I have come across the following problem, which may have a well known answer, but I was unable to locate one.
Consider two parameterized curves given by $$F(t)=(f_1(t), \dots, f_n(t), \quad G(t)= ( g_1(t), \dots, g_n(t)),$$ where $f_i(t)$ and $g_i(t)$ are polynomials of degree at most d. Assume that the Zariski closures of the images of $F$ and $G$ (which are irreducible varieties of dimension at most one) are distinct.
Is there a bound depending on d and n for the number of intersection points of the image of $F$ and the image of $G$?
Of course, when these curves are planar we have Bezout’s theorem. There seem to be results of similar flavor in intersection, but cannot find an exact statement. I would highly appreciate if someone could point me to the right reference.
Linearly project both curves to generic linearly embedded projective planes and apply Bezout's theorem to get an upper bound of $d^2$. You can't do better, since the curves could lie in a linearly embedded projective plane and have $d^2$ intersection points.