Incomplete quadratic exponential sums in several variables

Abstract We consider incomplete exponential sums in several variables of the form S ( f , n , m ) = 1 2 n ∑ x 1 ∈ { - 1 , 1 } ⋯ ∑ x n ∈ { - 1 , 1 } x 1 ⋯ x n e 2 π if ( x ) / p , where m > 1 is odd and f is a polynomial of degree d with coefficients in Z / m Z . 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.