On Tractable Exponential Sums

We consider the problem of evaluating certain exponential sums. These sums take the form Σx1,x2,...,xn∈ZN e2πi/N f(x1,x2,...,xn), where each xi is summed over a ring ZN, and f(x1,x2,...,xn) is a multivariate polynomial with integer coefficients. We show that the sum can be evaluated in polynomial time in n and log N when f is a quadratic polynomial. This is true even when the factorization of N is unknown. Previously, this was known for a prime modulus N. On the other hand, for very specific families of polynomials of degree ≥ 3 we show the problem is #P-hard, even for any fixed prime or prime power modulus. This leads to a complexity dichotomy theorem -- a complete classification of each problem to be either computable in polynomial time or #P-hard -- for a class of exponential sums. These sums arise in the classifications of graph homomorphisms and some other counting CSP type problems, and these results lead to complexity dichotomy theorems. For the polynomial-time algorithm, Gauss sums form the basic building blocks; for the hardness result we prove group-theoretic necessary conditions for tractability.