We consider incomplete exponential sums in several variables of the form
S(f,n,m) = \frac{1}{2^n} \sum_{x_1 \in \{-1,1\}} ... \sum_{x_n \in \{-1,1\}}
x_1 ... x_n e^{2\pi i f(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 grows to infinity. The
conjecture is known to hold in the case when m=3 and d=2, but existing methods
for studying incomplete exponential sums appear to be insufficient to resolve
the question for an arbitrary odd modulus m, even when 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 of the
conjecture. We show that a bound of the required sort holds for almost all
quadratic polynomials, a stronger form of the conjecture holds for all
quadratic polynomials with no more than 10 variables, and for arbitrarily many
variables the conjecture is true for a class of quadratic polynomials having a
special form.Comment: 31 pages (minor corrections from original draft, references to new
results in the subject, publication information