Wronskian of a rational map ($W=p^{\prime }q-p{q}^{\prime }$)

For a reduced rational map $R=p/q$, its derivative is $W/q^2$ with $W=p^{\prime }q-p{q}^{\prime }$. Away from target-coordinate poles, the zeros of this Wronskian identify its ramification points of a holomorphic map; at a pole use the reciprocal coordinate $q/p$. A pole of order $m$ contributes ramification order $m-1$, and infinity must also be checked in local coordinates. For degree $N$ the Riemann-Hurwitz formula gives total ramification $2N-2$. The affine polynomial $W$ can have fewer finite zeros because some ramification lies at infinity. In the rational map approximation for Skyrmions, these directions have zero angular baryon density.