Consider the following bivariate polynomial $$P (x,y) = \sum_{m=0}^{2}\sum_{n=0}^{2} a_{m,n} x^{m} y^{n}$$ where the coefficients $a_{m,n}$ can be complex numbers.
I was wondering if there are any theories that can indicate that this polynomial can be factorized as $$P (x,y) = \left( \sum_{m=0}^{1} \sum_{n=0}^{1} b_{m,n} x^{m} y^{n} \right) \left( \sum_{m=0}^{1} \sum_{n=0}^{1} c_{m,n} x^{m} y^{n} \right),$$
and whether it can be extended to bivariate polynomials with higher degrees.
I found the following description in Wikipedia https://en.wikipedia.org/wiki/Polynomial
All polynomials with coefficients in a unique factorization domain (for example, the integers or a field) also have a factored form in which the polynomial is written as a product of irreducible polynomials and a constant. This factored form is unique up to the order of the factors and their multiplication by an invertible constant. In the case of the field of complex numbers, the irreducible factors are linear.
I wonder if this can explain my problem?
No. I don't think all such polynomials can be factored. Heuristically, the space of such polynomials has "dimension" 9 (there are 9 different coefficients $a_{m,n}$), while the space of factorizations has "dimension" at most 8 (there are 4 different coefficients $b_{m,n}$ and 4 different coefficients $c_{m,n}$), so there exists some polynomial that can't be factored in this way.