Invariant theory as a study of properties of polynomials under translational transformations is developed. 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. The connection with equations for hyper-elliptic Weierstrass and hyper-elliptic Jacobian functions is found. Algorithm of calculation of eigenvalues of the polynomials based on the evolution process is elaborated. Elements of the generalized dynamics with n-order characteristic polynomials are built.