Inversion of a mapping associated with the Aomoto-Forrester system

TL;DR

This paper studies a class of Hamiltonian systems related to root systems, computes Birkhoff series near stationary points, and derives diagrammatic expansions for inverses of a rational map introduced by Aomoto and Forrester.

Contribution

It extends the analysis of Aomoto-Forrester systems by computing Birkhoff series and providing new diagrammatic expansion series for the inverses of associated rational maps.

Findings

Computed Birkhoff series near stationary points.

Derived diagrammatic expansion series for inverse rational maps.

Extended the understanding of Hamiltonian systems related to root systems.

Abstract

This article is devoted to the study of a general class of Hamiltonian systems which extends the Calogero systems with external quadratic potential associated to any root system. The interest for such a class comes from a previous article of Aomoto and Forrester. We consider first the one-degree of freedom case and compute the Birkhoff series defined near each of its stationary points. In general, the analysis of the system motivates finding some expression for the inverses of a rational map introduced by Aomoto and Forrester. We derive here some diagrammatic expansion series for these inverses.