On the number of intersection points of lines and circles in ${\mathbb{R}}^3$

Abstract

We consider the following question: Given $n$ lines and $n$ circles in ${\mathbb{R}}^3$, what is the maximum number of intersection points lying on at least one line and on at least one circle of these families. We prove that if there are no $n^{1/2}$ curves (lines or circles) lying on an algebraic surface of degree at most two, then the number of these intersection points is $O(n^{3/2})$.

12 pages