Example
As an example consider where over . As has degree 5 and the roots are all distinct, is a curve of genus . Its graph is depicted in Figure 1.
From this picture it is immediately clear that we cannot use the chords and tangents method to define a group law on the set of points of a hyperelliptic curve. The group law on elliptic curves is based on the fact that a straight line through two points lying on an elliptic curve has a unique third intersection point with the curve. Note that this is always true since lies on the curve. From the graph of it is clear that this does not need to hold for an arbitrary hyperelliptic curve. Actually, Bézout's theorem states that a straight line and a hyperelliptic curve of genus 2 intersect in 5 points. So, a straight line through two point lying on does not have a unique third intersection point, it has three other intersection points.
Read more about this topic: Imaginary Hyperelliptic Curve
Famous quotes containing the word example:
“Our intellect is not the most subtle, the most powerful, the most appropriate, instrument for revealing the truth. It is life that, little by little, example by example, permits us to see that what is most important to our heart, or to our mind, is learned not by reasoning but through other agencies. Then it is that the intellect, observing their superiority, abdicates its control to them upon reasoned grounds and agrees to become their collaborator and lackey.”
—Marcel Proust (1871–1922)