Incomplete Quadratic Exponential Sums in Several Variables
- DOI: 10.1016/j.jnt.2005.04.005
- arXiv: math/0412063
Abstract
We consider incomplete exponential sums in several variables of the form S(f,n,m) = frac12 n sumx1 in -1,1 ... sumxn in -1,1 x1 ... xn e 2pi 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.
Author-supplied keywords
Sign up today - FREE
Mendeley saves you time finding and organizing research. Learn more
- All your research in one place
- Add and import papers easily
- Access it anywhere, anytime