I hadn't seriously attempted this problem until just now (I saw it last month and got scared

). But I just solved it without too much trouble!

Let

, the field of rational functions in

. Consider

as elements of

(which is a P.I.D.). Then it is easy to see that

are relatively prime in

. Hence we can find rational functions

such that

. Multiplying through by a common denominator, we find that there exist polynomials

such that

. The right side of this equation can vanish for only a finite number of possible values of

, hence there are only a finite number of values of

which are candidates for an intersection point of the curves

and

. Reversing the roles of

, we arrive at the same conclusion for

; hence there are only a finite number of possible intersection points of the two curves.