I am interested in such a question.
What are necessary and sufficient conditions for representation of an algebraic polynomial in several variables with real coefficients in the form of the product of linear polynomials?
For example, let us that we have a polynomial in three variables $P(x,y,z)$.
I mean the following its representation: $$ P(x,y,z)=(a_{1}+b_{1}x+c_{1}y+d_{1}z)(a_{2}+b_{2}x+c_{2}y+d_{2}z)\ldots (a_{m}+b_{m}x+c_{m}y+d_{m}z). $$
I will be grateful for any help, for any references.
Thank you very much in advance!