Towards the Reciprocity of Quartic Theta-Weyl Sums, and Beyond

Abstract

In the van der Corput method, one of the three principal methods of exponential sums, one treats the vdC reciprocal function f*(y) = f(xy) − yxy (f′(xy) = y). In the case where f(x) is a polynomial of degree K, i.e. the Weyl sum, one encounters a situation similar to the elliptic transformation and the quadratic and cubic polynomial cases were successfully treated by the author. The main idea is to introduce the decomposition x = nK−1 + m and think of n as the global variable of the function f(n) = K−1/KαxK/K−1F((Ξ)). Then the k-th vdC reciprocal function f*(y) given by (2.9) is essentially of the similar form to f (in terms of n1, y = n 1 K−1 + m1) and the length of interval of summation remains the same order, establishing the k-th reciprocation (under elliptic transformations). The behavior of f* under parabolic transformation is postponed for later researches. Instead, the inductive representation of f* is also given, with the concrete examples of the quartic case.