Multilinear Polynomials Modulo Composites

Understanding the power of constant-depth circuits that are allowed to use MODm gates, where m is an arbitrary but fixed positive integer, is a fundamental and inviting problem in theoretical computer science. Despite intensive efforts for more than twenty five years, this problem remains wide open. In this column, we focus our attention on the related, but much simpler, model of computing a boolean function by multilinear polynomials over the ring Z(m), when m is a composite number. As widely known, it is essential to understand this model in order to make progress with constant-depth circuits with MOD gates. We survey some recent results in this natural model that yield superpolynomial lower bounds on the size of some restricted circuits with MODm gates. The ingredients that get used in these results are perhaps more interesting. Some natural next steps emerge from these results that are also of independent mathematical interest. It is hoped that progress along these lines is feasible and would provide further insight into the general problem.