The minimal stage, energy preserving Runge-Kutta method for polynomial Hamiltonian systems is the Averaged Vector Field method

No Runge-Kutta method can be energy preserving for all Hamiltonian systems. But for problems in which the Hamiltonian is a polynomial, the averaged vector field (AVF) method can be interpreted as a Runge-Kutta method whose weights $ b_i$ and abscissae $ c_i$ represent a quadrature rule of degree at least that of the Hamiltonian. We prove that when the number of stages is minimal, the Runge-Kutta scheme must in fact be identical to the AVF scheme.