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Let $f(x,y)=0$ and $g(x,y)=0$ be curves in $\mathbb R^2$. Assume that the origin $(0,0)\in \mathbb R^2$ is a $d$-fold point of $f$ and an $e$-fold point of $g$, respectively. Let $f_d(x,y)$ be the sum of the terms of degree $d$ in $f(x,y)$, $g_e(x,y)$ be the sum of the terms of degree $e$ in $g(x,y)$. If $f_d(x,y)$ and $g_e(x,y)$ have a common factor of positive degree, then the intersection multiplicity $I_O(f,g)>de.$

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mfn -- 1. what is the question? 2. there is a much better chance of getting a good answer if you make the question easier to read (hint: dollars). –  algori Oct 12 '11 at 3:51
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@algori: the dollar hint makes you sound like you're asking for a bribe. Of course, if you'd asked for latex instead, who knows what it would make it sound like... –  Thierry Zell Oct 12 '11 at 12:11
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1 Answer

This is proved in Fulton's "Algebraic Curves", available online here. The precise reference is Section 3.3, property (5) on p. 37.

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