The Basic Theory - Elliptic Curves: Number Theory and Cryptography - page 19

infinity on an elliptic curve will soon be identified with one of these points at
infinity in P 2 K .
The two-dimensional a ne plane over K is often denoted

A 2 K = { ( x, y ) ∈ K × K } .

We have an inclusion

A 2 K → P 2 K

given by

( x, y ) → ( x : y :1) .

In this way, the a ne plane is identified with the finite points in P 2 K . Adding
the points at infinity to obtain P 2 K can be viewed as a way of “compactifying”
the plane (see Exercise 2.10).
A polynomial is homogeneous of degree n if it is a sum of terms of the
form ax i y j z k with a ∈ K and i + j + k = n . For example, F ( x, y, z )=

2 x 3 − 5 xyz +7 yz 2 is homogeneous of degree 3. If a polynomial F is homoge-

neous of degree n then F ( λx, λy, λz )= λ n F ( x, y, z ) for all λ ∈ K . It follows
that if F is homogeneous of some degree, and ( x 1 ,y 1 ,z 1 ) ( x 2 ,y 2 ,z 2 ), then

F ( x 1 ,y 1 ,z 1 ) = 0 if and only if F ( x 2 ,y 2 ,z 2 ) = 0. Therefore, a zero of F in P 2 K
does not depend on the choice of representative for the equivalence class, so
the set of zeros of F in P 2 K is well defined.
If F ( x, y, z ) is an arbitrary polynomial in x, y, z , then we cannot talk about
apointin P 2 K where F ( x, y, z ) = 0 since this depends on the representative
( x, y, z ) of the equivalence class. For example, let F ( x, y, z )= x 2 +2 y − 3 z .
Then F (1 , 1 , 1) = 0, so we might be tempted to say that F vanishes at (1 : 1 :
1). But F (2 , 2 , 2) = 2 and (1 : 1 : 1) = (2 : 2 : 2). To avoid this problem, we
need to work with homogeneous polynomials.
If f ( x, y ) is a polynomial in x and y , then we can make it homogeneous by
inserting appropriate powers of z . For example, if f ( x, y )= y 2

∼

−x 3

−Ax−B ,

then we obtain the homogeneous polynomial F ( x, y, z )= y 2 z − x 3 − Axz 2

−

Bz 3 .If F is homogeneous of degree n then

F ( x, y, z )= z n f ( x

z , y z )

and

f ( x, y )= F ( x, y, 1) .

We can now see what it means for two parallel lines to meet at infinity. Let

y = mx + b 1 , y = mx + b 2

be two nonvertical parallel lines with b 1 = b 2 . They have the homogeneous

forms

y = mx + b 1 z, y = mx + b 2 z.