Geometry of Rational Functions and Nonlinear Integrable Systems

Abstract

A complete parametrization of the space ${\text{Rat}}_\rho (n)$ of rational functions of degree n and fixed denominator $p(z)$ in terms of a nonlinear integrable system is established. The space ${\text{Rat}}_p (n)$ can be viewed as the moduli space of certain controllable and observable linear dynamical systems. It is proved that ${\text{Rat}}_p (n)$ is difleomorphic to the moduli (or parameter) space of solutions of a natural generalization of the celebrated finite Toda equation called the cyclic-Toda hierarchy. The original Toda flow is identified with one of the connected components of ${\text{Rat}}_p (n)$, where $p(z)$ is the characteristic polynomial of a Jacobi matrix. To prove the correspondence we use an exponential of cyclic matrix polynomials and its $QR$ factorization which induces isospectral deformations.