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Bézout’s theorem is a fundamental statement in intersection theory. It asserts that the number of intersection points of two plane algebraic curves of degrees [latex]m[/latex] and [latex]n[/latex] is exactly [latex]mn[/latex], provided that one works in a projective plane over an algebraically closed field, counts points with multiplicity, and includes points at infinity where parallel asymptotes meet.
Bézout’s theorem elegantly quantifies the intersection of curves. In the standard affine plane, the number of intersections can be less than [latex]mn[/latex] for several reasons. First, some solutions might have complex coordinates. Second, lines that are parallel in the affine plane can be thought of as meeting at a ‘point at infinity’; moving to the projective plane [latex]\mathbb{P}^2[/latex] systematically includes these points. Third, some intersection points might be ‘degenerate’, such as a line being tangent to a circle. In this case, the single point of tangency must be counted with a multiplicity of two for the theorem to hold. The concept of intersection multiplicity is a crucial and subtle part of the theory that makes the count exact.
For example, a parabola ([latex]y=x^2[/latex], degree 2) and a line ([latex]y=ax+b[/latex], degree 1) should intersect at [latex]2 \times 1 = 2[/latex] points. This is clear when the line cuts through the parabola. When the line is tangent, there is one point, but it has multiplicity 2. If the line doesn’t intersect the parabola in the real plane, there are two intersection points with complex coordinates. The theorem generalizes to higher dimensions, stating that [latex]n[/latex] hypersurfaces of degrees [latex]d_1, \dots, d_n[/latex] in [latex]\mathbb{P}^n[/latex] intersect in exactly [latex]d_1 \cdots d_n[/latex] points, again, when counted properly.
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