Symmetry groups and deformations of sums of exponentials

TL;DR

The paper investigates planar curves formed as images of the unit circle under weighted sums of exponentials, unifying rosette curves, Rhodonea curves, and Spirograph-type trajectories through Laurent-series maps $f(S^1)$. It develops a comprehensive theory of symmetries, relating them to permutations of zeros and poles and classifying the symmetry groups as cyclic or dihedral, with irrational exponents yielding finite or dense groups; it also proves density results in annuli for irrational exponent configurations. The work extends to algebraic-geometry aspects via Quine's framework, showing real-algebraic varieties containing the image and analyzing how symmetries constrain these varieties; it further analyzes the evolution under the one-dimensional wave equation, preserving symmetry and producing time-dependent changes in winding, cusps, and self-intersections in the binomial case. Collectively, the results provide a practical framework for constructing curves with prescribed numbers of cusps, self-intersections, and winding, and for understanding how such curves behave under dynamic evolution, with broad connections to rosette theory and complex polynomial imagery.

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.

Figures (7)

• Figure 1: Two polynomials of symmetry of type $(2, 5).$ On the left, $p(z) = z^2 + z^7 + z^{12}$ that has real coefficients. The symmetry group of $p(S^1)$ contains both a rotational symmetry of order $5$ and a mirror symmetry along the $x$-axis. On the right, the polynomial $p(z)=2 z^2-2iz^7+iz^{12}$ has no mirror symmetries, i.e. the image is chiral, hence the symmetry group of $p(S^1)$ contains no elements of order two
• Figure 2: Image of the unit circle under the polynomial $p(z) = z \bigl (z- \frac{1}{2} \bigr )(z-i)$
• Figure 3: The rosette for $p(z) = 1+ z + z^2 + z^3 + z^4 + z^5$, that crosses $5$ times the origin and $6$ times the point $(1, 0)$
• Figure 4: On the left, the evolution of $p(z) = 26 z^2 + z^{10}$ where self-intersections are absent for all times. On the right, the plots of $\psi$ and $\phi$ that never intersect
• Figure 5: Evolution of $p(z) = 6z+z^6$ with $c=1/6$ and period $T=6.$ For $t=0$, the rosette has five cusps and self-intersections occur, and disappear, at later times $0<t \leq 1.$

Theorems & Definitions (44)

• definition 1
• definition 2
• definition 3: pausinger2021symmetry
• proposition 1
• proof
• remark 1
• lemma 1
• proof
• theorem 1
• proof