Given a monomial ideal , where m _i are monomials, and a polynomial f as an arithmetic circuit the monomial Ideal Membership Problem is to test if f ∈ I. We show that the problem has a randomized polynomial time algorithm for constant k. Furthermore, if k is constant and f is computed by a ΣΠ Σ circuit with output gate of bounded fanin then we can test whether f ∈ I in deterministic polynomial time.