I am looking to find the intersection number of the affine plane curves $F=Y^2-X^3+X$ and $G=(X^2+Y^2)^3-4X^2Y^2$, and I need to do it with the order function of the local ring of $F$ at the origin.   So my attempt goes as follows:  $$k[X,Y] \twoheadrightarrow \Gamma(F)=k[X,Y]/(F) \hookrightarrow \mathcal{O}_{(0,0)}(F)$$ $$ G \mapsto \bar{G} = G\mod (F) \mapsto \frac{\bar{G}}{1} $$  Then I get the intersection number as $I((0,0),F\cap G) = \text{Ord}_{(0,0)}^F(\bar{G})$.  For the map onto the coordinate ring of $F$, I wrote $X\mapsto x, Y\mapsto y : y^2=x^3-x$. From there I thought that $\bar{G} = (x^2+(x^3-x))^3 - 4x^2(x^3-x)$. The map into the local ring (which I know should be a DVR as the origin is a simple point of $F$) is a natural injection, so I thought I should be able to write $\bar{G} = u\cdot t^n$ as one can do in a DVR. I can't! I'm not sure what is going wrong.
Firstly, I apologise that this question is specific to two polynomials. I understand that this post will not help a lot of people, and for that I am sorry. I will make it up to you all by anwsering lots more questions in the future! However, as my lecturers enjoy time off on the weekend, I am unable to ask this to anyone else who would know what the heck I'm talking about. Sorry if my notation is not widely used, I can elaborate more if required. Please excuse my poor Latex too.  Let:  $B = Y^2 - X^3 + X$, and   $F = (X^2 + Y^2)^3 - 4X^2Y^2$.  I wish to find the intersection number of $B$ and $F$ at $P = (0,0)$  i.e $I(P,F \cap B). $  However, I want to use the following proposition to do this:  If $P$ is simple on $B$ then $I(P,F \cap B) = ord_P^B(F) $.  I will omit proving that $P$ is simple on $B$ as I hope it is clear to see it is.  I know that $I(P,F \cap B) &gt; M_p(F) * M_p(B) = 1 * 4 $, where $M_P(G)$ is the multiplicity of $P$ on $G$.  Now, consider the Image of $X$ and $Y$:  $K[X,Y] \longrightarrow \frac{K[X,Y]}{Y^2 - X^3 + X} \longrightarrow O_P(Y^2 - X^3 + X)$  $X \longrightarrow x$  $Y \longrightarrow y$  Where lower case denotes the image.  Then $ord_P^{Y^2 - X^3 + X}((X^2 + Y^2)^3 - 4X^2Y^2)$  Can be found by substituting $y^2 = x^3 - x$ into $(x^2 + y^2)^3 - 4x^2y^2$  Which gives:  $ord_P^{Y^2 - X^3 + X}(x^9+3x^8 -5x^6 -4x^5 +3x^4 +3x^3)=3$  i.e $I(P,F \cap B) = 3$, a contradiction to  $I(P,F \cap B) &gt; 4$ as above.  The correct answer I know is 6, so where have I gone wrong?
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