Consider a homogeneous polynomial $F(X,Y)\in\mathbb C[X,Y]$. Why can we always write it as: $$F(X,Y)=\prod(a_iX+b_iY)^{r_i}\ ?$$
I can't find a proof of this fact.
Many thanks in advance.
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10Fundamental Theorem of Algebra. Divide by $Y^n$, where $n$ is the degree. We get a polynomial in the variable $X/Y$. This polynomial is a product of linear factors. – André Nicolas Aug 08 '14 at 13:35
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if $m=\deg F$, then $F(X,Y)=Y^mF(X/Y,1)=Y^mQ(X/Y)$ where $Q(T)\in \Bbb C[T]$ ($\deg Q=r\leq m$), so $Q(T)=\prod (a_iT+b_i)^{r_i}$ ($\sum r_i=r$), now we have
$$F(X,Y)=Y^mQ(X/Y)=Y^m\prod(a_i(X/Y)+b_i)^{r_i}=Y^{m-r}\prod(a_iX+b_i Y)^{r_i}$$ ($\sum r_i=r $).
Hamou
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Dear Hamou, a little nitpick: you should modify your answer slightly if $F$ is divisible by $Y$, because then $Q$ is of degree $\lt m$. See here – Georges Elencwajg Oct 04 '15 at 18:35
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