# How does one write the factorization of a general multivariable polynomial?

For a general univariate polynomial of order $$n$$, $$P_n(x)$$, I can express it succinctly as $$P_n(x) = C\prod_{i = 1}^n (x - r_i),$$ where $$C$$ is a constant and $$r_i$$ are the roots.

Is there a succinct way to write a general multivariable polynomial in this way? I am not looking for a form that includes redundancies, such as $$\sum_{i,j} A_{ij}x_i x_j$$ for two variables.

• I actually think you should consider the summation approach a bit more. After all, you can write a sum in terms of products. Imagine a function f(w) (w is a vector). f(w) = PiProd(stuff). Then log(f(w)) = log(PiProd(stuff)) = log(PiProd(stuff)_term1)+log(PiProd(stuff)_term2+...) = Sum(PiProd(stuff_termk)), where PiProd is your product function. So you should be able to work your way backwards to the product expression that you seek. I've only given it a cursory thought, so maybe I'm missing something. Best wishes, Danny! All is well.
– user564215
Nov 18, 2020 at 0:50
• No, there is not. For example, the polynomial $x^2+y^2-1$ is irreducible over $\mathbb C$ and can not be written as a product of linear factors with coefficients in $\mathbb C$. You can always write it as a product of its irreducible factors (over a given field), but these factors need not be linear. Nov 18, 2020 at 1:19