Oh yes, of course. See this documentation on bezier equation. http://www.tinaja.com/glib/cubemath.pdf
Also, I have seen scripts on this very forum, I think, which position and orient objects on a path.
Of course, with the note that Illustrator's curves are Bezier curves, and for Beziers, there may be more than one Y for a given X point (I know - the maths is horrible.)
Good thing most illustrators don't like to use the most extreme looped and self-intersecting curves (Illustrator itself has no qualms in inserting these in, for example, Live Traced results!)
@Math – I found the following enlightened (just to get an idea what we are talking about):
figure 5.1 Traversing a curve using de Casteljau's algorithm.
and figure 6.1: Bézier curve splitting. Curve order can be lowered/elevated.