Determination of the intersection of projective plane P2 (K) curves based on the Bezout’s theorem

Abstract

The question on the number of points of intersection of curves had been the subject of conjecture for many years. One of the theorems that arise from such conjectures is the Bezout’s theorem. Bezout gives a rigorous proof that when two polynomials in two variables are set equal to 0 simultaneously, one of degree m and the other of degree n, then there cannot be more thanmn solutions unless the two polynomials have a common factor. This is a first form of Bezout’s theorem. In order to have a chance of obtaining a full complement of mn solutions, there have been several adjustments to the first form of Bezout’s theorem, which allow complex solutions instead of just real solutions and considering projective plane curves instead of ordinary plane curves to allow for solutions at infinity. This condition can be realized in the example of two lines, which have no points of intersection on the affine plane if they are parallel. However, in the projective planes, parallel lines do intersect, at a point of the line at infinity. This motivates generalization of the Bezout’s theorem in the number of intersection of projective plane curves under certain conditions. In the case of two variables conics in the affine plane, the resultant can be applied to solve the intersection points. The notion and properties of intersection multiplicity is then applied on each of the intersection points including the points at infinity by considering the homogenization variable. By letting the homogenization variable equals to 1, the properties of intersection multiplicity can be applied to determine the multiplicity of the affine intersection points. By letting the homogenization variable equals to 0, the intersection multiplicity of the points at infinity can also be determined, thus the implementation of the Bezout’s theorem has been illustrated using selected examples.