The ideal membership problem and polynomial identity testing

Given a monomial ideal I = 〈m1,m2, · · · ,mk〉 where mi are monomials and a polynomial f as an arithmetic circuit the Ideal Membership Problem is to test if f ∈ I . We study this problem and show the following results. (a) If the ideal I = 〈m1,m2, · · · ,mk〉 for a constant k then there is a randomized polynomial-time membership algorithm to test if f ∈ I . This result holds even for f given by a black-box, when f is of small degree. (b) When I = 〈m1, m2, · · · ,mk〉 for a constant k and f is computed by a ΣΠΣ circuit with output gate of bounded fanin we can test whether f ∈ I in deterministic polynomial time. This generalizes the Kayal-Saxena result [KS07] of deterministic polynomial-time identity testing for ΣΠΣ circuits with bounded fanin output gate. (c) When k is not constant the problem is coNP-hard. However, the problem is upper bounded by coAM over the field of rationals, and by coNPModpP over finite fields. (d) Finally, we discuss identity testing for certain restricted depth 4 arithmetic circuits. For ideals I = 〈f1, · · · , f`〉 where each fi ∈ F[x1, · · · , xk] is an arbitrary polynomial but k is a constant, we show similar results as (a) and (b) above.