= Solution
A <rational function> is a quotient $r=P/Q$ of <polynomials> with $Q\ne0$. Cancel common factors. It determines a map on the <Riemann sphere> by assigning infinity at poles and assigning at infinity the limit in the sphere. Equivalently homogenize $P,Q$ to the same degree $d=\max(\deg P,\deg Q)$ and set $f([Z:W])=[P_h(Z,W):Q_h(Z,W)]$. They have no common projective zero: finite ones would contradict coprimality, and at infinity at least one has a nonzero leading coefficient. This also proves the map is <holomorphic>.
The degree of the nonconstant <rational function> is $d$. The degree of a <holomorphic map> of compact <Riemann surfaces> is the number of preimages of any target point, counted with local mapping multiplicity. For target $[a:b]$, the homogeneous equation $bP_h-aQ_h=0$ is a nonzero homogeneous <polynomial> of degree $d$. Over $\mathbb C$ it factors into $d$ linear factors, with multiplicity, giving exactly $d$ projective zeros. Locally its zero order is precisely the map's multiplicity, since the other target-coordinate denominator is nonzero. \b[The two degrees are equal.]
There is a <Möbius transformation> sending any ordered triple of distinct points to $0,1,\infty$. When all are finite one can use $T_A(z)=(z-a_1)(a_2-a_3)/[(z-a_3)(a_2-a_1)]$; the formulas for infinity are its limits. Define $T_B$ in the same way. Then $T_B^{-1}\circ T_A$ sends $A$ to $B$ and restricts to a <biholomorphism> $X\setminus A\to X\setminus B$.
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