Let $q(y, z) = u\_1 + u\_2y + u\_3 z + u\_4y^2 + u\_5yz + u\_6z^2 + u\_7y^3 + u\_8y^2z + u\_9yz^2 +$ $\hspace{2.55cm}u\_{10}y^3z + u\_{11}y^2z^2 + u\_{12}y^3z^2$


Can $q(y, z)$ be factorized as 
\begin{equation}
q(y, z) =(v\_1+v\_2y)(v\_3+v\_4y+v\_5z+v\_6yz)(v\_7+v\_8y+v\_9z+v\_{10}yz)?
\end{equation}


Here, {$u$} and {$v$} are complex numbers.


Are there any general principles to factorize a bivariate polynomial?


Thanks.



Factorization of bivariate polynomial
I answered your first question in the comments. To answer your second question, yes, these general principles go under the name of "Hensel lifting", see for example these [lecture notes](http://people.csail.mit.edu/madhu/FT98/lect7.ps).


For a computer algebra implementation, you could try [Sage](http://www.sagemath.org/doc/constructions/polynomials.html). But you will not find any nontrivial factorization of your polynomial for arbitrary complex coefficients $u\_n$.


