Past exam of the mathematics course of the University of Cambridge / 2002 / ib / Paper 4 / 13G / d

Write the nonconstant bijective map as $p/q$ with coprime polynomials and set $d=\max (\mathrm{deg}p,\mathrm{deg}q)$. For a finite value $w$, its finite preimages solve $p(z)-wq(z)=0$. Except for a possible leading-coefficient cancellation value, this polynomial has degree $d$.

Multiple roots can occur only at zeros of $p^{\prime }q-p{q}^{\prime }$, since at a root $q\ne 0$ by coprimality. This derivative numerator is a nonzero polynomial for a nonconstant rational map. There are therefore only finitely many finite critical values to exclude. Choose $w$ outside them and outside the leading cancellation value. The fundamental theorem of algebra then gives exactly $d$ distinct finite preimages. Bijectivity forces $d=1$.

Hencewhich is a Möbius transformation. Conversely, the inverse is $(d_{0}w-b)/(a-cw)$, interpreted at its pole and at infinity, proving bijectivity on the Riemann sphere. The argument also establishes the generic-preimage interpretation of the degree of a rational map of the Riemann sphere.