I study coding theory and we use the textbook Fundamentals of Error-Correcting Codes . In the section related to Algebraic Geometry Code, we need to compute Intersection Multiplicity of two curve in a given point . The book did not define IM , instead they wrote
" We do not formally define Intersection multiplicity because the definitions is too technical.... we can compute multiplicity similarly to the way multiplicity of zeros is computed for polynomials in one variable ".
So its easy to know how to find the Intersection multiplicity of the elliptic curve determined by $x^3+xz^2+z^3+y^2z+yz^2=0$ and the curve $x=0$ in the point $(0,1,0)$ is equal to 1 . but if we change the point to $(0,1,1)$ , how can we compute the Intersection multiplicity in this point ?
Remark:- $f(x,y,z)=x^3+xz^2+z^3+y^2z+yz^2 \in \mathbb{F}_2[x,y,z].$
It is quite easy to define the intersection multiplicity of your elliptic curve with a line at any point. Indeed, let $f\in\mathbb F[x,y,z]$ be a homogeneous polynomial and $\ell\in\mathbb F[x,y,z]$ a linear homogeneous one. In your case, $\ell=x$ and $f$ as you wrote it. Let $p=(a,b,c)$ be any point with $\ell(p)=0$ and $p_0=(a_0,b_0,c_0)$ a point different from $p$ with $\ell(p_0)=0$. The point $p_0$ serves as our point at infinity for the projective line given by $\ell$. Set $$h(T) := f(a+a_0T,b+b_0T,c+c_0T)\in\mathbb F[T]$$ Then, the intersection multiplicity of the line and curve at $p$ is defined to be the maximal $n\in\mathbb Z$ such that $T^n$ divides $h(T)$, $$\mu_p(\ell,f) := \max\{ n\in\mathbb Z \mid T^n \text{ divides } h(T) \}$$
Intuition: As $T$ approaches zero, $h(T)$ approaches $f(p)$ "along" the line given by $\ell$.
A little more formal: Set $L:=Z(\ell)$ and $E:=Z(f)$, both as subsets of the projective plane $\mathbb P^2$. The polynomial $h$ is the restriction of $f$ to the line $L\setminus \{ p_0 \}\cong \mathbb F$. Indeed, $L$ consists of points $[ta+sa_0:tb+sb_0:tc+sc_0]$ with $[s:t]\in\mathbb P^2$ and the only point with $t=0$ is $p_0$. Hence, $L\setminus\{ p_0 \}$ is just points of the form $[a+sa_0:b+sb_0:c+sc_0]$ for $s\in\mathbb F$. Thus, $\mu_p(\ell,f)$ is the multiplicity of the zero of $h$ at the origin, which is $p$. In summary, $\mu_p(\ell,f)$ is the vanishing order of $f|_L$ at $p$.