THE BLACK-BOX QUERY COMPLEXITY OF POLYNOMIAL SUMMATION

For any given Boolean formula phi(x(1),..., x(n)), one can efficiently construct (using arithmetization) a low-degree polynomial p(x(1),..., x(n)) that agrees with phi over all points in the Boolean cube {0, 1}(n); the constructed polynomial p can be interpreted as a polynomial over an arbitrary field F. The problem # SAT (of counting the number of satisfying assignments of phi) thus reduces to the polynomial summation Sigma(x is an element of{0,1}n) p(x). Motivated by this connection, we study the query complexity of the polynomial summation problem: Given (oracle access to) a polynomial p(x(1),..., x(n)), compute Sigma(x is an element of{0,1}n) p(x). Obviously, querying p at all 2(n) points in {0, 1}(n) suffices. Is there a field F such that, for every polynomial p is an element of F[x(1),..., x(n)], the sum Sigma(x is an element of{0,1}n) p(x) can be computed using fewer than 2(n) queries from F(n)? We show that the simple upper bound 2(n) is in fact tight for any field F in the black-box model where one has only oracle access to the polynomial p. We prove these lower bounds for the adaptive query model where the next query can depend on the values of p at previously queried points. Our lower bounds hold even for polynomials that have degree at most 2 in each variable. In contrast, for polynomials that have degree at most 1 in each variable (i.e., multilinear polynomials), we observe that a single query is sufficient over any field of characteristic other than 2. We also give query lower bounds for certain extensions of the polynomial summation problem.