polynomials - Test if two curves intersect before finding roots

If you think about it, there does not seem to be a way around eliminating two variables:
$$ exists t_1, t_2, x_1(t_1)=x_2(t_2) & y_1(t_1)=y_2(t_2) $$
I.e., the $x$ and $y$ coordinates coincide, but each point is obtained at a different moment in time (or again, in terms of mobile points, the two trajectories intersect, but the two mobile points do not necessarily collide.)

Over the reals, it's the cylindrical algebraic decomposition that does that for you (explained for instance in the book by Basu-Pollack-Roy which is freely available). Any decent computer algebra system should be able to solve this.