Bachelor of Arts
Algebraic geometry, Plane curves, Intersection number, Bezouts theorem
In algebraic geometry, seemingly geometric problems can be solved using algebraic techniques. Some of the most basic geometric objects we can study are polynomial curves in the plane. In this paper we focus on the intersections of two curves. We address both the number of times two curves intersect at a given point, counting multiplicity (whatever that means), and the total number of intersections of the curves, again counting multiplicity. The former is known as the intersection number of the curves at the point. This concept, although geometrically motivated, can be described in algebraic terms; it is this relationship which makes it such a powerful concept. The paper concludes with an important application of the intersection number, Bezout's Theorem. This ubiquitous theorem gives a beautifully concise solution to the total number of intersections, given sufficiently nice assumptions on the curves and the ambient space.
Nichols, Margaret E., "Intersection Number of Plane Curves" (2013). Honors Papers. 335.