Sorry for long post...
In Example 9.3 of the lecture notes of MA40188 Algebraic Curves (2016) by Ziyu Zhang from ShanghaiTech (Lecture notes), he goes like By the method of in the proof of Theorem 8.8
Example 9.3
Theorem 8.8 and its proof
My understanding of the computation method
I understood that the computation method in the proof of Theorem 8.8 that he meant as the following way; given the found linear factorisation;
$f(x, y, -ax/c - by/c) = (r_1 x + s_1 y) \cdots (r_d x + s_d y) = 0$, each factor $(r_i x + s_i y)$ determines a solution $[x : y] = [-s_i : r_i]$ which gives the point $p_i = [- s_i : r_i : a s_i / c - b r_i / c] \in L \cap D$ where $L$ and $D$ follow from the proof (See L1 and L3 of the proof).
My question
Now, back on the example 9.3, I tried to reproduce the part the third intersection point R of AB and C to be R = [0 : 1 : 1].
Then, as in the method in proof of Theorem 8.8, he replaced $z$ w/h $-ax / c - by / c$ so I tried to solve $y^2z - x^3 + 4xz^2 - z ^3 = 0$ w.r.t $x$ to find the form of $x$ in terms of y and z. But I'm stuck at $- x ^3 + 4xz^ = z^3 - y^2 z$ and don't know how to go from here...
Could someone help me?
What they're doing here is using the linear form to eliminate a variable, so that we're left with a form in two variables, which can be factored.
In this case, the linear form is $4y - 4z$, which has the same zero set as $y - z$, so to solve this, we just plug in $z = y$ into the cubic form and factor.
Explicitly, our equation becomes $$y^3 - x^3 + 4xy^2 - y^3 = 4xy^2 - x^3.$$ But now, we can factor this as $x(4y^2 - x^2) = x(2y - x)(2y + x)$. This will then give us the intersection points $[0: 1 : 1], [2: 1: 1],$ and $[-2 : 1 : 1]$ as required.