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.