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I'm working Problem H of section I of chapter V of Miranda's book. The problem says

Let $X$ be the smooth projective plane curve defined by $y^2z=x^3-xz^2$. Compute the intersection divisors of the lines $x=0$, $y=0$ and $z=0$ with $X$.

Roughly speaking, take for example the intersection with $y=0$: $$ \begin{cases} y^2z=x^3-xz^2 \\ x=0 \end{cases} \quad \Longrightarrow \quad x(x^2-z^2)=0 \quad \Longrightarrow x=0, x=\pm z $$ so I get the points $p_1=[0:0:1]$, $p_2=[1:0:1]$ and $p_3=[1:0:-1]$. So I'm expect to get $$div(y)=p_1+p_2+p_3$$ and it works in this case: I take for $p_1$ the homogeneous polynomial of degree 1 $H=z$, so $H(p_1)\ne 0$ and I can compute $$div(y)(p_1)=ord_{p_1}(x/z)=ord_{p_1}(x)-ord_{p_1}(z)=1-0=1.$$ I repeat the computation also for $p_2$ and $p_3$ and it's all ok.

The problem come up with $x=0$: $$\begin{cases} y^2z=x^3-xz^2 \\ x=0 \end{cases} \quad \Longrightarrow y^2z=0 $$ so I get $p_1=[0:0:1]$ (counted two times, like in the affine real picture) and $p_2=[0:1:0]$. So I expect that my divisor will be in the form $$div(x)=2p_1+p_2.$$ Like before, I take $H=z$ so $H(p_1)\ne 0$ and compute using the definition: $$div(x)(p_1)=ord_{p_1}(x/z)=1.$$ Where am I wrong? This it makes me going crazy.

$y^2=x^3-x$

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    Why do you think $ord_{p_1}(x/z)$ is $1$? You need to calculate this in local coordinates; namely find transfer the meromorphic function $x/z$ to aopen neighborhood in $\mathbb C$ and calculate the degree of the vanishing. Miranda, page 26. It seems you calculated the first two cases not quite rigorously, but ended up with the right answer. I agree with what you think you the hyperplane divisors should be though. – 2017-02-13
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    Uhm...ok. I could use local coordinate, say dividing by $z$, so $[X,Y,1]=[x/z:y/z:1]$. Now, I get (?) $$ord_{p_1}(x/z)=ord_{(0,0)}(X)$$ Then? I feel a bit confused, because $ord_{(0,0)}(X)=1$. – 2017-02-13
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    No, that's not what I'm saying. $X$ is a complex manifold of dimension $1$, by implicit function theorem. Since it's a manifold, you can cover $X$ by open sets homeomorphic to open sets in $\mathbb C$, where the transition functions are holomorphic. Say $U$ is an open subset in $\mathbb C$ corresponding to an open neighborhood $V$ of the originwith $\phi: U \to V$ sending $p$ to the origin. The $\phi$ followed by $x/z$ is a meromorphic function. Calculate Laurent expansion of $x/z \circ \phi$ at $p$, and then you will see the order. This is the definition of order on Miranda page 26. – 2017-02-13
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    You're claiming the order is 1, but I don't see a Laurent expansion argument, unless you're using some other fact. – 2017-02-13
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    Everything is clear now, except that I can figure out a polynomial expression for $\phi$, any idea? – 2017-02-14

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