polynomials - Commas under the Sigma notation

This might be a trivial notation question but I am having trouble understanding the following equation, describing a polynomial as sum of monomials:

$P(x_1, x_2, \dots, x_n) = \sum\limits_{i_1, i_2, \dots, i_l} a_{i_1, i_2, \dots, i_l} x_{i_1} x_{i_2} \dots x_{i_l}$.

I am not sure what this notation implies as it's the first time I have come across it. Is it saying:

$P(x_1, x_2, \dots, x_n) = a_{i_1}x_{i_1} + a_{i_2}x_{i_2}+ \dots + a_{i_l}x_{i_l}$ ??

But later the text says that the monomials are of the form: $x_{i_1} x_{i_2} \dots x_{i_l}$.

Thanks!

Answer

The notation $$\sum_{i, j}$$ means "the sum over the variables $i$ and $j$ independently". For example, $$\sum_{i, j} x_{i} x_{j} = (x_1 x_1 + x_1 x_2 + \dots + x_1 x_n) + (x_2 x_1 + x_2 x_2 + \dots + x_2 x_n) + \dots$$