A prolongation-projection algorithm for computing the finite real variety of an ideal

We provide a real algebraic symbolic-numeric algorithm for computing the real variety VR(I) of an ideal I ⊆ R[x], assuming VR(I) is finite (while VC(I) could be infinite). Our approach uses sets of linear functionals on R[x], vanishing on a given set of polynomials generating I and their prolongations up to a given degree, as well as on polynomials of the real radical ideal R √ I obtained from the kernel of a suitably defined moment matrix assumed to be positive semidefinite and of maximum rank. We formulate a condition on the dimensions of projections of these sets of linear functionals, which serves as stopping criterion for our algorithm. This algorithm is based on standard numerical linear algebra routines and semidefinite optimization and combines techniques from previous work of the authors together with an existing algorithm for the complex variety. This results in a unified methodology for the real and complex cases.