Visualizing Bézout’s theorem

Bézout's theorem says that two polynomials of degrees n and m intersect in nm points. But if we graph xy = 1 and y = x2 it looks like they only intersect at the single point (1,1) instead of at four points.

Intersection 

To see all four points of intersection we need to look at the corresponding homogeneous curves xy = z2 and yz = x2. Here's what this looks like:

INtersection2 

Intersection-4 

  • Santi Martínez

    The graphic only shows two of the four points (in the spherical projection each point appears two times).
    The other two points are, in fact, complex (corresponding to cubic roots of 1).
    Note that: xy=1, y=x^2, so replacing y in the first equation, we get: x*x^2=1, so, there are three affine intersection points, of which only one of them (1,1) is real.
    The fourth point (which is seen using the spherical projection) is a point at infinity, so we can use the homogenouse curves xy=z^2 and yz=x^2 to compute it.
    Since it is a point at infinity, we need to assign z=0, so the equations are now xy=0 and 0=x^2. Clearly, x=0, so the fourth intersection point is [0:1:0] (which is the same as any other [0:k:0], as long as k =/= 0).

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