Rank-one matrix completion is solved by the sum-of-squares relaxation of order two

This note studies the problem of nonsymmetric rank-one matrix completion. We show that in every instance where the problem has a unique solution, one can recover the original matrix through the second round of the sum-of-squares/Lasserre hierarchy with minimization of the trace of the moments matrix. Our proof system is based on iteratively building a sum of N - 1 linearly independent squares, where N is the number of monomials of degree at most two, corresponding to the canonical basis (z^α - z₀^α)². Those squares are constructed from the ideal I generated by the constraints and the monomials provided by the minimization of the trace.