Real forms of the complex Neumann system: A method for finding real roots of polynomial [formula omitted

Abstract

The topology of Arnold–Liouville level sets of the real forms of the complex generic Neumann system depends indirectly on the positions of the roots of the special polynomial US(λ). 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 method for checking the existence and positions of the real roots of the polynomials US(λ) is given. The method and algorithm are based on searching of a positive solution of a system of linear equations. In case n=2, we provide a complete solution to the problem of existence of real roots for all special polynomials and determine the topology of the Arnold–Liouville level sets.