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

Let f(x,y)=0 and g(x,y)=0 be curves in R^2. Assume that the origin (0,0) is a d-fold point of f and an e-fold point of g. 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). Then, 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.

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
mfn
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Intersection of curves

Let f(x,y)=0 and g(x,y)=0 be curves in R^2. Assume that the origin (0,0) is a d-fold point of f and an e-fold point of g. 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). Then, 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.