Index Theorems for Polynomial Pencils

We survey index theorems counting eigenvalues of linearized Hamiltonian systems and characteristic values of polynomial operator pencils. We present a simple common graphical interpretation and generalization of the index theory using the concept of graphical Krein signature. Furthermore, we prove that derivatives of an eigenvector u= u(\lambda) of an operator pencil L(\lambda) satisfying L(\lambda) u(\lambda)= \mu(\lambda) u(\lambda) evaluated at a characteristic value of L(\lambda) do not only generate an arbitrary chain of root vectors of L(\lambda) but the chain that carries an extra information.