Evaluation of a rational function in the elliptic curve unit element

Question

I'm reading on divisors and elliptic curve pairing. For a field $F$ and a rational function $f(x,y) \in F(x,y)$ it's often written $f(P)$ for points $P$ on the curve. But what is $f(P)$ when $P = \infty$, i.e. the unit element for elliptic curve addition?

Answer 1

You have to first apply homogenization to $f$ by $Z^n f(\frac{X}{Z}, \frac{Y}{Z})$ to get $f_{\textrm{homog}}$ which we will also denote by $f$. Then $f(\infty) = f(0, 1, 0)$.

For example lets say $f = x^2 - y + 2$, then homogenizing we get the projective curve $f = X^2 - YZ + 2Z$, and evaluation $f(\infty) = 0$.