Notes on rational maps and Popov vortices

with the condition that an, bm 6= 0 as other wise the polynomials would be of a lower degree than claimed. We will also want to impose that one of a0, b0 is no zero, if they were both zero then the two polynomials would have a common zero at z = 0 and thus would not be co-prime. We also need to mention that both P and Q are not the zero polynomial. If P is the zero polynomial then R is the constant function zero and if Q is the zero polynomial then R is the constant function ∞. A rational map has a degree defined in terms of the degrees of the polynomials deg(R) = max{deg(P ), deg(Q)}. (1.2)