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We provide a real algebraic symbolic-numeric algorithm for computing the realvariety $V_R(I)$ of an ideal $I$, assuming it is finite while $V_C(I)$ may notbe. Our approach uses sets of linear functionals on $R[X]$, vanishing on agiven set of polynomials generating $I$ and their prolongations up to a givendegree, as well as on polynomials of the real radical ideal of $I$, obtainedfrom the kernel of a suitably defined moment matrix assumed to be positivesemidefinite and of maximum rank. We formulate a condition on the dimensions ofprojections of these sets of linear functionals, which serves as stoppingcriterion for our algorithm. This algorithm, based on standard numerical linearalgebra routines and semidefinite optimization, combines techniques fromprevious work of the authors together with an existing algorithm for thecomplex variety. This results in a unified methodology for the real and complexcases.

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