Let and be polynomials with real coefficients. Show that if have no common factors, then the curves and will intersect in at most finitely
many points. ("at most finitely many" means either none or finitely many!)
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.