Cramer-Euler Paradox

The Cramer-Euler paradox is the apparent conflict between the number of points generally needed to determine a plane algebraic curve and the number of intersections of two such curves. A plane algebraic curve of degree is generally determined by points. In particular, a conic section generally requires five points and a cubic curve nine. However, the Maclaurin-Bézout theorem states that two plane algebraic curves of degree with no common component have intersections in the complex projective plane, counted with multiplicity. Two cubics can therefore intersect in nine distinct points. The point conditions in this case are not independent, so nine points do not always determine a unique cubic curve (Bradley and Stemkoski).