Incomplete Quadratic Exponential Sums in Several Variables

Details Incomplete Quadratic Exponential Sums in Several Variables Incomplete Quadratic Exponential Sums in Several Variables

2004-12-02

arxiv.org/abs/math/0412063v3

Journal of Number Theory: Volume 116, Issue 1, January 2006, pages 168-199

Mathematics

Number Theory

31 pages (minor corrections from original draft, references to new results in the subject, publication information)

Scientific paper

10.1016/j.jnt.2005.04.005

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.

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