Evolution of Coe±cients and Eigenvalues of Polynomials and Related Generalized Dynamics (pp.401-418)
Authors: (Robert M. Yamaleev, Facultad de Estudios Superiores, Universidad Nacional Autonoma de Mexico, Cuautitl¶an Izcalli, Campo 1, C.P.54740, M¶exico. Joint Institute for Nuclear Research, LIT, Dubna, Russia)
Abstract: Invariant properties of the polynomials are studied. Class of polynomials with congruent
set of eigenvalues is introduced. Evolution equations for eigenvalues and coefficients
remaining the polynomial within proper class of polynomials are formulated.
Evolution equations are given by equations forWeierstrass and Jacobian hyper-elliptic
functions. The link between evolution equations for the coefficients of the polynomials
and dynamical evolution equations of physical systems is established. Within the
framework of generalized mechanics the inner and outer coordinates of momenta are
interrelated as roots and coefficients of the characteristic polynomials. In particularly,
in the case of n = 2 degree we deal with the relativistic dynamics. The case of n = 3
degree we arrive to new dynamics beyond the relativistic one. As an application the
algorithm of finding of roots of the polynomials based on the evolution process is built.