Regular Polygonal Vortex Filament Evolution and Exponential Sums

Abstract

In this paper, we give a rigorous proof for the expression of the angle between adjacent sides in the skew polygons appearing at rational times in the evolution of regular polygons of $M$ sides under the vortex filament equation. The proof depends on showing that some exponential sums with arithmetic content are purely imaginary.

9 pages, 1 figure