Semidefinite characterization and computation of zero-dimensional real radical ideals
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.
|Keywords||Algebraic geometry - Zero-dimensional ideal - (Real) radical ideal - Semidefinite programming|
|MSC||Real algebraic sets (msc 14P05)|
|THEME||Logistics (theme 3)|
|Journal||Foundations of Computational Mathematics|
|Project||Semidefinite programming and combinatorial optimization|
Lasserre, J.B, Laurent, M, & Rostalski, P. (2008). Semidefinite characterization and computation of zero-dimensional real radical ideals. Foundations of Computational Mathematics, 8(5), 607–647.