Lissajous Varieties
Algebraic Geometry
2026-02-19 v2
Abstract
This paper studies affine algebraic varieties parametrized by sine and cosine functions, generalizing algebraic Lissajous figures in the plane. We show that, up to a combinatorial factor, the degree of these varieties equals the volume of a polytope. We deduce defining equations from rank constraints on a matrix with polynomial entries. We discuss applications of Lissajous varieties in dynamical systems, in particular the Kuramoto model. This leads us to study connections with convex optimization and Lissajous discriminants.
Cite
@article{arxiv.2509.06844,
title = {Lissajous Varieties},
author = {Francesco Maria Mascarin and Simon Telen},
journal= {arXiv preprint arXiv:2509.06844},
year = {2026}
}
Comments
To appear in Advances in Geometry. 24 pages, 8 figures