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Real forms of the complex Neumann system: a method for finding real roots of polynomial \(U_{\mathcal{S}} ( \lambda )\). (English) Zbl 1457.37079

Summary: 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 \(U_{\mathcal{S}} ( \lambda )\). 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 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. 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.

MSC:

37J39 Relations of finite-dimensional Hamiltonian and Lagrangian systems with topology, geometry and differential geometry (symplectic geometry, Poisson geometry, etc.)
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
15A06 Linear equations (linear algebraic aspects)
26C10 Real polynomials: location of zeros
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References:

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