Special curves and Hamiltonian systems from polynomial eigenfunctions of an elliptic operator

TL;DR

This paper explores special algebraic curves and Hamiltonian systems derived from polynomial eigenfunctions associated with elliptic operators, focusing on their geometric properties and bifurcation behaviors.

Contribution

It introduces new classes of maximising and free plane curves related to polynomial eigenfunctions and analyzes bifurcations in corresponding Hamiltonian dynamical systems.

Findings

Identification of new algebraic curves linked to polynomial eigenfunctions

Analysis of bifurcation phenomena in polynomial Hamiltonian systems

Characterization of geometric properties of associated hypersurfaces

Abstract

A family of polynomials linked to the set of the deltoid tangents and its associated algebraic hypersurfaces has been presented in recent years. In this paper we study some related maximising and free plane curves. We also analyse the bifurcations on polynomial Hamiltonian dynamical systems defined from such a family.