  This paper discusses the split feasibility problem with polynomials. The sets
are semi-algebraic, defined by polynomial inequalities. They can be either
convex or nonconvex, either feasible or infeasible. We give semidefinite
relaxations for representing the intersection of the sets. Properties of the
semidefinite relaxations are studied. Based on that, a semidefinite relaxation
algorithm is given for solving the split feasibility problem. Under a general
condition, we prove that: if the split feasibility problem is feasible, we can
get a feasible point; if it is infeasible, we can obtain a certificate for the
infeasibility. Some numerical examples are given.
