Improper intersection of algebraic curves

Abstract

Bezout's theorem gives an upper bound on the degree of the intersection of properly intersecting algebraic varieties. In spaces of dimension higher than two, however, intersections between many algebraic varieties such as curves are improper. Bezout's theorem cannot be directly used to bound the number of points at which these curves intersect. In this paper an algebrogeometric technique is developed for obtaining an upper bound on the number of intersection points of two irreducible algebraic curves in k-dimensional space. The theorems obtained are applied to the specific case of intersecting algebraic space curves in three-dimensional space, and a number of examples are analyzed in this regard. The implications of the derived results for computer-aided geometric design are discussed.