Real transformations with polynomial invariants

This paper seeks to generalise one aspect of classical Krein theory for linear Hamiltonian systems by examining how the existence of a non-trivial, homogeneous, polynomial W of degree m ≥ 2 with (Ax, ∇W(x)) = 0,x ∈ ℝ^N, affects the spectrum of a real linear transformation A on ℝ^N. Amongst other things it is shown that (i) such a W exists if, and only if, the spectrum of A is linearly dependent over the natural numbers, and (ii) there exists such a W which is non-degenerate if, and only if, all the eigenvalues of A are imaginary and semi-simple. In classical Krein theory W is quadratic. Our enquiry is motivated by a theory of topological invariants for dynamical systems which have a first integral. Degenerate Hamiltonian systems are a special class where the present considerations are relevant.