I read in my course book that asymptotes have at least two intersection with algebraic curve at infinity. How can I take this fact in my head in a visualized way? And what does that at least means? What are these multiple intersections on infinity ?
One can say that this is the definition of an asymptote; since an asymptote is a tangent to the curve at a point at infinity, the order of contact between the curve and the asymptote is higher than zero, which can be viewed as a multiple intersection. See the two references here.