Incomplete quadratic exponential sums in several variables

We consider incomplete exponential sums in several variables of the form S(f, n, m) = 1/2(n) Sigma(x1 epsilon{-1,1}) center dot center dot center dot Sigma(xn epsilon{-1,1}) x1...xn e(2 pi if(x)/p) where m > 1 is odd and f is a polynomial of degree d with coefficients in Z/mZ. We investigate the conjecture, originating in a problem in computational complexity, that for each fixed d and m the maximum norm of S(f, n, m) converges exponentially fast to 0 as n tends to infinity; we also investigate the optimal bounds for these sums. Previous work has verified the conjecture when m = 3 and d = 2. In the present paper we develop three separate techniques for studying the problem in the case of quadratic f, each of which establishes a different special case. We show that a bound of the required sort holds for almost all quadratic polynomials, the conjecture holds for all quadratic polynomials with n <= 10 variables (and the conjectured bounds are sharp), and for arbitrarily many variables the conjecture is true for a class of quadratic polynomials having a special form. (c) 2005 Elsevier Inc. All rights reserved.