Factoring homogeneous polynomials in two variables.

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.


Solution 1:

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 $).