Vortex Solutions of the Liouville Equation

The most general vortex solution of the Liouville equation (which arises in nonrelativistic Chern–Simons theory) is associated with rational functions, f(z)=P(z) Qz), where P(z) and Q(z) are both polynomials, deg P N. This allows us to prove that the solution depends on 4N parameters without the use of an index theorem, as well as the flux quantization =-N(sign k).