About multivariable quadratic polynomials Say one has a polynomial function $f : \mathbb{C}^n \rightarrow \mathbb{C}$ such that it is quadratic in any of its variables $z_i$ (for $i \in \{ 1,2,..,n\}$). Then it follows that for any $i$ one can rewrite the polynomial as $f = A_i (z_i - a_i)(z_i - b_i)$, where the constants depend on the $i$ chosen. 


*

*But does it also follow that one can write the function in the form, $f = \prod_{i=1}^n (B_i (z_i - c_i)(z_i - d_i)  )$ ? 


If the above is not true then what is the simplest decomposition that can be written for such a $f$? 


*

*Like is it possible to redefine the basis in $\mathbb{C}^n$ to some $Z_i$ such that $f = \sum_{i} (a_i Z_i^2 + b_i Z_i + c_i)$ ? 

*Like any extra leverage that can be gotten if one knows that $f$ is real-rooted? 

You can't expect the $A_i,a_i,b_i$ to be constant. One could have $$f=z_1^2+z_2^2=(z_1+iz_2)(z_1-iz_2).$$
What about if $$f=z_1^2+z_2^2+z_3^2$$
