The ideal membership problem and polynomial identity testing

Abstract

Given a monomial ideal I = 〈 m 1 , m 2 , … , m k 〉 where m i are monomials and a polynomial f by an arithmetic circuit, the Ideal Membership Problem is to test if f ∈ I . We study this problem and show the following results. (a) When the ideal I = 〈 m 1 , m 2 , … , m k 〉 for a constant k, we can test whether f ∈ I in randomized polynomial time. This result holds even for f given by a black-box, when f is of small degree. (b) When I = 〈 m 1 , m 2 , … , m k 〉 for a constant k and f is computed by a Σ Π Σ circuit with output gate of bounded fanin, we can test whether f ∈ I in deterministic polynomial time. This generalizes the Kayal–Saxena result [11] of deterministic polynomial-time identity testing for Σ Π Σ circuits with bounded fanin output gate. (c) When k is not constant the problem is coNP-hard. We also show that the problem is upper bounded by coMA PP over the field of rationals, and by coNP ModpP over finite fields. (d) Finally, we discuss identity testing for certain restricted depth 4 arithmetic circuits. For ideals I = 〈 f 1 , … , f ℓ 〉 where each f i ∈ F [ x 1 , … , x k ] is an arbitrary polynomial but k is a constant, we show similar results as (a) and (b) above.