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In the book "Basic Agebraic Geometry I (third edition, 2013)" at page 14 Shafarevich says, about plane curves, what it follows:

If $P=(0,0)$ and the leading terms (note:by leading terms I suppose he means the terms of lowest degree) in the equation of the curve have degree $r$, then $r$ is called the multiplicity of $P$.

Now I'm confused. It seems that he defines the multiplicity only for the point $(0,0)$, what about the other points of $\mathbb A^2_k$?

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    $\begingroup$ Yes, you are right, he defines multiplicity at $(0,0)$ only. However If you want to define the multiplicity of the curve $f(x,y)=0$ at $(a,b)$, just make the change of variables $X=x-a,Y=y-b$ and study the multiplicity of the curve $g(X,Y)=f(X+a,Y+b)=0$ at the point $X=0, Y=0$, which Shafarevich explained how to compute. $\endgroup$ – Georges Elencwajg Sep 29 '13 at 18:30

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