Real Forms of the Complex Neumann System: Real Roots of Polynomial $U_{\cal S}(\lambda)$

TL;DR

This paper introduces a new method for analyzing the real roots of a special polynomial related to the Neumann system, aiding in understanding the topology of Liouville sets for certain real forms.

Contribution

A novel algorithm based on linear equations for checking the existence and positions of real roots of $U_{\

Findings

Complete solution for the existence of real roots when n=2.

New algorithm improves understanding of Liouville set topology.

Method facilitates analysis of polynomial roots in integrable systems.

Abstract

The topology of Liouville sets of the real forms of the complex generic Neumann system depends indirectly on the roots of the special polynomial $U_{\cal S}(\lambda)$. For certain polynomials, the existence and positions of the real roots, according to the suitable parameters of the system, is not obvious. In the paper, a novel method for checking the existence and positions of the real roots of the polynomials $U_{\cal S}(\lambda)$ is given. The method and algorithm are based on searching of a positive solution of a system of linear equations. We provide a complete solution to the problem of existence of real roots for all special polynomials in case $n=2$. This is a step closer to determining the topology of the Liouville sets.