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Elliptic Tales by Robert Gross, Avner Ash

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Chapter 3

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THE PROJECTIVE PLANE

Road Map

Adding elements to our number systems is not sufficient to make the two definitions of degree agree. Next, we must add elements to the plane, changing the affine plane to the projective plane, to force there to be more intersection points between a line and a curve.

1. Points at Infinity

The story so far: If we have a polynomial f(x, y) in two variables of degree d, and if we intersect the graph of f = 0 with a parametrized line given by the simultaneous equations x = at + b, y = ct + e, we get a number N of intersection points. This number N is equal to the number of solutions of the polynomial equation ...

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