ideas from [75]. When p is the constant polynomial 1 and K is defined by the equations h 1 = 0, .…”

“…, h m0 = 0, then K min p is the set of all common real roots of the h j 's. The paper [75] explains in detail how the moment methodology applies to finding real roots, and [76] extends this to complex roots.…”