For Bezout's theorem, you have some inaccurate claims: the line x+y=0 and x+y=1 never interset. We need to move to projective space for the degree of the intersection to be exactly mn. | For Bezout's theorem, you have some inaccurate claims: the line x+y=0 and x+y=1 never interset. We need to move to projective space for the degree of the intersection to be exactly mn. |