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I have the following situation: Let $L_1,L_2,L_3$ and $L_4$ be linear forms in $\mathbb{C}^k$ such that $\frac{L_1L_2}{L_3}=L_4$. Can we conclude from here that $L_3$ is a multiple of $L_1$ or of $L_2$? This is equivalent to ask if the factorization of $L_1L_2$ is unique.

I read in a commentary of another question that this is true but there was no proof. Thanks for the help!

Diego
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  • What do you mean by a linear form? A linear function $f : \mathbb{C}^k \to \mathbb{C}$ which takes the form $f(z_1, ..., z_n) = f(z) = \mu z-\lambda$, for some $\mu, \lambda \in \mathbb{C}$? –  Apr 11 '18 at 09:00
  • Yes, but with $\lambda=0$. In other words, every $L_i$ is in the dual of $\mathbb{C}^k$. – Diego Apr 11 '18 at 09:20

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