Let the elliptic curve $E: Y^2 = X^3+2X^2-3X$ and the function $g: Y+X-1$.
I found that the zeros of $g$ on the curve $E$ (in projective coordinates) are $S(1:0:1)$ with order $o_S = 2$ and $P(-1:2:1)$ with order $o_P = 1$.
How about the poles? There exists any pole of $g$ on the curve $E$ ?
My thinking: $g \rightarrow \infty$ when $X \rightarrow \infty$ or $Y \rightarrow \infty$ or both. But when we take a look on the curve equation $E$:
- if $X \rightarrow \infty$ then $X^3 \rightarrow \infty$ and it means $Y^2 \rightarrow \infty$ with the "same speed" as $X^3$ (because the equation holds). So in fact both $X$ and $Y$ must tend to infinity to have a pole. But what is exactly would be that pole or maybe it does not exist.
Edit:
Maybe I am wrong, but I would say that the only pole is the point at infinity $O(0:1:0)$ because $g = \frac{Y+X-Z}{Z}$ in projective coordinates. So $g \rightarrow \infty$ when $Z = 0$, which means that the only pole is $O$. Am I right ?