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.