The rationality of the Fourier coefficients of certain Eisenstein series on tube domains (I)

Liang-Chi Tsao

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Jordan obstruction to the integrability of Hamiltonian systems with homogeneous potentials

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In this paper, we consider the natural complex Hamiltonian systems with homogeneous potential $V (q)$ , $q \in ℂ^{n}$ , of degree $k \in ℤ^{☆}$ . The known results of Morales and Ramis give necessary conditions for the complete integrability of such systems. These conditions are expressed in terms of the eigenvalues of the Hessian matrix $V^{''} (c)$ calculated at a non-zero point $c \in ℂ^{n}$ , such that $V^{'} (c) = c$ . The main aim of this paper is to show that there are other obstructions for the integrability which appear if the matrix $V^{''} (c)$ is not diagonalizable....