Nonexistence Of Rational First Integrals For Polynomial Systems

In the study of dynamical system, particularly in the study of Hamiltonian system, an important and basic problem is that how to judge whether is integrable. In general, an autonomous ordinary differential equation systemdx/dt=f(x),x(x₁,…,x_n)∈Cⁿ, (1.1)where f(x)=(f₁(x),…,f_n(x)),is called integrable, if it has asufficiently rich set of first integrals such that its solution can be expressed by these integrals.Here a single-valued functionΦ(x) is called a firstintegral of system (1.1) if it is constant along any solution curve of system (1.1). IfΦ(x) is differentiate, then this definition can be written as the condition...