Reading about homogeneous polynomials in an introduction to invariant theory, I'm trying to better understand some notation. In particular, every homogeneous polynomial of degree $d$ can be written as
$$f =\sum_{|\alpha|=d}a_\alpha x^\alpha ,$$
where $\alpha= (\alpha_1, \dots, \alpha_n) \in \mathbb{N}^n$, $a_{\alpha} \in \mathbb{C}$ and $x^{\alpha}:=x_1^{\alpha_1}\cdots x_n^{\alpha_n}$. Now what does it exactly mean to take the sum over $|\alpha|=d$? is it just an abbreviation for saying that we take the sum over $\alpha$ such that the condition holds? Also by replacing the definition of $x^\alpha$ in the above equation what I obtain is:
$$f =\sum_{\alpha_1 + \dots + \alpha_n=d}a_{\alpha_1} x_1^{\alpha_1}\cdots a_{\alpha_n}x_n^{\alpha_n}$$
but I'm not really sure about how to provide examples and say what is what.