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

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 $U_{\mathcal{S}}(\lambda )$. 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 $U_{\mathcal{S}}(\lambda )$ 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.

23 pages, 7 figures, 3 tables