The Black-Box Query Complexity of Polynomial Summation

Abstract.

For any given Boolean formula $\phi(x_1, \ldots , x_n)$, one can efficiently construct (using arithmetization) a low-degree polynomial $p(x_1, \ldots , x_n)$ that agrees with ϕ over all points in the Boolean cube {0, 1}ⁿ; the constructed polynomial p can be interpreted as a polynomial over an arbitrary field ${\mathbb{F}}$. The problem #SAT (of counting the number of satisfying assignments of ϕ) thus reduces to the polynomial summation $\sum_{x \in \{0,1\}^n} p(x)$. Motivated by this connection, we study the query complexity of the polynomial summation problem: Given (oracle access to) a polynomial p(x₁, ... , x _n ), compute $\sum_{x \in \{0,1\}^n} p(x)$. Obviously, querying p at all 2ⁿ points in {0, 1}ⁿ suffices. Is there a field ${\mathbb{F}}$ such that, for every polynomial $p \in {\mathbb{F}}[x_1, \ldots , x_n]$, the sum $\sum_{x \in \{0,1\}^n} p(x)$ can be computed using fewer than 2ⁿ queries from ${\mathbb{F}}^n?$ We show that the simple upper bound 2ⁿ is in fact tight for any field ${\mathbb{F}}$ 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.