  We consider numerical certification of approximate solutions to a system of
polynomial equations with more equations than unknowns by first certifying
solutions to a square subsystem. We give several approaches that certifiably
select which are solutions to the original overdetermined system. These
approaches each use different additional information for this certification,
such as liaison, Newton-Okounkov bodies, or intersection theory. They may be
used to certify individual solutions, reject nonsolutions, or certify that we
have found all solutions.
