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Computing the real solutions to a system of polynomial equations is achallenging problem, particularly verifying that all solutions have beencomputed. We describe an approach that combines numerical algebraic geometryand sums of squares programming to test whether a given set is "complete" withrespect to the real solution set. Specifically, we test whether the Zariskiclosure of that given set is indeed equal to the solution set of the realradical of the ideal generated by the given polynomials. Examples with finitelyand infinitely many real solutions are provided, along with an example havingpolynomial inequalities.

Numerically validating the completeness of the real solution set of a system of polynomial equations