Symmetry groups and deformations of sums of exponentials

TL;DR

This paper investigates the symmetry, deformation, and geometric properties of planar curves generated by sums of exponentials, providing new insights and methods for constructing curves with specific features.

Contribution

It generalizes previous results on exponential sums, relates them to other curve classes, and introduces a systematic approach to construct curves with desired properties.

Findings

Analyzed symmetry groups and winding numbers of exponential sum curves.

Established a unified method for constructing curves with prescribed features.

Connected exponential sum curves to broader classes of geometric curves.

Abstract

We study the symmetry groups and winding numbers of planar curves obtained as images of weighted sums of exponentials. More generally, we study the image of the complex unit circle under a finite or infinite Laurent series using a particular parametrization of the circle. We generalize various previous results on such sums of exponentials and relate them to other classes of curves present in the literature. Moreover, we consider the evolution under the wave equation of such curves for the case of binomials. Interestingly, our methods provide a unified and systematic way of constructing curves with prescribed properties, such as the number of cusps, the number of intersection points or the winding number.