Resonance set of a polynomial and problem of formal stability

Let 𝑓𝑛(𝑥) be a monic polynomial of degree with real coefficients 𝑓𝑛(𝑥) def = + 𝑎1𝑥𝑛-1 + 𝑎2𝑥𝑛-2 + · · · + 𝑎𝑛. The space Π ≡ R𝑛 of its coefficients 𝑎1,... is called the coefficient space of 𝑓𝑛(𝑥). A pair of roots 𝑡𝑖, 𝑡𝑗, 𝑖, = 1,..., 𝑛, 𝑖 ̸= 𝑗, of the polynomial 𝑓𝑛(𝑥) is called : 𝑞-commensurable if : = : 𝑞. Resonance set ℛ𝑝:𝑞(𝑓𝑛) of the polynomial 𝑓𝑛(𝑥) is called the set of all points of Π at which 𝑓𝑛(𝑥) has at least a pair of : 𝑞-commensurable roots, i.e. ℛ𝑝:𝑞(𝑓𝑛) = {𝑃 ∈ Π : ∃ 𝑖, = 1,..., 𝑛, : = : 𝑞}. The chain Ch(𝑘) 𝑝:𝑞 (𝑡𝑖) of : 𝑞-commensurable roots of length is called the finite part of geometric progression with common ratio 𝑝/𝑞 and scale factor 𝑡𝑖, each member of which is a root of the polynomial 𝑓𝑛(𝑥). The value is called the generating root of the chain. Any partition of degree of 𝑓𝑛(𝑥) defines a certain structure of its : 𝑞-commensurable roots and it corresponds to some algebraic variety 𝑙, = 1,..., 𝑝𝑙(𝑛) of dimension in the coefficient space Π. The number of such varieties of dimension is equal to 𝑝𝑙(𝑛) and total number of all varieties consisting the resonance set ℛ𝑝:𝑞(𝑓𝑛) is equal to 𝑝(𝑛) - 1. Algorithm for parametric representation of any variety from the resonance set ℛ𝑝:𝑞(𝑓𝑛) is based on the following Theorem. Let 𝒱𝑙, dim = 𝑙, be a variety on which 𝑓𝑛(𝑥) has different chains roots and the chain Ch(𝑚) 𝑝:𝑞 (𝑡1) has length > 1. Let r𝑙(𝑡1, 𝑡2,..., 𝑡𝑙) is a parametrization of variety 𝒱𝑙. Therefore the following formula r𝑙(𝑡1,..., 𝑡𝑙, 𝑣) = r𝑙(𝑡1,..., 𝑡𝑙)+ 𝑝(𝑣 - 𝑝𝑚-1𝑡1) 𝑡1(𝑝𝑚 - 𝑞𝑚) [r𝑙(𝑡1,..., 𝑡𝑙) - r𝑙((𝑞/𝑝)𝑡1,..., 𝑡𝑙)] gives parametrization of the part of variety 𝒱𝑙+1, on which there exists Ch(𝑚-1) 𝑝:𝑞 (𝑡1), simple root and other chains of roots are the same as on the initial variety 𝒱𝑙. From the geometrical point of view the Theorem means that part of variety 𝒱𝑙+1 is formed as ruled surface of dimension 𝑙+1 by the secant lines, which cross its directrix at two points defined by such values of parameters 𝑡11 and 𝑡21 that 𝑡11 : 𝑡21 = : 𝑝. If 𝑓𝑛(𝑥) has on the variety 𝒱𝑙+1 pairs of complex-conjugate roots it is necessary to make continuation of obtained parametrization r𝑙(𝑡1,..., 𝑡𝑙, 𝑣). Resonance set of a cubic polynomial 𝑓3(𝑥) can be used for solving the problem of formal stability of a stationary point (SP) of a Hamiltonian system with three degrees of freedom. Let Hamiltonian function 𝐻(z) expand in SP 𝐻(z) = Σ︀∞ =2 𝐻𝑖(z), where z = (q, p), q and p - are coordinates and momenta, 𝐻𝑖(z) - are homogeneous functions of degree 𝑖. Characteristic polynomial 𝑓( ) of the linearized system ˙z = 𝐽𝐴z, = Hess𝐻2, can be considered as a monic cubic polynomial. Resonance sets ℛ𝑝:𝑞(𝑓𝑛) for = 1, 4, 9, 16, = 1, give the boundaries of subdomains in Π, where Bruno's Theorem of formal stability [9] can be applied.