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.

  • $\begingroup$ 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. $\endgroup$
    – user564215
    Nov 18, 2020 at 0:50
  • 1
    $\begingroup$ 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. $\endgroup$ Nov 18, 2020 at 1:19


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