Incomplete quadratic exponential sums in several variables Original Research Article

We consider incomplete exponential sums in several variables of the form imagewhere m>1 is odd and f is a polynomial of degree d with coefficients in image. 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 nless-than-or-equals, slant10 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.